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THEORYOF GAMES

AND ECONOMICBEHAVIOR)))

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THEORY OFGAMES

AND ECONOMICBEHAVIOR))

By JOHNVON NEUMANN,and

OSKARMORGENSTERN))

PRINCETON

PRINCETONUNIVERSITYPRESS

1953)))

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Copyright 1944,by PrincetonUniversity PressPRINTED IN THE UNITED STATESOP AMERICA

Second printing (SECOND EDITION), 1947Third printing, 1948

Fourth printing, 1950Fifth printing (THIRD EDITION), 1953

Sixth printing, 1955))

LONDON: GEOFFREY CUMBERLEGE OXFORD UNIVERSITY PRESS)))

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PREFACETO FIRSTEDITION

This bookcontainsan expositionand various applicationsof a mathe-matical theory of games.The theory has beendevelopedby one of ussince1928and is now publishedfor the first time in its entirety. Theapplicationsareof two kinds:On the one hand to gamesin the propersense,on the other hand to economicand sociologicalproblemswhich, as we hopeto show,arebest approachedfrom this direction.

Theapplicationswhich we shall make to gamesserveat leastas muchto corroboratethe theory as to investigatethesegames.Thenature of thisreciprocalrelationship will becomeclearas the investigation proceeds.Our major interestis, of course,in the economicand sociologicaldirection.Herewe can approachonly the simplestquestions. However,theseques-tions areof a fundamental character.Furthermore,our aim is primarilyto show that thereis a rigorousapproach to thesesubjects,involving, asthey do, questionsof parallelor oppositeinterest,perfect or imperfect infor-mation, free rational decisionor chanceinfluences.

JOHN VON NEUMANN

OSKAR MORGENSTERN.

PRINCETON, N. J.January, 1943.))

PREFACETO SECONDEDITIONThe secondedition differs from the first in some minor respectsonly.

We have carriedout as completean elimination of misprintsas possible,andwish to thank several readerswho have helpedus in that respect.We haveaddedan Appendix containing an axiomatic derivation of numerical utility.This subjectwas discussedin considerabledetail, but in the main qualita-tively, in Section3. A publication of this proof in a periodicalwas promisedin the first edition,but we found it more convenient to add it as an Appendix.Various Appendiceson applicationsto the theory of location of industriesand on questionsof the four and five persongameswere alsoplanned,buthad to be abandonedbecauseof the pressureof other work.

Sincepublication of the first edition several papers dealing with thesubjectmatter of this bookhave appeared.

The attention of the mathematically interestedreadermay be drawnto the following: A. Wald developeda new theory of the foundations ofstatisticalestimation which is closelyrelatedto, and drawson, the theory of)))

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vi PREFACE TO SECONDEDITION

the zero-sumtwo-persongame (\"StatisticalDecisionFunctionsWhichMinimize the Maximum Risk,\"Annals of Mathematics, Vol. 46 (1945)pp. 265-280).Healso extendedthe main theorem of the zero-sumtwo-persongames(cf. 17.6.)to certaincontinuous-infinite-cases,(\"Generalizationof a Theoremby von Neumann ConcerningZero-SumTwo-PersonGames,\"Annals of Mathematics,Vol. 46 (1945),pp. 281-286.)A new, very simpleand elementaryproof of this theorem (which coversalso the more generaltheorem referredto in footnote 1on page 154)was given by L.H.Loomis,(\"Ona Theoremofvon Neumann,\"Proc.Nat.Acad.,Vol.32(1946)pp.213-215).Further,interestingresultsconcerningthe role of pure and of mixedstrategiesin the zero-sum two-persongame wereobtainedby /. Kaplanski,(\"A Contributionto von Neumann'sTheory of Games,\"Annals of Mathe-matics,Vol. 46 (1945),pp. 474-479). We alsointend to comeback to vari-ous mathematical aspectsof this problem. Thegroup theoreticalproblemmentioned in footnote 1on page 258 wassolvedby C.Chevalley.

Theeconomicallyinterestedreadermay find an easierapproach to theproblemsof this book in the expositionsof L. Hururicz, (\"TheTheory ofEconomicBehavior,\"American EconomicReview, Vol. 35 (1945),pp.909-925) and of J. Marschak(\"Neumann'sand Morgenstern'sNew Approachto Static Economics,\"Journal of PoliticalEconomy, Vol. 54, (1946),pp.97-115).

JOHN VON NEUMANN

OSKAR MORGENSTERNPRINCETON, N. J.

September,1946.)))

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PREFACETO THIRDEDITION

TheThird Editiondiffersfrom the SecondEditiononly in the eliminationof such further misprintsas have come to our attention in the meantime,and we wish to thank several readerswho have helpedus in that respect.

Sincethe publicationof the SecondEdition,the literature on this subjecthas increasedvery considerably. A completebibliographyat this writingincludesseveralhundred titles. We aretherefore not attempting to giveone here. We will only list the following bookson this subject:

(1) H.W. Kuhn and A. W. Tucker(eds.),\" Contributionsto the Theoryof Games, I,\" Annals of MathematicsStudies,No.24, Princeton(1950),containing fifteen articlesby thirteen authors.

(2) H.W. Kuhn and A. W. Tucker(eds.),\" Contributionsto the Theoryof Games,II,\"Annals of MathematicsStudies,No. 28,Princeton(1953),containing twenty-one articlesby twenty-two authors.

(3) J'.McDonald,Strategy in Poker,Businessand War, New York(1950).

(4) J. C.C.McKinsey,Introduction to the Theory of Games, NewYork (1952).

(5) A. Wald, StatisticalDecisionFunctions,New York (1950).(6) J.Williams, The CompleatStrategyst,Beinga Primeron the Theory

of Gamesof Strategy, New York (1953).Bibliographieson the subjectarefound in all of the above booksexcept

(6). Extensivework in this field has beendoneduring the last yearsby thestaff of the RAND Corporation,Santa Monica,California. A bibliographyof this work can be found in the RAND publicationRM-950.

In the theory of n-persongames,therehave beensomefurther develop-ments in the direction of \" non-cooperative\"games.In this respect,particularly the work of J.F.Nash, \" Non-cooperativeGames,\"Annals ofMathematics, Vol. 54, (1951),pp. 286-295,must be mentioned.Furtherreferencesto this work arefound in (1),(2),and (4).

Of various developmentsin economicswe mention in particular \"linearprogramming\" and the \" assignmentproblem\"which also appear to beincreasinglyconnectedwith the theory of games. The readerwill find

indicationsof this $gain in (1),(2),and (4).Thetheory of utility suggestedin Section1.3.,and in the Appendix to the

SecondEdition has undergoneconsiderabledevelopmenttheoretically,aswell as experimentally,and in various discussions. In this connection,thereadermay consultin particular the following:

M. Friedman and L.J.Savage,\"TheUtility Analysis of ChoicesInvolv-ing Risk,\" Journalof PoliticalEconomy,Vol. 56,(1948),pp.279-304.

vii)))

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viii PREFACE TO THIRDEDITION

J.Marschak,\"Rational Behavior, UncertainProspects,and MeasurableUtility/' Econometrica,Vol. 18,(1950),pp. 111-141.

F.Mostellerand P.Nogee,\" An ExperimentalMeasurementof Utility,\"Journalof Political Economy,Vol. 59, (1951),pp. 371-404.

M. Friedman and L.J.Savage, \"TheExpected-Utility Hypothesisandthe Measurability of Utility,\" Journal of Political Economy, Vol. 60,(1952),pp.463-474.

Seealso the Symposiumon CardinalUtilitiesin Econometrica,Vol. 20,(1952):

H.Wold, \"OrdinalPreferencesor CardinalUtility?\"A. S. Manne, \"The Strong IndependenceAssumption Gasoline

Blendsand ProbabilityMixtures.\"P.A. Samuelson,\"Probability,Utility, and the IndependenceAxiom.\"E. Malinvaud, \"Noteon von Neumann-Morgenstem'sStrongInde-

pendenceAxiom.\"

In connectionwith the methodologicalcritique exercisedby some of thecontributors to the last-mentionedsymposium,we would like to mentionthat we appliedthe axiomatic method in the customaryway with the cus-tomary precautions.Thus the strict,axiomatic treatmentof the conceptof utility (in Section3.6.and in the Appendix)is complementedby anheuristic preparation (in Sections3.1.-3.5.).The latter'sfunction is toconvey to the readerthe viewpoints to evaluate and to circumscribethevalidity of the subsequent axiomatic procedure.In particular our dis-cussionand selectionof \"natural operations\"in thosesectionscoverswhatseemsto us the relevant substrate of the Samuelson-Malinvaud\"inde-pendenceaxiom.\"

JOHN VON NEUMANN

OSKAR MORGENSTERN

PRINCETON, N. J.January, 1953.)))

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TECHNICALNOTE

Thenature of the problemsinvestigatedand the techniquesemployedin this booknecessitatea procedurewhich in many instancesis thoroughlymathematical.Themathematical devicesusedareelementaryin the sensethat no advancedalgebra,or calculus,etc.,occurs. (With two, rather unim-portant, exceptions:Partof the discussionof an examplein 19.7.etsequ.anda remark in A.3.3.makeuseof somesimpleintegrals.) Conceptsoriginatingin settheory, linear geometry and group theory play an important role,butthey areinvariably taken from the early chaptersof thosedisciplinesand aremoreover analyzed and explainedin specialexpositorysections.Neverthe-lessthe bookis not truly elementary becausethe mathematical deductionsarefrequently intricate and the logicalpossibilitiesare extensivelyexploited.

Thus no specificknowledgeof any particularbodyof advancedmathe-matics is required. However,the readerwho wants to acquaint himselfmore thoroughly with the subjectexpoundedhere,will have to familiarizehimself with the mathematical way of reasoning definitely beyond itsroutine, primitive phases. Thecharacterof the procedureswill be mostlythat of mathematical logics,settheory and functional analysis.

We have attempted to presentthe subjectin sucha form that a readerwho is moderatelyversedin mathematicscanacquirethe necessarypracticein the courseof this study. We hope that we have not entirely failed inthis endeavour.

In accordancewith this, the presentationis not what it would bein astrictly mathematical treatise. All definitions and deductionsare con-siderablybroaderthan they would bethere. Besides,purely verbal dis-cussionsand analysestake up a considerableamount of space. We havein particular tried to give, whenever possible,a parallelverbal expositionfor every major mathematical deduction. It is hoped that this procedurewill elucidatein unmathematical language what the mathematical techniquesignifies and will also show where it achievesmore than can be donewithout it.

In this, as well as in our methodologicalstand, we aretrying to followthe best examplesof theoreticalphysics.

Thereaderwho is not specificallyinterestedin mathematicsshouldatfirst omit thosesectionsof the book which in his judgmentaretoo mathe-matical.We prefer not to give a definite list of them, sincethis judgmentmust necessarilybe subjective. However,those sectionsmarked with anasteriskin the tableof contentsaremostlikelyto occurto the average readerin this connection.At any ratehe will find that theseomissionswill littleinterfere with the comprehensionof the early parts, although the logical)))

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TECHNICAL NOTE

chain may in the rigorous sensehave suffered an interruption. As heproceedsthe omissionswill graduallyassumea more seriouscharacterandthe lacunaein the deductionwill becomemore and more significant. Thereaderis then advised to start again from the beginningsincethe greaterfamiliarity acquiredis likelyto facilitate a betterunderstanding.

ACKNOWLEDGMENT

Theauthorswish to expresstheir thanks to PrincetonUniversity and tothe Institutefor Advanced Study for their generoushelp which renderedthis publicationpossible.

They arealsogreatly indebtedto the PrincetonUniversity Presswhichhas made every effort to publishthis book in spiteof wartime difficulties.The publisherhas shown at all times the greatestunderstandingfor theauthors'wishes.)))

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CONTENTSPREFACE v

TECHNICAL NOTE ix

ACKNOWLEDGMENT x

CHAPTER IFORMULATIONOF THEECONOMICPROBLEM

1.THE MATHEMATICAL METHODIN ECONOMICS 11.1.Introductory remarks 11.2.Difficulties of the application of the mathematical method 21.3.Necessarylimitations of the objectives 61.4.Concluding remarks 7

2.QUALITATIVE DISCUSSIONOFTHE PROBLEM OF RATIONAL BEHAV-IOR 8

2.1.Theproblem of rational behavior 82.2.\"Robinson Crusoe\" economy and socialexchangeeconomy 92.3.Thenumber of variables and the number of participants 122.4. Thecaseof many participants: Freecompetition 132.5.The \"Lausanne\" theory 15

3.THE NOTION OF UTILITY 153.1.Preferencesand utilities 153.2.Principles of measurement: Preliminaries 163.3.Probability and numerical utilities 173.4. Principles of measurement: Detaileddiscussion 203.5.Conceptual structure of the axiomatic treatment of numerical

utilities 243.6.Theaxioms and their interpretation 263.7. Generalremarks concerning the axioms 283.8.Therole of the conceptof marginal utility 29

4. STRUCTURE OF THE THEORY: SOLUTIONSAND STANDARDS OFBEHAVIOR 31

4.1.Thesimplest conceptof a solution for one participant 314.2. Extension to all participants 334.3. Thesolution as a set of imputations 344.4. Theintransitive notion of \"superiority\" or \"domination\" 374.5. Theprecisedefinition of a solution 394.6. Interpretation of our definition in terms of \"standards of behavior\" 404.7. Gamesand socialorganizations 434.8. Concluding remarks 43

CHAPTER IIGENERAL FORMALDESCRIPTIONOF GAMESOF STRATEGY

5. INTRODUCTION 465.1.Shift of emphasis from economicsto games 465.2.Generalprinciples of classification and of procedure 46)))

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6.THE SIMPLIFIED CONCEPTOF A GAME 486.1.Explanation of the termini technici 486.2.Theelementsof the game 496.3.Information and preliminary 516.4. Preliminarity, transitivity, and signaling 51

7. THE COMPLETECONCEPTOF A GAME 557.1.Variability of the characteristicsof eachmove 557.2. Thegeneraldescription 57

8.SETSAND PARTITIONS 608.1.Desirability of a set-theoreticaldescription of a game 608.2.Sets,their properties,and their graphical representation 618.3.Partitions, their properties,and their graphical representation 638.4. Logisticinterpretation of setsand partitions 66

*9. THE SET-THEORETICAL DESCRIPTIONOF A GAME 67*9.1.Thepartitions which describea game 67*9.2. Discussionof thesepartitions and their properties 71

*10.AXIOMATIC FORMULATION 73*10.1.Theaxioms and their interpretations 73*10.2.Logisticdiscussion of the axioms 76*10.3.Generalremarks concerning the axioms 76*10.4.Graphical representation 77

11.STRATEGIES AND THEFINAL SIMPLIFICATION OFTHE DESCRIPTIONOF A GAME 7911.1.Theconceptof a strategy and its formalization 7911.2.The final simplification of the description of a game 8111.3.Theroleof strategiesin the simplified form of a game 8411.4.Themeaning of the zero-sum restriction 84

CHAPTER IIIZERO-SUMTWO-PERSONGAMES: THEORY

12.PRELIMINARY SURVEY 8512.1.Generalviewpoints 8512.2.Theone-persongame 8512.3.Chanceand probability 8712.4.Thenext objective 87

13.FUNCTIONAL CALCULUS 8813.1.Basicdefinitions 8813.2.Theoperations Max and Min 8913.3.Commutativity questions 9113.4.Themixed case. Saddlepoints 9313.5.Proofsof the main facts 95

14.STRICTLY DETERMINED GAMES 98141.Formulation of the problem 9814.2.Theminorant and the majorant games 10014.3.Discussionof the auxiliary games 101)))

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CONTENTS14.4. Conclusions 10514.5.Analysis of strict determinateness 10614.6.Theinterchange of players. Symmetry 10914.7.Non strictly determined games 11014.8.Program of a detailed analysis of strict determinateness 111

*15.GAMES WITH PERFECT INFORMATION 112*15.1. Statement of purpose. Induction 112*15.2.Theexactcondition (First step) 114*15.3.Theexactcondition (Entire induction) 116*15.4.Exact discussion of the inductive step 117*15.5.Exact discussion of the inductive step (Continuation) 120*15.6.Theresult in the caseof perfect information 123*15.7.Application to Chess 124*15.8.Thealternative, verbal discussion 126

16.LINEARITY AND CONVEXITY 12816.1.Geometricalbackground 12816.2.Vector operations 12916.3.Thetheorem of the supporting hyperplanes 13416.4.Thetheorem of the alternative for matrices 138

17.MIXED STRATEGIES. THE SOLUTIONFOR ALL GAMES 14317.1.Discussionof two elementary examples 14317.2.Generalization of this viewpoint 14517.3.Justification of the procedureas applied to an individual play 14617.4.Theminorant and the majorant games. (For mixed strategies) 14917.5.Generalstrict determinateness 15017.6.Proofof the main theorem 15317.7.Comparison of the treatment by pure and by mixed strategies 15517.8.Analysis of generalstrict determinateness 15817.9.Further characteristicsof goodstrategies 16017.10.Mistakes and their consequences.Permanent optimality 16217.11.Theinterchange of players. Symmetry 165

CHAPTER IV

ZERO-SUMTWO-PERSONGAMES: EXAMPLES18.SOME ELEMENTARY GAMES 169

18.1.Thesimplest games 16918.2.Detailedquantitative discussion of thesegames 17018.3.Qualitative characterizations 17318.4.Discussionof somespecificgames. (Generalizedforms of Matching

Pennies) 17518.5.Discussionof someslightly more complicatedgames 17818.6.Chanceand imperfect information 18218.7.Interpretation of this result 185

*19.POKER AND BLUFFING 186*19.1.Descriptionof Poker 186*19.2.Bluffing 188*19.3.Descriptionof Poker (Continued) 189*19.4.Exact formulation of the rules 190)))

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*19.5.Descriptionof the strategy 191*19.6.Statement of the problem 195*19.7.Passagefrom the discreteto the continuous problem 196*19.8.Mathematical determination of the solution 199*19.9.Detailedanalysis of the solution 202*19.10.Interpretation of the solution 204*19.11.Moregeneral forms of Poker 20719.12.Discretehands 208*19.13.m possiblebids 209*19.14.Alternate bidding 211*19.15.Mathematical descriptionof all solutions 216*19.16.Interpretation of the solutions. Conclusions 218

CHAPTERV

ZERO-SUMTHREE-PERSONGAMES20.PRELIMINARY SURVEY 220

20.1.Generalviewpoints 22020.2.Coalitions 221

21.THE SIMPLE MAJORITY GAME OF THREE PERSONS 22221.1.Definition of the game 22221.2.Analysis of the game:Necessityof \"understandings\" 22321.3.Analysis of the game:Coalitions. The role of symmetry 224

22.FURTHER EXAMPLES 22522.1.Unsymmetric distributions. Necessityof compensations 22522.2.Coalitionsof different strength. Discussion 22722.3.An inequality. Formulae 229

23.THE GENERAL CASE 23123.1.Detaileddiscussion. Inessentialand essentialgames 23123.2.Completeformulae 232

24. DISCUSSIONOF AN OBJECTION 23324.1.Thecaseof perfectinformation and its significance 23324.2. Detaileddiscussion. Necessityof compensationsbetween three or

more players 235CHAPTERVI

FORMULATIONOF THE GENERAL THEORY:ZERO-SUMn-PERSONGAMES

25.THE CHARACTERISTIC FUNCTION 23825.1.Motivation and definition 23825.2.Discussionof the concept 24025.3.Fundamental properties 24125.4. Immediate mathematical consequences 242

26.CONSTRUCTIONOF A GAME WITH A GIVEN CHARACTERISTICFUNCTION 24326.1.The construction 24326.2.Summary 245)))

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27. STRATEGIC EQUIVALENCE. INESSENTIAL AND ESSENTIAL GAMES 24527.1.Strategic equivalence.Thereducedform 24527.2.Inequalities. Thequantity y 24827.3. Inessentiality and essentiality 24927.4. Various criteria. Non additive utilities 25027.5.The inequalities in the essentialcase 25227.6.Vector operationson characteristicfunctions 253

28.GROUPS,SYMMETRY AND FAIRNESS 25528.1.Permutations, their groups and their effecton a game 25528.2.Symmetry and fairness 258

29.RECONSIDERATION OF THE ZERO-SUM THREE-PERSONGAME 26029.1.Qualitative discussion 26029.2.Quantitative discussion 262

30.THE EXACT FORM OP THE GENERAL DEFINITIONS 26330.1.Thedefinitions 26330.2.Discussionand recapitulation 265

*30.3. Theconceptof saturation 26630.4. Threeimmediate objectives 271

31.FIRST CONSEQUENCES 27231.1.Convexity, flatness, and somecriteria for domination 27231.2.Thesystem of all imputations. Oneelement solutions 27731.3.The isomorphism which correspondsto strategic equivalence 281

32.DETERMINATION OFALL SOLUTIONSOFTHE ESSENTIAL ZERO-SUMTHREE-PERSONGAME 28232.1.Formulation of the mathematical problem. The graphical method 28232.2.Determination of all solutions 285

33.CONCLUSIONS 28833.1.Themultiplicity of solutions. Discrimination and its meaning 28833.2.Staticsand dynamics 290

CHAPTERVIIZERO-SUMFOUR-PERSONGAMES

34.PRELIMINARY SURVEY 29134.1.Generalviewpoints 29134.2. Formalism of the essentialzerosum four person games 29134.3. Permutations of the players 294

35.DISCUSSIONOF SOME SPECIAL POINTSIN THE CUBEQ 29535.1.Thecorner /. (and V., VI., VII.) 29535.2.Thecorner VIII.(and //.,///.,7F.,).Thethree persongame and

a \"Dummy\" 29935.3.Someremarks concerning the interior of Q 302

36.DISCUSSIONOP THE MAIN DIAGONALS 30436.1.Thepart adjacentto the corner VIII.'Heuristic discussion 30436.2.Thepart adjacentto the corner VIII.:Exact discussion 307

*36.3. Other parts of the main diagonals 312)))

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37.THE CENTER AND ITS ENVIRONS 31337.1.First orientation about the conditions around the center 31337.2.The two alternatives and the roleof symmetry 31537.3.The first alternative at the center 31637.4. Thesecondalternative at the center 31737.5.Comparisonof the two central solutions 31837.6.Unsymmetrical central solutions 319

J8.A FAMILY OPSOLUTIONSFOR A NEIGHBORHOODOPTHE CENTER 321*38.1.Transformation of the solution belonging to the first alternative at

the center 321*38.2. Exactdiscussion 322*38.3. Interpretation of the solutions 327

CHAPTERVIIISOME REMARKS CONCERNINGn ^ 5 PARTICIPANTS

39.THE NUMBER OF PARAMETERS IN VARIOUS CLASSESOF GAMES 33039.1.The situation for n - 3, 4 33039.2.The situation for all n ^ 3 330

10.THE SYMMETRIC FIVE PERSON GAME 33240.1.Formalism of the symmetric five person game 33240.2. The two extreme cases 33240.3. Connectionbetween the symmetric five persongame and the 1,2, 3-

symmetric four person game 334

CHAPTERIXCOMPOSITIONAND DECOMPOSITIONOF GAMES

11.COMPOSITIONAND DECOMPOSITION 33941.1.Searchfor n-persongames for which all solutions can be determined 33941.2.The first type. Composition and decomposition 34041.3.Exactdefinitions 34141.4.Analysis of decomposability 34341.5.Desirability of a modification 345

12.MODIFICATION OF THE THEORY 34542.1.No completeabandonment of the zerosum restriction 34542.2. Strategicequivalence. Constant sum games 34642.3. Thecharacteristicfunction in the new theory 34842.4. Imputations, domination, solutions in the new theory 35042.5. Essentiality, inessentiality and decomposability in the new theory 351

13.THE DECOMPOSITIONPARTITION 35343.1.Splitting sets. Constituents 35343.2. Propertiesof the system of all splitting sets 35343.3. Characterizationof the system of all splitting sets. Thedecomposi-

tion partition 35443.4. Propertiesof the decompositionpartition 357

14.DECOMPOSABLEGAMES. FURTHER EXTENSION OF THE THEORY 35844.1.Solutions of a (decomposable)game and solutions of its constituents 35844.2. Composition and decompositionof imputations and of setsof impu-

tations 359)))

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44.3. Composition anddecomposition of solutions. Themain possibilitiesand surmises 361

44.4. Extension of the theory. Outsidesources 36344.5. Theexcess 36444.6. Limitations of the excess.Thenon-isolatedcharacterof a game in

the new setup 36644.7. Discussionof the new setup. E(eQ)t F(e ) 367

45. LIMITATIONS OF THE EXCESS.STRUCTURE OF THE EXTENDEDTHEORY 36845.1.The lower limit of the excess 36845.2. Theupper limit of the excess.Detachedand fully detachedimputa-

tions 36945.3. Discussionof the two limits, |r|i,|r|2. Their ratio 37245.4. Detachedimputations and various solutions. The theorem con-

necting E(e ), F(e ) 37545.5. Proof of the theorem 37645.6. Summary and conclusions 380

46.DETERMINATION OF ALL SOLUTIONSOF A DECOMPOSABLE GAME 38146.1.Elementary propertiesof decompositions 38146.2. Decompositionand its relation to the solutions: First results con-

cerning F( ) 38446.3. Continuation 38646.4. Continuation 38846.5. Thecompleteresult in F(e ) 39046.6. Thecompleteresult in E(e ) 39346.7. Graphical representation of a part of the result 39446.8. Interpretation: Thenormal zone. Heredity of various properties 39646.9. Dummies 39746.10.Imbedding of a game 39846.11.Significance of the normal zone 40146.12.First occurrenceof the phenomenon of transfer: n - 6 402

47. THE ESSENTIAL THREE-PERSONGAME IN THE NEW THEORY 40347.1.Needfor this discussion 40347.2. Preparatory considerations 40347.3. Thesix casesof the discussion. Cases(I)-(III) 40647.4. Case(IV) :First part 40747.5. Case(IV): Secondpart 40947.6. Case(V) 41347.7. Case(VI) 41547.8. Interpretation of the result: The curves (one dimensional parts) in

the solution 41647.9. Continuation: Theareas (two dimensional parts) in the solution 418

CHAPTER XSIMPLEGAMES

48.WINNING AND LOSINGCOALITIONSAND GAMES WHERE THEYOCCUR 42048.1.Thesecondtype of 41.1.Decisionby coalitions 42048.2. Winning and Losing Coalitions 421)))

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49. CHARACTERIZATION OP THE SIMPLE GAMES 42349.1.Generalconceptsof winning and losing coalitions 423'49.2. Thespecialrole of one element sets 42549.3. Characterization of the systems TF, L of actual games 42649.4. Exact definition of simplicity 42849.5. Someelementary propertiesof simplicity 42849.6. Simple gamesand their W, L. TheMinimal winning coalitions:Wm 42949.7. Thesolutions of simple g^mes 430

50.THE MAJORITY GAMES AND THE MAIN SOLUTION 43150.1.Examples of simple games:Themajority games 43150.2. Homogeneity 43350.3.A more direct use of the conceptof imputation in forming solutions 43550.4. Discussionof this direct approach 43650.5.Connections with the general theory. Exact formulation 43850.6. Reformulation of the result 44050.7. Interpretation of the result 44250.8.Connection with the Homogeneous Majority game. 443

51.METHODSFOR THE ENUMERATION OF ALL SIMPLE GAMES 44551.1.Preliminary Remarks 44551.2.Thesaturation method: Enumeration by means of W 44651.3.Reasonsfor passing from W to Wm. Difficulties of using Wm 44851.4.Changed Approach: Enumeration by means of Wm 45051.5.Simplicity and decomposition 45251.6.Inessentiality, Simplicity and Composition. Treatment of the excess45451.7.A criterium of decomposability in terms of Wm 455

52.THE SIMPLE GAMES FOR SMALL n 45752.1.Program, n = 1,2 play no role. Disposalof n = 3 45752.2.Procedurefor n ^ 4:Thetwo element setsand their role in classify-

ing the Wm 45852.3.Decomposability of casesC*, Cn_2, Cn_i 45952.4. Thesimple games other than [1, , 1,n 2]* (with dummies):

TheCasesCk, k 0, 1, , n - 3 46152.5.Disposalof n = 4, 5 462

53.THE NEW POSSIBILITIESOF SIMPLE GAMES FOR n ^ 6 46353.1.TheRegularities observedfor n ^ 6 46353.2.Thesix main counter examples(for n * 6, 7) 464

54. DETERMINATION OF ALL SOLUTIONSIN SUITABLE GAMES 47054.1.Reasonsto considerother solutions than the main solution in simple

games 47054.2. Enumeration of those games for which all solutions areknown 47154.3. Reasonsto considerthe simple game [1, , 1,n 2]A 472

*55.THE SIMPLE GAME [1, , 1,n - 2]h 473*55.1.Preliminary Remarks 473*55.2. Domination. Thechief player. Cases(I) and (II) 473*55.3. Disposalof Case(I) 475*55.4. Case(II):Determination of Y 478*55.5. Case(II):Determination of V 481*55.6. Case(II):a and S* 484)))

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*55.7. Case(II')and (II\.")Disposalof Case(IF) 485*55.8^Case(II\:")a and V. Domination 487*55.9. Case(II\:")Determination of V 488*55.10.Disposalof Case(II\") 494*55.11.Reformulation of the completeresult 497*55.12.Interpretation of the result 499

CHAPTERXIGENERAL NON-ZERO-SUMGAMES

56.EXTENSION OF THE THEORY 50456.1.Formulation of the problem 50456.2.The fictitious player. Thezerosum extension r 50556.3.Questionsconcerning the characterof r 50656.4. Limitations of the use of r 50856.5.The two possibleprocedures 51056.6.Thediscriminatory solutions 51156.7. Alternative possibilities 51256.8.Thenew setup . 51456.9.Reconsiderationof the casewhen T is a zerosum game 51656.10.Analysis of the conceptof domination 52056.11.Rigorous discussion 52356.12.Thenew definition of a solution 526

57.THE CHARACTERISTIC FUNCTION AND RELATED TOPICS 52757.1.The characteristicfunction: The extendedand the restrictedform 52757.2.Fundamental properties 52857.3.Determination of all characteristicfunctions 53057.4. Removable setsof players 53357.5.Strategicequivalence.Zero-sum and constant-sum games 535

58.INTERPRETATION OF THE CHARACTERISTIC FUNCTION 53858.1.Analysis of the definition 53858.2.Thedesireto make a gain vs. that to inflict a loss 53958.3.Discussion 541

59.GENERAL CONSIDERATIONS 54259.1.Discussionof the program 54259.2.Thereducedforms. The inequalities 54359.3.Various topics 546

60.THE SOLUTIONSOF ALL GENERAL GAMES WITH n ^ 3 54860.1.Thecasen - 1 54860.2.Thecasen - 2 54960.3.Thecasen = 3 55060.4. Comparison with the zerosum games 554

61.ECONOMICINTERPRETATION OF THE RESULTSFOR n = 1,2 55561.1.Thecasen - 1 55561.2.Thecasen = 2. The two person market 55561.3.Discussionof the two person market and its characteristicfunction 55761.4.Justification of the standpoint of 58 55961.5.Divisiblegoods. The \"marginal pairs\" 56061.6.Theprice. Discussion 562)))

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CONTENTS

62.ECONOMICINTERPRETATION OF THE RESULTSFOR n = 3:SPECIAL

CASE 56462.1.Thecasen = 3, specialcase. Thethree personmarket 56462.2.Preliminary discussion 56662.3.Thesolutions: First subcase 56662.4. Thesolutions:Generalform 56962.5. Algebraical form of the result 57062.6.Discussion 571

63.ECONOMICINTERPRETATION OFTHE RESULTSFOR n = 3:GENERAL

CASE 57363.1.Divisible goods 57363.2. Analysis of the inequalities 57563.3. Preliminary discussion 57763.4. Thesolutions 57763.5. Algebraical form of the result 58063.6. Discussion 581

64. THE GENERAL MARKET 58364.1.Formulation of the problem 58364.2. Somespecialproperties. Monopoly and monopsony 584

CHAPTER XIIEXTENSIONOF THECONCEPTSOF DOMINATION

AND SOLUTION65.THE EXTENSION. SPECIAL CASES 587

65.1.Formulation of the problem 58765.2.Generalremarks 58865.3.Orderings, transitivity, acyclicity 58965.4. Thesolutions: For a symmetric relation. For a completeordering 59165.5.Thesolutions: For a partial ordering 59265.6.Acyclicity and strict acyclicity 59465.7.Thesolutions: For an acyclicrelation 59765.8.Uniqueness 'of solutions, acyclicity and strict acyclicity 60065.9.Application to games:Discretenessand continuity 602

66.GENERALIZATION OF THE CONCEPTOF UTILITY 60366.1.Thegeneralization. Thetwo phasesof the theoretical treatment 60366.2. Discussionof the first phase 60466.3.Discussionof the secondphase 60666.4. Desirability of unifying the two phases 607

67. DISCUSSIONOF AN EXAMPLE 60867.1.Descriptionof the example 60867.2. Thesolution and its interpretation 61167.3.Generalization:Different discreteutility scales 61467.4. Conclusions concerning bargaining 616

APPENDIX: THE AXIOMATIC TREATMENT OF UTILITY 617INDEX OF FIGURES 633INDEX OF NAMES 634INDEX OF SUBJECTS 635)))

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CHAPTER IFORMULATIONOF THE ECONOMICPROBLEM

1.The Mathematical Method in Economics1.1.Introductory Remarks

1.1.1.Thepurposeof this bookis to presenta discussionof somefunda-mental questionsof economictheory which requirea treatmentdifferentfrom that which they have found thus far in the literature.Theanalysisis concernedwith somebasic problemsarising from a study of economicbehavior which have beenthe centerof attention of economistsfor a longtime. They have their origin in the attempts to find an exactdescriptionof the endeavorof the individual to obtain a maximum of utility, or,in thecaseof the entrepreneur,a maximum of profit. It is well known whatconsiderableand in fact unsurmounted difficulties this task involvesgiven even a limited number of typical situations, as, for example,in thecaseof the exchangeof goods,director indirect,between two or morepersons,of bilateralmonopoly, of duopoly,of oligopoly,and of free compe-tition. It will be madeclearthat the structureof theseproblems,familiarto every student of economics,is in many respectsquite different from theway in which they are conceivedat the presenttime. It will appear,furthermore, that their exactpositingand subsequentsolutioncan only beachievedwith the aid of mathematical methodswhich divergeconsiderablyfrom the techniquesapplied by olderor by contemporarymathematicaleconomists.

1.1.2.Our considerationswill leadto theapplicationof the mathematicaltheory of \"gamesof strategy\" developedby one of us in severalsuccessivestagesin 1928and 1940-1941.1 After the presentationof this theory, itsapplication to economicproblems in the senseindicated above will beundertaken. Itwill appearthat it providesa new approachto a numberofeconomicquestionsas yet unsettled.

We shall first have to find in which way this theory of gamescan bebrought into relationshipwith economictheory, and what their commonelementsare. This can bedonebestby stating briefly the nature of somefundamental economicproblems so that the common elementswill beseenclearly. It will then becomeapparent that thereis not only nothingartificial in establishingthis relationshipbut that on the contrary this

1The first phasesof this- work were published:J.von Neumann, \"Zur TheoriederGesellschaftsspiele,\"Math. Annalen, vol. 100 (1928),pp. 295-320. The subsequentcompletion of the theory, as well as the more detailed elaboration of the considerationsof loc.cit.above,arepublished here for the first time.

1)))

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2 FORMULATIONOF THEECONOMIC PROBLEM

theory of gamesof strategy is the properinstrument with which to developa theory of economicbehavior.

One would misunderstandthe intent of our discussionsby interpretingthem as merely pointing out an analogy betweenthesetwo spheres.We

hopeto establishsatisfactorily,after developinga few plausibleschematiza-tions, that the typical problems of economicbehavior becomestrictlyidenticalwith the mathematical notions of suitablegamesof strategy.

1.2.Difficulties of the Application of the Mathematical Method

1.2.1.It may be opportuneto beginwith someremarksconcerningthenature of economictheory and to discussbriefly the questionof the rolewhich mathematicsmay take in its development.

Firstletus be aware that thereexistsat presentno universal systemofeconomictheory and that, if one should ever be developed,it will veryprobablynot be during our lifetime. The reasonfor this is simply thateconomicsis far too difficult a scienceto permit its constructionrapidly,especiallyin view of the very limited knowledgeand imperfect descriptionof the facts with which economistsaredealing. Only thosewlio fail toappreciatethis condition arelikelyto attempt the constructionof universalsystems. Even in scienceswhich arefar more advancedthan economics,likephysics,thereis no universal systemavailable at present.

To continue the simile with physics: It happens occasionallythat aparticular physical theory appears to provide the basis for a universalsystem,but in all instancesup to the presenttime this appearancehas notlasted more than a decadeat best. The everyday work of the researchphysicistis certainly not involved with such high aims, but ratheris con-cernedwith specialproblemswhich are\" mature.\" Therewould probablybeno progressat all in physicsif a seriousattempt weremade to enforcethat super-standard.The physicistworks on individual problems,someof greatpracticalsignificance, others of less.Unifications of fields whichwereformerly dividedand far apart may alternatewith this type of work.However,such fortunate occurrencesarerareand happen only after eachfield has beenthoroughly explored.Consideringthe fact that economicsis much more difficult, much lessunderstood,and undoubtedlyin a muchearlierstageof its evolution as a sciencethan physics,oneshouldclearlynotexpectmore than a developmentof theabove type in economicseither.

Secondwe have to noticethat the differences in scientific questionsmake it necessaryto employ varying methodswhich may afterwardshaveto bediscardedif betteronesoffer themselves.This has a doubleimplica-tion:In somebranchesof economicsthe most fruitful work may bethat ofcareful, patient description;indeedthis may be by far the largestdomainfor the presentand for some time to come. In othersit may be possibleto developalready a theory in a strictmanner, and for that purposetheuseof mathematicsmay berequired.)))

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THEMATHEMATICALMETHOD IN ECONOMICS 3

Mathematicshas actually beenused in economictheory, perhapsevenin an exaggeratedmanner. In any caseits usehas not beenhighly suc-cessful. This is contrary to what one observesin othersciences:Theremathematicshas beenappliedwith greatsuccess,and most sciencescouldhardly get along without it. Yet the explanationfor this phenomenon isfairly simple.

1.2.2.It is not that thereexistsany fundamental reason why mathe-maticsshouldnot beused in economics.Theargumentsoften heard thatbecauseof the human element,of the psychologicalfactors etc.,or becausethere is allegedly no measurementof important factors, mathematicswill find no application,can all bedismissedas utterly mistaken. Almostall these objectionshave been made, or might have been made, manycenturiesago in fields where mathematicsis now the chief instrument ofanalysis. This \"

might have been\"is meant in the following sense:Letus try to imagine ourselvesin the periodwhich precededthe mathematicalor almost mathematical phase of the developmentin physics,that is the16th century, or in chemistry and biology, that is the 18thcentury.Taking for granted the skepticalattitude of thosewho objectto mathe-matical economicsin principle,the outlook in the physicaland biologicalsciencesat theseearly periodscan hardly have beenbetterthan that ineconomicsmutatis mutandis at present.

As to the lack of measurementof the most important factors, theexampleof the theory of heat is mostinstructive ; before the developmentofthe mathematical theory the possibilitiesof quantitative measurementswere lessfavorable there than they are now in economics.The precisemeasurementsof the quantity and quality of heat (energyand temperature)were the outcomeand not the antecedentsof the mathematical theory.This ought to be contrastedwith the fact that the quantitative and exactnotions of prices,money and the rateof interestwere already developedcenturiesago.

A further group of objectionsagainst quantitative measurementsineconomics,centersaround the lack of indefinite divisibility of economicquantities. This is supposedlyincompatiblewith the use of the infini-

tesimalcalculusand hence( !)of mathematics.It is hard to seehow suchobjectionscan be maintained in view of the atomic theoriesin physicsandchemistry,the theory of quanta in electrodynamics,etc.,and the notoriousand continuedsuccessof mathematical analysiswithin thesedisciplines.

At this point it is appropriateto mention another familiar argument ofeconomicliteraturewhich may be revived as an objectionagainst themathematical procedure.

1.2.3.In orderto elucidatethe conceptionswhich we areapplying toeconomics,we have given and may give again some illustrations from

physics. Thereare many socialscientistswho objectto the drawing ofsuch parallels on various grounds, among which is generally found theassertionthat economictheory cannot bemodeledafter physicssinceit is a)))

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4 FORMULATIONOF THEECONOMICPROBLEM

scienceof social,of human phenomena,has to takepsychologyinto account,etc. Suchstatementsareat leastpremature. It is without doubt reason-ableto discoverwhat has led to progressin other sciences,and to investigatewhether the applicationof the same principlesmay not lead to progressin economicsalso.Shouldthe needfor the applicationof different principlesarise,it could be revealedonly in the courseof the actual developmentof economictheory. This would itself constitutea major revolution.But sincemost assuredlywe have not yet reachedsuch a state and it isby no means certainthat there ever will be need for entirely differentscientific principles it would be very unwise to consideranything elsethan the pursuit of our problemsin the manner which has resulted in theestablishmentof physicalscience.

1.2.4*The reason why mathematicshas not been more successfulineconomicsmust, consequently,be found elsewhere.The lack of realsuccessis largely due to a combination of unfavorable circumstances,someof which can be removed gradually. Tobeginwith, the economicproblemswerenot formulated clearlyand areoften stated in suchvague terms as tomake mathematical treatmenta prioriappear hopelessbecauseit is quiteuncertain what the problemsreally are. Thereis no point in usingexactmethodswhere thereis no clarity in the conceptsand issuesto which theyareto beapplied.Consequentlythe initial task is to clarify the knowledgeof the matterby further careful descriptive work. But even in thoseparts of economicswhere the descriptiveproblemhas been handled moresatisfactorily,mathematical tools have seldom been used appropriately.They wereeitherinadequatelyhandled,as in the attempts to determineageneraleconomicequilibrium by the merecounting of numbersof equationsand unknowns,or they led to meretranslations from a literary form ofexpressioninto symbols,without any subsequentmathematical analysis.

Next,the empiricalbackgroundof economicscienceis definitely inade-quate. Our knowledgeof the relevant facts of economicsis incomparablysmaller than that commandedin physics at the time when the mathe-matization of that subjectwas achieved. Indeed,the decisivebreakwhichcamein physics in the seventeenth century, specificallyin the field ofmechanics,was possibleonly becauseof previousdevelopmentsin astron-omy. It was backedby severalmillennia of systematic,scientific,astro-nomical observation,culminating in an observer of unparalleledcaliber,TychodeBrahe. Nothing of thissorthasoccurredin economicscience.Itwould have been absurd in physicsto expectKeplerand Newton withoutTycho, and there is no reasonto hope for an easierdevelopment in

economics.Theseobvious comments should not be construed,of course,as a

disparagementof statistical-economicresearchwhich holdsthe realpromiseof progressin the properdirection.

It is due to the combination of the above mentionedcircumstancesthat mathematical economicshasnot achievedvery much. Theunderlying)))

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THEMATHEMATICALMETHOD IN ECONOMICS 5

vaguenessand ignorance has not beendispelledby the inadequate andinappropriate use of a powerful instrument that is very difficult tohandle.

Inthelight of theseremarkswemay describeour own positionasfollows:The aim of this book liesnot in the directionof empiricalresearch.Theadvancement of that side of economicscience,on anything like the scalewhich was recognizedabove as necessary,is clearly a task of vast propor-tions. It may behopedthat as a result of the improvements of scientifictechniqueand of experiencegained in other fields, the developmentof

descriptiveeconomicswill not take as much timeas the comparisonwith

astronomy would suggest. But in any casethe task seemsto transcendthe limits of any individually plannedprogram.

We shallattempt to utilize only somecommonplaceexperienceconcern-ing human behavior which lends itself to mathematical treatmentandwhich is of economicimportance.

We believethat the possibilityof a mathematical treatmentof thesephenomenarefutes the \"fundamental'1 objectionsreferredto in 1.2.2.

It will be seen,however, that this processof mathematization is notat all obvious. Indeed,the objectionsmentioned above may have theirroots partly in the rather obvious difficulties of any directmathematicalapproach. We shall find it necessaryto draw upon techniquesof mathe-maticswhich have not beenusedheretofore in mathematical economics,andit is quitepossiblethat further study may resultin the future in the creationof new mathematical disciplines.

To conclude,we may alsoobservethat part of the feeling of dissatisfac-tion with the mathematical treatment of economictheory derives largelyfrom the fact that frequently one is offered not proofs but mere assertionswhich arereally no betterthan the sameassertionsgiven in literary form.Very frequently the proofs arelackingbecausea mathematical treatmenthas beenattempted of fields which areso vast and so complicatedthat fora long time to come until much more empiricalknowledgeis acquiredthere is hardly any reasonat all to expectprogressmore mathematico.Thefact that thesefields have beenattackedin this way as for examplethe theory of economicfluctuations, the time structureof production,etc.indicates how much the attendant difficulties arebeing underestimated.They areenormousand we arenow in no way equippedfor them.

1.2.6.We have referred to the nature and the possibilitiesof thosechangesin mathematical technique in fact, in mathematicsitself which

a successfulapplicationof mathematics to a new subjectmay produce.It is important to visualize thesein theirproperperspective.

It must not beforgotten that thesechangesmay bevery considerable.Thedecisivephaseof the applicationof mathematicsto physics Newton'screationof a rational disciplineof mechanics brought about, and can

hardly be separatedfrom, the discovery of the infinitesimal calculus.(Thereareseveralother examples,but none strongerthan this.))))

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6 FORMULATIONOF THEECONOMICPROBLEM

The importanceof the socialphenomena,the wealth and multiplicityof theii manifestations, and the complexityof their structure,aieat leastequal to thosein physics. It is therefore to be expectedor feared thatmathematical discoveriesof a staturecomparableto that of calculuswill

be neededin ordei to producedecisivesuccessin this field. (Incidentally,it is in this spirit that our presentefforts must be discounted.)A fortioriit is unlikely that a mererepetitionof the tricks which servedus so well in

physicswill do for the socialphenomenatoo. Theprobabilityis very slimindeed,sinceit will be shown that we encounter in our discussionssomemathematical problemswhich arequite different from thosewhich occurinphysicalscience.

Theseobservationsshouldberememberedin connectionwith the currentoveremphasison the use of calculus,differential equations, etc.,as themain toolsof mathematical economics.

1.3.NecessaryLimitations of the Objectives1.3.1.We have to return, therefore, to the positionindicated earlier:

It is necessaryto begin with those problemswhich aredescribedclearly,even if they shouldnot be as important from any other point of view. Itshouldbeadded,moreover, that a treatment of thesemanageableproblemsmay leadto results which are already fairly well known, but the exactproofs may neverthelessbe lacking. Before they have been given therespectivetheory simplydoesnot existas a scientific theory. The move-ments of the planetswere known long before their courseshad beencalcu-lated and explainedby Newton's theory, and the sameapplies in manysmaller and less dramatic instances. And similarly in economictheory,certainresults say the indeterminatenessof bilateral monopoly may beknown already. Yet it is of interestto derive them again from an exacttheory. The same could and should be said concerningpractically allestablishedeconomictheorems.

1.3.2.It might be added finally that we do not propose to raisethequestion of the practicalsignificance of the problemstreated. This fallsin line with what was said above about the selectionof fields for theory.Thesituation is not different herefrom that in othersciences.Theretoothe most important questionsfrom a practicalpoint of view may have beencompletelyout of reachduring long and fruitful periodsof their develop-ment. This is certainly still the casein economics,where it is of utmostimportance to know how to stabilize employment,how to increasethenational income, or how to distribute it adequately. Nobody can reallyanswer thesequestions,and we need not concernourselveswith the pre-tensionthat therecan be scientific answersat present.

Thegreatprogressin every sciencecamewhen, in the study of problemswhich weremodestas comparedwith ultimate aims, methodsweredevel-opedwhich couldbe extendedfurther and furthei. Thefree fall is a verytrivial physicalphenomenon,but it wasthestudy of this exceedinglysimple)))

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THEMATHEMATICALMETHOD IN ECONOMICS 7

fact and its comparisonwith the astronomicalmaterial, which brought forthmechanics.

It seemsto us that the samestandard of modestyshouldbeappliedin

economics.It is futile to try to explain and \" systematically1' at thateverything economic.The sound procedureis to obtain first utmostprecisionandmasteryin a limited field,and then to proceedto another, some-what wider one,and so on. This would also do away with the unhealthypracticeof applyingso-calledtheoriesto economicor socialreform wherethey arein no way useful.

We believethat it is necessaryto know as much as possibleabout thebehavior of the individual and about the simplestforms of exchange.Thisstandpointwas actually adopted with remarkablesuccessby the foundersof the marginal utility school,but neverthelessit is not generallyaccepted.Economistsfrequently point to much larger,more \"

burning\" questions,and

brush everything aside which prevents them from making statementsabout these. The experienceof more advanced sciences,for examplephysics,indicatesthat this impatiencemerely delays progress,includingthat of the treatment of the \"

burning\" questions. Thereis no reasonto

assumethe existenceof shortcuts.

1.4.Concluding Remarks

1.4.It is essentialto realize that economistscan expectno easierfatethan that which befell scientistsin otherdisciplines.It seemsreasonableto expectthat they will have to take up first problemscontainedin the verysimplestfacts of economiclife and try to establishtheorieswhich explainthem and which really conform to rigorousscientific standards. We canhave enough confidence that from then on the scienceof economicswill

grow further, gradually comprisingmatters of more vital impoitancethanthose with which one has to begin.1

The field coveredin this book is very limited, and we approach it inthis senseof modesty. We do not worry at all if the results of oui studyconform with views gained recently or held for a long time,for what isimportant is the gradual developmentof a theory, based on a carefulanalysisof the ordinary everyday interpretation of economicfacts. Thispreliminary stageis necessarilyheuristic, i.e.the phase of transition fromunmathematical plausibility considerationsto the formal procedureofmathematics.The theory finally obtainedmust bemathematically rigor-ous and conceptuallygeneral. Its first applications are necessarilytoelementary problemswhere the result has never been in doubt and notheory is actually required. At this early stagethe applicationselvestocorroboratethe theory. Thenext stagedevelopswhen the theoryisapplied

1Thebeginning is actually of a certain significance, becausethe forms of exchangebetween a few individuals are the sameas those observedon someof the most importantmarkets of modern industry, or in the caseof barter exchangebetween statesin inter-national trade.)))

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8 FORlVgJLATION OF THEECONOMICPROBLEM

to somewhat more complicatedsituationsin which it may alreadylead to acertainextent beyond the obvious and the familiar. Heretheory andapplicationcorroborateeachother mutually. Beyondthis liesthe field ofrealsuccess:genuine prediction by theory. It is well known that allmathematized scienceshave gone through these successivephases ofevolution.

2.Qualitative Discussionof the Problemof Rational Behavior

2.1.TheProblem of Rational Behavior

2.1.1.The subjectmatter of economictheory is the very complicatedmechanism of pricesand production,and of the gaining and spendingofincomes. In the courseof the development of economicsit has beenfound, and it is now well-nighuniversally agreed,that an approachto thisvast problemis gainedby the analysisof the behavior of the individualswhich constitutethe economiccommunity. This analysishas beenpushedfairly far in many respects,and while therestill existsmuch disagreementthe significanceof the approachcannot be doubted,no matter how greatits difficultiesmay be. The obstaclesareindeedconsiderable,even if theinvestigation shouldat first be limited to conditionsof economicsstatics,asthey well must be. One of the chief difficulties lies in properly describingthe assumptionswhich have to be madeabout the motives of the individual.Thisproblemhas beenstated traditionally by assumingthat the consumerdesiresto obtain a maximum of utility or satisfaction and the entrepreneura maximum of profits.

The conceptualand practicaldifficulties of the notion of utility, andparticularly of the attempts to describeit as a number, arewell known andtheir treatment is not among the primary objectivesof this work. We shallneverthelessbe forced to discussthem in someinstances,in particular in

3.3.and 3.5.Let it be saidat once that the standpointof the presentbookon this very important and very interestingquestionwill be mainly oppor-tunistic. We wish to concentrateon one problem which is not that ofthe measurement of utilities and of preferences and we shall thereforeattempt to simplify all other characteristicsas far as reasonablypossible.We shall therefore assumethat the aim of all participants in the economicsystem, consumersas well as entrepreneurs,is money, or equivalently asinglemonetary commodity. This issupposedto beunrestrictedlydivisibleand substitutable,freely transferable and identical,even in the quantitativesense,with whatever \" satisfaction\"or \"

utility\" is desiredby eachpar-ticipant. (Forthe quantitative characterof utility, cf. 3.3.quotedabove.)

It is sometimesclaimedin economicliterature that discussionsof thenotions of utility and preferencearealtogetherunnecessary,sincethesearepurely verbal definitions with no empirically observableconsequences,i.e.,entirely tautological.It doesnot seemto us that thesenotions arequali-tatively inferior to certainwell establishedand indispensablenotions in)))

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THEPROBLEMOF RATIONAL BEHAVIOR 9

physics, like force, mass, charge,etc. That is, while they are in theirimmediate form merely definitions, they becomesubjectto empiricalcontrolthrough the theorieswhich arebuilt upon them and in no other way.Thus the notion of utility is raisedabove the status of a tautology by sucheconomictheoriesas makeuseof it and the resultsof which canbecomparedwith experienceor at least with common sense.

2.1.2.The individual who attempts to obtain theserespectivemaximais also said to act \"rationally.\" But it may safely be stated that thereexists,at present,no satisfactory treatment of the question of rationalbehavior. Theremay, for example,existseveralways by which to reachthe optimum position;they may dependupon the knowledgeand under-standing which the individual has and upon the paths of action open tohim. A study of all thesequestionsin qualitative terms will not exhaustthem, becausethey imply, as must be evident, quantitative relationships.It would, therefore, be necessaryto formulate them in quantitative termsso that all the elementsof the qualitative descriptionaretaken into con-sideration. This is an exceedinglydifficult task, and we can safely saythat it has not beenaccomplishedin the extensiveliteratureabout thetopic. The chief reasonfor this lies,no doubt, in the failure to developand apply suitable mathematical methods to the problem; this wouldhave revealedthat the maximum problemwhich is supposedto correspondto the notion of rationality is not at all formulated in an unambiguous way.Indeed,a more exhaustive analysis (to be given in 4.S.-4.5.)revealsthatthe significant relationshipsaremuch more complicatedthan the popularand the \" philosophical\"use of the word \" rational\" indicates.

A valuable qualitative preliminary descriptionof the behavior of theindividual is offered by the Austrian School,particularly in analyzing theeconomyof the isolated \" RobinsonCrusoe.\"We may have occasiontonote also some considerationsof Bohm-Bawerkconcerningthe exchangebetweentwo or more persons. Themore recentexpositionof the theory ofthe individual'schoicesin the form of indifference curve analysisbuildsupon the very samefacts or allegedfacts but usesa method which isoften heldto be superiorin many ways. Concerningthis we refer to thediscussionsin2.1.1.and 3.3.

We hope,however, to obtain a realunderstandingof the problemofexchangeby studyingit from an altogetherdifferent angle;this is, from theperspectiveof a \"gameof strategy.\" Our approach will becomeclearpresently,especiallyafter some ideaswhich have beenadvanced,say byBohm-Bawerk whoseviews may beconsideredonly as a prototypeof thistheory aregiven correctquantitative formulation.

2.2.\"Robinson Crusoe\" Economy and SocialExchange Economy

2.2.1.Let us look more closelyat the type of economy which is repre-sentedby the \"RobinsonCrusoe\"model,that is an economyof an isolatedsinglepersonor otherwiseorganizedunder a singlewill. This economy is)))

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10 FORMULATIONOF THEECONOMICPROBLEM

confronted with certainquantitiesof commoditiesand a number of wantswhich they may satisfy. Theproblemis to obtain a maximum satisfaction.This is consideringin particular our above assumptionof the numericalcharacterof utility indeed an ordinary maximum problem,its difficulty

dependingappaiently on the number of variablesand on the nature of thefunction to be maximized;but this is more of a practicaldifficulty than atheoreticalone.1 If one abstracts from continuous productionand fromthe fact that consumption too stretchesover time (and often usesdurableconsumers'goods),one obtains the simplest possible model.It wasthought possibleto use it as the very basis for economic theory, but thisattempt notably a feature of the Austrian version was often contested.The chief objectionagainstusing this very simplified model of an isolatedindividual for the theory of a socialexchangeeconomy is that it does notrepresentan individual exposedto the manifold socialinfluences. Hence,it is said to analyze an individual who might behave quitedifferently if hischoiceswere made in a socialworld where he would be exposedto factorsof imitation, advertising,custom,and so on. Thesefactors certainly makea greatdifference,but it is to be questionedwhether they changethe formalpropertiesof the processof maximizing. Indeedthe latterhas never beenimplied,and sincewe areconcernedwith this problemalone,we can leavethe above socialconsiderationsout of account.

Someother differencesbetween\" Crusoe\" and a participant in a socialexchangeeconomy will not concernus either. Suchis the non-existenceofmoney as a meansof exchangein the first casewhere thereisonly a standardof calculation,for which purposeany commodity can serve. Thisdifficultyindeedhas beenploughedunder by our assumingin 2.1.2.a quantitativeand even monetary notion of utility. We emphasizeagain:Our interestlies in the fact that even after all thesedrastic simplifications Crusoeisconfronted with a formal problemquite different from the one a participantin a socialeconomy faces.

2.2.2.Crusoeis given certain physicaldata (wants and commodities)and his task is to combineand apply them in such a fashion as to obtaina maximum resultingsatisfaction. Therecan be no doubt that he controlsexclusively all the variables upon which this result depends say theallotting of resources,the determination of the usesof the samecommodityfor different wants,etc.2

Thus Crusoefaces an ordinary maximum problem, the difficulties ofwhich areof a purely technical and not conceptualnature, aspointedout.

2.2.3.Considernow a participant in a socialexchangeeconomy. Hisproblemhas, of course,many elementsin common with a maximum prob-

1It is not important for the following to determine whether its theory is completeinall its aspects.

2Sometimes uncontrollable factors also intervene, e.g.the weather in agriculture.Thesehowever arepurely statistical phenomena. Consequently they can be eliminatedby the known proceduresof the calculus of probabilities:i.e.,by determining the prob-abilities of the various alternatives and by introduction of the notion of \" mathematicalexpectation.\" Cf.however the influence on the notion ofutility, discussedin 3.3.)))

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THEPROBLEMOF RATIONAL BEHAVIOR 11lem. But it also contains some,very essential,elementsof an entirelydifferent nature. Hetoo tries to obtain an optimum result. But in orderto achieve this, he must enterinto relationsof exchangewith others. Iftwo or more personsexchangegoodswith eachother,then the result foreachone will depend in generalnot merely upon his own actions but onthoseof the others as well. Thus eachparticipant attempts to maximizea function (hisabove-mentioned\"result\")of which he doesnot control allvariables. This is certainly no maximum problem,but a peculiarand dis-~concertingmixture of severalconflictingmaximum problems. Every parti-cipant is guidedby another principleand neither determinesall variableswhich affect his interest.

This kind of problemis nowhere dealt with in classicalmathematics.We emphasizeat the risk of beingpedanticthat this is no conditional maxi-mum problem, no problem of the calculus of variations, of functionalanalysis, etc. It arises in full clarity, even in the most \" elementary\"situations,e.g.,when all variablescan assumeonly a finite number of values.

A particularly striking expressionof the popular misunderstandingabout this pseudo-maximumproblemis the famous statementaccordingtowhich the purposeof socialeffort is the \"greatestpossiblegood for thegreatestpossible number.\" A guiding principle cannot be formulatedby the requirementof maximizing two (ormore) functions at once.

Sucha principle,taken literally, is self-contradictory, (in generalonefunction will have no maximum where the other function has one.) It isno betterthan saying, e.g.,that a firm should obtain maximum pricesat maximum turnover, or a maximum revenue at minimum outlay. Ifsomeorderof importanceof these principlesor someweightedaverage ismeant, this shouldbe stated. However,in thesituationof the participantsin a socialeconomy nothing of that sort is intended, but all maxima aredesiredat once by various participants.

One would be mistaken to believe that it can be obviated, like thedifficulty in the Crusoecasementioned in footnote 2 on p.10,by a mererecourseto the devicesof the theory of probability. Every participantcandeterminethe variableswhich describehis own actionsbut not thoseof theothers. Neverthelessthose \"alien\" variablescannot,from his point of view,be describedby statistical assumptions. This is becausethe othersareguided,just as he himself, by rational principles whatever that may mean

and no modus procedendican be correctwhich doesnot attempt to under-stand thoseprinciplesand the interactionsof the conflicting interestsof allparticipants.

Sometimessomeof theseinterests run more 01 less parallel then wearenearerto a simplemaximum problem.But they can just as well beopposed.The generaltheory must cover all thesepossibilities,all inter-mediarystages,and all their combinations.

2.2.4.The difference betweenCrusoe'sperspectiveand that of a par-ticipant in a socialeconomy can alsobeillustratedin this way:Apart from)))

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12 FORMULATIONOF THEECONOMIC PROBLEM

those variableswhich his will controls,Crusoeis given a number of datawhich are \"dead\";they are the unalterable physicalbackgroundof thesituation. (Even when they are apparently variable, cf. footnote 2 onp. 10,they are really governed by fixed statistical laws.) Not a singledatum with which hehas to deal reflectsanother person'swill or intentionof an economickind basedon motives of the samenature as his own. A

participant in a socialexchangeeconomy,on the otherhand, faces dataof this last type as well:they arethe productof other participants'actionsand volitions (likeprices).Hisactionswill be influencedby hisexpectationof these,and they in turn reflect the other participants'expectationof hisactions.

Thus the study of the Crusoeeconomy and the use of the methodsapplicableto it, is of much more limited value to economictheory thanhasbeenassumedheretofore even by themostradicalcritics. Thegroundsfor this limitation lie not in the field of thosesocial relationshipswhichwe have mentionedbefore although we donot questiontheir significancebut rather they arise from the conceptualdifferences betweenthe original(Crusoe's)maximum problemand the more complexproblemsketchedabove.

We hope that the readerwill be convinced by the above that we facehereand now a really conceptual and not merely technical difficulty.And it is this problemwhich the theory of \" gamesof strategy\"is mainlydevisedto meet.

2.3.TheNumber of Variables and the Number of Participants

2.3.1.Theformal set-upwhich we used in the precedingparagraphstoindicatethe events in a socialexchangeeconomymadeuse of a number of\" variables\"which describedthe actionsof the participantsin this economy.Thus every participantis allotteda setof variables,\"his\" variables,whichtogethercompletelydescribehis actions,i.e.expresspreciselythe manifes-tations of his will. We call thesesetsthe partial setsof variables. Thepartial setsof all participantsconstitutetogetherthe setof all variables,tobecalledthe total set. Sothe total numberof variablesis determinedfirst

by thenumberof participants,i.e.of partialsets,and secondby the numberof variablesin every partial set.

From a purely mathematical point of view there would be nothingobjectionablein treatingall the variablesof any onepartial setas a singlevariable, \"the\" variable of the participant correspondingto this partialset. Indeed,this is a procedurewhich we aregoing to use frequently inour mathematical discussions;it makes absolutely no difference con-ceptually,and it simplifiesnotationsconsiderably.

Forthe moment, however, we proposeto distinguishfrom eachotherthevariables within eachpartial set. The economicmodelsto which one isnaturally led suggestthat procedure;thus it is desirableto describeforevery participantthe quantity of every particulargoodhewishesto acquireby a separatevariable, etc.)))

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THEPROBLEM OF RATIONAL BEHAVIOR 132.3.2.Now we must emphasizethat any increaseof the number of

variables inside a participant'spartial set may complicateour problemtechnically, but only technically. Thus in a Crusoeeconomy wherethereexistsonly oneparticipant and only one partial setwhich then coin-cideswith the total set this may make the necessarydeterminationof amaximum technicallymore difficult, but it will not alterthe \"puremaxi-mum \" characterof the problem.If, on the otherhand, the number ofparticipants i.e.,of the partial setsof variables is increased,somethingof a very different nature happens. To use a terminology which will turnout to be significant, that of games,this amounts to an increasein thenumber of players in the game. However,to takethe simplestcases,athree-persongame is very fundamentally different from a two-persongame,a four-persongame from a three-persongame,etc. The combinatorialcomplicationsof the problem which is, as we saw, no maximum problemat all increasetremendouslywith every increasein the number of players,

as our subsequentdiscussionswill amply show.We have gone into this matter in such detail particularly becausein

mostmodelsof economicsa peculiarmixture of thesetwo phenomenaoccurs.Whenever the number of players,i.e.of participants in a socialeconomy,increases,the complexityof the economicsystem usually increasestoo;e.g.the number of commoditiesand services exchanged,processesofproductionused,etc. Thus the number of variablesin every participant'spartial set is likely to increase.But the number of participants, i.e.ofpartialsets,has increasedtoo. Thus both of the sourceswhich we discussedcontributeparipassuto the total increasein the number of variables. It isessentialto visualize eachsourcein its properrole.

2.4.TheCaseof Many Participants :Free Competition

2.4.1.In elaborating the contrast between a Crusoeeconomy and asocialexchangeeconomy in 2.2.2.-2.2.4.,we emphasizedthose featuresof thelatterwhich becomemore prominent when thenumber of participants

while greaterthan 1 is of moderatesize. The fact that every partici-pant is influenced by the anticipated reactionsof the others to his ownmeasures,and that this is true for eachof the participants,ismoststrikinglythe cruxof the matter (as far as the sellersareconcerned)in the classicalproblemsof duopoly, oligopoly,etc. When the number of participantsbecomesreally great,somehope emergesthat the influence of every par-ticularparticipant will becomenegligible,and that the above difficultiesmay recedeand a more conventional theory becomepossible.Theseare,of course,the classicalconditionsof \"free competition.\"Indeed,thiswas the starting point of much of what is best in economictheory. Com-pared with this caseof greatnumbers free competition the casesof smallnumbers on the side of the sellersmonopoly, duopoly,oligopoly wereeven consideredto be exceptionsand abnormities. (Even in thesecasesthe number of participants is still very largein view of the competition)))

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14 FORMULATIONOF THEECONOMICPROBLEM

among the buyers. Thecasesinvolving really smallnumbersarethose ofbilateralmonopoly, of exchangebetweena monopoly and an oligopoly,ortwo oligopolies,etc.)

2.4.2.In all fairnessto the traditional point of view this much oughtto besaid:It is a well known phenomenon in many branchesof the exactand physicalsciencesthat very greatnumbersareoften easierto handlethan thoseof medium size. An almost exacttheory of a gas,containingabout 1026 freely moving particles,is incomparablyeasierthan that of thesolarsystem,madeup of 9 major bodies;and still more than that of a mul-

tiplestar of threeor four objectsof about the samesize. Thisis, of course,due to the excellentpossibilityof applyingthe laws of statistics and prob-abilitiesin the first case.

Thisanalogy, however, is far from perfectfor our problem. Thetheoryof mechanicsfor 2, 3, 4, bodies is well known, and in its generaltheoretical(asdistinguishedfrom its specialand computational) form is thefoundation of the statistical theory for great numbers. For the socialexchangeeconomy i.e.for the equivalent \" gamesof strategy\" the theoryof 2, 3,4, participants was heretofore lacking. It is this need thatour previous discussionswere designedto establishand that our subsequentinvestigationswill endeavor to satisfy. In other words, only after thetheory for moderatenumbersof participantshas beensatisfactorilydevel-oped will it be possibleto decidewhether extremelygreatnumbersof par-ticipants simplify the situation. Let us say it again:We share the hopechiefly becauseof the above-mentionedanalogy in otherfields! that suchsimplifications will indeed occur. The currentassertionsconcerningfreecompetition appear to be very valuable surmisesand inspiringanticipationsof results. But they arenot resultsand it is scientifically unsoundto treatthem as such as long as the conditionswhich we mentionedabove arenotsatisfied.

Thereexistsin the literaturea considerableamount of theoreticaldis-cussionpurportingto show that the zonesof indeterminateness(of ratesofexchange) which undoubtedlyexistwhen the number of participants issmall narrow and disappearas the number increases.This then wouldprovidea continuous transition into the idealcaseof free competition fora very greatnumber of participants where all solutionswould be sharplyand uniquely determined.While it is to be hopedthat this indeedturns outto be the casein sufficient generality,one cannot concedethat anythinglike this contention has beenestablishedconclusively thus far. Thereisno gettingaway from it:The problem must be formulated, solved andunderstoodfor smallnumbersof participantsbefore anything canbe provedabout the changesof its characterin any limiting caseof largenumbers,suchas free competition.

2.4.3.A really fundamental reopeningof this subject is the moredesirablebecauseit is neithercertainnor probablethat a mereincreaseinthe number of participants will always lead in fine to the conditions of)))

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THE NOTION OF UTILITY 15free competition.Theclassicaldefinitions of free competitionall involvefurther postulatesbesidesthe greatnessof that number. E.g.,it is clearthat if certaingreatgroupsof participantswill for any reasonwhatsoeveract together,then the great number of participants may not becomeeffective; the decisiveexchangesmay take placedirectly between large\" coalitions,\" l few in number, and not betweenindividuals, many in number,actingindependently. Our subsequentdiscussionof \" gamesof strategy\"will show that the roleand sizeof \" coalitions\" is decisivethroughout theentiresubject. Consequentlythe above difficulty though not new stillremains the crucial problem.Any satisfactory theory of the \"

limitingtransition \" from small numbersof participants to largenumberswill haveto explainunder what circumstancessuch big coalitionswill orwill not beformed i.e.when the largenumbersof participants will becomeeffectiveand lead to a more or lessfree competition.Which of thesealternativesislikelyto arisewill dependon the physicaldata of the situation. Answeringthis questionis,we think, the real challengeto any theoryof free competition.

2.5.The \"Lausanne\" Theory

2.6.This sectionshould not be concludedwithout a referenceto theequilibrium theory of the LausanneSchooland alsoof various othersystemswhich take into consideration \" individual planning \" and interlockingindividual plans. All thesesystemspay attention to the interdependenceof the participantsin a socialeconomy. This, however, is invariably doneunder far-reaching restrictions.Sometimesfree competitionis assumed,after the introduction of which the participants face fixed conditionsandact like a number of RobinsonCrusoes solelybent on maximizing theirindividual satisfactions,which undertheseconditionsareagain independent.In othercasesother restrictingdevicesareused, all of which amount toexcludingthe free play of \" coalitions\" formed by any or all types of par-ticipants. Thereare frequently definite, but sometimeshidden, assump-tions concerningthe ways in which their partly paralleland partly oppositeinterestswill influence the participants,and causethem to cooperateor not,as the casemay be. We hopewehave shown that sucha procedureamountsto a petitio principii at least on the planeon which we shouldlike to putthe discussion.Itavoids the real difficulty and dealswith a verbal problem,which is not the empiricallygiven one. Of coursewe do not wish to ques-tion the significance of theseinvestigations but they do not answerourqueries.

3.TheNotion of Utility

3.1.Preferencesand Utilities

3.1.1.We have stated alreadyin 2.1.1.in what way we wish to describethe fundamental conceptof individual preferencesby the use of a rather

1 Such as trade unions, consumers'cooperatives,industrial cartels,and conceivablysomeorganizations more in the political sphere.)))

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far-reaching notion of utility. Many economistswill feel that we areassumingfar too much (cf.the enumeration of the propertieswe postulatedin 2.1.1.),and that our standpointis a retrogressionfrom the more cautiousmodern techniqueof \" indifferencecurves.\"

Before attempting any specificdiscussionlet us state as a generalexcusethat our procedureat worst is only the applicationof a classicalpreliminary deviceof scientific analysis:To divide the difficulties, i.e.toconcentrateon one (the subjectproper of the investigation in hand), andto reduceall othersas far as reasonablypossible,by simplifying and schema-tizing assumptions. We shouldalso add that this high handed treatmentof preferencesand utilities is employedin the main body of our discussion,but weshall incidentally investigate to a certainextentthe changeswhich anavoidance of the assumptionsin questionwould causein our theory (cf. 66.,67.).

We feel, however, that onepart of our assumptionsat least that oftreatingutilities as numerically measurable quantities is not quite asradical as is often assumedin the literature.We shall attempt to provethis particular point in the paragraphswhich follow. It is hopedthat thereaderwill forgive us for discussingonly incidentally in a condensedforma subjectof so greata conceptualimportanceas that of utility. It seemshowever that even a few remarks may be helpful, becausethe questionof the measurability of utilities is similar in characterto correspondingquestionsin the physicalsciences.

3.1.2.Historically,utility was fiist conceivedas quantitatively measur-able,i.e.as a number. Valid objectionscan be and have beenmadeagainstthis view in its original, naive form. It is clearthat every measurementor rather every claim of measurability must ultimately be basedon someimmediate sensation, which possiblycannot and certainly need not beanalyzed any further. 1 In the caseof utility the immediatesensationofpreference of one objector aggregateof objectsas against anotherprovidesthis basis. But this permitsus only to say when for one persononeutility is greaterthan another. It is not in itself a basisfor numericalcomparisonof utilities for one person nor of any comparisonbetweendifferent persons. Sincethereis no intuitively significant way to add twoiUtilities for the same person, the assumption that utilities are of non-Jnumericalcharactereven seemsplausible. Themodern methodof indiffer-encecurve analysisis a mathematical procedureto describethis situation.

3.2.Principlesof Measurement :Preliminaries

3.2.1.All this is strongly reminiscentof the conditionsexistantat thebeginning of the theory of heat:that too wasbasedon the intuitively clearconceptof one bodyfeeling warmer than another, yet therewas no immedi-ate way to expresssignificantly by how much, or how many times, or inwhat sense.

1Such as the sensations of light, heat, muscular effort, etc.,in the correspondingbranchesof physics.)))

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THENOTION OFUTILITY 17

Thiscomparisonwith heat alsoshowshow little onecanforecasta prioriwhat the ultimate shapeof sucha theory will be. Theabove crudeindica-tions do not discloseat all what, as we now know, subsequentlyhappened.It turned out that heat permitsquantitative descriptionnot by one numbeibut by two:the quantity of heat and temperature.The former is ratherdirectly numerical becauseit turned out to be additive and also in anunexpectedway connectedwith mechanical energy which was numericalanyhow. The latter is also numerical, but in a much more subtle way;it is not additive in any immediatesense,but a rigidnumerical scalefor it

emergedfrom the study of the concordantbehavior of idealgases,and therole of absolutetemperaturein connectionwith theentropy theorem.

3.2.2.The historicaldevelopmentof the theory of heat indicatesthatone must be extremelycareful in making negative assertionsabout anyconceptwith the claim to finality. Even if utilities lookvery unnumericaltoday, the history of the experiencein the theory of heat may repeatitself,and nobodycan foretell with what ramifications and variations. 1 And itshould certainly not discouragetheoreticalexplanationsof the formalpossibilitiesof a numerical utility.

3.3.Probability and Numerical Utilities

3.3.1.We can go even onestep beyond the above doublenegationswhich were only cautionsagainstprematureassertionsof the impossibilityof a numerical utility. Itcan beshown that under the conditionson whichthe indifference curve analysisis based very littleextraeffort is neededtoreacha numerical utility.

Ithasbeenpointedout repeatedlythat a numerical utility is dependentupon the possibilityof comparing differencesin utilities. Thismay seemand indeedis a more far-reaching assumptionthan that of a mereabilityto statepreferences.But it will seemthat the alternatives to which eco-nomic preferencesmust be appliedaresuchas to obliteratethis distinction.

3.3.2.ILetus for the moment acceptthe pictureof an individual whosesystem ol preferencesis all-embracmg^md-eemplete,i^e^who^forany twoobjectsor rather for any two imaginedevents,possessesa clearintuition ofpreference.

Morepreciselywe expecthim, for any two alternative eventswhich areput before him as possibilities,to beable to tell which of the two heprefers.

It is a very natural extensionof this pictureto permitsuchan individualto comparenot only events, but even combinationsof events with statedprobabilities.2

By a combination of two events we mean this:Let the two events bedenotedby B and C and use, for the sakeof simplicity,the probability

1 A goodexampleof the wide variety of formal possibilities is given by the entirelydifferent development of the theory of light, colors,and wave lengths. All thesenotionstoo becamenumerical, but in an entirely different way.

2Indeedthis is necessaryif he is engagedin economicactivities which are explicitlydependenton probability. Of.the example of agriculture in footnote 2 on p. 10.)))

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50%-50%.Then the \" combination\"is the prospect of seeingB occurwith a probabilityof 50%and (if B doesnot occur)C with the (remaining)probability of 50%. We stressthat the two alternatives are mutuallyexclusive,so that no possibilityof complementarity and the like exists.Also, that an absolutecertainty of the occurrenceof eitherB or C exists.

Torestateour position. We expectthe individual under considerationto possessa clearintuition whether he prefers the event A to the 50-50combination of B or C, or conversely. It is clearthat if he prefersA to Band also to C, then he will prefer it to the above combination as well;similarly, if he prefersBaswellas C to A, then he will prefei the combinationtoo. But if he shouldprefer A to, say B,but at the sametime C to A, thenany assertionabout his preferenceof A against the combination containsfundamentally new information. Specifically:If he now prefersA to the50-50combination of Band C, this providesa plausiblebasefor the numer-ical estimatethat his preferenceof A over B is in excessof his preferenceofC over A. 1-2

If this standpoint is accepted,then there is a criterionwith which tocomparethe preferenceof C over A with the preferenceof A over B. It iswell known that therebyutilities or rather differencesof utilities becomenumerically measurable.

That the possibilityof comparisonbetweenA, B,and C only to thisextent is already sufficient for a numerical measurementof \" distances\"was first observedin economicsby Pareto. Exactly the same argumenthas beenmade,however, by Euclid for the positionof points on a line infact it is the very basisof his classicalderivation of numerical distances.

The introduction of numerical measurescan be achievedeven moredirectly if use is made of all possibleprobabilities.Indeed:Considerthreeevents, C, A, B,for which the orderof the individual'spreferencesis the onestated. Let a bea realnumber between and 1,such that A

is exactlyequallydesirablewith the combinedevent consistingof a chanceof probability1 a for Band the remaining chanceof probabilitya.for C.Thenwe suggestthe use of a as a numerical estimatefor the ratio of thepreferenceof A over B to that of C over B.8 An exactand exhaustive

1Togive a simple example:Assume that an individual prefers the consumption of aglassof tea to that of a cup of coffee,and the cup of coffeeto a glassof milk. If we nowwant to know whether the last preferencei.e.,difference in utilities exceedsthe former,it suffices to placehim in a situation where he must decidethis: Doeshe prefera cup ofcoffeeto a glassthe content of which will bedetermined by a 50%-50%chancedeviceastea or milk.

1Observethat we have only postulated an individual intuition which permits decisionas to which of two \"events\" is preferable. But we have not directly postulated anyintuitive estimate of the relative sizesof two preferencesi.e.in the subsequent termi-nology, of two differences of utilities.

This is important, sincethe former information ought to beobtainablein a reproduci-bleway by mere \"questioning.\"1This offers a goodopportunity for another illustrative example. Theabovetech-nique permits a direct determination of the ratio q of the utility of possessing1unit of acertain goodto the utility of possessing2 units of the samegood. Theindividual must)))

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THENOTION OF UTILITY 19elaborationof theseideasrequiresthe useof theaxiomaticmethod. A sim-ple treatment on this basis is indeed possible.We shall discuss it in3.5-3.7.

3.3.3.To avoid misunderstandingslet us state that the \" events\"which wereused above as the substratum of preferencesareconceivedasfuture events so as to make all logically possible alternatives equallyadmissible. However, it would be an unnecessarycomplication,as faras our presentobjectivesareconcerned,to getentangledwith the problemsof the preferencesbetweenevents in different periodsof the future. 1 Itseems,however, that such difficulties can be obviated by locating all\"events\"in which we areinterestedat one and the same, standardized,moment, preferably in the immediatefuture.

Theabove considerationsareso vitally dependentupon the numericalconceptof probability that a few words concerningthe latter may beappropriate.

Probability has often been visualized as a subjective conceptmoreor lessin the nature of an estimation. Sincewe proposeto use it in con-structing an individual, numerical estimationof utility, the above view ofprobabilitywould not serveour purpose. Thesimplestprocedureis, there-fore, to insistupon the alternative, perfectly well founded interpretationofprobabilityas frequency in long runs. This gives directly the necessarynumerical foothold.2

3.3.4.This procedurefor a numerical measurementof the utilitiesof theindividual depends,of course,upon the hypothesisof completenessin thesystem of individual preferences.8 It is conceivable and may even in away be more realisticto allow for caseswhere the individual is neitherable to statewhich of two alternativeshe prefersnor that they areequallydesirable.In this casethe treatment by indifference curves becomesimpracticabletoo.4

Howrealthis possibilityis, both for individuals and for organizations,seemsto be an extremelyinterestingquestion,but it is a questionof fact.It certainlydeservesfurther study. We shall reconsiderit briefly in 3.7.2.

At any ratewe hopewe have shown that the treatmentby indifferencecurves implieseithertoo much or too little:if the preferencesof the indi-

begiven the choiceof obtaining 1 unit with certainty or of playing the chanceto get twounits with the probability <*, or nothing with the probability 1 a. If he prefers the

former, then a < 5; if he prefers the latter, then a > g; if he cannot state a preferenceeither way, then a = q.1It is well known that this presents very interesting, but as yet extremely obscure,connectionswith the theory of saving and interest, etc.

2If one objectsto the frequency interpretation of probability then the two concepts(probability and preference)can be axiomatized together. This too leadsto a satis-factory numerical conceptof utility which will bediscussedon another occasion.

8 We have not obtained any basis for a comparison, quantitatively or qualitatively,of the utilities of different individuals.

4Theseproblems belong systematically in the mathematical theory of orderedsets.The abovequestion in particular amounts to asking whether events, with respecttopreference,form a completely or a partially orderedset. Cf.65,3.)))

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vidual arenot all comparable,then the indifference curves do not exist.1If theindividual'spreferencesareall comparable,then we can even obtaina(uniquely defined) numeiicalutility which renders the indifference curvessuperfluous.

All this becomes,of course,pointless for the entrepreneurwho cancalculatein terms of (monetary)costsand profits.

3.3.5.Theobjectioncouldbe raised that it is not necessaryto go intoall these intricatedetails concerningthe measurabilityof utility, sinceevidently the common individual, whosebehavior onewants to describe,doesnot measurehis utilities exactly but rather conducts his economicactivitiesin a sphereof considerablehaziness. Thesameis true,of course,for much of his conductregardinglight, heat,musculareffort, etc. But inorderto build a scienceof physicsthesephenomenahad to be measured.And subsequentlythe individual hascometo usethe resultsof suchmeasure-ments directly or indirectly even in his everyday life. The samemayobtain in economicsat a future date. Once a fuller understanding ofeconomicbehavior has beenachievedwith the aid of a theory which makesuseof this instrument, the life of the individual might bematerially affected.It is, therefore, not an unnecessarydigressionto study theseproblems.

3.4.Principlesof Measurement:DetailedDiscussion3.4.1.The readermay feel, on the basis of the foregoing, that we

obtaineda numerical scaleof utility only by beggingthe principle,i.e.byreally postulatingthe existenceof such a scale. We have argued in 3.3.2.that if an individual prefersA to the 50-50combination of B and C (whilepreferring C to A and A to JB), this providesa plausiblebasisfor the numer-ical estimatethat this preferenceof A over B exceedsthat of C over A.Are we not postulatinghere or taking it for granted that one preferencemay exceedanother, i.e.that such statementsconvey a meaning? Sucha view would be a completemisunderstandingof our procedure.

3.4.2.We arenot postulating or assuming anything of the kind. Wehave assumedonly one thing and for this thereis goodempiricalevidence

namely that imagined events can be combinedwith probabilities. Andtherefore the same must be assumedfor the utilities attachedto them,whatever they may be. Or to put it in more mathematical language:

Therefrequently appear in sciencequantities which are a priorinotmathematical, but attached to certain aspectsof the physical world.Occasionallythesequantities can be grouped togetherin domainswithin

which certainnatural, physicallydefined operationsare possible.Thusthephysicallydefined quantity of \"mass\"permitsthe operationof addition.Thephysico-geometricallydefined quantity of \"distance\"2 permitsthesame

1Points on the same indifference curve must be identified and are therefore noinstances of incomparability.

f Let us, for the sakeof the argument, view geometry as a physical discipline, asufficiently tenable viewpoint. By \"geometry\" we mean equally for the sakeof theargument Euclidean geometry.)))

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THE NOTION OF UTILITY 21operation.On the other hand, the physico-geometricallydefined quantityof \" position\"doesnot permit this operation,1but it permitsthe operationof forming the \" centerof gravity\" of two positions.2 Again other physico-geometricalconcepts,usually styled \" vectorial\" likevelocity andaccelera-tion permit the operationof \" addition.\"

3.4.3.In all thesecaseswhere such a \" natural\" operationis given aname which is reminiscentof a mathematical operation like the instancesof \" addition\"above one must carefully avoid misunderstandings. Thisnomenclature is not intended as a claim that the two operationswith thesamename areidentical, this is manifestly not the case;it only expressesthe opinion that they possesssimilar traits, and the hope that somecor-respondencebetweenthem will ultimately be established.Thisof coursewhen feasibleat all is done by finding a mathematical model for thephysicaldomain in question,within which those quantitiesaredefined bynumbers,so that in the model the mathematical operationdescribesthesynonymous \" natural\" operation.

To return to our examples:\" energy\"and \"mass\"becamenumbersinthe pertinent mathematical models,\"natural\"additionbecomingordinaryaddition. \"Position\"as well as the vectorial quantitiesbecametriplets3 ofnumbers,calledcoordinatesor componentsrespectively. The \"natural\"conceptof \"centerof gravity\" of two positions{#1,x%, x3) and \\x' ly x'2, z'a}/with the \"masses\"a, 1 a (cf. footnote 2 above),becomes

{ax,+ (1- a)x(,ax,+ (1- a)*J,ax,+ (1- <*X).5

The\"natural\" operationof \"addition\"of vectors{zi,x2, x*\\ and [x(,z,x'z \\

becomes{xi+ x[,x2 + x2, x* + ZgJ.6

What was said above about \"natural\"and mathematical operationsappliesequally to natural and mathematical relations. The various con-ceptsof \"greater\"which occurin physics greaterenergy, force, heat,velocity, etc. aregoodexamples.

These\"natural\" relationsare the best base upon which to constructmathematical modelsand to correlatethe physicaldomain with them.7'8

1 We arethinking ofa \"homogeneous\" Euclidean space,in which no origin or frame ofreferenceis preferred aboveany other.

2 With respectto two given massesa, occupying those positions. It may be con-venient to normalize sothat the total massis the unit, i.e.**1 *.

3 We are thinking of three-dimensional Euclidean space.4 We arenow describing them by their three numerical coordinates.8 This isusually denotedby a (xi,z2,z8 1 + (1-a)js|,xj, *',) Cf.(16:A:c)in 16.2.1.8 This is usually denotedby (xi, x*, xs\\ -f (z'i, zj, xj|. Cf.the beginning of 16.2.1.7 Not the only one. Temperature is a good counter-example. The \"natural\" rela-

tion of \"greater,\" would not have sufficed to establish the present day mathematicalmodel, i.e.the absolute temperature scale. Thedevicesactually used weredifferent.Cf.3.2.1.

8 We do not want to give the misleading impression of attempting here a completepicture of the formation of mathematical models,i.e.of physical theories. It should beremembered that this is a very varied processwith many unexpectedphases. An impor-tant one is, e.g.,the disentanglement of concepts:i.e.splitting up something which at)))

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3.4.4.Herea further remarkmust bemade. Assumethat a satisfactorymathematical model for a physicaldomain in the above sensehas beenfound, and that the physical quantities under considerationhave beencorrelatedwith numbers. In this caseit is not true necessarilythat thedescription (of the mathematical model) provides for a unique way ofcorrelatingthe physicalquantitiesto numbers;i.e.,it may specifyan entirefamily of such correlationsthe mathematical name is mappings anyoneof which can beusedfor the purposesof the theory. Passagefrom oneof thesecorrelationsto another amounts to a transformation of the numericaldata describingthe physicalquantities. We then say that in this theorythe physicalquantities in questionaredescribedby numbers up to thatsystemof transformations. Themathematical name ofsuchtransformationsystemsis groups.1

Examplesof such situationsarenumerous. Thus the geometricalcon-ceptof distanceis a number, up to multiplication by (positive)constantfactors.2 The situation concerningthe physical quantity of mass is thesame. Thephysicalconceptof energyis a number up to any linear trans-formation, i.e.addition of any constant and multiplication by any (posi-tive) constant.8 Theconceptof positionis defined up to an inhomogeneousorthogonal linear transformation. 4-B The vectorial conceptsare definedup to homogeneoustiansformationsof the samekind.5'6

3.4.6.It is even conceivablethat a physicalquantity is a number up toany monotone transformation. This is the casefor quantities for which

only a \"natural\" relation \" greater\" exists and nothing else. E.g.thiswas the casefor temperatureas long as only the conceptof \" warmer \" wasknown;7 it appliesto the Mohs'scaleof hardnessof minerals;it appliesto))

superficial inspection seemsto beone physical entity into severalmathematical notions.Thus the \"disentanglement\" of forceand energy, of quantity of heat and temperature,weredecisivein their respectivefields.

It is quite unforeseeablehow many such differentiations still lie ahead in economictheory.

1 We shall encounter groups in another context in 28.1.1,where referencesto theliterature arealsofound.

*I.e.there is nothing in Euclidean geometry to fix a unit of distance.3I.e.there is nothing in mechanics to fix a zeroora unit ofenergy. Cf.with footnote 2

above. Distancehas a natural zero, the distance of any point from itself.4I.e.|*i,x,X||areto bereplacedby {xi*,xa*,x9*\\ where))

-f OU.TI 4-013X3 + 61,*i* - 0*1X1 -f 022X1 -f 023X3 -f 62,^i* - 031X1 -f as2X S + 033X3 + &3,))

the a</, bi being constants, and the matrix (a,/) what is known asorthogonal.I.e.there is nothing in geometry to fix either origin or the frame of referencewhenpositions are concerned;and nothing to fix the frame of referencewhen vectors areconcerned.

f I.e.the bi in footnote 4 above. Sometimes a wider conceptof matrices ispermissible, all those with determinants ^ 0. We neednot discussthesematters here.'But no quantitatively reproducible method of thermometry .)))

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THENOTION OF UTILITY 23

the notion of utility when this is based on the conventional idea of prefer-ence. In thesecasesonemay betemptedto taketheviewthat thequantityin questionis not numerical at all, consideringhow arbitrary thedescriptionby numbersis. It seemsto be preferable,however, to refrain from suchqualitative statements and to stateinstead objectivelyup to what systemof transformations the numerical description is determined.The casewhen the system consistsof all monotone transformations is, of course,aratherextremeone;various graduationsat the otherend of the scalearethe transformation systems mentioned above:inhomogeneousor homo-geneousorthogonal linear transformations in space,linear transformationsof one numerical variable, multiplication of that variable by a constant.1In fine, the caseeven occurswhere the numerical descriptionis absolutelyrigorous,i.e.where no transformations at all needbe tolerated.2

3.4.6.Given a physicalquantity, the system of transformations up towhich it is describedby numbersmay vary in time, i.e.with the stageofdevelopmentof the subject. Thus temperaturewas originally a numberonly up to any monotone transformation. 8 With the developmentofthermometry particularly of the concordantideal gas thermometry thetransformations were restrictedto the linearones,i.e.only the absolutezero and the absolute unit were missing. Subsequentdevelopmentsofthermodynamicseven fixed the absolute zero so that the transformationsystemin thermodynamicsconsistsonly of the multiplication by constants.Examplescouldbe multipliedbut thereseemsto be no needto go into thissubjectfurther.

For utility the situation seemsto be of a similar nature. One maytake the attitude that the only \"natural\" datum in this domain is therelation \"greater,\"i.e.the conceptof preference.In this caseutilitiesarenumerical up to a monotone transformation. This is, indeed,thegenerallyacceptedstandpoint in economicliterature,best expressedin the techniqueof indifference curves.

To narrow the systemof transformations it would be necessaryto dis-cover further \"natural\"operationsor relations in the domain of utility.Thus it was pointed out by Pareto4 that an equality relationfor utilitydifferences would suffice; in our terminology it would reducethe transfor-mation system to the linear transformations.6 However,sinceit doesnot

1Onecould alsoimagine intermediate casesof greater transformation systems thanthesebut not containing all monotone transformations. Various forms of the theory ofrelativity give rather technical examplesof this.

2 In the usual language this would hold for physical quantities where an absolutezeroas well as an absoluteunit can bedefined. This is, e.g.,the casefor the absolutevalue(not the vector!)of velocity in such physical theories as those in which light velocityplays a normative role:Maxwellian electrodynamics, specialrelativity.

8 As long asonly the conceptof \" warmer\" i.e.a \"natural\" relation \"greater\" wasknown. We discussedthis in extenao previously.

4 V. Pareto, Manuel d'EconomiePolitique, Paris, 1907,p. 264.'Thisis exactly what Euclid did for position on a line. The utility conceptof

\" preference\" correspondsto the relation of \"lying to the right of\" there, and the (desired)

relation of the equality of utility differences to the geometrical congruenceof intervals.)))

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24 FORMULATIONOF THEECONOMIC PROBLEM

seemthat this relation is really a \"natural\"one i.e.onewhich canbeinterpretedby reproducibleobservations the suggestiondoesnot achievethe purpose.

3.6.Conceptual Structure of the Axiomatic Treatment of Numerical Utilities

3.6.1.Thefailure of oneparticulardeviceneednot excludethepossibilityof achieving the sameend by anotherdevice. Our contention is that thedomain of utility containsa \"natural\" operationwhich narrows the systemof transformations to preciselythe same extentas the otherdevicewouldhave done. Thisis the combination of two utilitieswith two given alterna-tive probabilities a, 1 a, (0 < a < 1) as describedin 3.3.2.Theprocessis so similar to the formation of centersof gravity mentionedin3.4.3.that it may be advantageousto use the same terminology. Thuswe have for utilitiesu, v the \"natural\"relation u > v (read:u is preferableto v), and the \"natural\"operation an + (1 a)v, (0 < a < 1),(read:centerof gravity of u, v with the respectiveweightsa, 1 a;or:combina-tion of u, v with the alternative probabilities ,! ). If the existenceand reproducibleobservability of theseconceptsis conceded,then ourway is clear:We must find a correspondencebetweenutilitiesand numberswhich carriesthe relation u > v and the operation au + (1 a)v forutilities into the synonymousconceptsfor numbers.

Denotethe correspondenceby

u ->p = v(w),

u beingthe utility and v(u) the number which the correspondenceattachesto it. Our requirementsarethen:

(3:l:a) u > v implies v(u) > v(v),(3:l:b) v(au + (1- a)v) = av(u) + (1- a)v(y).1

If two suchcorrespondences(3:2:a) u-+p= v(u),(3:2:b) u - p' = v'(u),shouldexist,then they setup a correspondencebetweennumbers

(3:3) p+P',for which we may alsowrite

(3:4) P'))

Since(3:2:a),(3:2:b)fulfill (3:1:a),(3:1:b), the correspondence(3:3),i.e.the function 0(p)in (3:4)must leave therelationp > cr

2 and the operation

^Observethat in in eachcase the left-hand side has the \"natural\" conceptsforutilities, and the right-hand sidethe conventional onesfor numbers.

1Now theseareapplied to numbers p, o-l)))

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THENOTION OF UTILITY 25

ap + (1 )<r unaffected (cf footnote 1 on p.24). I.e.(3:5:a) p > <r implies <f>(p) > <(<r),(3:5:b) <t>(ap + (1- a)r) = a*(p)+ (1- a)0(<r).Hence</>(p) must bea linear function, i.e.(3:6) p' = <(p) es o> p + i,

where w , i arefixed numbers(constants)with w >0.Sowe see:If such a numerical valuation of utilities1existsat all, then

it is determinedup to a lineartransformation. 2'8 I.e.then utility is anumberup to a linear transformation.

In orderthat a numerical valuation in the above senseshouldexistitis necessaryto postulate certainpropertiesof the relation u > v and theoperation au + (1 ct)v for utilities. The selectionof thesepostulatesor axioms and their subsequentanalysis leadsto problemsof a certainmathematical interest. In what follows we give a generaloutline of thesituation for the orientation of the reader;a completediscussionis found inthe Appendix.

3.5.2.A choiceof axiomsis not a purely objectivetask. It is usuallyexpectedto achieve somedefinite aim somespecifictheorem or theoremsare to be derivable from the axioms and to this extent the problemisexactand objective.But beyond this there arealways other importantdesiderataof a lessexactnature:Theaxiomsshouldnot be too numerous,their systemis to be as simpleand transparent as possible,and eachaxiomshouldhave an immediateintuitive meaning by which its appropriatenessmay be judgeddirectly.4 In a situation like ours this last requirementisparticularly vital, in spite of its vagueness:we want to make an intuitiveconceptamenable to mathematical treatmentand to seeas clearly aspossiblewhat hypothesesthis requires.

The objectivepart of our problemis clear:the postulatesmust implythe existenceof a correspondence(3:2:a)with the properties (3:l:a),(3:l:b)as described in 3.5.1.The further heuristic,and even estheticdesiderata,indicated above, do not determinea unique way of findingthis axiomatic treatment.In what follows we shall formulate a set ofaxiomswhich seemsto be essentiallysatisfactory.

1 I.e.a correspondence(3:2:a)which fulfills (3:1:a),(3:1:b).8 I.e.one of the form (3:6).3 Remember the physical examplesof the samesituation given in 3.4.4. (Ourpresent

discussion is somewhat more detailed.) We do not undertake to fix an absolute zeroand an absoluteunit of utility.

4The first and the last principle may represent at leastto a certain extent oppositeinfluences: If we reducethe number of axioms by merging them as far as technicallypossible,we may losethe possibility of distinguishing the various intuitive backgrounds.Thus we could have expressedthe group (3:B)in 3.6.1.by a smaller number of axioms,but this would have obscuredthe subsequent analysis of 3.6.2.

Tostrike a properbalanceis a matter of practical and to someextent evenestheticjudgment.)))

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26 FORMULATIONOF THEECONOMICPROBLEM

3.6.TheAxioms and Their Interpretation

3.6.1.Our axioms arethese:We considera system U of entities1 u, v, w, . In V a relation is

given, u > v, and for any number a, (0 < a < 1),an operation

au + (1 a)v = w.

Theseconceptssatisfy the following axioms:

(3:A) u > v is a completeordering of f/.2

This means:Write u < v when v > u. Then:

(3:A:a) Forany two u y v one and only one of the threefollowingrelationsholds:

u = v t u > v, u < v.

(3:A:b) u > v, v > w imply u > w.z

(3:B) Orderingand combining.4

(3:B:a) u < v impliesthat u < au + (1 a)v.(3:B:b) u > v impliesthat u > au + (1 a)v.(3:B:c) u < w < v impliesthe existenceof an a with

au + (1 a)v < w.

(3:B:d) u > w > v impliesthe existenceof an a with

au + (1 a)v > w.

(3:C) Algebra of combining.

(3:C:a) au + (1- a)v = (1- a)v + au.(3:C:b) a(ftu + (1- fiv) + (1- a)v = yu + (1- y)v

where 7 = aft.

Onecan show that theseaxioms imply the existenceof a correspondence(3:2:a)with the properties(3:1:a), (3:1:b) as describedin 3.5.1.Hencethe conclusionsof 3.5.1.hold good:The system U i.e.in our presentinterpretation,the systemof (abstract)utilities is one of numbersup toa linear transformation.

The construction of (3:2:a)(with (3:1:a), (3:1:b) by means of theaxioms (3:A)-(3:C))is a purely mathematical task which is somewhatlengthy, although it runs along conventional lines and presents no par-

1 This is, of course,meant to be the system of (abstract) utilities, to becharacterizedby our axioms. Concerning the general nature of the axiomatic method, cf.the remarksand referencesin the last part of 10.1.1.

*For a more systematic mathematical discussion of this notion, cf. 65.3.1.Theequivalent conceptof the completenessof the system of preferenceswas previously con-sideredat the beginning of 3.3.2.and of 3.4.6.

8 Theseconditions (3:A:a),(3:Aft) correspondto (65:A:a),(65:A:b)in 65.3.1.4 Remember that the a, 0,y occurring herearealways > 0, < 1.)))

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THENOTION OF UTILITY 27

ticular difficulties. (Cf.Appendix.)It seemsequallyunnecessaryto carry out the usual logisticdiscussion

of theseaxioms1on this occasion.We shall however say a few more wordsabout the intuitive meaningi.e.the justification of eachone of our axioms(3:A)-(3:C).3.6.2.Theanalysisof our postulatesfollows:

(3:A:a*) This is the statement of the completenessof the systemofindividual preferences.It is customaryto assumethis whendiscussingutilities or preferences,e.g.in the \" indifferencecurveanalysismethod.\" Thesequestionswere alreadyconsideredin3.3.4.and 3.4.6.

(3:A:b*) This is the \"transitivity \" of preference,a plausibleand

generally acceptedproperty.(3:B:a*) We statehere:If v is preferableto u, then even a chance

1 a of v alternatively to u ispreferable. Thisislegitimatesinceany kind of complementarity (or the opposite)has beenexcluded,cf. the beginning of 3.3.2.

(3:B:b*) Thisis the dual of (3:B:a*),with \"lesspreferable\"in placeof\" preferable.\"

(3:B:c*) We statehere:If w is preferableto u, and an even morepreferablev is also given, then the combination of u with achance1 a of v will not affect w'& preferability to it if thischanceis small enough. I.e.:Howeverdesirablev may be initself, one can makeits influence as weakas desiredby givingit a sufficiently small chance. This is a plausible\"

continuity\"assumption.

(3:B:d*) Thisis the dual of (3:B:c*),with \"lesspreferable\" in place.of\"preferable.\"

(3:C:a*) This is the statement that it is irrelevant in which ordertheconstituentsu, v of a combination arenamed. It is legitimate,particularly sincethe constituents are alternative events, cf.(3:B:a*)above.

(3:C:b*) This is the statement that it is irrelevant whether a com-bination of two constituents is obtained in two successivesteps, first the probabilitiesa, 1 a, then the probabilities0,1 /J; or in one operation, the probabilities7, 1 y where7 = a.2 Thesame things can be said for this as for (3:C:a*)above. It may be,however, that this postulatehas a deepersignificance, to which one allusion is made in 3.7.1.below.

1A similar situation is dealt with more exhaustively in 10.;those axioms describeasubject which is more vital for our main objective. Thelogistic discussion is indicatedthere in 10.2.Someof the general remarks of 10.3.apply to the present casealso.

2This is of coursethe correctarithmetic of accounting for two successiveadmixturesof v with u.)))

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28 FORMULATIONOF THEECONOMICPROBLEM

3.7.GeneralRemarks Concerning the Axioms

3.7.1.At this point it may be well to stop and to reconsiderthe situa-tion. Have we not shown too much? We can derive from the postulates(3:A)-(3:C)the numerical characterof utility in the senseof (3:2:a)and(3:1:a), (3:1:b) in 3.5.1.;and (3:1:b) statesthat the numerical values ofutility combine (with probabilities)like mathematical expectations!And

yet the conceptof mathematical expectationhas beenoften questioned,and its legitimatenessis certainly dependentupon some hypothesiscon-cerning the nature of an \" expectation.\"1 Have we not then beggedthequestion? Do not our postulates introduce, in some oblique way, thehypotheseswhich bring in the mathematical expectation?

More specifically:May there not exist in an individual a (positiveornegative) utility of the mereact of \"

taking a chance,\"of gambling, wRich

the use of the mathematical expectationobliterates?Howdid our axioms (3:A)-(3:C)getaround this possibility?As far as we can see,our postulates(3:A)-(3:C)do not attempt to avoid

it. Even that one which gets closestto excludinga \"utility of gambling\"(3:C:b)(cf. its discussionin 3.6.2.),seemsto be plausibleand legitimate,unlessa much more refined systemof psychologyis used than the one nowavailable for the purposesof economics.The fact that a numerical utilitywith a formula amounting to the use of mathematical expectationscanbe built upon (3:A)-(3:C),seemsto indicate this:We have practicallydefined numerical utility as being that thing for which the calculusofmathematical expectationsis legitimate.2 Since(3:A)-(3:C)securethatthe necessaryconstructioncan be carried out, conceptslike a \" specificutility of gambling\" cannot be formulated free of contradiction on thislevel.3

3.7.2.As we have stated,the last time in 3.6.1.,our axioms arebasedon the relation u > v and on the operation au + (1 a)v for utilities.It seemsnoteworthy that the lattermay be regardedas more immediatelygiven than the former:One can hardly doubt that anybody who couldimagine two alternative situations with the respectiveutilities u, v couldnot also conceive the prospect of having both with the given respectiveprobabilities ,! . On the other hand one may questionthe postulateof axiom (3:A:a)for u > v, i.e.the completenessof this ordering.

Let us considerthis point for a moment. We have concededthat onemay doubt whether a person^canalways decidewhich of two alternatives

1 Cf. Karl Menger: Das Unsicherheitsmoment in der Wertlehre, Zeitschrift ftir

National6konomie, vol. 5, (1934)pp.459ff. and Gerhard Tintner: A contribution to thenon-static Theory of Choice,Quarterly Journal of Economics, vol. LVI, (1942)pp.274ff.

1Thus Daniel Bernoulli's well known suggestion to \"solve\" the \"St. PetersburgParadox\" by the useof the so-called\"moral expectation\" (instead of the mathematicalexpectation)means defining the utility numerically as the logarithm of one'smonetarypossessions.

9 This may seemto bea paradoxicalassertion. But anybody who has seriously triedto axiomatize that elusive concept,will probably concurwith it.)))

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THENOTION OF UTILITY 29

with the utilities u, v he prefers.1 But, whatever the merits of thisdoubt are, this possibility i.e.the completenessof the system of (indi-vidual) preferences must beassumedeven for thepurposesof the\"indiffer-encecurve method\" (cf. our remarks on (3:A:a)in 3.6.2.).But if thisproperty of u > v 2 is assumed,then our useof the much lessquestionableau + (1 ot)v

* yieldsthe numerical utilitiestoo!4

If the generalcomparabilityassumptionis not made,5 a mathematicaltheory based on au + (1 <x)v togetherwith what remainsof u > v

is still possible.6 It leadsto what may bedescribedas a many-dimensionalvector conceptof utility. This is a more complicatedand lesssatisfactoryset-up,but we do not proposeto treatit systematicallyat this time.

3.7.3.This brief expositiondoesnot claim to exhaust the subject,butwe hope to have conveyedthe essentialpoints. To avoid misunderstand-ings,the followingfurther remarksmay be useful.

(1) We re-emphasizethat we areconsideringonly utilitiesexperiencedby oneperson. Theseconsiderationsdo not imply anything concerningthecomparisonsof the utilitiesbelongingto different individuals.

(2) Itcannot bedeniedthat the analysisof themethodswhich makeuseof mathematical expectation(cf. footnote 1on p.28 for the literature)isfar from concludedat present. Our remarks in 3.7.1.liein this direction,but much more shouldbe said in this respect. Therearemany interestingquestions involved, which however lie beyond the scopeof this work.Forour purposesit sufficesto observethat the validity of the simpleandplausibleaxioms(3:A)-(3:C)in 3.6.1.for the relation u > v and the oper-ation au + (1 a)v makesthe utilitiesnumbersup to a linear transforma-tion in the sensediscussedin thesesections.

3.8.TheRole of the Conceptof Marginal Utility .3.8.1.The precedinganalysismade it clearthat we feel free to make

use of a numerical conceptionof utility. On the otherhand, subsequent1Or that he can assertthat they arepreciselyequally desirable.2I.e.the completenesspostulate (3:A:a).1I.e.the postulates (3:B),(3:C)together with the obvious postulate (3:A:b).4 At this point the readermay recallthe familiar argument accordingto which the

unnumerical (\"indifference curve\") treatment of utilities is preferableto any numericalone, becauseit is simpler and basedon fewer hypotheses. This objection might belegitimate if the numerical treatment werebasedon Pareto'sequality relation for utilitydifferences (cf. the end of 3.4.6.).This relation is, indeed, a stronger and more compli-catedhypothesis, added to the original onesconcerning the general comparability ofutilities (completenessof preferences).

However, we used the operation au + (1 ) instead, and wehopethat the readerwill agreewith us that it representsan even safer assumption than that of the complete-nessof preferences.

We think therefore that our procedure,as distinguished from Pareto's,is not opento the objectionsbasedon the necessityof artificial assumptions and a lossof simplicity.

6This amounts to weakening (3:A:a)to an (3:A:a')by replacing in it \"one and onlyone\" by \"at most one/' Theconditions (3:A:a')(3:A:b)then correspondto (65:B:a),(65:B:b).

6 In this casesome modifications in the groups of postulates (3:B),(3:0)are alsonecessary.)))

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30 FORMULATIONOF THEECONOMICPROBLEM

discussionswill show that we cannot avoid the assumptionthat all subjectsof the economy under considerationare completelyinformed about thephysicalcharacteristicsof the situation in which they operateand areableto perform all statistical, mathematical, etc.,operationswhich this knowl-edgemakespossible.Thenature and importanceof this assumptionhasbeengiven extensiveattention in the literatureand the subjectis probablyvery far from being exhausted. We proposenot to enterupon it. Thequestionis too vast and too difficult and we believethat it is best to \" dividedifficulties.\" I.e.we wish to avoid this complicationwhich, while interest-ing in its own right, should be consideredseparately from our presentproblem.

Actually we think that our investigations although they assume\" completeinformation\" without any further discussion do make a con-tribution to the study of this subject. It will beseenthat many economicand socialphenomenawhich areusually ascribedto the individual'sstateof\" incompleteinformation\" make their appearancein our theory and can besatisfactorilyinterpreted with its help. Sinceour theory assumes \" com-pleteinformation,\" we concludefrom this that those phenomenahavenothing to do with the individual's \" incompleteinformation.\" Someparticularly striking examplesof this will be found in the conceptsof\" discrimination\"in 33.1.,of \"incompleteexploitation\"in 38.3.,and of the\"transfer\"or \"tribute\"in 46.11.,46.12.

On thebasisof the above we would even venture to questionthe impor-tanceusually ascribedto incompleteinformation in its conventional sense1in economicand socialtheory. It will appear that somephenomenawhichwould prima facie have to be attributed to this factor, have nothing to dowith it.2

3.8.2.Let us now consideran isolatedindividual with definite physicalcharacteristicsand with definite quantities of goodsat his disposal.Inview of what wassaid above, he is in a positionto determinethe maximum

utility which can be obtained in this situation. Sincethe maximum is awell-definedquantity, the sameis true for the increasewhich occurswhen aunit of any definite goodis added to the stockof all goodsin the possessionof the individual. This is, of course,the classicalnotion of the marginalutility of a unit of the commodity in question.8

Thesequantities areclearly of decisiveimportancein the \"RobinsonCrusoe\"economy. The above marginal utility obviously correspondsto

1We shall seethat the rules of the games consideredmay explicitly prescribethatcertain participants should not possesscertain piecesof information. Cf. 6.3.,6.4.(Gamesin which this doesnot happen arereferredto in 14.8.and in (15:B)of15.3.2.,andarecalledgameswith \" perfectinformation.\") We shall recognizeand utilize this kind of\"incomplete information\" (according to the above, rather to be called \"imperfectinformation\.") But we rejectall other types, vaguely defined by the use of conceptslike complication, intelligence, etc.

2Our theory attributes these phenomena to the possibility of multiple \"stablestandards of behavior\" cf 4.6.and the end of 4.7.

*Moreprecisely:the so-called\"indirectly dependent expectedutility.\)

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SOLUTIONSAND STANDARDSOF BEHAVIOR 31the maximum effort which he will be willing to make if he behavesaccord-ing to the customary criteria of rationality in orderto obtain a furtherunit of that commodity.

It is not clearat all, however, what significance it has in determiningthe behavior of a participant in a socialexchangeeconomy. We saw thatthe principlesof rational behavior in this casestill await formulation, andthat they arecertainly not expressedby a maximum requirementof theCrusoetype. Thus it must be uncertain whether marginal utility has anymeaning at all in this case.1

Positivestatements on this subjectwill be possibleonly after we havesucceededin developinga theory of rational behavior in a socialexchangeeconomy, that is, as was stated before, with the help of the theory of\"gamesof strategy.\" It will be seenthat marginal utility does,indeed,play an important role in this casetoo,but in a more subtle way than isusually assumed.

4.Structureof the Theory :Solutionsand Standardsof Behavior

4.1.TheSimplest Conceptof a Solution for OneParticipant

4.1.1.We have now reachedthe point where it becomespossibletogive a positive descriptionof our proposedprocedure.This means pri-marily an outline and an accountof the main technicalconceptsanddevices.

As we stated before, we wish to find the mathematically completeprincipleswhich define \" rational behavior\" for the participants in a socialeconomy, and to derive from them the generalcharacteristicsof thatbehavior. And while the principlesought to be perfectly general i.e.,valid in all situations we may be satisfiedif we can find solutions,for themoment, only in somecharacteristicspecialcases.

First of all we must obtain a clearnotion of what can be acceptedas asolution of this problem;i.e.,what the amount of information is which asolution must convey, and what we should expectregardingits formalstructure, A preciseanalysisbecomespossibleonly after thesemattershave beenclarified.

4.1.2.Theimmediateconceptof a solutionis plausiblya setof rulesforeachparticipantwhich tell him how to behavein every situation which mayconceivably arise. Onemay objectat this point that this view is unneces-sarily inclusive. Sincewe want to theorize about \" rational behavior,\"thereseemsto be no needto give the individual advice as to his behavior insituations other than those which arisein a rational community. Thiswould justify assumingrational behavior on the part of the othersas well,in whatever way we are going to characterizethat. Sucha procedurewould probablylead to a unique sequenceof situationsto which alone ourtheory needrefer.

1All this is understood within the domain of our severalsimplifying assumptions. Ifthey are relaxed,then various further difficulties ensue.)))

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32 FORMULATIONOF THEECONOMIC PROBLEM

This objectionseemsto be invalid for two reasons:First,the \"rulesof the game,\"i.e.the physicallaws which give the

factual backgroundof the economicactivities under considerationmay beexplicitlystatistical Theactionsof the participantsof the economy maydeterminethe outcomeonly in conjunctionwith events which depend onchance(with known probabilities),cf. footnote 2 on p.10and 6.2.1.Ifthis is taken into consideration,then the rulesof behavior even in a perfectlyrational community must providefor a greatvariety of situations someofwhich will be very far from optimum.1

Second,and this iseven more fundamental, the rulesof rational behaviormust providedefinitely for the possibilityof irrational conducton the partof others. In otherwords:Imaginethat we have discovereda setof rulesfor all participants to be termedas \"optimal\"or \"rational\"eachofwhich is indeed optimal provided that the other participants conform.Then the questionremainsas to what will happenif someof the participantsdonot conform. If that shouldturn out to beadvantageousfor them and,quite particularly, disadvantageousto the conformists then the above\"solution\"would seemvery questionable.We arein no positionto give apositivediscussionof thesethings as yet but we want to make it clearthat under such conditionsthe \"solution,\"or at leastits motivation, mustbe consideredas imperfect and incomplete.In whatever way we formulatethe guidingprinciplesand the objectivejustification of \"rational behavior,\"provisoswill have to be made for every possibleconductof \"theothers.\"Only in this way can a satisfactoryand exhaustive theory be developed.But if the superiorityof \"rational behavior\" over any other kind is to beestablished,then its description must include rules of conduct for allconceivable situations including those where \"the others\" behavedirrationally, in the senseof the standardswhich the theory will setfor them.

4.1,3.At this stagethe readerwill observea greatsimilarity with theeveryday conceptof games.We think that this similarity is very essential;indeed,that it is more than that. Foreconomicand socialproblemsthegamesfulfill or shouldfulfill the samefunction which various geometrico-mathematical modelshave successfullyperformed in the physicalsciences.Suchmodelsaretheoreticalconstructswith a precise,exhaustive and nottoo complicateddefinition; and they must be similar to reality in thoserespectswhich are essentialin the investigation at hand. To reca-pitulate in detail:The definition must be preciseand exhaustive inorderto make a mathematical treatmentpossible.The constructmustnot be unduly complicated,so that the mathematical treatmentcan bebrought beyond the mereformalism to the point where it yieldscompletenumerical results. Similarity to reality is neededto make the operationsignificant. And this similarity must usually berestrictedto a few traits

1That a unique optimal behavior is at all conceivablein spite of the multiplicity ofthe possibilities determined by chance,is ofcoursedue to the useof the notion of \"mathe-matical expectation.1' Cf.loc.cit. above.)))

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SOLUTIONSAND STANDARDSOF BEHAVIOR 33

deemed\"essential\"pro tempore sinceotherwisethe above requirementswould conflict with eachother.1

It is clearthat if a model of economicactivities is constructedaccordingto theseprinciples,the descriptionof a game results. This is particularlystriking in the formal descriptionof markets which areafter all the coreof the economicsystem but this statement is true in all casesand withoutqualifications.

4.1.4.We describedin 4.1.2.what we expecta solution i.e.a character-ization of \" rational behavior \" to consistof. Thisamounted to a completeset of rules of behavior in all conceivablesituations. This holds equiv-alently for a social economy and for games. The entire result in theabove senseis thus a combinatorial enumeration of enormous complexity.But we have accepteda simplifiedconceptof utility accordingto which allthe individual strives for is fully describedby one numerical datum (cf.2.1.1.and 3.3.).Thus the complicatedcombinatorial catalogue whichwe expectfrom a solution permitsa very brief and significant summariza-tion:the statementof how much 2- 3 the participantunder considerationcanget if he behaves \" rationally. \" This \"canget\"is, of course,presumedtobe a minimum; he may get more if the others make mistakes (behaveirrationally).

It ought to be understoodthat all this discussionis advanced,as itshouldbe,preliminary to the building of a satisfactorytheory along thelines indicated. We formulate desideratawhich will serve as a gauge ofsuccessin our subsequentconsiderations;but it is in accordancewith theusual heuristic procedureto reason about thesedesiderata even beforewe are able to satisfy them. Indeed,this preliminary reasoningis anessentialpart of the processof finding a satisfactorytheory.4

4.2.Extension to All Participants

4.2.1.We have consideredso far only what the solution ought to be forone participant. Let us now visualize all participants simultaneously.I.e.,let us considera socialeconomy, or equivalently a game of a fixednumber of (sayn) participants. The completeinformation which a solutionshould convey is, as we discussedit, of a combinatorial nature. It wasindicated furthermore how a single quantitative statement contains thedecisivepart of this information, by stating how much eachparticipant

1E.g.,Newton's description of the solar system by a small number of \"masspomts.\"Thesepoints attract eachother and move like the stars;this is the similarity in the essen-tials, while the enormous wealth of the other physical features of the planets has beenleftout of account.

2 Utility; for an entrepreneur, profit ; for a player, gain or loss.8 We mean, of course,the \"mathematical expectation,\" if there is an explicit element

of chance. Cf.the first remark in 4.1.2.and alsothe discussionof 3.7.1.4Thosewho are familiar with the development of physics will know how important

such heuristic considerations canbe. Neither general relativity nor quantum mechanicscould have beenfound without a \"T>rA-thpnrptip*i ;/ Hiannaainn nf thft HpaidemtA concern-ing the theory-to-be.)))

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34 FORMULATIONOF THEECONOMICPROBLEM

obtainsby behaving rationally. Considertheseamounts which the severalparticipants''obtain.\" If the solution didnothing more in the quantitativesensethan specifythese amounts,1 then it would coincidewith the wellknown conceptof imputation:it would just statehow the total proceedsareto be distributedamong the participants.2

We emphasizethat the problemof imputation must be solvedbothwhen the total proceedsarein fact identicallyzero and when they arevari-able. This problem,in its generalform, has neither beenproperlyformu-latednor solvedin economicliterature.

4.2.2.We can seeno reason why one should not be satisfiedwith asolutionof this nature, providing it can befound: i.e.a singleimputationwhich meetsreasonablerequirementsfor optimum (rational) behavior.(Of coursewe have not yet formulated theserequirements.Foran exhaus-tive discussion,cf. loc.cit.below.) Thestructure of the societyundercon-siderationwould then be extremelysimple:Therewould existan absolutestateof equilibrium in which the quantitative shareof every participantwould be preciselydetermined.

It will be seenhowever that such a solution, possessingall necessaryproperties,does not exist in general. The notion of a solution will haveto bebroadenedconsiderably,and it will be seenthat this is closelycon-nectedwith certaininherent features of socialorganization that arewellknown from a \" common sense\" point of view but thus far have not beenviewed in properperspective.(Cf.4.6.and 4.8.1.)

4.2.3.Our mathematical analysisof the problemwill show that thereexists,indeed,a not inconsiderablefamily of gameswhere a solution can bedefined and found in the above sense:i.e.as one singleimputation. Insuch casesevery participant obtains at leastthe amount thus imputed tohim by just behaving appropriately,rationally. Indeed,he getsexactlythis amount if the other participantstoo behave rationally; if they do not,he may geteven more.

Thesearethe gamesof two participantswhere the sum of all paymentsis zero. While thesegamesare not exactly typical for majoreconomicprocesses,they contain someuniversally important traits of all gamesandthe resultsderivedfrom them arethe basisof the generaltheory of games.We shalldiscussthem at length in Chapter III.

4.3.TheSolution asa Setof Imputations

4.3.1.If eitherof the two above restrictionsis dropped,the situation isalteredmaterially.

1And of course,in the combinatorial sense,as outlined above,the procedurehow toobtain them.

*In games as usually understood the total proceedsare always zero;i.e.oneparticipant can gain only what the others lose. Thus there is a pure problem of distri-bution i.e.imputation and absolutely none of increasing the total utility, the \"socialproduct.\" In all economicquestions the latter problem arisesas well, but the questionof imputation remains. Subsequently we shall broaden the conceptof a game by drop-ping the requirement of the total proceedsbeing zero(cf.Ch.XI).)))

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SOLUTIONSAND STANDARDSOF BEHAVIOR 35

The simplestgamewhere the secondrequirementis oversteppedis atwo-persongame where the sum of all payments is variable. This cor-respondsto a socialeconomy with two participants and allows both fortheir interdependenceand for variability of total utility with their behavior.1As a matter of fact this is exactly the caseof a bilateral monopoly (cf.6L2.-61.6.).The well known \"zone of uncertainty \" which is found incurrent efforts to solve the problemof imputation indicatesthat a broaderconceptof solution must be sought. This casewill be discussedloc.cit.above. Forthe moment we want to use it only as an indicatorof the diffi-

culty and pass to the other casewhich is more suitableas a basisfor a firstpositive step.

4.3.2.Thesimplestgamewhere the first requirementis disregardedis athree-persongame where the sum of all paymentsis zero. In contrast tothe above two-persongame, this doesnot correspondto any fundamentaleconomicproblembut it representsneverthelessa basicpossibilityin humanrelations. Theessentialfeature is that any two playerswho combineandcooperateagainst a third can thereby securean advantage. Theproblemis how this advantage shouldbe distributedamong the two partners in thiscombination. Any such schemeof imputation will have to take intoaccountthat any two partnerscan combine;i.e.while any onecombinationis in the processof formation, eachpartner must considerthe fact that hisprospectiveally could break away and join the third participant.

Of coursethe rules of the game will prescribehow the proceedsof acoalition should be divided between the partners. But the detaileddis-cussion to be given in 22.1.shows that this will not be, in general,thefinal verdict. Imaginea game (of threeor more persons) in which twoparticipants can form a very advantageouscoalition but where the rulesof the gameprovide that the greatestpart of the gain goesto the firstparticipant. Assume furthermore that the secondparticipant of thiscoalition can alsoentera coalition with the third one,which is lesseffectivein toto but promiseshim a greaterindividual gain than the former. Inthis situation it is obviously reasonablefor the first participant to transfera part of the gainswhich he couldgetfrom the first coalition to the secondparticipant in order to save this coalition. In other words: One mustexpectthat under certainconditionsoneparticipant of a coalition will bewilling to pay a compensationto his partner. Thus the apportionmentwithin a coalition depends not only upon the rules of the game butalso upon the above principles,under the influence of the alternativecoalitions.2

Common sensesuggeststhat one cannot expectany theoreticalstate-ment as to which alliance will beformed3 but only information concerning

1It will beremembered that we make useofa transferable utility, cf.2.1.1.*This doesnot mean that the rules of the game areviolated, sincesuch compensatory

payments, if madeat all, aremadefreely in pursuance ofa rational consideration.1Obviously three combinations of two partners eachare possible. In the example

to be given in 21.,any preferencewithin the solution for a particular alliancewill be a)))

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36 FORMULATIONOF THEECONOMIC PROBLEM

how the partners in a possiblecombination must divide the spoilsin orderto avoid the contingency that any oneof them desertsto form a combinationwith the third player. All this will bediscussedin detailand quantitativelyin Ch.V.

It suffices to statehere only the result which the above qualitativeconsiderationsmakeplausibleand which will be establishedmore rigorouslyloc.cit. A reasonableconceptof a solution consistsin this caseof a systemof three imputations. Thesecorrespond to the above-mentionedthreecombinationsor alliancesand expressthe division of spoilsbetweenrespec-tive allies.

4.3.3.The last result will turn out to be the prototype of the generalsituation. We shall seethat a consistenttheory will result from lookingfor solutions which are not single imputations, but rather systems ofimputations.

It is clearthat in the above three-persongameno single imputationfrom the solution is in itself anything like a solution. Any particularalliancedescribesonly one particularconsiderationwhich entersthe mindsof the participants when they plan their behavior. Even if a particularallianceis ultimately formed, the division of the proceedsbetweenthe allieswill be decisivelyinfluenced by the other allianceswhich eachone mightalternatively have entered.Thus only the three alliancesand theirimputations togetherform a rational whole which determinesall of itsdetailsand possessesa stability of its own. It is, indeed,this whole whichis the really significant entity, more so than its constituent imputations.Even if one of theseis actually applied, i.e.if one particular allianceisactually formed, the others arepresent in a \"virtual\" existence:Althoughthey have not materialized,they have contributedessentiallyto shapinganddeterminingthe actualreality.

In conceiving of the generalproblem,a socialeconomy or equivalentlya gameof n participants,we shall with an optimismwhich can be justifiedonly by subsequentsuccessexpectthe samething:A solution shouldbeasystemof imputations1possessingin its entirety somekind of balanceandstability the nature of which we shall try to determine.We emphasizethat this stability whatever it may turn out to be will be a propertyof the systemas a whole and not of the singleimputationsof which it iscomposed.Thesebrief considerationsregarding the three-persongamehave illustratedthis point.

4.3.4.Theexactcriteriawhich characterizea systemof imputationsas asolutionof our problemare,of course,of a mathematical nature. Forapreciseand exhaustive discussionwe must therefore refer the readerto thesubsequentmathematical developmentof the theory. Theexactdefinition

limine excludedby symmetry. I.e.the game will besymmetric with respectto all threeparticipants. Of.however 33.1.1.

1They may again include compensations between partners in a coalition, asdescribedin 4.3.2.)))

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SOLUTIONSAND STANDARDSOF BEHAVIOR 37

itself is stated in 30.1.1.We shallneverthelessundertaketo give a prelimi-nary, qualitative outline. We hopethis will contributeto the understandingof the ideas on which the quantitative discussionis based. Besides,theplaceof our considerationsin the generalframework of socialtheory will

becomeclearer.

4.4.TheIntransitive Notion of \"Superiority\" or \"Domination\"

4.4.1.Let us return to a more primitive conceptof thesolutionwhich weknow already must be abandoned. We mean the ideaof a solution as asingleimputation. If this sort of solutionexistedit would have to be animputation which in someplausiblesensewassuperiorto all otherimputa-tions. This notion of superiority as between imputations ought to beformulated in a way which takesaccountof the physicaland socialstruc-ture of the milieu. That is, one should define that an imputation x issuperiorto an imputation y whenever this happens:Assume that society,i.e.the totality of all participants,has to considerthe questionwhether ornot to \"accept\"a staticsettlementof all questionsof distributionby theimputation y. Assumefurthermore that at this moment the alternativesettlementby the imputation x is alsoconsidered.Then this alternative xwill suffice to excludeacceptanceof y. By this we mean that a sufficientnumber of participantsprefer in their own interestx to i/, and areconvincedor can be convinced of the possibilityof obtaining the advantages of x.In this comparisonof x to y the participants shouldnot be influenced bythe considerationof any third alternatives (imputations).I.e.we conceivethe relationshipof superiorityas an elementaryone,correlatingthe twoimputations x and y only. The further comparisonof three or moreultimately of all imputations is the subjectof the theory which mustnow follow, as a superstructureerectedupon the elementaryconceptofsuperiority.

Whether the possibilityof obtaining certainadvantagesby relinquishingy for x y as discussedin the above definition, can be madeconvincing to theinterestedpartieswill dependupon the physicalfacts of the situation inthe terminology of games,on the rulesof the game.

We prefer to use,insteadof \" superior\"with its manifold associations,aword more in the nature of a terminus technicus. When the above describedrelationshipbetween two imputations x and y exists,1 then we shall saythat x dominates y.

If one restatesa little more carefully what shouldbe expectedfrom asolution consistingof a singleimputation, this formulation obtains:Suchan imputation should dominate all others and be dominated bynone.

4.4.2.The notion of domination as formulated or rather indicatedabove is clearly in the nature of an ordering,similar to the question of

1 That is, when it holds in the mathematically preciseform, which will be given in30.1.1.)))

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preference,or of size in any quantitative theory. The notion of a singleimputation solution1 correspondsto that of the first elementwith respectto that ordering.2

Thesearchfor sucha first elementwould be a plausibleone if the order-ing in question, i.e.our notion of domination, possessedthe importantpropertyof transitivity ; that is, if it weretrue that whenever x dominatesy and y dominatesz, then alsox dominatesz. In this caseone might proceedas follows:Starting with an arbitrary x, look for a y which dominatesa:;ifsuch a y exists,chooseone and lookfor a z which dominatesy\\ if sucha zexists,chooseone and lookfor a u which dominatesz, etc. Inmost practicalproblemsthereis a fair chancethat this processeitherterminatesafter afinite number of steps with a w which is undominatedby anything else,orthat the sequencex, y, z, u, , goeson ad infinitum, but that thesex,y, z, u, tend to a limiting positionw undominatedby anything else.And, due to the transitivity referredto above,the final w will in eithercasedominateall previously obtainedx, y, z, w, .

We shall not go into more elaboratedetails which could and shouldbegiven in an exhaustive discussion.Itwill probablybe clearto the readerthat the progress through the sequence#, y, z, u, - correspondstosuccessive\" improvements \" culminating in the \"

optimum,\" i.e.the \"first\"

elementw which dominatesall othersand is not dominated.All this becomesvery different when transitivity does not prevail.

In that caseany attempt to reachan \"optimum\" by successiveimprove-mentsmay be futile. It can happenthat x is dominatedby y y y by z, andz in turn by x.8

4.4.3.Now the notion of domination on which we rely is, indeed,nottransitive. In our tentative descriptionof this conceptwe indicatedthat xdominatesy when thereexistsa group of participants eachoneof whomprefershis individual situation in x to that in y, and who areconvincedthat they areable as a group i.e.as an alliance to enforce their prefer-ences. We shall discussthesematters in detail in 30.2.This group ofparticipantsshall becalledthe \"effectiveset\"for the domination of x over y.Now when x dominatesy and y dominatesz, the effectivesets for thesetwodominationsmay be entirely disjunct and therefore no conclusionscan bedrawn concerningthe relationshipbetweenz and x. It can even happenthat z dominates x with the help of a third effective set,possiblydisjunct'from both previous ones.

1We continue to use it asan illustration although we have shown already that it is aforlorn hope. Thereasonfor this is that, by showing what is involved if certain complica-tions did not arise, we can put thesecomplications into better perspective. Our realinterest at this stageliesof coursein thesecomplications, which are quite fundamental.

1Themathematical theory of ordering is very simple and leadsprobably to a deeperunderstanding of theseconditions than any purely verbal discussion. The necessarymathematical considerations will be found in 65.3.

8 In the caseof transitivity this is impossible because if a proof bewanted x neverdominates itself. Indeed,if e.g.y dominates x, z dominates y, and x dominates z, thenwe can infer by transitivity that x dominates x.)))

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Thislackof transitivity, especiallyin the above formalistic presentation,may appear to be an annoying complicationand it may even seemdesirableto makean effort to rid the theory of it. Yet the readerwho takes anotherlookat the last paragraphwill notice that it really containsonly a circum-locution of a most typical phenomenon in all socialorganizations. Thedomination relationshipsbetweenvarious imputationsz, i/, z, i.e.betweenvarious statesof society correspondto the various ways in whichthese can unstabilize i.e.upset each other. That various groups ofparticipants acting as effective sets in various relationsof this kind maybring about \"cyclical\"dominations e.g.,y over x,z over y, and x over zis indeedone of the most characteristicdifficulties which a theory of thesephenomenamust face.

4.5.ThePreciseDefinition of a Solution

4.6.1.Thus our task is to replacethe notion of the optimum i.e.of thefirst element by something which can take over its functions in a staticequilibrium. This becomesnecessarybecausethe original concepthasbecomeuntenable. We first observedits breakdownin the specificinstanceof a certainthree-persongame in 4.3.2.-4.3.3.But now we have acquireda deeperinsight into the causeof its failure:it is the nature of our conceptofdomination, and specificallyits intransitivity.

This type of relationshipis not at all peculiarto our problem. Otherinstancesof it arewell known in many fields and it is to be regrettedthatthey have never receiveda genericmathematical treatment.We mean allthoseconceptswhich arein the generalnature of a comparisonof preferenceor \"superiority,\"or of order, but lack transitivity: e.g.,the strength ofchessplayersin a tournament, the \"paperform\" in sports and races,etc.1

4.5.2.Thediscussionof the three-persongame in 4.3.2.-4.3.3.indicatedthat the solution will be,in general,a set of imputationsinsteadof a singleimputation. That is, the conceptof the \"first element\"will have to bereplacedby that of a set of elements(imputations)with suitableproperties.In the exhaustive discussionof this game in 32.(cf. also the interpreta-tion in 33.1.1.which callsattention to somedeviations)the systemof threeimputations,which was introducedas the solutionof the three-persongamein4.3.2.-4.3.3.,will be derivedin an exactway with the helpof the postulatesof 30.1.1.Thesepostulateswill bevery similar to those which character-ize a first element.They are,of course,requirementsfor a setof elements(imputations),but if that set shouldturn out to consistof a singleelementonly, then our postulates go over into the characterization of the firstelement(in the total systemof all imputations).

We do not give a detailedmotivation for thosepostulatesas yet, but weshallformulate them now hoping that the readerwill find them to besome-

1Someof these problems have beentreated mathematically by the introduction ofchanceand probability. Without denying that this approachhas a certain justification,we doubt whether it is conducive to a completeunderstanding even in those cases. Itwould bealtogether inadequate for our considerations ofsocialorganization.)))

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40 FORMULATIONOF THEECONOMIC PROBLEM

what plausible. Somereasonsof a qualitative nature, or ratheronepossibleinterpretation,will be given in the paragraphsimmediatelyfollowing.

4.5.3.The postulatesareas follows:A setSof elements(imputations)is a solutionwhen it possessesthesetwo properties:

(4:A:a) No y containedin S is dominatedby an x containedin S.(4:A:b) Every y not containedin S is dominatedby somex con-

tained in S.(4:A:a)and (4:A:b)can be stated as a singlecondition:

(4:A:c) Theelementsof S arepreciselythose elementswhich areundominated by elementsof S.1

The*readerwho is interestedin this type of exercisemay now verifyour previous assertionthat for a setS which consistsof a singleelementxthe above conditionsexpresspreciselythat x is the first element.

4.5.4.Partof the malaisewhich the precedingpostulatesmay causeatfirst sight is probablydue to their circularcharacter.This is particularlyobvious in the form (4:A:c),where the elementsof S arecharacterizedby arelationshipwhich is again dependent upon S. It is important not tomisunderstandthe meaning of this circumstance.

Sinceour definitions (4:A:a)and (4:A:b),or (4:A:c),arecircular i.e.implicit for S, it is not at all clearthat there really existsan S whichfulfills them, nor whether if thereexistsone the S is unique. Indeedthesequestions,at this stagestill unanswered,arethe main subjectof thesubsequenttheory. What is clear,however, is that thesedefinitions tellunambiguously whether any particular S is or is not a solution. If oneinsists on associatingwith the conceptof a definition the attributes ofexistenceand uniquenessof the objectdefined, then one must say: Wehave not given a definition of S,but a definition of a property of S wehave not defined the solution but characterizedall possible solutions.Whether the totality of all solutions,thus circumscribed,contains no S,exactlyoneS,or several<S's,is subjectfor further inquiry.2

4.6.Interpretation of Our Definition in Termsof \"Standards of Behavior\"

4.6.1.Thesingleimputation is an often used and well understoodcon-ceptof economictheory, while the setsof imputations to which we havebeenled areratherunfamiliar ones. It is therefore desirableto correlatethem with somethingwhich has a well establishedplacein our thinkingconcerningsocialphenomena.

1Thus (4:A:c)is an exactequivalent of (4:A:a)and (4:A:b)together. It may impressthe mathematically untrained readeras somewhat involved, although it is really astraightforward expressionof rather simple ideas.

2It should be unnecessary to say that the circularity, or rather implicitness, of(4:A:a)and (4:A:b),or (4:A:c),doesnot at all mean that they are tautological. Theyexpress,of course,a very seriousrestriction of S.)))

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SOLUTIONSAND STANDARDSOF BEHAVIOR 41Indeed,it appearsthat the setsof imputations S which we areconsider-

ing correspondto the \"standard of behavior \" connectedwith a socialorganization. Let us examine this assertionmore closely.

Let the physicalbasis of a social economy be given, or, to take abroaderview of the matter, of a society.1 According to all tradition andexperiencehuman beingshave a characteristicway of adjustingthemselvesto sucha background. This consistsof not setting up one rigidsystemofapportionment,i.e.of imputation, but rather a variety of alternatives,which will probably all expresssomegeneralprinciplesbut neverthelessdiffer among themselvesin many particular respects.2 This system ofimputations describesthe \" establishedorder of society\" or \" acceptedstandard of behavior/1

Obviously no random grouping of imputations will do as sucha \" stand-ard of behavior \":it will have to satisfy certain conditionswhich character-ize it as a possibleorderof things. Thisconceptof possibilitymust clearlyprovide for conditionsof stability. The readerwill observe,no doubt,that our procedurein the previous paragraphsis very much in this spirit:The setsS of imputations x, y, z, correspondto what we now call\"standardof behavior/'and the conditions(4:A:a)and (4:A:b),or (4:A:c),which characterizethe solution S express,indeed,a stability in the abovesense.

4.6.2.The disjunctioninto (4:A:a)and (4:A:b)is particularly appropri-ate in this instance. Recall that domination of y by x means that theimputation x, if taken into consideration,excludesacceptanceof theimputation y (this without forecasting what imputation will ultimately beaccepted,cf. 4.4.1.and 4.4.2.).Thus (4:A:a)expressesthe fact that thestandard of behavior is free from inner contradictions:No imputation y

belongingto S i.e.conforming with the \"acceptedstandard of behavior\"can be upset i.e.dominated by another imputation x of the samekind.

On the other hand (4:A:b)expressesthat the \"standardof behavior \" canbe used to discreditany non-conforming procedure:Every imputation y

not belonging to S can be upset i.e.dominated by an imputation x

belongingto S.Observethat we have not postulatedin 4.5.3.that a y belonging to S

shouldnever be dominatedby any x.3 Ofcourse,if this shouldhappen,thenx would have to be outsideof S, due to (4:A:a). In the terminology ofsocialorganizations:An imputation y which conforms with the \"accepted

1 In the caseof a game this means simply as we have mentioned before that therules of the game are given. But for the present simile the comparison with a socialeconomy is more useful. We suggest therefore that the readerforget temporarily theanalogy with gamesand think entirely in terms of socialorganization.

2Theremay be extreme, or to use a mathematical term, \"degenerate\" specialcaseswhere the setup is of such exceptional simplicity that a rigid single apportionment canbeput into operation. But it seemslegitimate to disregard them asnon-typical.

8 It can be shown, cf. (31:M)in 31.2.3.,that such a postulate cannot be fulfilledin general;i.e.that in all really interesting casesit is impossible to find an Swhich satisfiesit together with our other requirements.)))

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standard of behavior\" may be upset by another imputation x,but in thiscaseit iscertain that x doesnot conform.l Itfollowsfrom our other require-ments that then x is upset in turn by a third imputation z which againconforms. Sincey and z both conform, z cannot upset y a further illustra-tion of the intransitivity of \" domination. \"

Thus our solutionsS correspondto such \" standards of behavior ' ashave an inner stability: oncethey are generally acceptedthey overruleeverything elseand no part of them can be overruled within the limits ofthe acceptedstandards. This is clearly how things are in actual socialorganizations, and it emphasizesthe perfect appropriatenessof the circularcharacterof our conditionsin 4.5.3.

4.6.3.We have previously mentioned, but purposelyneglectedto dis-cuss,an important objection:That neither the existencenor the uniquenessof a solution S in the senseof the conditions(4:A:a)and (4:A:b),or (4:A:c),in 4.5.3.is evident or established.

Therecan be, of course,no concessionsas regards existence.If itshould turn out that our requirementsconcerninga solution S are,in anyspecialcase,unfulfillable, this would certainly necessitatea fundamentalchange in the theory. Thus a general proof of the existenceof solutionsSfor all particular cases2 is most desirable.It will appear from our subse-quent investigations that this proof has not yet beencarried out in full

generality but that in all casesconsideredso far solutionswere found.As regardsuniquenessthe situation is altogetherdifferent. The often

mentioned \" circular\"characterof our requirements makes it ratherprobablethat the solutionsarenot in generalunique. Indeedwe shall inmost casesobservea multiplicity of solutions.3 Consideringwhat we havesaidabout interpretingsolutionsas stable \" standardsof behavior\" this hasa simple and not unreasonablemeaning, namely that given the samephysicalbackgrounddifferent \" establishedordersof society\"or \" acceptedstandards of behavior\" can be built, all possessingthose characteristicsofinner stability which we have discussed.Sincethis conceptof stabilityis admittedly of an \"inner\" nature i.e.operative only under the hypothesisof generalacceptanceof the standard in question thesedifferent standardsmay perfectly well be in contradictionwith eachother.

4.6.4.Our approach should be compared with the widely held viewthat a social theory is possibleonly on the basis of some preconceivedprinciplesof socialpurpose. Theseprincipleswould includequantitativestatementsconcerningboth the aims to be achieved in toto and the appor-tionments betweenindividuals. Oncethey areaccepted,a simplemaximum

problemresults.

1 We use the word \"conform\" (to the \"standard of behavior'')temporarily as asynonym for being contained in S,and the word \"upset\" asa synonym for dominate.

*In the terminology of games:for all numbers of participants and for all possiblerules of the game.1An interesting exceptionis 65.8.)))

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SOLUTIONSAND STANDARDSOF BEHAVIOR 43

Let us note that no such statement of principlesis ever satisfactoryper se, and the argumentsadduced in its favor areusually eitherthose ofinner stability or of less clearly defined kinds of desirability,mainly con-cerning distribution.

Little can be said about the lattertype of motivation. Our problemis not to determine what ought to happen in pursuanceof any set ofnecessarilyarbitrary a priori principles,but to investigate where theequilibrium of forceslies.

As far as the first motivation is concerned,it has been our aim to givejust thosearguments preciseand satisfactoryform, concerningboth globalaims and individual apportionments. This made it necessaryto takeupthe entirequestionof inner stabilityasa problemin its own right. A theorywhich is consistentat this point cannot fail to give a preciseaccountof theentireinterplay of economicinterests,influence and power.

4.7.Gamesand SocialOrganizations

4.7.It may now be opportuneto revive the analogy with games,whichwe purposelysuppressedin the previous paragraphs (cf. footnote 1 onp. 41). The parallelismbetween the solutionsS in the senseof 4.5.3.onone hand and of stable \" standards of behavior \" on the other can be usedfor corroborationof assertionsconcerningtheseconceptsin both directions.At leastwe hope that this suggestionwill have someappeal to the reader.We think that the procedureof the mathematical theory of games ofstrategy gainsdefinitely in plausibility by the correspondencewhich existsbetween its conceptsand those of social organizations. On the otherhand, almostevery statement which we or for that matter anyone elseever made concerningsocial organizations, runs afoul of some existingopinion. And, by the very nature of things, most opinionsthus far couldhardly have been proved or disprovedwithin the field of social theory.It is therefore a greathelpthat all our assertionscan be borneout by specificexamplesfrom the theory of gamesof strategy.

Suchis indeed one of the standard techniquesof using modelsin thephysicalsciences.This two-way procedurebrings out a significant func-tion of models,not emphasizedin their discussionin 4.1.3.

Togive an illustration:The question whether several stable \" ordersof society\" or \" standards of behavior \" based on the samephysicalback-ground are possibleor not, is highly controversial.Thereis little hopethat it will be settledby the usual methodsbecauseof the enormouscom-plexity of this problemamong otherreasons.But we shall give specificexamplesof gamesof threeor four persons,where one gamepossessesseveralsolutionsin the senseof 4.5.3.And someof theseexampleswill be seento be modelsfor certainsimpleeconomicproblems.(Cf.62.)

4.8.Concluding Remarks

4.8.1.In conclusionit remainsto makea few remarksof a more formalnature.)))

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We beginwith this observation:Our considerationsstarted with singleimputations which were originally quantitative extractsfrom moredetailed combinatorial setsof rules. From thesewe had to proceedtosetsS of imputations,which undercertainconditionsappearedas solutions.Sincethe solutionsdo not seem to be necessarilyunique, the completeanswer to any specific problemconsistsnot in finding a solution, but in

determiningthe setof all solutions. Thus the entity for which we look in

any particularproblemis really a setof setsof imputations. Thismay seemto be unnaturally complicatedin itself;besidesthereappearsno guaranteethat this processwill not have to be carriedfurther, conceivably becauseof laterdifficulties. Concerningthesedoubts it sufficesto say: First,themathematical structureof the theory of gamesof strategyprovidesa formaljustification of our procedure.Second,the previously discussedconnectionswith \" standards of behavior \" (correspondingto setsof imputations)andof the multiplicity of \" standards of behavior \" on the samephysicalback-ground (correspondingto setsof setsof imputations)makejust this amountof complicatednessdesirable.

Onemay criticize our interpretationof setsof imputationsas \" standardsof behavior.\" Previously in 4.1.2.and 4.1.4.we introduceda more ele-mentary concept,which may strike the readeras a direct formulation of a\" standard of behavior\":this was the preliminary combinatorial conceptof a solution as a setof rulesfor eachparticipant,telling him how to behavein every possiblesituation of the game. (Fromthese rules the singleimputations were then extractedas a quantitative summary, cf. above.)Sucha simpleview of the \" standard of behavior\" could be maintained,however, only in gamesin which coalitionsand the compensationsbetweencoalition partners (cf. 4.3.2.)play no role,sincethe above rules do notprovide for thesepossibilities. Gamesexistin which coalitionsand compen-sationscan be disregarded:e.g.the two-persongame of zero-summentionedin 4.2.3.,and more generally the \" inessential\" games to be discussedin27.3.and in (31:P)of 31.2.3.But the general,typical game in particularall.significantproblemsofa socialexchangeeconomy cannot betreatedwith-out thesedevices. Thus the samearguments which forcedus to considersetsof imputations instead of singleimputationsnecessitatethe abandonmentof that narrow conceptof \" standard of behavior.\" Actually we shall callthesesetsof rules the \" strategies\"of the game.

4.8.2.Thenext subjectto be mentioned concernsthe staticor dynamicnature of the theory. We repeatmost emphatically that our theory isthoroughly static. A dynamic theory would unquestionably be morecompleteand therefore preferable. But thereis ampleevidencefrom otherbranchesof sciencethat it is futile to try to build one as long as the staticside is not thoroughly understood. On the other hand, the readermayobjectto somedefinitely dynamic arguments which weremadein the courseof our discussions.This appliesparticularly to all considerationsconcern-ing the interplay of various imputationsunder the influence of \"domina-)))

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tion,\" cf. 4.6.2.We think that this is perfectly legitimate. A statictheory dealswith equilibria.1 The essentialcharacteristicof an equilibriumis that it has no tendencyto change,i.e.that it is not conducive to dynamicdevelopments. An analysis of this feature is, of course,inconceivablewithout the useof certain rudimentary dynamic concepts.Theimportantpoint is that they arerudimentary. In other words:Forthe real dynamicswhich investigatesthe precisemotions, usually far away from equilibria,amuch deeperknowledgeof thesedynamic phenomenais required.2'3

4.8.3.Finally let us note a point at which the theory of socialphenomenawill presumablytake a very definite turn away from the existingpatternsofmathematical physics. Thisis,of course,only a surmiseon a subjectwheremuch uncertainty and obscurityprevail.

Our static theory specifiesequilibria i.e.solutionsin the senseof 4.5.3.which aresetsof imputations. A dynamic theory when one is found

will probablydescribethe changesin termsof simplerconcepts:of a singleimputation valid at the moment under consideration or somethingsimilar. Thisindicatesthat the formal structureof this part of the theorythe relationshipbetweenstaticsand dynamics may be genericallydifferentfrom that of the classicalphysicaltheories.4

All these considerationsillustrate oncemore what a complexity oftheoreticalforms must be expectedin socialtheory. Our static analysisalone necessitatedthe creationof a conceptualand formal mechanism whichis very different from anything used,for instance,in mathematical physics.Thus the conventional view of a solution as a uniquely defined number oraggregateof numberswas seento be too narrow for our purposes,in spiteof its successin other fields. The emphasison mathematical methodsseemsto be shifted more towardscombinatoricsand settheory and awayfrom the algorithm of differential equationswhich dominate mathematicalphysics.

1Thedynamic theory dealsalsowith inequilibria even if they aresometimes called\"dynamic equilibria.\"1Theabovediscussion of statics versus dynamics is, ofcourse,not at all a constructionad hoc. The readerwho is familiar with mechanics for instance will recognizein it areformulation of well known features of the classicalmechanical theory of statics anddynamics. What we do claim at this time is that this is a general characteristic ofscientific procedureinvolving forcesand changesin structures.

3Thedynamic conceptswhich enter into the discussion ofstatic equilibria areparallelto the \"virtual displacements

\" in classicalmechanics. Thereadermay alsoremember atthis point the remarks about \"virtual existence1' in 4.3.3.

4 Particularly from classicalmechanics. Theanalogies of the type used in footnote 2above,ceaseat this point.)))

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CHAPTER IIGENERAL FORMAL DESCRIPTIONOF GAMESOF STRATEGY

5.Introduction5.1.Shift of Emphasis from Economicsto Games

5.1.It shouldbe clearfrom the discussionsof Chapter I that a theoryof rational behavior i.e.of the foundations of economicsand of the mainmechanismsof socialorganization requiresa thorough study of the \" gamesof strategy/' Consequentlywe must now takeup the theory of gamesas anindependentsubject. In studying it as a problem in its own right, ourpoint of view must of necessityundergoa seriousshift. In Chapter I ourprimary interestlay in economics.It was only after having convincedourselvesof the impossibilityof making progressin that field without aprevious fundamental understanding of the games that we graduallyapproachedthe formulations and the questionswhich arepartial to thatsubject. But the economicviewpoints remainedneverthelessthe dominantonesin all of Chapter I. From this ChapterIIon, however, we shallhaveto treatthe gamesas games.Therefore we shall not mind if somepointstaken up have no economicconnectionswhatever, it would not be possibleto do full justiceto the subjectotherwise. Of coursemost of the mainconceptsarestill those familiar from the discussionsof economicliterature(cf. the next section)but the details will often bealtogetheralien to itand details,as usual, may dominate the expositionand overshadowtheguiding principles.

5.2.GeneralPrinciplesof Classification and of Procedure

5.2.1.Certain aspectsof \" games of strategy\" which were alreadyprominent in the last sectionsof ChapterI will not appear in the beginningstagesof the discussionswhich we are now undertaking. Specifically:Therewill be at first no mention of coalitionsbetween players and thecompensationswhich they pay to each other. (Concerningthese, cf.4.3.2.,4.3.3.,in ChapterI.) We give a brief accountof the reasons,whichwill alsothrow somelight on our generaldispositionof the subject.

An important viewpoint in classifying gamesis this:Is the sum of allpaymentsreceivedby all players (at the end of the game)always zero;oris this not thecase? If it is zero, then one can say that the playerspay onlyto eachother,and that no productionor destructionof goodsis involved.All gameswhich areactually playedfor entertainmentareof this type. Butthe economicallysignificant schemesaremost essentiallynot such. Therethe sum of all payments, the total socialproduct, will in generalnot be

46)))

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zero, and not even constant. I.e.,it will dependon the behavior of theplayers the participants in the socialeconomy. This distinction wasalreadymentioned in 4.2.1.,particularly in footnote 2, p.34. We shall callgamesof the first-mentioned type zero-sumgames,and thoseof the lattertype non-zero-sumgames.

We shall primarily constructa theory of the zero-sum games,but it will

be found possibleto dispose,with its help,of all games,without restriction.Precisely:We shall show that the general(hencein particular the variablesum) n-person game can be reduced to a zero-sumn + 1-persongame.(Cf.56.2.2.)Now the theory of the zero-sumr?-persongame will be basedon the specialcaseof the zero-sum two-persongame. (Cf.25.2.)Henceour discussionswill beginwith a theory of thesegames,which will indeedbe carriedout in ChapterIII.

Now in zero-sum two-person games coalitions and compensationscan play no role.1 The questionswhich areessentialin thesegamesareof a different nature. Theseare the main problems:How does eachplayer plan his course i.e.how doesone formulate an exactconceptof astrategy? What information is available to eachplayer at every stageof the game? What is the roleof a playerbeinginformed about the otherplayer'sstrategy? About the entire theory of the game?

5.2.2.All these questionsareof courseessentialin all games,for anynumber of players,even when coalitions and compensationshave comeintotheir own. But for zero-sum two-persongames they are the only oneswhich matter, as our subsequentdiscussionswill show. Again, all thesequestionshave beenrecognizedas important in economics,but wethink thatin the theory of gamesthey arisein a more elementary as distinguishedfrom composite fashion. They can, therefore, be discussedin a preciseway and as we hope to show be disposedof. But in the processof thisanalysisit will be technically advantageous to rely on picturesand exampleswhich arerather remotefrom the field of economicsproper, and belongstrictly to the field of gamesof the conventional variety. Thus the dis-cussions which follow will be dominated by illustrations from Chess,\" MatchingPennies,\"Poker,Bridge,etc.,and not from the structure of

cartels,markets,oligopolies,etc.At this point it is also opportuneto recallthat we considerall trans-

actionsat the end of a game as purely monetary ones i.e.that we ascribeto all playersan exclusivelymonetary profit motive. Themeaning of thisin termsof the utility conceptwas analyzed in 2.1.1.in ChapterI. Forthe

present particularly for the \" zero-sum two-persongames\"to be discussed

1The only fully satisfactory \"proof\" of this assertion lies in the construction of acompletetheory of all zero-sum two-person games, without use of those devices. Thiswill be done in Chapter III,the decisiveresult being contained in 17. It ought to beclearby common sense,however, that \"understandings\" and \"coalitions\" can have norolehere:Any such arrangement must involve at leasttwo players hencein this caseall

players for whom the sum of payments is identically zero. I.e.th re areno opponentsleft and no possibleobjectives.)))

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first (cf. the discussionof 5.2.1.)it is an absolutelynecessarysimplifi-cation. Indeed,we shall maintain it through most of the theory, althoughvariants will be examinedlateron. (Cf.ChapterXII,in particular66.)

6.2.3.Our first task is to give an exactdefinition of what constitutesagame. As long as the conceptof a game has not beendescribedwith

absolute mathematical combinatorial precision, we cannot hope togive exactand exhaustive answersto the questionsformulated at the endof 5.2.1.Now while our first objectiveis as was explainedin 5.2.1.thetheory of zero-sum two-persongames,it is apparent that the exactdescrip-tion of what constitutesa game neednot be restrictedto this case. Conse-quently we can begin with the descriptionof the generaln-persongame.In giving this descriptionwe shall endeavorto do justiceto all conceivablenuancesand complicationswhich can arisein a game insofar as they arenot of an obviously inessentialcharacter.In this way we reach in severalsuccessivesteps a rather complicatedbut exhaustive and mathematicallyprecisescheme.And then we shall seethat it is possibleto replacethisgeneralschemeby a vastly simplerone,which is neverthelessfully andrigorously equivalent to it. Besides,the mathematical device whichpermits this simplification is also of an immediate significance for ourproblem:It is the introduction of the exactconceptof a strategy.

It shouldbe understoodthat the detour which leads to the ultimate,simple formulation of the problem,over considerablymore complicatedones is not avoidable. It is necessary to show first that all possiblecomplicationshave beentaken into consideration,and that the mathe-matical devicein questiondoesguaranteethe equivalence of the involvedsetup to the simple.

All this can and must be donefor all games,of any number of play-ers. But after this aim has been achievedin entire generality,the nextobjectiveof the theory is as mentioned above to find a completesolutionfor the zero-sum two-persongame. Accordingly, this chapter will dealwith all games,but the next one with zero-sumtwo-persongamesonly. Afterthey aredisposedof and someimportant exampleshave beendiscussed,weshall beginto re-extendthe scopeof the investigation first to zero-sum n-persongames,and then to all games.

Coalitionsand compensationswill only reappearduring this latterstage.

6.TheSimplifiedConceptof a Game

6.1.Explanation of the Termini Technici

6.1.Beforean exactdefinition of the combinatorial conceptof a gamecan be given, we must first clarify the useof sometermini. Therearesome notions which are quite fundamental for the discussionof games,but the useof which in everyday languageis highly ambiguous. Thewordswhich describethem areusedsometimesin one sense,sometimesin another,and occasionally worst of all as if they were synonyms. We must)))

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therefore introducea definite usageof termini technici, and rigidly adhereto it in all that follows.

First,one must distinguishbetween the abstract conceptof a game,and the individual plays of that game. The game is simply the totalityof the rules which describeit. Every particular instance at which thegame is played in a particularway from beginning to end,is a play.1

Second,the correspondingdistinctionshouldbe made for the moves,which arethe componentelementsof the game. A move is the occasionof a choicebetweenvarious alternatives, to bemade eitherby one of theplayers, or by some devicesubjectto chance,under conditionspreciselyprescribedby the rulesof the game. Themove is nothing but this abstract\" occasion/'with the attendant details of description, i.e.a componentof the game. The specificalternative chosenin a concreteinstance i.e.in a concreteplay is the choice. Thus the moves are relatedto thechoicesin the same way as the game is to the play. The game consistsof a sequenceof moves, and the play of a sequenceof choices.2

Finally, the rulesof the game shouldnot be confusedwith the strategiesof the players. Exactdefinitions will be given subsequently,but thedistinction which we stress must be clearfrom the start. Each playerselectshis strategy i.e.the generalprinciplesgoverning his choices freely.While any particular strategy may be good or bad provided that theseconceptscan be interpreted in an exactsense(cf. 14.5.and 17.8-17.10.)it is within the player's discretionto useor to rejectit. Therules of thegame,however, areabsolutecommands. If they areever infringed, thenthe whole transactionby definition ceasesto be the game describedby thoserules. In many casesit is even physicallyimpossibleto violate them.3

6.2.TheElements of the Game

6.2.1.Let us now considera game F of n playerswho, for the sake ofbrevity, will be denotedby 1, , n. Theconventional pictureprovidesthat this game is a sequenceof moves, and we assumethat both the numberand the arrangementof thesemoves is given ab initio. We shallseelaterthat these restrictionsare not really significant, and that they can beremoved without difficulty. For the presentlet us denotethe (fixed)number of moves in F by v this is an integerv = 1,2, . Themovesthemselveswe denoteby 3TCi, , 3TC,, and we assumethat this is thechronologicalorderin which they areprescribedto takeplace.

1In most gameseveryday usagecallsa play equally a game ; thus in chess,in poker,in many sports, etc. In Bridge a play correspondsto a \"rubber,\" in Tennis to a \"set,\"but unluckily in thesegames certain components of the play are again called\"games.\"The French terihinology is tolerably unambiguous: \"game\" * \"jeu,\" \"play\"\"partie.\"1In this sensewe would talk in chessof the first move, and of the choice\"E2-E4.\"

1E.g.:In Chessthe rules of the game forbid a player to move his king into a positionof \"check.\" This is a prohibition in the sameabsolutesensein which he may not move apawn sideways. But to move the king into a position where the opponent can \"check-mate\" him at the next move is merely unwise, but not forbidden.)))

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Every move 3TC,, K = 1, , v, actually consists of a number ofalternatives, among which the choice which constitutes the move 9TC,takesplace. Denotethe number of these alternatives by a< and thealternativesthemselvesby GLK(l), , &()

The moves areof two kinds. A move of the first kind, or a personalmove, is a choicemadeby a specificplayer,i.e.dependingon his free decisionand nothing else. A move of the secondkind, or a chance move, is a choicedependingon somemechanicaldevice,which makesits outcomefortuitouswith definite probabilities.1 Thus for every personal move it must bespecifiedwhich player's decisiondeterminesthis move, whose move it is.We denotethe playerin question(i.e.hisnumber)by kK. Sofc, = 1, ,n. Fora chancemove we put (conventionally) fc<

= 0. In this casetheprobabilitiesof the various alternativesa(l), , &(<*)must be given.We denotetheseprobabilitiesby p(l), , p(a)respectively.26.2.2.In a move 3TC, the choiceconsists of selectingan alternative(*(!), ' , Ct(<O>i.e.its number!,,.We denotethe numberso chosenby <r,. Thus this choiceis characterizedby a number <r = 1,

, a,. And the completeplay is describedby specifyingall choices,correspondingto all moves OTli, , 9R,. I.e.it isdescribedby a sequence

Now the rule of the game T must specifywhat the outcomeof the playis for eachplayerk = !,- , n, if the play isdescribedby a given sequence<TI, ov I.e.what payments every player receiveswhen the play iscompleted.Denotethe payment to the playerk by $k ($k > if k receivesa payment, $k < if he must make one, $* = if neitheris the case).Thus each$* must be given as a function of the <TI, , <r,:

$k =SF*(<ri, , <r v), k = 1, , n.

We emphasizeagain that the rules of the gameT specifythe function$k(*i) ' ' ' , <r>) only as a function,3 i.e.the abstractdependenceof each5u on the variables<TI, , <r . But all the time eachcr, is a variable,with the domain of variability !,-,,.A specificationof particularnumerical values for the <r<, i.e.the selectionof a particular sequence<n,

, <r,, is no part of the game T. It is, as we pointed out above, thedefinition of a play.

1E.g.,dealing cardsfrom an appropriately shuffled deck,throwing dice,etc. It isevenpossibleto include certain gamesofstrength and skill, where \"strategy\" plays a role,e.g.Tennis, Football, etc. In thesethe actions of the players are up to a certain pointpersonalmoves i.e.dependent upon their free decision and beyond this point chancemoves, the probabilities being characteristicsof the player in question.Sincethe p(l), , pK(aK) are probabilities, they are necessarilynumbers *z 0.Sincethey belong to disjunct but exhaustive alternatives, their sum (for a fixed K) mustbeone. I.e.:

3For a systematic exposition of the conceptof a function cf. 13.1.)))

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6.3.Information and Preliminarity

6.3.1.Our descriptionof the gameF is not yet complete.We havefailed to include specificationsabout the state of information of everyplayer at eachdecisionwhich he has to make, i.e.whenever a personalmove turns up which is his move. Therefore we now turn to this aspectof the matter.

Thisdiscussionis best conductedby followingthe moves 9fTli, , 2fTl,,as the correspondingchoicesaremade.

Let us therefore fix our attention on a particular move 3fH. If this3TC< is a chancemove, then nothing more needbesaid:the choiceis decidedby chance;nobody's will and nobody's knowledgeof other things caninfluence it. But if $TC,is a personalmove, belongingto the playerfc, thenit is quite important what kK'a stateof information is when he forms hisdecisionconcerningTO, i.e.his choiceof <r.

Theonly things he can beinformed about arethe choicescorrespondingto the moves preceding9fR the moves SfTCi, , 9fTC_i. I.e.he may knowthe values of <TI, , <rK-.i.But he need not know that much. It is animportant peculiarityof F, justhow much information concerning(TI, ,<rK-i the player kK is granted, when he is called upon to choose<? K. Weshall soonshow in severalexampleswhat the nature of such limitations is.

Thesimplesttype of rule which describeskK'sstateof information at 3fn,

is this:a set A consistingof somenumbersfrom among X = 1, , K 1,is given. It is specifiedthat kK knowsthe values of the a\\ with X belong-ing to A, and that he is entirely ignorant of the <r\\ with any otherX.

In this casewe shall say, when X belongsto A,, that X is preliminaryto /c. This impliesX = 1, , K 1,i.e.X < K, but neednot be impliedby it. Or, if we consider,insteadof X, K, the correspondingmoves SfTlx, 3Tl:Preliminarity impliesanteriority, 1 but neednot be impliedby it.

6.3.2.In spite of its somewhat restrictive character,this conceptofpreliminarity deservesa closerinspection. In itself, and in its relationshipto anteriority (cf. footnote 1 above), it gives occasionto various combina-torial possibilities. Thesehave definite meaningsin thosegamesin whichthey occur,and we shall now illustrate them by someexamplesof particu-larly characteristicinstances.

6.4.Preliminarity, Transitivity, and Signaling

6.4.1.We begin by observing that there existgames in which pre-liminarity and anteriority arethe samething. I.e.,where the playerskK

who makes the (personal)move 9TC,is informed about the outcome of thechoicesof all anteriormoves 3Tli,

* , 9Tl<_ i. Chessisa typicalrepresenta-tive of this classof gamesof \"perfect\"information. They aregenerallyconsideredto be of a particularly rational character.We shall seein 15.,specificallyin 15.7.,how this can be interpreted in a preciseway.

1 In time, X < K means that 9H\\ occursbeforeWLK .)))

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Chesshas the further feature that all its moves arepersonal. Now itis possibleto conservethe first-mentioned property the equivalenceofpreliminarity and anteriority even in gameswhich contain chancemoves.Backgammonis an exampleof this.1 Somedoubt might be entertainedwhether the presenceof chancemoves doesnot vitiate the \" rational char-acter\"of the gamementioned in connectionwith the precedingexamples.

We shallseein 15.7.1.that this is not so if a very plausibleinterpretationof that \" rational character\"is adhered to. It is not important whetherall moves arepersonalor not; the essentialfact is that preliminarity andanteriority coincide.

6.4.2.Let us now considergames where anteriority doesnot implypreliminarity. I.e.,where the player fc who makesthe (personal)move 9TC,

is not informed about everything that happenedpreviously. Thereis alarge family of gamesin which this occurs.Thesegamesusually con-tain chancemoves as well as personalmoves. Generalopinion considersthem as being of a mixed character:while their outcomeis definitelydependent on chance,they are also strongly influenced by the strategicabilitiesof the players.

Pokerand Bridgearegoodexamples.Thesetwo gamesshow,further-more, what peculiar features the notion of preliminarity can present,onceit has beenseparated from anteriority. This point perhapsdeservesa little more detailedconsideration.

Anteriority, i.e.the chronological ordering of the moves, possessesthe property of transitivity. 2 Now in the presentcase,preliminarityneednot be transitive. Indeedit is neither in Pokernor in Bridge,and theconditionsunder which this occursarequite characteristic.

Poker:Let 3TCM be the deal of his \"hand\" to player 1 a chancemove;3Tlxthe first bid of player1 a personalmove of 1; 9fTC the first (subsequent)bid of player2 a personalmove of 2. Then 3flfl M is preliminary to 9fTlx and3Tlx to 3fll but 2fTl M is not preliminary to 3TC,.3 Thus we have intransitivity,but it involves both players. Indeed,it may first seemunlikely thatpreliminarity couldin any game be intransitive among the personalmovesof one particularplayer. Itwould requirethat this playershould\"forget\"betweenthe moves 9fTl\\ and 3TC,the outcomeof the choiceconnectedwith

3TV and it is difficult to seehow this \"forgetting\" couldbe achieved,and

1Thechancemoves in Backgammon are the dicethrows which decidethe total num-berof stepsby which eachplayer'smen may alternately advance. Thepersonalmovesare the decisionsby which eachplayer partitions that total number of stepsallotted tohim among his individual men. Also his decisionto double the risk, and his alternativeto acceptor to give up when the opponent doubles. At every move, however, the out-comeof the choicesof all anterior moves arevisible to all on the board.

2I.e.:If 3TC M is anterior to 9Tlx and 311x to 9K then 3TC M is anterior to 9TC. Specialsitua-tions where the presenceor absenceof transitivity was of importance, were analyzed in4.4.2.,4.6.2.of Chapter Iin connection with the relation of domination.

3I.e.,1 makes his first bid knowing his own \"hand\"; 2 makes his first bid knowing1's(preceding) first bid; but at the same time 2 is ignorant of 1's\"hand.\"4 We assumethat 9TC M is preliminary to 9fR\\ and 3Tlxto 3TZ* but 9TC M not to 3TC*.)))

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even enforced1 Neverthelessour next exampleprovidesan instance ofjust this.

Bridge:Although Bridge is played by 4 persons, to be denoted byA,B,C,D,it shouldbe classifiedas a two-persongame. Indeed,A and Cform a combination which is more than a voluntary coalition, and so doB and D. ForA to cooperatewith B (orD) instead of with C would be\" cheating,\"in the samesensein which it would be \"cheating\"to lookintoJB'scardsor failing to followsuit during the play. I.e.it would be a viola-tion of the rulesof the game. If three(or more)personsplay poker,thenit is perfectly permissiblefor two (or more) of them to cooperateagainstanother player when their interests areparallel but in BridgeA and C(and similarly B and D) must cooperate,while A and B areforbiddentocooperate.Thenatural way to describethis consistsin declaringthat A

and C arereally one player 1,and that B and D arereally one player2.Or,equivalently:Bridgeis a two-persongame,but the two players1and 2do not play it themselves. 1actsthrough two representativesA and Cand2 through two representativesB and D.

Considernow the representativesof 1,A and C. Therulesof thegamerestrictcommunication, i.e.the exchangeof information, between them.E.g.:let 9TC M be the deal of his \"hand\" to A a chancemove; 9TCx the firstcard playedby A a personalmove of 1; 3TI, the card playedinto this trickby C a personalmove of 1. Then 9TIM is preliminary to 3fllx and 2fllx to 9TC,

but 2(TI M is not preliminary to 9TC,.1 Thus we have again intransitivity, butthis time it involves only one player. It is worth noting how the necessary\"forgetting\" of 3TIM between3Tlx and 3Tl was achievedby \"splittingthepersonality\"of 1into A and C.

6.4.3.The above examplesshow that intransitivity of the relation ofpreliminarity correspondsto a very well known componentof practicalstrategy: to the possibilityof \"signaling.\"If no knowledgeof 3fTC M isavailable at 3fTC,but if it is possibleto observe9Hx'soutcomeat 3TC,and 3Tlxhas been influenced by 9fTl M (by knowledgeabout 9TC/S outcome),thenOTx is really a signal from 9TIM to 3TC, a devicewhich (indirectly)relaysinformation. Now two oppositesituationsdevelop,accordingto whether3flflx and 9TC,aremoves of the sameplayer,or of two different players.

In the first case which, as we saw, occursin Bridge the interestofthe player (who is fc x = fc,).lies in promoting the \"signaling,\" i.e.thespreading of information \"within his own organization.\" This desirefinds its realization in the elaboratesystem of \"conventional signals\"inBridge.2 Theseareparts of the strategy, and not of the rulesof the game

1I.e.A plays his first cardknowing his own \"hand\"; Ccontributes to this trick know-ing the (initiating) cardplayed by A ;but at the sametime Cis ignorant of A's \"hand/ 1

1Observethat this \"signaling

\" is consideredto be perfectly fair in Bridge if it iscarriedout by actionswhich areprovided for by the rules of the game. E.g.it is correctfor A and C (the two components of player 1,cf.6.4.2.)to agree beforethe play begins!

that an \"original bid\" of two trumps\" indicates\" a weakness of the other suits. But

it is incorrect i.e.\"cheating11 to indicate a weakness by an inflection of the voiceat

bidding, or by tapping on the table, etc.)))

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54 DESCRIPTIONOF GAMESOF STRATEGY

(cf. 6.1.),and consequentlythey may vary, 1 while the game of Bridgeremainsthe same.

In the secondcase which, as we saw, occursin Poker the interestof the player (we now mean k\\, observethat herek\\ ^ kK) liesin preventingthis \"signaling,\" i.e.the spreadingof information to the opponent (&<).This is usually achieved by irregular and seeminglyillogical behavior(when making the choiceat 3Tlx) this makes it harder for the opponentto draw inferencesfrom the outcomeof 3fTlx (which he sees)concerningtheoutcomeof 91ZM (of which he has no directnews). I.e.this proceduremakesthe \"signal\"uncertain and ambiguous. We shallseein 19.2.1.that this isindeedthe function of \"bluffing\" in Poker.2

We shallcall thesetwo proceduresdirectand inverted signaling. Itoughtto beadded that inverted signaling i.e.misleadingthe opponent occursin almostall games,including Bridge.This is so sinceit is based on theintransitivity of preliminarity when severalplayersareinvolved, which iseasy to achieve. Directsignaling,on the otherhand, is rarer;e.g.Pokercontainsno vestigeof it. Indeed,as we pointedout before,it impliestheintransitivity of preliminarity when only one player is involved i.e.itrequiresa well-regulated\" forgetfulness\" of that player,which isobtainedinBridgeby the deviceof \"splitting the playerup\" into two persons.

At any rate Bridge and Pokerseemto be reasonablycharacteristicinstancesof these two kinds of intransitivity of directand of invertedsignaling,respectively.

Both kindsof signalinglead to a delicateproblemof balancingin actualplaying, i.e.in the processof trying to define \"good,\"\"rational\"playing.Any attempt to signalmore or to signal lessthan \"unsophisticated\"playingwould involve, necessitatesdeviations from the \"unsophisticated\"way ofplaying. And this is usually possibleonly at a definite cost,i.e.its directconsequencesarelosses.Thus the problemis to adjust this \"extra\"signal-ing so that its advantages by forwarding or by withholding informationoverbalancethe losseswhich it causesdirectly. Onefeelsthat this involvessomethinglikethesearchfor an optimum, although it isby no meansclearlydefined. We shall seehow the theory of the two-persongametakes carealreadyof this problem,and we shall discussit exhaustively in one charac-teristicinstance. (Thisis a simplifiedform of Poker. Cf. 19.)

Let us observe, finally, that all important examplesof intransitivepreliminarity aregamescontaining chancemoves. Thisis peculiar,becausethere is no apparent connectionbetween these two phenomena.3- 4 Our

1They may even be different for the two players,i.e.for A and Con one hand andB and D on the other. But \"within the organization'

1 of one player, e.g.for A and C,they must agree.1And that \"bluffing\" is not at all an attempt to secureextra gains in any directsense when holding a weak hand. Cf.loc.cit.

*Cf.the corresponding question when preliminarity coincideswith anteriority, andthus is transitive, as discussedin 6.4.1.As mentioned there, the presenceor absenceofchancemoves is immaterial in that case.

4 \"Matching pennies\" is an examplewhich has a certain importance in this connec-tion. This and other relatedgameswill bediacussedin 18.)))

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(COMPLETECONCEPTOF A GAME 55

subsequentanalysiswill indeedshow that the presenceor absenceof chancemoves scarcelyinfluences the essentialaspects of the strategies in thissituation.

7. The CompleteConceptof a Game

7.1.Variability of the Characteristicsof Each Move

7.1.1.We introducedin 6.2.1.the a* alternatives &*(!), , Gt<(a)of the move STl*. Also the index fc* which characterizedthe move as apersonalor chanceone,and in the first casethe player whose move it is;and in the secondcasethe probabilitiesp*(l), * * , PK (<X K) of the abovealter-natives. We describedin 6.3.1.the conceptof preliminarity with the helpof the setsA,, this beingthe setofall X (from amongthe X = 1, ,* !)which are preliminary to K. We failed to specify,however, whether allthese objectsa*, k f , A and the Ct(<r), p*(a) for o- = 1, , a* dependsolelyon K or alsoon other things. These\" other things

\" can, of course,only be the outcome of the choicescorrespondingto the moves which areanterior to 3fn. I.e.the numbers <n, , <r_i. (Cf.6.2.2.)

This dependencerequiresa more detaileddiscussion.First,the dependenceof the alternatives (%((?) themselves(as distin-

guishedfrom their number a!)on <n, , o-,_i is immaterial. We mayas well assumethat the choicecorrespondingto the move 311,is made notbetweenthe Ct,(<r) themselves,but betweentheir numbersa. Infine, it isonly the <r of 311*,i.e.cr, which occursin the expressionsdescribingthe out-come of the play, i.e.in the functions SF*(<M, , cr*), k = 1, , n.1(Cf.6.2.2.)

Second,all dependences(on en, , (7<_i) which arisewhen 3TI*turnsout to be a chancemove i.e.when kK

= (cf. the end of 6.2.1.)causenocomplications. They do not interfere with our analysisof the behavior ofthe players. This disposes,in particular, of all probabilitiesp(<r),sincethey occuronly in connection with chancemoves. (TheA,, on the otherhand, never occurin chancemoves.)

Third, we must considerthe dependences(on <n, , <r_i) of thekK, A, when 2KI, turns out to be a personalmove.2 Now this possibility

is indeeda sourceof complications. And it is a very real possibility.3 Thereasonis this.

1 Theform and nature of the alternatives GLK(<r) offered at 3TI,might, of course,conveyto the player kK (if WLK is a personal move) some information concerning the anterior<TX, , <r_i values, if the a(<r) dependon those. But any such information shouldbe specifiedseparately, as information available to A; at 2fTl*. We have disfcussed thesimplest schemesconcerning the subject of information in 6.3.1.,and shall completethediscussion in 7.1.2.Thediscussion of , kK, A*, which follows further below, is charac-teristic alsoas far as the roleof the a,t(<r) aspossiblesourcesof information is concerned.

8 Whether this happens for a given K, will itself depend on kK and hence indirectlyon <n, , <7-i sinceit is characterizedby kK & (cf.the endof6.2.1.).

3E.g.:In Chessthe number ofpossiblealternatives at 9R K dependson the positionsof the men, i.e.the previous courseof the play. In Bridge the player who plays the first)))

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56 DESCRIPTIONOF GAMES OF STRATEGY

7.1.2.The player kK must be informed at 311,of the values of a,,kK,A* sincethesearenow part of the rulesof the game which he must observe.Insofaras they dependupon ai, , <r_i, he may draw from them certainconclusionsconcerningthe values of <TI, , <7_i. But he is supposedto know absolutelynothing concerningthe <r\\ with X not in AK! It is hardto seehow conflicts can be avoided.

To be precise:Thereis no conflict in this specialcase:Let A, be inde-pendentof all (T\\,

- - - , <7\\_], and let K, kK dependonly on the <r\\ with X in A*.Then the player kK can certainly not get any information from aK, kK, A

beyond what he knowsanyhow (i.e.the values of the <r\\ with X in A*). Ifthis is the case,we say that we have the specialform of dependence.

But do we always have the specialform of dependence?To take anextremecase:What if AK is always empty i.e.kK expectedto be completelyuninformed at 3TC, and yet e.g.a* explicitlydependent on some of the<n, , <r._i!

This is clearly inadmissible. We must demandthat all numerical con-clusionswhich can be derived from the knowledgeof a*, & AK, must beexplicitlyand ab initio specifiedas information available to the player k K

at 3TC,. It would be erroneous,however, to try to achieve this by includingin A* the indicesX of all theseo-x, on which a,,fc,, A, explicitlydepend. Inthe first placegreatcaremust be exercisedin orderto avoid circularity inthis requirement,as far as A is concerned.1 But even if this difficulty doesnot arise,becauseA, dependsonly on K and not on cri, , o-^i i.e.if theinformation available to every player at every moment is independentofthe previous courseof the play the above proceduremay still be inadmis-sible. Assume, e.g.,that aK dependson a certain combination of some <r\\

from among the X = l, -,* 1,and that the rules of the gamedoindeedprovidethat the playerkK at y(l K shouldknow the value of this com-bination,but that it doesnot allow him to know more (i.e.the values of theindividual ai, , ov-i). E.g.:Hemay know the value of cr M + <r\\ whereM, X are both anteriorto K (ju, X < K), but he is not allowed to know theseparatevalues of <T M and a\\.

One could try various tricks to bring back the above situation to ourearlier,simpler,scheme,which describesfc/s stateof information by meansof the set A,.2 But it becomescompletelyimpossibleto disentanglethevarious componentsof fc/s information at 3TI,, if they themselvesoriginatefrom personal moves of different players, or of the same player but in

card to the next trick, i.e.kK at 911*, is the onewho took the last trick, i.e.again dependentupon the previous courseof the play. In someforms of Poker, and someother relatedgames,the amount of information available to a player at a given moment, i.e.A* at 9HK,dependson what he and the others did previously.1The <r\\ on which, among others, A, dependareonly defined if the totality of all A,for all sequences<n,

- , <r<_i, is considered.Should every A* contain theseX?2 In the aboveexample one might try to replacethe move 9TIM by a new one in which

not o>is chosen,but <? -f *\\. SHI* would remain unchanged. Then k*at 9R K would beinformed about the outcome of the choiceconnectedwith the new 3TCu onlv.)))

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COMPLETE CONCEPTOF A GAME 57

different stagesof information. In our above examplethis happens iffc M 5^ fc\\, or if fc^

= fc x but the stateof information of this player is not thesameat 311,,and at SfTCx.

1

7.2.TheGeneralDescription7.2.1.Therearestill various, more or lessartificial, tricksby which one

couldtry to circumvent thesedifficulties. But the most natural procedureseemsto be to admit them, and to modify our definitions accordingly.

This is doneby sacrificing the A, as a meansof describingthe stateofinformation. Instead,we describethe stateof information of the playerfc,at the time of his personalmove 311,explicitly:By enumerating thosefunc-tionsof the variable <r\\ anterior to this move i.e.of the <TI, , <r_i thenumerical values of which he is supposedto know at this moment. This isa systemof functions, to be denotedby <,.

So$,is a setof functions))

Sincethe elementsof < describethe dependenceon <TI, , <r_i, so<,itselfis fixed,i.e.dependingon K only.2 , k K may dependon <n, , <r_i, andsincetheir values areknown to kK at 91Z,, thesefunctions))

must belong to $. Of course,whenever it turns out that k K= (for a

specialsetof <ri, , cr_i values),then the move 3fTl is a chanceone(cf.above),and no usewill bemadeof <!>* but this doesnot matter.

Our previous mode of description,with the A*, is obviously a specialcaseof the presentone,with the $*.8

7.2.2.At this point the readermay feel a certaindissatisfactionaboutthe turn which the discussionhas taken. It is true that the discussionwasdeflectedinto this directionby complicationswhich arosein actual andtypical games (cf. footnote 3 on p.55). But the necessityof replacingthe AK by the <$ originated in our desireto maintain absolute formal(mathematical) generality. Thesedecisive difficulties, which caused usto take this step (discussedin 7.1.2.,particularly as illustrated by thefootnotes there)werereally extrapolated.I.e.they werenot characteristic))

1 In the instance offootnote 2on p.56,this means :If fr M ?*fc\\, there isno player to whomthe new move 3!Z M (where <rM 4-<rx is chosen, and which ought to be personal)can beattributed. If & M

= k\\ but the stateof information varies from 3TIM to $TCx,then no stateof information canbesatisfactorily prescribedfor the new move 9Tt M.1This arrangement includes nevertheless the possibility that the stateof informationexpressedby * dependson <n, , <rK_\\. This is the case if, e.g.,all functionsh(vi, , <r_0 of <J> show an explicit dependenceon o>for onesetof values of <r\\, while

being independent of o>for other values of <r\\. Yet * is fixed.1If ** happens to consist of all functions of certain variables <r\\ say of those for

which X belongsto a given set M* and of no others, then the $* description specializesback to the A one:A, being the aboveset M. But we have seenthat we cannot, in

general,count upon the existenceof such a set.)))

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58 DESCRIPTIONOF GAMES OF STRATEGY

of the original examples,which areactual games. (E.g.Chessand Bridgecan be describedwith the helpof the A*.)

Thereexistgameswhich requirediscussionby meansof the $. But inmost of them onecould revert to the A, by meansof various extraneoustricks and the entiresubjectrequiresa ratherdelicateanalysisupon whichit does not seemworth while to enterhere.1 Thereexistunquestionablyeconomicmodelswhere the $ arenecessary.2

Themost important point, however, is this.In pursuitof the objectiveswhich we have setourselveswe must achieve

the certainty of having exhaustedall combinatorialpossibilitiesin connec-tion with the entireinterplay of the various decisionsof the players, theirchanging statesof information, etc. Theseareproblems,which have beendwelt uponextensivelyin economicliterature.We hopeto show that theycan be disposedof completely. But for this reasonwe want to be safeagainst any possibleaccusationof having overlookedsomeessentialpossi-bility by unduespecialization.

Besides,it will be seenthat all the formal elementswhich we areintro-ducing now into the discussiondo not complicateit ultima analyst. I.e.they complicateonly the present,preliminary stage of formal descrip-tion. The final form of the problemturns out to be unaffected by them.(Cf.11.2.)

7.2.3.Thereremainsonly onemore point to discuss:The specializingassumptionformulated at the very start of this discussion(at the beginningof 6.2.1.)that both the number and thearrangementof the moves aregiven(i.e.fixed) ab initio. We shall now seethat this restrictionis not essential.

Considerfirst the \"arrangement\"of the moves. The possiblevaria-bility of the nature of eachmove i.e.of its fc has already receivedfull

consideration(especiallyin 7.2.1.).Theorderingof the moves SHI,, k = 1,, v, was from the start simply the chronologicalone. Thus there is

nothing left to discusson this score.Considernext the number of moves v. This quantity too could be

variable, i.e.dependent upon the courseof the play.3 In describingthisvariability of v a certainamount of caremust beexercised.

1We mean card games where players may discard some cardswithout uncoveringthem, and are allowed to take up or otherwise use openly a part of their discardslater.Thereexistsalsoa game of double-blind Chess sometimes called\" Kriegsspiel\" which

belongsin this class. (For its description cf.9.2.3.With referenceto that description:Each player knows about the \"

possibility\" of the other's anterior choices,withoutknowing those choicesthemselves and this \"possibility\" is a function of all anteriorchoices.)

1Let a participant be ignorant of the full detailsof the previous actionsof the others,but let him beinformed concerning certain statistical resultants of those actions.

3It is, too, in most games:Chess,Backgammon, Poker,Bridge. In the caseof Bridgethis variability is due first to the variable length of the \"bidding\" phase,and secondtothe changing number of contracts neededto make a \"rubber\" (i.e.a play). Examplesof gameswith a fixed v areharder to find: we shall seethat we can make v fixed in everygame by an artifice, but games in which v is ab initio fixed are apt to be monotonous.)))

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COMPLETECONCEPT OF A GAME 59

The courseof the play is characterizedby the sequence(of choices)*i,' ' ' , <r (cf. 6.2.2.).Now one cannot statesimply that v may be afunction of the variables <TI, , <r,, becausethe full sequence<TI, , a,cannot be visualized at all, without knowing beforehand what its length v

is going to be.1 Thecorrectformulation is this:Imaginethat the variables0*1,0*2,ffs,

* arechosenone after the other.2 If this successionof choicesis carriedon indefinitely, then the rulesof the game must at someplace v

stop the procedure.Then v for which the stopoccurswill, of course,dependon all the choicesup to that moment. It is the number of moves in thatparticular play.

Now this stop rule must be suchas to give a certainty that every con-ceivable play will be stopped sometime.I.e.it must be impossibletoarrange the successivechoicesof a\\, <r 2, <r 3, in sucha manner (subjectto the restrictionsof footnote 2 above) that the stop never comes.Theobvious way to guarantee this is to devise a stop rule for which it iscertain that the stop will comebefore a fixed moment, say *>*. I.e.thatwhile v may dependon o-i, a*, a^ - , it is sure to be v ^ v* where v*does not dependon <TI, <r2, <r3, . If this is the casewe say that thestop rule is boundedby v*. We shall assumefor the gameswhich we con-sider that they have stop rulesboundedby (suitable,but fixed) numbers

* 3,4

1I.e.one cannot say that the length of the game dependson all choicesmade in con-nection with all moves, sinceit will depend on the length of the game whether certainmoves will occurat all. Theargument is clearly circular.

*Thedomain of variability of <r\\ is 1, , on. Thedomain of variability of <TI is1, , at, and may depend on <n: a2 = az(<ri). Thedomain of variability of <r is1, , ai, and may dependon <TI, <r z: i ai(<n, <rt). Etc.,etc.

8 This stop rule is indeed an essentialpart of every game. In most games it is easyto find v's fixed upper bound v*. Sometimes, however, the conventional form of therules of the game doesnot excludethat the play might under exceptionalconditions goon ad infinitum. In all these casespractical safeguards have been subsequently incor-porated into the rules of the game with the purpose of securing the existenceof thebound v*. It must be said, however, that these safeguards are not always absolutelyeffective although the intention is clearin every instance, and even where exceptionalinfinite plays exist they are of little practical importance. It is nevertheless quiteinstructive, at least from a mathematical point ofview, to discussa few typical examples.

We give four examples,arranged according to decreasingeffectiveness.ficarte\": A play is a \"rubber,\" a \"rubber\" consistsof winning two \"games\" out of

three (cf.footnote 1 on p. 49),a \"game\" consistsof winning five \"points,\" and each\"deal\" gives one player or the other oneor two points. Hencea \"rubber\" is completeafter at most three \"games,\" a \"game\" after at most nine \"deals,\" and it is easytoverify that a \"deal\" consistsof 13,14or 18moves. Hencev* - 3 ^9 18-486.

Poker:A priori two players could keep \"overbidding\" eachother ad infinitum. It istherefore customaiy to add to the rules a proviso limiting the permissible number of\"overbids.\" (Theamounts of the bids are also limited, soas to make the number ofalternatives a at thesepersonal moves finite.) This of coursesecuresa finite v*.

Bridge:Theplay is a \"rubber\" and this could go on forever if both sides(players)invariably failed to make their contract. It is not inconceivable that the sidewhich is in

danger of losing the \"rubber,\" should in this way permanently prevent a completion ofthe play by absurdly high bids. This is not done in practice,but there is nothing explicitin the rules of the game to prevent it. In theory, at any rate, somestop rule should beintroduced in Bridge.

Chess:It is easy to construct sequencesof choices(in the usual terminology:)))

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60 DESCRIPTIONOF GAMESOFSTRATEGY

Now we can makeuseof this bound v* to getentirelyrid of thevariabil-ity of v.

This is donesimplyby extendingthe schemeof the gameso that therearealways v* moves 2fEi, , 9TI,*. Forevery sequence<TI,<r, 0-3,

everything is unchangedup to the move 3TI,,and all moves beyond2fTl,are\"dummy moves.\" I.e.if we considera move 3TC,, K = 1, , v*, for asequenceo-i,era, <TS, for which v < K, then we make 9TC a chancemovewith one alternative only1 i.e.one at which nothing happens.

Thus the assumptionsmade at the beginningof 6.2.1.particularlythat v is given ab initio arejustified ex post.

8.Setsand Partitions8.1.Desirability of a Set-theoreticalDescriptionof a Game

8.1.We have obtained a satisfactory and generaldescriptionof theconceptof a game,which couldnow berestatedwith axiomatic precisionand rigidity to serveas a basisfor the subsequentmathematical discussion.It is worth while, however, before doingthat, to passto a different formula-tion. This formulation is exactlyequivalent to the one which we reachedin the precedingsections,but it is more unified, simplerwhen stated in ageneral form, and it leadsto more elegantand transparent notations.

In orderto arrive at this formulation we must use the symbolismofthe theory of sets and more particularly of partitions more extensivelythan we have doneso far. This necessitatesa certainamount of explana-tion and illustration, which we now proceedto give.

\"moves\") particularly in the \"end game\" which can go on ad infinitum without everending the play (i.e.producing a \"checkmate\.") Thesimplest onesareperiodical,i.e.indefinite repetitions of the samecycleof choices,but there exist non-periodical onesaswell. All of them offer a very real possibility for the player who is in danger of losing tosecuresometimes a \" tie.\" For this reasonvarious \"tie rules \" i.e.stop rules arein usejust to prevent that phenomenon.

Onewell known \"tie rule\" is this: Any cycleof choices(i.e.\"moves\,") when threetimes repeated,terminates the play by a \"tie.\" This rule excludesmost but not allinfinite sequences,and henceis really not effective.

Another \"tie rule\" is this: If no pawn has beenmoved and no officer taken (theseare \"irreversible\" operations,which cannot be undone subsequently) for 40moves, thenthe play is terminated by a \"tie.\" It is easyto seethat this rule is effective, although thev* is enormous.

4 From a purely mathematical point of view, the following question could be asked:Let the stop rule be effective in this senseonly, that it is impossible so to arrange thesuccessivechoices<TI, at, at, that the stop never comes. I.e.let there always be afinite v dependent upon <n, <rj, <rj, . Doesthis by itself securethe existenceof afixed, finite v*bounding the stop rule? I.e.such that all v ^ v*?

The question is highly academicsinceall practical game rules aim to establish a v*directly. (Cf., however, footnote 3 above.) It is nevertheless quite interestingmathematically.

The answer is \".Yes,\" i.e.v* always exists. Cf.e.g.D.K&nig: tJber eine Schluss-weiseaus dem Endlichen ins Unendliche, Acta Litt. acScient.Univ. Szeged,Sect.Math.Vol. III/II(1927)pp.121-130;particularly the Appendix, pp.12&-130.

1Thismeans, ofcourse,that = 1,kK= 0,and p*(l) *= 1.)))

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SETSAND PARTITIONS 61

8.2.Sets,Their Properties,and Their Graphical Representation

8.2.1.A set is an arbitrary collectionof objects,absolutelyno restrictionbeing placed on the nature and number of these objects,the elementsof the set in question. Theelementsconstitute and determinethe setassuch,without any orderingor relationshipof any kind betweenthem. I.e.if two setsA, B aresuch that every elementof A is alsoone of Bandviceversa, then they areidentical in every respect,A = B. The relationshipof a beingan elementof the setA is alsoexpressedby sayingthat a belongsto A. 1

We shallbe interestedchiefly, although not always,in finite setsonly,i.e.setsconsistingof a finite number of elements.Given any objectsa,0,7, we denotethe setof which they arethe

elementsby (a,0,7, ) It is alsoconvenient to introducea setwhichcontainsno elementsat all, the empty set.2 We denotethe emptysetby .We can,in particular,form setswith preciselyone element,one-elementsets.Theone-elementset (a), and its unique elementa, arenot the same thingand shouldnever be confused.3

We re-emphasizethat any objectscan be elementsof a set. Of coursewe shall restrictourselvesto mathematical objects.But the elementscan, for instance,perfectly well be setsthemselves(cf. footnote 3), thusleadingto setsof sets,etc. Theselatteraresometimescalledby someother

equivalent name, e.g.systems or aggregatesof sets.But this is notnecessary.

8.2.2.Themain conceptsand operationsconnectedwith setsarethese:

(8:A:a) A is a subset of B, or B a supersetof A, if every elementofA is alsoan elementof B. In symbols:A Bor B 2A. A\\s

a propersubsetof B,or B a propersupersetof A, if the above istrue, but if B containselementswhich arenot elementsof A.In symbols:A cB or B => A. We see:If A is a subsetof B andB is a subset of A , then A = B. (This is a restatementof theprincipleformulated at the beginningof 8.2.1.)Also: A is aproper subset of B if and only if A is a subset of B withoutA = B.

1 Themathematical literature of the theory of sets is very extensive. We make nouse of it beyond what will be said in the text. The interested readerwill find moreinformation on set theory in the good introduction: A. Fraenkel: Einleitung in die Men-genlehre, 3rd Edit. Berlin 1928;conciseand technically excellent:F.Hausdorff: Mengen-lehre, 2nd Edit. Leipzig1927.

*If two sets A, B are both without elements, then we may say that they have thesame elements. Hence,by what we said above,A ~ B. I.e.there exists only oneempty set.

This reasoning may sound odd,but it isneverthelessfaultless.8 Thereare some parts of mathematics where (a) and a can be identified. This is

then occasionallydone, but it is an unsound practice. It is certainly not feasible in

general. E.g.,let a be something which is definitely not a one-elementset, i.e.atwo-element set(,0),or the empty setQ. Then (a) and a must bedistinguished, since(a) is a one-elementset while a is not.)))

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62 DESCRIPTIONOF GAMES OF STRATEGY

(8:A:b) The sum of two setsA, B is the set of all elementsof A

togetherwith all elementsof B, to bedenotedby A u B. Simi-larly the sums of more than two setsareformed.1

(8:A:c) Theproduct, or intersection,of two setsA, B is the setof allcommon elementsof A and of B, to be denoted by A n B.Similarlythe productsof more than two setsareformed.l

(8:A:d) Thedifferenceof two setsA, B (A the minuend, B the subtra-hend) is the setof all thoseelementsof A which donot belongtoB, to be denotedby A -B.1

(8:A:e) When B is a subsetof A, we shallalsocall A B the comple-ment of B in A. Occasionallyit will be so obvious which setA is meant that we shallsimplywrite B and talk about thecomplementof B without any further specifications.

(8:A:f) Two setsA, B aredisjunct if they have no elementsin com-mon, i.e.if A n B = .

(8:A:g) A system (set)& of setsis said to be a systemof pairwisedis-junct setsif all pairsof different elementsof Q,aredisjunctsets,i.e.if for A, B belongingto a,A ^ B impliesA n B = .

8.2.3.At this point somegraphicalillustrationsmay be helpful.We denotethe objectswhich areelementsof setsin theseconsiderations

by dots (Figure1). We denotesets by encirclingthe dots (elements)))

Figure 1.which belong to them, writing the symbol which denotesthe set acrossthe encirclingline in oneor more places(Figure1). ThesetsA, C in thisfigure are,by the way, disjunct,while A, Barenot.

1This nomenclature of sums, products, differences, is traditional. It is basedoncertain algebraicanalogieswhich we shall not use here. In fact, the algebra of theseoperations U, n, alsoknown as Booleanalgebra, has a considerableinterest of its own.Cf.e.g.A. Tarski:Introduction to Logic,New York, 1941.Cf.further Garrett Birkhoff:LatticeTheory, New York 1940. This bookis of wider interest for the understanding ofthe modern abstract method. Chapt.VI. dealswith BooleanAlgebras. Further litera-ture is given there.)))

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SETSAND PARTITIONS)) 63))

With this devicewe can alsorepresentsums,productsand differencesofsets(Figure2). In this figure neither A is a subsetof B nor B one of A,henceneither the difference A B nor the difference JB A is a comple-ment. In the next figure, however, B is a subsetof A, and so A B is thecomplementof B in A (Figure3).))

Figure 2.)) Figure 3.))

8.3.Partitions, Their Properties,and Their Graphical Representation

8.3.1.Let a set ft and a system of setsCt be given. We say that ais a partition in ft if it fulfills the two followingrequirements:(8:B:a) Every elementA of a is a subset of ft, and not empty.(8:B:b) ft is a systemof pairwisedisjunctsets.

This concepttoo has been the subjectof an extensiveliterature.1

We say for two partitions ft, (B that a is a subpartition of (B, if they fulfill

this condition:(8:B:c) Every elementA of a is a subset of someelementB of (B.2

Observethat if a is a subpartitionof (B and (Ba subpartitionofa,then a = (B.3

Next we define:(8:B:d) Given two partitions a, (B, we form the systemof all those

intersectionsA n B A running over all elementsof ft andB over1 Cf. G. Birkhoff loc.cit. Our requirements (8:B:a),(8:B:b)are not exactly the

customary ones. Precisely:Ad (8:B:a):It is sometimes not required that the elementsA of a be not empty.

Indeed,we shall have to make oneexceptionin 9.1.3.(cf.footnote 4on p.69).Ad (8:B:b):It is customary to require that the sum of all elementsof d beexactly

the set ft. It is more convenient for our purposesto omit this condition.2SinceCt, (B are alsosets, it is appropriate to compare the subset relation (asfar as

Ct, (B are concerned)with the subpartition relation. Oneverifies immediately that if Ct

is a subset of (B then Q, is alsoa subpartition of (B, but that the conversestatement is not(generally) true.

3Proof:Consideran element A of a. It must be subset of an element B of (B,andB in turn subset of an element Ai of Ct. SoA, A\\ have common elements all thoseofthe not empty set A i.e.are not disjunct. Sincethey both belong to the partition d,this necessitatesA *= AI. SoA isa subsetofB and B oneof A (= AI). HenceA = J3,and thus A belongs to (B;I.e.:a is a subsetof (B. (Cf.footnote 2 above.) Similarly (B is a subset of a.Hencea = (B.)))

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64)) DESCRIPTIONOF GAMES OF STRATEGY))

all thoseof (B which arenot empty. This again is clearly apartition, the superpositionof Ot, (B.1

Finally, we alsodefine the above relationsfor two partitionsCt, (Bwithin

a given setC.(8:B:e) a is a subpartition of (B within C, if every A belongingto ft

which is a subset of C is also subset of someB belongingto (B

which is a subsetof C.(8:B:f) Ot is equal to (B within C if the samesubsetsof Careelements

of a and of (B.

Clearly footnote 3 on p. 63 applies again, mutatis mutandis. Also,the above conceptswithin ft arethe sameas the original unqualified ones.))

Figure 4.))

.S3v))

Figure 5.8.3.2.We give again somegraphicalillustrations,in the senseof 8.2.3.We begin by picturing a partition. We shall not give the elements

of the partition which aresets names,but denoteeachone by an encir-clingline (Figure4).

We picturenexttwo partitionsa,(Bdistinguishingthem by markingtheencirclinglinesof the elementsof aby and of the elementsof (B by

1It is easyto show that the superposition of a,(Bis a subpartition ofboth a and (Band that every partition C which is a subpartition of both a and (B is alsoone of theirsuperposition. Hencethe name. Cf.O.Birkhoff, loc.cit.Chapt.I-II.)))

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SETSAND PARTITIONS)) 65))

- (Figure5). In this figure a isa subpartitionof (B. In thefollowingoneneithera is a subpartition(B nor is & oneof a (Figure6). We leave itto the readerto determinethe superpositionof Ot, (Bin this figure.))

Figure 6.))

Figure 7.)) Figure 8.))

Figure 9.Another, more schematic,representationof partitionsobtainsby repre-

sentingthe set8by one dot, and every elementof the partition which is a)))

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66 DESCRIPTIONOF GAMES OF STRATEGY

subset of 12 by a line goingupwardfrom thisdot. Thus the partition aofFigure5 will be representedby a much simplerdrawing (Figure7). Thisrepresentationdoesnot indicatethe elementswithin the elementsof thepartition, and it cannot be used to representseveralpartitions in 12 simul-taneously, as was done in Figure6. However,this deficiency can beremoved if the two partitions ft, (B in 12arerelatedas in Figure5:If Q isasubpartitionof (B. In this casewe can represent12 again by a dot at thebottom, every elementof (B by a line going upward from this dot as in

Figure7 and every elementof ft as another line going further upward,beginningat the upperendof that line of (B, which representsthe elementof(B of which this elementof ft is a subset. Thus we can representthe twopartitions ft, (B of Figure5 (Figure8). This representationis again lessrevealing than the correspondingone of Figure5. But its simplicitymakesit possibleto extendit further than picturesin the vein of Figures4-6couldpracticallygo. Specifically:We can picture by this devicea sequenceofpartitions fti, , ft M, where eachoneis a subpartitionof its immediatepredecessor.We give a typicalexamplewith ju

= 5 (Figure9).Configurations of this type have beenstudied in mathematics,and are

known as trees.

8.4.LogisticInterpretation of Setsand Partitions

8.4.1.Thenotionswhich we have describedin 8.2.1.-8.3.2.will be usefulin the discussionof gameswhich follows,becauseof the logisticinterpreta-tion which can be put upon them.

Let us beginwith the interpretationconcerningsets.If 12 is a set of objectsof any kind, then every conceivableproperty

which some of theseobjectsmay possess,and others not can be fullycharacterizedby specifyingthe set of thoseelementsof 12 which have thisproperty. I.e.if two propertiescorrespondin this senseto the same set(thesamesubset of 12),then the sameelementsof 12 will possessthesetwoproperties,i.e.they areequivalent within 12, in the sensein which this termis understoodin logic.

Now the properties(of elementsof 12) arenot only in this simplecor-respondencewith sets(subsetsof 12),but the elementarylogical operationsinvolving propertiescorrespondto the setoperationswhich we discussedin8.2.2.

Thus the disjunction of two propertiesi.e.the assertionthat at leastone of them holds correspondsobviously to forming the sum of their sets,the operationA u B. Theconjunction of two properties i.e.the assertionthat both hold correspondsto forming the productof their sets, theoper-ation A n B. And finally, the negation of a property i.e.the assertionof the opposite correspondsto forming the complementof its set, theoperation A. 1

1Concerning the connection of set theory and of formal logiccf.,e.g.,O. Birkhoff,loc.cit.Chapt.VIII.)))

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SET-THEORETICALDESCRIPTION 67

Insteadof correlatingthe subsetsof ft to propertiesin ft as doneabovewe may equallywell correlatethem with all possiblebodiesof information

concerningan otherwiseundetermined elementof ft. Indeed,any suchinformation amounts to the assertion that this unknown elementof ft

possessesa certain specified property. It is equivalently representedby the setof all those elementsof ft which possessthis property; i.e.towhich the given information has narrowed the range of possibilitiesforthe unknown elementof ft.

Observe,in particular, that the empty set correspondsto a propertywhich never occurs,i.e.to an absurd information. And two disjunctsetscorrespondto two incompatibleproperties,i.e.to two mutually exclusivebodiesof information.

8.4.2.We now turn our attention to partitions.By reconsideringthe definition (8:B:a),(8:B:b)in 8.3.1.,and by restat-

ing it in our presentterminology, wesee:A partition is a systemof pairwisemutually exclusivebodiesof information concerningan unknown elementof ft none of which is absurd in itself. In otherwords:A partition is apreliminary announcement which stateshow much information will begiven laterconcerningan otherwiseunknown elementof ft; i.e.to whatextentthe range of possibilitiesfor this elementwill benarrowed later. Butthe actual information is not given by the partition, that would amount toselectingan elementof the partition, sincesuchan elementis a subsetof ft,i.e.actual information.

We can therefore say that a partition in ft is a pattern of information.As to the subsets of ft: we saw in 8.4.1.that they correspondto definiteinformati6n. In order to avoid confusion with the terminology used forpartitions,we shall use in this case i.e.for a subsetof ft the wordsactualinformation.

Considernow the definition (8:B:c)in 8.3.1.,and relateit to our presentterminology. This expressesfor two partitionsd,(B in ft the meaning of ft

beinga subpartitionof (B: it amounts to the assertionthat the informationannouncedby d includesall the information announcedby (B (and possiblymore) ; i.e.that the pattern of information a includesthe pattern of informa-tion (B.

Theseremarksput the significanceof the Figures4-9in 8.3.2.in a newlight. It appears,in particular,that the treeof Figure9 picturesa sequenceof continually increasingpatterns of information.

9.TheSet-theoreticalDescriptionof a Game9.1.ThePartitions Which Describea Game

9.1.1.We assumethe number of moves as we now know that wemayto be fixed. Denotethis number again by v, and the moves themselvesagain by 3TCi, , SKI,.

Considerall possibleplays of the game F, and form the set ft of which

they arethe elements.If we use the descriptionof the precedingsections,)))

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68 DESCRIPTIONOF GAMES OE STRATEGY

then all possibleplaysaresimplyall possiblesequences<n, , *.l Thereexistonly a finite number of suchsequences,2 and so ft is a finite set.

Thereare,however, also more directways to form ft. We can,e.g.,form it by describingeachplay as the sequenceof the v + 1consecutivepositions8 which ariseduring its course.In general,of course,a givenpositionmay not be followed by an arbitrary position,but the positionswhich arepossibleat a given moment arerestrictedby the previousposi-tions, in a way which must be preciselydescribedby the rulesof the game.4

Sinceour descriptionof the rulesof the game beginsby forming ft, it may beundesirableto let ft itself dependsoheavily on all the detailsof thoserules.We observe,therefore, that thereis no objectionto including in ft absurdsequencesof positionsas well.6 Thus it would be perfectly acceptableevento let ft consistof all sequencesof v -f 1successivepositions,without anyrestrictionswhatsoever.

Our subsequentdescriptionswill show how the really possibleplaysareto beselectedfrom this, possiblyredundant,set ft.

9.1.2.v and ft beinggiven, we enterupon the more elaboratedetailsofthe courseof a play.

Considera definite moment during this course,say that one whichimmediately precedesa given move 3TI,. At this moment the followinggeneralspecificationsmust be furnished by the rulesof the game.

Firstit is necessaryto describeto what extentthe events which haveled up to the move 9TC.6 have determinedthe courseof the play. Everyparticularsequenceof theseevents narrows the set ft down to a subset A K :this beingthe setof all thoseplaysfrom ft, the courseof which is, up to 3^,,the particular sequenceof events referredto. In the terminology of theearliersections,ft is as pointed out in 9.1.1.the set of all sequences<TI, , a,}then A t would be the setof those sequences<n,

- , a, forwhich the <TI, , <r,_i have given numerical values (cf.footnote 6 above).But from our presentbroaderpoint of view we needonly say that A K mustbea subset of ft.

Now the various possiblecoursesthe game may have taken up to 3TC,mustberepresentedby different setsA K. Any two suchcourses,if they aredifferent from eachother,initiate two entirely disjunct sets of plays; i.e.no play can have begun(i.e.run up to 9TC,)both ways at once. Thismeansthat any two different setsA x must be disjunct.

1Cf. in particular^ 6.2.2.The range of the <TI, , <rv is describedin footnote 2on p.59.

1Verification by means of the footnote referredto aboveis immediate.1Before3Jli, between 9lli and 9Rj, between 9R2 and 9flli, etc.,etc.,between 9TC,>_i and

3H,, after 3Hp.4 This is similar to the development of the sequence<n, , <r,, as describedin

footnote 2 on p. 59.*I.e.oneswhich will ultimately be found to be disallowed by the fully formulated

rules of the game.I.e.the choicesconnectedwith the anterior moves 9Tli, - , 9TC*-.i i.e.the numeri-calvalues <ri,)))

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SET-THEORETICALDESCRIPTION 69

Thus the completeformal possibilitiesof the courseof all conceivableplaysof our game up to 9TC,are describedby a family of pairwisedisjunctsubsetsof ft. This is the family of all the setsA K mentionedabove. Wedenotethis family by a.

Thesum of all setsA K containedin a,must contain all possibleplays.But sincewe explicitlypermitteda redundancyof 12 (cf. the end of 9.1.1.),this sum needneverthelessnot beequalto ft. Summing up:(9:A) ft is a partition in ft.

We couldalsosay that the partition &< describesthe pattern of informa-tion of a personwho knowseverything that happenedup to 3NI,;

1e.g.of anumpirewho supervisesthe courseof the play.29.1.3.Second,it must be known what the nature of the move 3fll isgoing to be. This is expressedby the k< of 6.2.1.:kK

= 1, , n if themove is personaland belongsto the playerfc,; fc, = if the move is chance.kt may dependupon the courseof the play up to 3TI,, i.e.upon the informa-tion embodiedin a,.3 This meansthat fc must be a constant within eachsetA K of Gfc,, but that it may vary from one A K to another.

Accordingly we may form for every fc = 0, 1, , n a set B*(k), whichcontainsall setsA K with kK

= k, the various BK(k) beingdisjunct. Thus theBK(k),k = 0,1, , n, form a family of disjunctsubsetsof ft. We denotethis family by (B,.

(9:B) (B, is again a partition in ft. Sinceevery A K of ft, is a subsetof someBK(k) of (B,therefore & is a subpartition of (B*.

But while therewas no occasionto specifyany particularenumerationof the setsA K of QK, it is not so with (B,. (B*consistsof exactlyn + 1setsBK(k),k = 0, 1, , n, which in this way appear in a fixed enumerationby means of the k = 0,1, , n.4 And this enumeration is essentialsinceit replacesthe function kK (cf. footnote 3 above).9.1.4.Third, the conditionsunder which the choiceconnectedwith themove 311,is to take placemust be describedin detail.

Assume first that 311,is a chancemove, i.e.that we arewithin the setjB(0). Then the significant quantitiesare:the number of alternativesaand the probabilitiesp*(l), , p*(ctt) of thesevarious alternatives (cf.the end of 6.2.1.).As was pointedout in 7.1.1.(this was the seconditem

1I.e.the outcome of all choicesconnectedwith the moves 9Ri, , 9TC-i. In ourearlier terminology: the values of <n, , <T*_I.

1It is necessaryto introduce such a person since,in general, no player will be in

possessionof the full information embodiedin a.8 In the notations of 7.2.1.,and in the senseof the preceding footnotes:k* -

fc(<ri, , <r,(_i).4 Thus (B is really not a set and not a partition, but a more elaborateconcept:it con-

sists of the sets(B,c(fc), A; = 0, 1, , n, in this enumeration.It possesses,however, the properties(8:B:a),(8:B:b)of 8.3.1.,which characterizea

partition. Yet even there an exceptionmust be made:among the sets (B(fc) there canbeempty ones.)))

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70 DESCRIPTIONOF GAMES OF STRATEGY

of the discussionthere),all thesequantities may depend upon the entireinformation embodiedin GLK (cf. footnote 3 on p.69),since3TI* is now achancemove. I.e.aK and the p(l), , />(<*) must be constant within

eachsetA K of QKl but they may vary from one A K to another.

Within eachone of theseA K the choiceamong the alternatives &(!),, &(*)takes place,i.e.the choiceof a aK

= 1, , aK (cf. 6.2.2.).This can be describedby specifyingaK disjunct subsets of A K which cor-respondto the restrictionexpressedby A K, plus the choiceof aK which hastaken place. We call thesesetsCK, and their system consistingof all CK

in all the A K which aresubsetsof BK(0) e(0). Thus6^(0)isa partition inBK(0). And sinceevery CK of G(0) is a subset of someA K of a,,thereforeC(0)is a subpartition of Q,K.

TheaK aredeterminedby (());2 hencewe neednot mention them anymore. For the p(l), , p(cOthis descriptionsuggestsitself:with

every CK of C^O)a number pK(C^) (its probability) must be associated,subjectto the equivalentsof footnote 2 on p.50.8

9.1.5.Assume,secondly,that 9ffl is a personalmove, say of the playerk = 1, , n, i.e.that we are within the set BK(k). In this casewemust specifythe stateof information of the playerk at 311*. In 6.3.1.thiswas describedby meansof the setA, in 7.2.1.by meansof the family offunctions <,the latterdescriptionbeingthe more generaland the final one.According to this descriptionk knowsat 3fTC the values of all functionsh(<n, , 0\"ic-i) of $ and no more. This amount of information operatesa subdivisionof BK(k) into several disjunct subsets, correspondingto thevarious possiblecontents of fc's information at ?RI K. We call these setsD,and their system >K(k). Thus ><(&)is a partition in BK(k).

Of coursefc'sinformation at STC*is part of the total information existingat that moment in the senseof 9.1.2.which is embodiedin ft, Hencein an A K of &,which is a subset of BK(k),no ambiguity can exist,i.e.thisA K cannot possesscommon elementswith mt>re than one DK of ><(&). Thismeansthat the A K in questionmust be a subset of a DK of ><(&). In otherwords:within BK(k) QK is a subpartitionof >(&)

In reality the courseof the play is narroweddown at am, within a setA K of GLX. But the player k whosemo've 9TIK is, does not know as much:as far as he is concerned,the play is merelywithin a setDK of >(fc). Hemust now make the choiceamong the alternatives&*(!), , d(cx)>i.e.the choiceof a <r = 1, , a,. As was pointed out in 7.1.2.and 7.2.1.(particularlyat the end of 7.2.1.),aK may well be variable, but it can onlydependupon the information embodiedin ><(&). I.e.it must be a constantwithin the setDK of 3D(fc) to which we have restrictedourselves.Thusthe choiceof a <T K

= 1, , a, can be describedby specifyinga disjunctsubsetsof D,,which correspondto the restrictionexpressedby DK, plus the

1We are within BK(Q),henceall this refersonly to AJs which aresubsetsof #(()).1 is the number of those C ofe(0)which aresubsetsof the given AK.*I.e.every p(C*);> 0, and for eachAK, and the sum extended over all C of e,t(0)

which aresubsetsof A t we have Sp.tC*) =1.)))

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SET-THEORETICALDESCRIPTION 71

choiceof<r,which hastakenplace. We call thesesetsC,and their systemconsistingof all CK in all the DK of )*(fc) <3(fc). Thus GK(k) isa partitionin BK(k). And sinceevery CK of QK(k) is a subsetof someDK of >*(fc),there-fore e(fc)is a subpartition of >(&).

The a* are determinedby e(fc);1 hencewe neednot mention themany more, a* must not be zero, i.e.,given a D<of $>K(k),someC of C(fc),which is a subsetof D,,must exist.2

9.2.Discussionof ThesePartitions and Their Properties9.2.1.We have completelydescribed in the precedingsectionsthe

situation at the moment which precedesthe move 3TC,. We proceednow todiscuss what happens as we go along these moves K = 1, -,*>.Itis convenient to add to these a K = v + 1,too, which correspondsto theconclusionof the play, i.e.followsafter the last move 311,.

ForK = 1, , v we have, as we discussedin the precedingsections,the partitionsa.,(B, = (B,(0),B.(l), , B.(n)),C.(0),C.(l), , C.(n),

>,(!), , 3>.(n).All of these,with the soleexceptionof &, refer to the move 3TC,, hencethey need not and cannot be defined for K = v + 1. But &+!has a per-fectly good meaning, as its discussionin 9.1.2.shows:It representsthefull information which can conceivably exist concerninga play, i.e.theindividual identity of the play.3

At this point two remarkssuggestthemselves:In the senseof the aboveobservations Oti correspondsto a moment at which no information isavailable at all. HenceCti shouldconsistof the one set0. On the otherhand, Ot,+i correspondsto the possibilityof actually identifying the playwhich has taken place. Hence<$+!is a systemof one-elementsets.

We now proceedto describe the transition from K to K + 1,whenK = 1, , v.

9.2.2.Nothing can be said about the changein the (B, e(fc), 5)(fc)when K is replacedby K + 1, our previous discussionshave shown thatwhen this replacementis madeanything may happen to those objects,i.e.to what they represent.

It is possible,however, to tell how QK+i obtainsfrom 6t.The information embodiedin &,+) obtains from that one embodied

in a by adding to it the outcome of the choiceconnectedwith the moveSHI*.

4 This ought to be clearfrom the discussionsof 9.1.2.Thus the

1 is the number of those CK of e(fc)which aresubsetsof the given A*.1We required this for k = 1, , n only, although it must be equally true for

Jc = with an AK, subset of #(0),in placeof our DK of 3)*(fc). But it is unnecessary tostate it for that case,becauseit is a consequenceof footnote 3 on p. 70;indeed, if no0*of the desiredkind existed,the Sp(C)of loc.cit, would be and not 1.

8 In the senseof footnote 1on p.69,the values of all <n, , <r,. And the sequence<TI, , <r,characterizes,asstated in 6.2.2.,the play itself.

4 In our earlier terminology: the value of <r.)))

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72 DESCRIPTIONOF GAMESOF STRATEGY

information in Ot,+i which goesbeyondthat in ft, ispreciselythe informationembodiedin the e(0),&(!), , C,(n).

This meansthat the partitionsa+iobtainsby superposingthe partitiona.with all partitions6(0),6(1), , <S>*(k). I.e.by forming the inter-sectionof every A , in a,with every C< in any C^O),C(l), , C(n),andthen throwing away the emptysets.

Owing to the relationshipof ft, and of the e(fc) to the setsBK(k) asdiscussedin the precedingsections we can say a little more about thisprocessof superposition.

InB(0),C(0)isa subpartitionof Ofc (cf.the discussionin 9.1.4.).Hencethere QK+i simply coincideswith G(0).In (&), k = 1, , n, G(fc)and ft, areboth subpartitionsof )(&)(cf.the discussionin 9.1.5.).Hencethere&<+i obtainsby first taking every DK of 3)(fc),then for every suchD,all A K of &x and all C of C,(fc)which aresubsetsof this DK, and forming allintersectionsA K n CK.

Every such set A K n C representsthoseplays which arise when theplayer fc, with the information of D beforehim, but in a situation which isreally in A K (a subset of D),makesthe choiceCK at the move 9TC,so as torestrictthings to C.

Sincethis choice,accordingto what was said before,is a possibleone,there existsuch plays. I.e.the set A K n CK must not be empty. Werestatethis:(9:C) If A K of Ot and C of C(fc)aresubsetsof the sameDK of >(&),

then the intersectionA K n C must not beempty.

9.2.3.Therearegamesin which onemight betemptedto set this require-ment aside. Thesearegames in which a player may make a legitimatechoicewhich turns out subsequentlyto bea forbiddenone;e.g.the double-blind Chessreferred to in footnote 1on p.58:herea playercan makeanapparentlypossiblechoice(\"move\")on his own board, and will (possibly)be told only afterwardsby the \"umpire\" that it is an \"impossible\"one.

This example is, however, spurious. The move in question is bestresolvedinto a sequenceof severalalternative ones. It seemsbest to givethe contemplatedrulesof double-blindChessin full.

Thegameconsistsof a sequenceof moves. At eachmove the \"umpire\"announcesto both playerswhether the precedingmove was a \"possible\"one. If it was not, the nextmove is a personalmove of the same playeras the precedingone;if it was, then the nextmove is the otherplayer'spersonalmove. At eachmove the player is informed about all of his ownanteriorchoices,about the entiresequenceof \"possibility\"or \"impossibil-ity\" of all anterior choicesof both players,and about all anterior instanceswhere either player threatenedcheckor took anything. But he knowsthe identity of his own lossesonly. In determiningthe courseof the game,the \"umpire\"disregardsthe \"impossible\"moves. Otherwisethe gameisplayed like Chess,with a stop rule in the senseof footnote 3 on p. 59,)))

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AXIOMATICFORMULATION 73

amplified by the further requirementthat no player may make (\"try\the samechoicetwice in any one uninterruptedsequenceof hisown personalmoves. (In practice,of course,the playersneedtwo chessboards out ofeachother'sview but both in the \" umpire's\" view to obtain thesecondi-tions of information.)

At any ratewe shall adhere to the requirementstated above. It will

beseenthat it is very convenient for our subsequentdiscussion(cf.11.2.1.).9.2.4.Only one thing remains:to reintroducein our new terminology,

the quantities SF*, k = 1, , n, of 6.2.2.SF* is the outcomeof the playfor the playerk. $k must be a function of the actual play which has takenplace.1 If we use the symbolir to indicate that play, then we may say:$k is a function of a variable IT with the domain of variability 12. I.e.:

$k = ^W, v in 12, k = 1, , n.

10.Axiomatic Formulation10.1.TheAxioms and Their Interpretations

10.1.1.Our descriptionof the generalconceptof a game,with the newtechnique involving the use of setsand of partitions, is now complete.All constructionsand definitions have been sufficiently explained in thepastsections,and we can therefore proceedto a rigorousaxiomatic definitionof a game. This is, of course,only a conciserestatementof the thingswhich we discussedmore broadlyin the precedingsections.

We give first the precisedefinition, without any commentary:2An n-persongame F, i.e.the completesystemof its rules,is determined

by the specificationof the following data:(10:A:a) A number v.

(10:A:b) A finite set 12.(10:A:c) Forevery k = 1, , n:A function

3k = $*(*), TTin 12.

(10:A:d) Forevery K = 1, , v, v + 1:A partition &K in 12.

(10:A:e) Forevery K = 1, , v: A partition (B, in 12. (B con-sists of n + 1setsBK(k), k = 0,1, , n, enumeratedinthis way.

(10:A:f) Forevery K = 1, , v and every k = 0,1, , n:A partition e(fc)in BK(k).

(10:A:g) Forevery K = 1, , v and every k = 1, , n:A

partition >(/;)in J3,(fc).(10:A:h) Forevery * = 1, , v and every C*of C(0):A number))

Theseentitiesmust satisfy the followingrequirements:(10:1:a) ft, is a subpartitionof (B,.(10:1:b) e(0)is a subpartitionof a.

1 In the old terminology, accordingly, wehad ff* - g*(ai, , o>). Cf.6.2.2.2For \"explanations\" cf. the end of 10.1.1.and the discussion of 10.1.2.)))

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74 DESCRIPTIONOF GAMES OF STRATEGY

(10:1:c) Fork = 1, , n:C,(fc)is a subpartitionof ><(fc).(10:1:d) Fork = 1, , n:Within B<(fc), 6t is a subpartitionof

ft(*).(10:1:e) Forevery K = 1, , ? and every .A* of Ot which is a

subset of 5(0):Forall CK of C(0) which aresubsets of this<4,P*(C*)^ 0,and for thesum extendedover them Sp^C,)= 1.

(10:1:f) Cti consistsof the one setQ.(10:1:g) Ot.,+1 consistsof one-elementsets.(10:l:h) ForK = 1, , v. QK+i obtainsfrom QK by superposingit

with all e,(*),k = 0, 1, , n. (Fordetails,cf. 9.2.2.)(10:1:i) For K = 1, , v: If A K of a, and CK of e*(fc), fc = 1,

, n aresubsetsof the same DK of >*(fc),then the inter-sectionA K n C, must not be empty.

(10:1:j) ForK = 1, , v and fc = 1, , n and every DK of>,(&):SomeC,(fc)of e,,which isa subsetof DK, must exist.

This definition shouldbe viewed primarily in the spirit of the modernaxiomatic method. We have even avoided giving names to the mathe-matical conceptsintroducedin (10:A:a)-(10:A:h)above, in order to estab-lishno correlation with any meaning which the verbal associationsof namesmay suggest. In this absolute \"purity\" theseconceptscan then be theobjectsof an exactmathematical investigation.1

This procedureis best suited to develop sharply defined concepts.The application to intuitively given subjectsfollows afterwards, whenthe exactanalysishas beencompleted.Cf. also what was said in 4.1.3.in Chapter I about the role of modelsin physics:The axiomatic modelsfor intuitive systemsareanalogousto the mathematical modelsfor (equallyintuitive) physicalsystems.

Once this is understood,however, there can be no harm in recallingthat this axiomatic definition was distilledout of the detailed empiricaldiscussionsof the sections,which precedeit. And it will facilitate its use,and make its structure more easilyunderstood,if we give the interveningconceptsappropriate names, which indicate,as much as possible,theintuitive background. And it is further useful to express,in the samespirit, the \"meaning\" of our postulates(10:l:a)-(10:l:j)i.e.the intuitiveconsiderationsfrom which they sprang.

All this will be,of course,merely a concisesummary of the intuitive con-siderationsof the precedingsections,which lead up to this axiomatization.

10.1.2.We statefirst the technicalnamesfor the conceptsof (10:A:a)-(10:A:h)in 10.1.1.

1 This is analogous to the present attitude in axiomatizing such subjects as logic,geometry, etc. Thus, when axiomatizing geometry, it is customary to state that thenotions of points, lines, and planes arenot to be a priori identified with anything intui-tive, they are only notations for things about which only the propertiesexpressedinthe axioms are assumed. Cf.,e.g.,D.Hilbert: DieGrundlagen der Geometrie,Leipzig1899,2rd Engl. Edition Chicago1910.)))

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AXIOMATICFORMULATION 75

(10:A:a*) v is the length of the game F.(10:A:b*) ft is the set of all playsof T.(10:A:c*) ^(TT)is the outcome of the play w for the player fc.

(10:A:d*) Ct is the umpire'spattern of information, an A of ft, is theumpire'sactual information at (i.e.immediately preceding)themove 3TC. (ForK = v + 1:At the end of the game.)

(10:A:e*) (B*is the pattern of assignment, a B*(k) of (B, is the actualassignment,of the move 9TC,.

(10:A:f*) <3*(fc) is the pattern of choice, a C,of 6<(fc) is the actualchoice, of the player & at the move 3TZ*. (For A; = 0:Ofchance.)

(10:A:g*) 3)(A;) is the player k'spattern of information, a DK of >*(&)the player k'sactual information, at the move 3TC.

(10:A:h*) p*(C*) is the probability of the actual choiceC at the(chance)move 3TI,.

We now formulate the \"meaning\" of the requirements (10:1:a)-(10:1:j) in the senseof the concludingdiscussionof 10.1.1with the useofthe above nomenclature.(10:l:a*) The umpire'spattern of information at the move 3TC

includesthe assignmentof that move.(10:l:b*) The pattern of choiceat a chancemove 9fTl includes the

umpire'spattern of information at that move.(10:1:c*) Thepattern of choiceat a personalmove 3TC,of the playerk

includesthe playerk'spattern of information at that move.(10:l:d*) The umpire'spattern of information at the move 9Tl

includes to the extentto which this is a personalmove of theplayerk the playerk'spattern of information at that move.

(10:l:e*) The probabilitiesof the various alternative choicesat achancemove 9TC< behave likeprobabilitiesbelongingto disjunctbut exhaustive alternatives.

(10:l:f*) The umpire'spattern of information at the first move isvoid.

(10:1:g*) Theumpire'spattern of information at the endof the gamedeterminesthe play fully.

(10:l:h*) The umpire'spattern of information at the move ST^+i(for K = v: at the end of the game)obtains from that one atthe move SfTl, by superposingit with the pattern of choiceatthe move 311,.

(10:l:i*) Leta move 2(11,begiven, which is a personalmove of theplayer k, and any actual information of the player k at thatmove also be given. Then any actual information of theumpireat that move and any actualchoiceof the playerk atthat move, which areboth within (i.e.refinements of) thisactual (player's)information, arealso compatiblewith eachOther T ** +.VIPV nnniir in flftiifll nlnvs)))

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76 DESCRIPTIONOF GAMES OF STRATEGY

(10:1:j*) Let a move SHI, be given, which is a personalmove of theplayer A;, and any actual information of the player k at thatmove also be given. Then the number of alternative actualchoices,available to the playerk, is not zero.

This concludesour formalization of the generalschemeof a game.10.2.Logistic Discussionof the Axioms

10.2.We have not yet discussedthosequestionswhich areconvention-ally associatedin formal logicswith every axiomatization:freedom from

contradiction,categoricity(completeness),and independenceof the axioms.1

Our systempossessesthe first and the last-mentionedproperties,but not thesecondone. Thesefacts areeasy to verify, and it is not difficult to seethat the situation isexactlywhat it shouldbe. Insumma:

Freedomfrom contradiction:Therecan be no doubt as to the existenceof games,and we did nothing but give an exactformalism for them. Weshall discussthe formalization of severalgameslater in detail, cf. e.g.theexamplesof 18.,19. From the strictly mathematical logistic point ofview, even the simplestgame can beused to establishthe fact of freedomfrom contradiction.But our real interestlies, of course,with the moreinvolved games,which arethe really interestingones.2

Categoricity (completeness):This is not the case,sincethere existmany different games which fulfill these axioms.Concerningeffectiveexamples,cf. the precedingreference.

The readerwill observethat categoricityis not intended in this case,sinceour axiomshave to define a classof entities(games)and not a uniqueentity.8

Independence:Thisis easyto establish,but we donot enterupon it.10.3.GeneralRemarks Concerning the Axioms

10.3.Therearetwo more remarkswhich ought to be madein connectionwith this axiomatization.

First,our procedurefollows the classicallines of obtaining an exactformulation for intuitively empirically given ideas. The notion of agame existsin generalexperiencein a practicallysatisfactory form, which isneverthelesstoo looseto be fit for exacttreatment. Thereaderwho hasfollowedour analysiswill have observedhow this imprecisionwas gradually

1 Cf.D.Hilbert, loc.cit.;0.Veblen & J.W. Young: ProjectiveGeometry, New York1910;H.Weyl: Philosophic der Mathematik und Naturwissenschaften, in Handbuch derPhilosophic, Munich, 1927.

1This is the simplest game:v 0, 12has only one element, sayiro. Consequentlyno (B,C(fc),5>(fc),exist,while the only O is di,consisting of 12alone. Define$(iro)for k = I,- , n. An obvious description of this game consistsin the statement thatnobody doesanything and that nothing happens. This alsoindicates that the freedomfrom contradiction is not in this casean interesting question,

8 This is an important distinction in the general logistic approachto axiomatization.Thus the axioms of Euclidean geometry describea unique object while those of grouptheory (in mathematics) or of rational mechanics (in physics) do not, sincethere existmany different groups and many different mechanical systems.)))

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AXIOMATIC FORMULATION 77

removed,the\"zoneof twilight\" successivelyreduced,and a preciseformula-

tion obtainedeventually.Second,it is hoped that this may serveas an exampleof the truth of a

much disputed proposition:That it is possible to describeand discussmathematically human actions in which the main emphasis lieson thepsychologicalside.In the presentcasethe psychologicalelementwasbrought in by the necessityof analyzing decisions,the information on thebasis of which they are taken, and the interrelatednessof such setsofinformation (at the various moves) with eachother. Thisinterrelatednessoriginatesin the connectionof the various setsof information in time,causation,and by the speculativehypothesesof the players concerningeachother.

Thereareof coursemany and most important aspectsof psychologywhich we have never touchedupon, but the fact remainsthat a primarilypsychologicalgroupof phenomenahas beenaxiomatized.

10.4.Graphical Representation

10.4.1.The graphicalrepresentationof the numerous partitions whichwe had to use to representa gameis not easy. We shall not attempt totreat this matter systematically:even relatively simple gamesseemtolead to complicatedand confusing diagrams,and so the usual advantagesofgraphicalrepresentationdo not obtain.

Thereare,however, somerestrictedpossibilitiesof graphicalrepresenta-tion, and we shall say a few wordsabout these.

In the first placeit is clearfrom (10:1:h) in 10.1.1.,(or equally by(10:1:h*)in 10.1.2.,i.e.by rememberingthe \"meaning,\") that a,+iis asubpartition of ft,. I.e.in the sequenceof partitions Cti, , ft,, ft,+ieachoneis a subpartitionof its immediatepredecessor.Consequentlythismuch canbepictured with the devicesof Figure9 in 8.3.2.,i.e.by a tree.(Figure9 is not characteristicin oneway: sincethe length of the gameF isassumedto be fixed, all branchesof the treemust continue to its full height.Cf. Figure10in 10.4.2.below.) We shall not attempt to add the ,(&),Cic(fc),3D(fc)to this picture.

Thereis, however, aclassof gameswherethesequenceCti, , ft,,ft,+itells practically the entire story. This is the important class alreadydiscussed in 6.4.1.,and about which more will be said in 15.wherepreliminarity and anteriority are equivalent. Its characteristicsfind asimpleexpressionin our presentformalism.

10.4.2.Preliminarity and anteriority areequivalent as the discussionsof 6.4.1.,6.4.2.and the interpretationof 6.4.3.show if and only if everyplayerwho makesapersonalmove knowsat that moment the entireanteriorhistory of the play. Let the player be fc, the move 9TC,. The assertionthat 9TI, is k'a personalmove means,then,that we arewithin BK(k). Hencethe assertion is that within BK(k) the player fc's pattern of informationcoincideswith the umpire'spattern of information; i.e.that 2D(fc)is equal to)))

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78)) DESCRIPTIONOF GAMES OF STRATEGY))

Q,K within BK(k). But >(&) is a partition in BK(k);hencethe above state-ment meansthat >,(&)simplyis that part of a* which liesin BK(k).

We restatethis:(10:B) Preliminarity and anteriority coincide i.e.every playerwho

makesa personalmove is at that moment fully informed aboutthe entireanterior history of the play if and only if >K(k) isthat part of (tK which lies in BK(k).

If this is the case,then we can argueon as follows:By (10:1:c)in 10.1.1.and the above, QK(k) must now be a subpartition of QK. This holds forpersonalmoves,i.e.for k = 1, , n, but for k = it follows immedi-))

Figure 10.ately from (10:1:b)in 10.1.1.Now (10:1:h)in 10.1.1.permitstheinferencefrom this (for detailscf. 9.2.2.)that a+icoincideswith QK(k) in BK(k) forall k = 0, 1, , n. (We could equally have used the correspondingpointsin 10.1.2.,i.e.the \" meaning \" of theseconcepts.We leave the verbalexpressionof the argument to the reader.) But QK(k) is a partition in BK(k) ;hencethe above statementmeans that C<(k) simply is that part of G^+iwhich lies in B^k).

We restatethis:(10:C) If the condition of (10:B)is fulfilled, then QK(k) is that part

of ft*+i which lies in B^k).Thus when preliminarity and anteriority coincide,then in our present

formalism the sequencetti, , a,,a,+iand the setsBx(k), k = 0,1,, n, for eachK = 1, , v, describethe gamefully. I.e.the picture)))

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STRATEGIESAND FINAL SIMPLIFICATION 79

of Figure9 in 8.3.2.must beamplified only by bracketingtogetherthoseelementsof each&,which belongto the sameset(&K(k). (Cf.however, theremark made in 10.4.1.)We can do this by encirclingthem with a line,acrosswhich the number k of BK(k) is written. SuchBK(k) as areemptycanbe omitted. We give an exampleof this for v = 5 and n = 3 (Figure10).

In many games of this class even this extradevice is not necessary,becausefor every K only one BK(k) is not empty. I.e.the characterof eachmove 311,is independentof the previous courseof the play.1 Then itsufficesto indicateat eachOt, the characterof the move 3fTC* i.e.the uniquek = 0,1, , n for which BK(k) j .11.Strategiesand the Final Simplification of the Descriptionof a Game

11.1.TheConceptof a Strategy and Its Formalization

11.1.1.Let us return to the courseof an actual play TTof the game F.Themoves SfTC* followeachotherin the order K = 1, , v. At each

move 9Tl a choiceis made,eitherby chance if the play is in B<(0) or by aplayerk = 1, , n if the play is in BK(k). Thechoiceconsistsin theselectionof a CK from QK (k) (k = or k = 1, , n, cf. above),to whichthe play is then restricted.If the choiceis madeby a player fc, then pre-cautionsmust be taken that this player's pattern of information shouldbeat this moment >*(&),as required. (That this can be a matter of somepracticaldifficulty is shown by such examplesas Bridge[cf. the end of6.4.2.]and double-blindChess[cf. 9.2.3.].)

Imaginenow that eachplayer k = 1, , n, instead of making eachdecisionas the necessityfor it arises,makesup his mind in advancefor allpossiblecontingencies;i.e.that the playerk beginsto play with a completeplan:$ plan which specifieswhat choiceshe will makein every possiblesitua-tion, for every possibleactual information which he may possessat thatmoment in conformity with the pattern of information which the rules ofthe gameprovide for him for that case. We call sucha plan a strategy.

Observethat if we requireeachplayerto start the game with a completeplan of this kind, i.e.with a strategy, we by no meansrestricthis freedomof action. In particular, we do not thereby force him to make decisionson the basisof lessinformation than therewould beavailable for him in eachpracticalinstancein an actualplay. This is becausethe strategy is sup-posed to specifyevery particular decisiononly as a function of just thatamount of actual information which would beavailable for this purposeinan actualplay. Theonly extraburden our assumptionputs on the playeris the intellectualone to be prepared with a rule of behavior for all even-tualities, although he is to go through one play only. But this isan innoc-uous assumption within the confines of a mathematical analysis. (Cf.also4.1.2.)

1This is true for Chess.The rules of Backgammon permit interpretations both)))

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80 DESCRIPTIONOF GAMESOF STRATEGY

11.1.2.The chancecomponent of the game can be treatedin the sameway.

Itis indeedobvious that it isnot necessaryto makethe choiceswhich areleft to chance,i.e.thoseof the chancemoves, only when thosemoves comealong. An umpire could make them all in advance, and disclosetheiroutcome to the playersat the various moments and to the varying extent,as the rulesof thegame provide about their information.

It is true that the umpire cannot know in advance which moves will bechanceones,and with what probabilities;this will in general dependupontheactual courseof the play. But as in the strategieswhich weconsideredabove he could provide for all contingencies:He coulddecidein advancewhat the outcome of the choicein every possiblechancemove shouldbe, forevery possibleanterior courseof the play, i.e.for every possibleactualumpire'sinformation at the move in question. Undertheseconditionstheprobabilitiesprescribedby the rulesof the game for eachone of the aboveinstanceswould be fully determined and so the umpire could arrange foreachone of the necessarychoicesto be effected by chance,with the appro-priate probabilities.

The outcomescouldthen be disclosedby the umpire to the players atthepropermoments and to the properextent as describedabove.

We call such a preliminary decisionof the choicesof all conceivablechancemoves an umpire'schoice.

We saw in the last sectionthat the replacementof the choicesof allpersonalmoves of the player k by the strategyof the playerk is legitimate ;i.e.that it does not modify the fundamental characterof the game F.Clearlyour present replacementof the choicesof all chancemoves by theumpire'schoiceis legitimate in the samesense.

11.1.3.It remains for us to formalize the conceptsof a strategy and ofthe umpire'schoice.The qualitative discussionof the two last sectionsmakesthis an unambiguous task.

A strategyof the player k doesthis:Considera move 3fTC,. Assumethatit has turned out to be a personalmove of the player &, i.e.assumethatthe play is within B(fc). Considera possibleactual information of theplayerk at that moment, i.e.considera D* of ><(&). Then the strategyin questionmust determinehis choiceat this juncture, i.e.a C of QK(k)which is a subsetof the above DK.

Formalized:(11:A) A strategy of the player A; is a function 2*(/c;DK) which is

defined for every * =!,,*/and every DK of >,(k),andwhosevalue

2*(*;D,) = C,has always theseproperties:CK belongsto 6(fc)and is a subsetofD,.

That strategies i.e.functions S*(K;D<) fulfilling the above requirementexistat all,coincidespreciselywith our postulate(10:1:j)in 10.1.1.)))

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STRATEGIESAND FINAL SIMPLIFICATION 81An umpire'schoicedoesthis:Considera move 9TC. Assume that it has turned out to be a chance

move, i.e.assume that the play is within J3(0).Considera possibleactual information of the umpire at this moment;i.e.consideran A K of ft,which is a subset of B(0).Then the umpire'schoicein question mustdeterminethe chancechoiceat this juncture, i.e.a C of <3(0) which isasubsetof the above A K.

Formalized:(11:B) An umpire'schoiceis a function 2o(*;A K) which is defined for

every K = 1, , v and every A K of QLK which is a subset ofBK(0)and whosevalue

So(*;A K) = C,

has always theseproperties:C belongsto <B(0) and is a subsetof A K.

Concerningthe existenceof umpire'schoices i.e.of functions 2 (*;^)fulfilling the above requirement cf. the remark after (11:A)above, andfootnote 2 on p.71.

Sincethe outcomeof the umpire'schoicedependson chance,the cor-respondingprobabilitiesmust be specified. Now the umpire'schoiceis anaggregateof independent chanceevents. There is such an event, asdescribedin 11.1.2.,for every K = 1, , v and every A K of G, which is asubset of J3(0).I.e.for every pair *, A< in the domain of definition ofSO(K;At). As far as this event is concernedthe probabilityof the particularoutcome2o(*;A K) = CK is p*(C^). Hencethe probability of the entireumpire'schoice,representedby the function 2 (*;A K) is the productof theindividual probabilitiespK(CK).1

Formalized:(11:C) The probabilityof the umpire'schoice, representedby the

function o(*; A K) is the product of the probabilitiesp(C),where 2 (*;A K) = C,and K, A K run over the entiredomain ofdefinition of 2o(*;A K) (cf. (11:B)above).

If we considerthe conditionsof (10:l:e)in 10.1.1.for all thesepairsK, A K, and multiply them all with eachother,then thesefacts result:Theprobabilitiesof (11:C)above areall ^ 0,and their sum (extendedover allumpire'schoices)is one. This is as it shouldbe,sincethe totality of allumpire'schoicesis a systemof disjunctbut exhaustive alternatives.

ll.).TheFinal Simplification of the Description of a Game11.2.1.If a definite strategy has beenadopted by eachplayer k = 1,

, n, and if a definite umpire'schoicehas beenselected,then thesedeterminethe entire courseof the play uniquely, and accordinglyits

1 Thechanceevents in question must be treated as independent.)))

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82 DESCRIPTIONOF GAMES OF STRATEGY

outcometoo,for eachplayer k = 1, , n. This shouldbe clearfromthe verbal descriptionof all theseconcepts,but an equally simpleformalproof can be given.

Denotethe strategiesin questionby 2*(/c;DK), k = 1, , n, and theumpire'schoiceby 2 (/c; A K). We shall determine the umpire'sactualinformation at all moments K = 1, , v} v + 1. In orderto avoidconfusing it with the above variable A^, we denoteit by A K.

A i is, of course,equalto 12itself. (Cf.(10:1:f) in 10.1.1.)Considernow a K = 1, , v, and assumethat the correspondingA K

is already known. Then A K is a subset of preciselyone BK(k),k = 0,1,- , n. (Cf.(10:1:a)in 10.1.1.)If k = 0,then 3TC,is a chancemove, and

so the outcomeof the choiceis 2o(*;A K). Accordingly A f +i = 2o(*;A K).(Cf. (10:1:h) in 10.1.1.and the details in 9.2.2.)If k = 1, , n, then3TC,is a personalmove of the playerk. A K is a subsetof preciselyone DK of

>,(fc). (Cf. (10:1:d)in 10.1.1.)Sothe outcomeof the choiceis Z*(K; DK).AccordinglyA f+i = A K n S*(K;DK). (Cf. (10:1:h)in 10.1.1.and the detailsin 9.2.2.)

Thus we determineinductively A\\, A^ A^ - , A vy A v+i in succession.But A 9+i is a one-elementset (cf. (10:1:g) in 10.1.1.);denoteits uniqueelementby if.

ThisTTis the actual play which took place.1 Consequentlythe outcomeof the play is *(*) for the playerk = 1, , n.

11.2.2.The fact that the strategiesof all players and the umpire'schoicedeterminetogetherthe actual play and so its outcomefor eachplayer opensup the possibilityof a new and much simplerdescriptionofthe gameF.

Considera given player k = 1, , n. Form all possiblestrategiesof his, 2*(*;DK), or for short 2*. While their number is enormous,it is obviously finite. Denoteit by Pk, and the strategiesthemselvesbyZJ, ' , 2&-

Form similarly all possibleumpire'schoices,2 (*;A K), or for short 2 .Again their numberis finite. Denoteit by /3o, and the umpire'schoicesby2j, , 2?o. Denotetheir probabilitiesby p1

, , p^o respectively.(Cf.(11:C)in 11.1.3.)All theseprobabilitiesare^ and their sum isone.(Cf.the end of 11.1.3.)

A definite choiceof all strategiesand of the umpire'schoices,say 2J* fork = 1, , n and for k = respectively,where

T* = 1, , /fo for k = 0,1, , n,

determinesthe play TT(cf. the end of 11.2.1.),and its outcome (TT)foreachplayerk = 1, , n. Write accordingly

(11:1) IF*(T) = S*(TO, TI, , r n) for k = 1, , n.1Theaboveinductive derivation of the Ai, ^2, j, , Av, A+i is just amathemat-

ical reproduction of the actual courseof the play. Thereadershould verify the parallel-ism of the stepsinvolved.)))

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STRATEGIESAND FINAL SIMPLIFICATION 83

The entire play now consistsof eachplayer k choosinga strategy SJ,i.e.a number 7 fc =!, , 0*;and of the chanceumpire'schoiceof TO

= 1,, 0o, with the probabilitiesp\\ , /A respectively.

The playerk must choosehis strategy,i.e.his r*, without any informationconcerningthe choicesof the otherplayers,or of the chanceevents (theumpire'schoice). This must be so sinceall the information he can at anytime possessisalreadyembodiedin his strategy 2* = SJ* i.e.in the function2jk = *(*; DK). (Cf. the discussionof 11.1.1.)Even if he holdsdefiniteviews as to what the strategiesof the otherplayersarelikely to be,theymust be alreadycontainedin the function 2* (K; D).11.2.3.All this means,however, that F has beenbrought back to thevery simplestdescription,within the leastcomplicatedoriginal framework ofthe sections6.2.1.-6.3.1.We have n + I moves, one chanceand onepersonalfor eachplayerk = 1, , n eachmove has a fixed number ofalternatives, /3 for the chancemove and0i, , ft for thepersonalonesand every player has to make this choicewith absolutelyno informationconcerningthe outcome of all otherchoices.1

Now we can get rid even of the chancemove. If the choicesof theplayers have taken place,the player k having chosenT*, then the totalinfluenceof the chancemove is this :Theoutcomeof the play for the playerk

may be any one of the numbers

9*(T0, TI, , Tn), 7 = 1,' ' ' , 00,with the probabilities p 1, , yA respectively. Consequently his\" mathematical expectation\" of the outcomeis))

(11:2) JC*(TI, ,rn) = PT.S*(TO, Tlf , r n).

r -lThe player's judgment must be directedsolelyby this \" mathematical

expectation/'becausethe various moves, and in particular the chancemove, are completelyisolated from eachother.2 Thus the only moveswhich matter arethe n personalmoves of the playersk = 1, n.

Thefinal formulation is therefore this:(11:D) Then persongame T, i.e.the completesystemof its rules,is

determinedby the specification of the followingdata:(ll:D:a) Forevery k = !, , n:A number /3*.(ll:D:b) Forevery k = 1, , n:A function

3C*= OC*(ri, , r n),r, = 1, , ft for j = 1, - , n.

1 Owing to this completedisconnectednessof the n -f 1 moves, it doesnot matterin what chronological order they areplaced.

2 We are entitled to use the unmodified \"mathematical expectation\" sincewe aresatisfied with the simplified conceptof utility, as stressedat the end of 5.2.2Thisexcludesin particular all those more elaborateconceptsof \"expectation,\" which arereally attempts at improving that naive conceptof utility. (E.g.D.Bernoulli's \"moralexpectation\" in the \"St. Petersburg Paradox.\))

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84 DESCRIPTIONOF GAMES OF STRATEGY

Thecourseof a play of T is this:Eachplayer k choosesa number r = 1, ,/}. Each

playermust makehis choicein absoluteignoranceof the choicesof the others. After all choiceshave been made, they aresubmittedto an umpire who determinesthat theoutcomeof theplay for the playerk is3C*(ri, , rn).

11.3.TheRoleof Strategiesin the Simplified Form of a Game

11.3.Observethat in this schemeno spaceis left for any kind of further11 strategy.\" Eachplayer has one move, and onemove only;and he mustmake it in absoluteignoranceof everything else.1 This completecrystal-lization of the problemin this rigid and final form was achievedby ourmanipulations of the sectionsfrom 11.1.1.on, in which the transition fromthe original moves to strategieswas effected. Sincewe now treat thesestrategiesthemselvesas moves, thereis no needfor strategiesof a higherorder.

11.4.TheMeaning of the Zero-sum Restriction

11.4.We concludetheseconsiderationsby determiningthe placeof thezero-sumgames(cf. 5.2.1.)within our final scheme.

That T is a zero-sumgame means,in the notation of 10.1.1.,this:n

(11:3) J 5k(*) = for all w of fl.Jb-l

If we passfrom ^(TT)to 8*(TO,r,, , rn), in the senseof 11.2.2.,then thisbecomes

n

(11:4) 9*(T ,Ti, , rn) =0 for all TO,TI, ,rn.*-i

And if we finally introduceJC*(TI, , r),in the senseof 11.2.3.,we obtain

n

(11:5) 5) OC*(ri, , rn) = for all n, , rn.*-i

Conversely,it isclearthat the condition(11:5)makesthegameF, which wedefined in 11.2.3.,oneof zero sum.

1Reverting to the definition of a strategy as given in 11.1.1.:In this game a player A;

has one and only one personalmove, and this independently of the courseof the play,the move 311*. And he must make his choiceat Sdl* with nil information. So his

strategy is simply a definite choicefor the move 9R*, no more and no less;i.e.preciselyr -1, , 0*.

We leaveit to the readerto describethis game in terms of partitions, and to comparethe abovewith the formalistic definition ofa strategy in (11:A) in 11.1.3.)))

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CHAPTER IIIZERO-SUMTWO-PERSONGAMES: THEORY

12.PreliminarySurvey12.1.GeneralViewpoints

12.1.1.In the precedingchapterwe obtained an all-inclusive formalcharacterizationof the generalgame of n persons(cf.10.1.).We followedup by developingan exactconceptof strategywhich permittedus to replacethe rather complicatedgeneralschemeof a gameby a much more simplespecialone,which was neverthelessshown to be fully equivalent to theformer (cf. 11.2.).In the discussionwhich follows it will sometimesbemore convenient to use one form, sometimesthe other. It is thereforedesirableto give them specifictechnicalnames. We will accordinglycallthem the extensiveand the normalizedform of the game,respectively.

Sincethesetwo forms arestrictly equivalent, it is entirely within ourprovince to use in eachparticular casewhichever is technicallymore con-venient at that moment. We propose,indeed,to make full use of thispossibility,and must therefore re-emphasizethat this doesnot in the leastaffect the absolutelygeneralvalidity of all our considerations.

Actually the normalized form is bettersuited for the derivation ofgeneraltheorems,while the extensiveform is preferablefor the analysisofspecialcases;i.e.,the former can be used advantageouslyto establishpro-pertieswhich arecommon to all games,while the latterbringsout charac-teristicdifferences of games and the decisive structural features whichdeterminethesedifferences. (Cf.for the former 14.,17.,and for the lattere.g.15.)12.1.2.Sincethe formal descriptionof all gameshas beencompleted,we must now turn to buildup a positivetheory. It is to be expectedthata systematicprocedureto this end will have to proceedfrom simplergamesto more complicatedgames.It is therefore desirable to establish anorderingfor all gamesaccordingto their increasingdegreeof complication.

We have already classifiedgamesaccordingto the number of partici-pants a game with n participants being calledan n-persongame andalsoaccordingto whether they areor arenot of zero-sum.Thus we mustdistinguishzero-sum n-persongames and general n-persongames. It will

beseenlaterthat the generaln-persongameis very closelyrelatedto thezero-sum (n + l)-persongame, in fact the theory of the former will

obtain as a specialcaseof the theory of the latter. (Cf.56.2.2.)12.2.TheOne-personGame

12.2.1.We begin with someremarks concerningthe one-persongame.In the normalized form this gameconsists of the choiceof a number

85)))

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86 ZERO-SUMTWO-PERSONGAMES: THEORY

r = 1, , 0,after which the (only) player1 getsthe amount 3C(r).1 Thezero-sum caseis obviously void2 and thereis nothing to say concerningit.The generalcasecorrespondsto a generalfunction 3C(r)and the \"best\"or \" rational\" way of acting i.e.of playing consistsobviously of this:Theplayer1will chooser = 1, , so as to makeJC(r)a maximum.

This extremesimplification of the one-persongame is, of course,dueto the fact that our variable r representsnot a choice(in a move) butthe player's strategy; i.e.,it expresseshis entire\"theory\"concerningthehandling of all conceivablesituationswhich may occurin the courseof theplay. It shouldbe rememberedthat even a one-persongame can be of avery complicatedpattern:Itmay contain chancemoves as well as personalmoves (of the only player),eachone possiblywith numerous alternatives,and the amount of information available to the player at any particularpersonalmove may vary in any prescribedway.

12.2.2.Numerousgoodexamplesof many complicationsand subtletieswhich may arisein this way aregiven by the various gamesof \"Patience\"or \"Solitaire.\"Thereis, however, an important possibilityfor which, tothe best of our knowledge,examplesare lacking among the customaryone-persongames.This is the caseof incompleteinformation, i.e.of non-equivalenceof anteriority and preliminarity of personal moves of theunique player (cf.6.4.).Forsuch an absenceof equivalenceit would benecessarythat the player have two personalmoves 311*and 3n\\ at neitherof which he is informed about the outcomeof the choiceof the other.Sucha stateof lackof information is not easy to achieve, but we discussedin 6.4.2.how it can bebrought about by \"splitting\" the player into twoor more personsof identicalinterestand with imperfect communications.We saw loc.cit.that Bridgeis an exampleof this in a two-persongame;it would be easy to constructan analogousone-persongame but unluckilythe known forms of \"solitaire\"arenot such.8

This possibilityis neverthelessa practicalone for certaineconomicsetups:A rigidly establishedcommunistic society, in which the structureof the distributionschemeisbeyonddispute(i.e.where thereis no exchange,but only oneunalterable imputation) would be such sincethe interestsof all the membersof sucha societyarestrictly identical4 this setup must betreatedas a one-persongame. But owing to the conceivableimperfectionsof communications among the members,all sorts of incompleteinformationcan occur.

Thisis then the casewhich, by a consistentuseof the conceptof strategy(i.e.of planning),is naturally reducedto a simplemaximum problem. Onthe basis of our previous discussionsit will therefore be apparent now

Cf.(ll:D:a),(ll:D:b)at the end of 11.2.3.We suppressthe index 1.8 Then3C(r)= 0, cf.11.4.8 The existing \"double solitaires\" are competitive games between the two partici-

pants, i.e.two-person games.4The individual members themselves cannot be consideredas players, since all

possibilitiesof conflict among them, as well as coalitions of some of them against theothers, are excluded.)))

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PRELIMINARY SURVEY 87

that this and this only is the casein which the simplemaximum formu-lation i.e.the \"RobinsonCrusoe\"form of economicsis appropriate.12.2.3.These considerationsshow also the limitations of the puremaximum i.e.the \"RobinsonCrusoe\" approach. The above exampleof a societyof a rigidly establishedand unquestioneddistributionschemeshowsthat on this planea rational and criticalappraisalof the distributionschemeitself is impossible. In order to get a maximum problem it wasnecessaryto placethe entireschemeof distributionamong the rulesof thegame, which are absolute, inviolable and above criticism.In order tobring them into the sphere of combat and competition i.e.the strategyof the game it is necessaryto considern-persongames with n ^ 2 andthereby to sacrifice the simplemaximum aspectof the problem.

12.3.Chanceand Probability

12.3.Before going further, we wish to mention that the extensiveliterature of \"mathematical games\" which was developed mainly inthe 18thand 19thcenturies dealsessentiallyonly with an aspectof thematter which we have already left behind. This is the appraisal of theinfluence of chance. This was, of course,effected by the discoveryandappropriate application of the calculus of probability and particularlyof the conceptof mathematical expectations.In our discussions,theoperationsnecessaryfor this purposewereperformedin 11.2.3.1*2

Consequentlywe are no longerinterestedin thesegames,where themathematical problemconsistsonly in evaluating the roleof chance i.e.in computing probabilitiesand mathematical expectations.Suchgameslead occasionallyto interesting exercisesin probability theory;8 but wehope that the readerwill agreewith us that they do not belong in thetheory of gamesproper.

12.4.TheNext Objective12.4.We now proceedto the analysis of more complicatedgames.

The generalone-persongame having beendisposedof, the simplestoneof the remaining games is the zero-sumtwo-persongame. Accordinglywe aregoing to discussit first.

Afterwards there is a choiceof dealing either with the generaltwo-persongame or with the zero sum three-persongame. It will be seenthatour techniqueof discussionnecessitatestakingup the zero-sumthree-person

1We do not in the leastintend, of course,to detractfrom the enormous importanceof thosediscoveries.It is just becauseof their great power that we arenow in a positionto treat this sideof the matter asbriefly aswedo. We areinterested in thoseaspectsofthe problem which are not settled by the conceptof probability alone;consequentlytheseand not the satisfactorily settled onesmust occupyour attention.

2 Concerning the important connection between the use of mathematical expectationand the conceptof numerical utility, cf.3.7.and the considerations which precedeit.

8 Somegameslike Roulette areof an even more peculiar character. In Roulette themathematical expectation of the players is clearly negative. Thus the motives for

participating in that game cannot be understood if one identifies the monetary returnwith utility.)))

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88 ZERO-SUMTWO-PERSONGAMES: THEORY

game first. After that we shallextendthe theory to the zero-sumn-persongame (for all n = 1,2,3, ) and only subsequentlyto this will it befound convenient to investigatethe generaln-persongame.

13.Functional Calculus13.1.BasicDefinitions

13.1.1.Our next objective is as stated in 12.4.the exhaustivedis-cussionof the zero-sumtwo-persongames. In orderto do this adequately,it will be necessaryto use the symbolismof the functional calculus or atleastof certainparts of it more extensivelythan we have done thus far.Theconceptswhich we needarethose of functions, of variables,of maximaand minima, and of the use of the two latteras functional operations.All

this necessitatesa certainamount of explanationand illustration, whichwill be given here.

After that is done,we will prove some theoremsconcerningmaxima,minima, and a certaincombination of thesetwo, the saddlevalue. Thesetheoremswill play an important part in the theory of the zero-sumtwo-persongames.

13.1.2.A function </> is a dependencewhich stateshow certainentities%, y, ' ' ' calledthe variables of <t> determinean entity u calledthevalue of </>. Thus u is determinedby <f> and by the x, y, - , and thisdetermination i.e.dependencewill be indicatedby the symbolicequation

u = </>(*, y, ).In principleit is necessaryto distinguishbetween the function <t> itselfwhich is an abstractentity, embodyingonly the generaldependenceofu = <t>(x, y, - - - ) on the x,y,

- - and its value <t>(x, y, - ) for anyspecificx,y, . In practicalmathematical usage,however, it is oftenconvenient to write <t>(x, y, - - - ) but with x, y, - - - indeterminateinstead of (cf. the examples(c)-(e)below;(a), (b) areeven worse,cf.footnote 1below).

In ordertodescribethe function < it is of coursenecessary amongother things to specify the number of its variables x, y, - - - . Thusthereexistone-variablefunctions <t>(x), two-variablefunctions <t>(x, y), etc.

Someexamples:(a) Thearithmetical operationsx + 1and x2areone-variablefunctions.l

(b) Thearithmetical operationsof additionand of multiplication x + y

and xy, aretwo-variable functions.1

(c) Forany fixed k the *(*) of 9.2.4.is a one-variable function (of w).But it can alsobe viewed as a two-variable function (of fc, w).

(d) Forany fixed k the S*(K, DJ of (11:A) in 11.1.3.is a two-variablefunction (of ic, D).2

(e) Forany fixed k the3C*(ri, , r5) of 11.2.3.isa n-variable function(of n, , rn).1

1 Although they donot appearin the abovecanonicalforms <t>(x), <t>(x, y).1We couldalsotreat k in (d) and k in (e)like k in (c),i.e.asa variable.)))

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FUNCTIONALCALCULUS 89

13.1.3.It is equally necessary,in orderto describea function < tospecify for which specificchoicesof its variables x, y, the value<t>(%> y>

' ' ' ) is defined at all. Thesechoices i.e.thesecombinations ofx,y, form the domain of <t>.

Theexamples(a)-(e)showsomeof the many possibilitiesfor the domainsof functions:They may consistof arithmetical or of analytical entities,aswell as of others. Indeed:

(a) We may considerthe domain to consistof all integernumbers,or equallywell of all realnumbers.

(b) All pairsof eithercategoryof numbersusedin (a),form thedomain.(c) Thedomain is the set Q of all objects* which representthe plays

of the gameT (cf. 9.1.1.and 9.2.4.).(d) Thedomain consistsof pairs of a positiveintegerK and a setD.(e) Thedomain consistsof certain systemsof positiveintegers.A function <f> is an arithmetical function if its variables are positive

integers;it is a numerical function if its variablesarerealnumbers;it is aset-functionif its variablesaresets(as,e.g.,D< in (d)).

Forthe moment we aremainly interestedin arithmetical and numericalfunctions.

We concludethis sectionby an observationwhich is a natural conse-quenceof our view of the conceptof a function. This is, that the numberof variables,the domain, and the dependenceof the value on the varia-bles, constitute the function as such:i.e.,if two functions <, \\l/ have thesame variablesx, y, and the same domain, and if <t>(x, y, ) =

y>' * * ) throughout this domain, then <, \\l/ areidenticalin all respects.1

13.2.TheOperationsMax and Min

13.2.1.Considera function <t> which has realnumbersfor values))

Assumefirst that is a one-variable function. If its variable can bechosen,say as x = z so that <f>(x ) ^ <t>(x') for all otherchoicesx',then wesay that <t> has themaximum <t>(x<>) andassumesit at x = x .

Observe that this maximum </>(z ) is uniquely determined;i.e.,themaximum may beassumedat x = XQ for severalz , but they must all fur-nish the samevalue <t>(xo).

2 We denotethis value by Max <t>(x), the maxi-mum value of <t>(x).

If we replace by ^, then the conceptof </>'s minimum, <A(x ), obtains,and of XQ where </> assumesit. Again theremay beseveralsuchx , but theymust all furnish the samevalue <t>(x Q). We denotethis value by Min <t>(x),

the minimum value of </>.

1 Theconceptof a function is closelyallied to that of a set,and the aboveshould beviewed in parallel with the exposition of 8.2.

*Proof:Considertwo such a? , say x'and x'J. Then 4(si) *(x'')and <K*'o')Hence4>(x'>))))

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90 ZERO-SUMTWO-PERSONGAMES: THEORY

Observethat there is no a priori guaranteethat eitherMax <t>(x) orMin <(>(x)exist.1

If, however, the domain of </> over which the variable x may run

consistsonly of a finite number of elements,then the existenceof bothMax <t>(x) and Min <t>(x) is obvious. This will actually be the casefor mostfunctions which we shall discuss.2 For the remaining ones it will be aconsequenceof their continuity togetherwith the geometricallimitations oftheir domains.8 At any ratewe arerestrictingour considerationsto suchfunctions, for which Max and Min exist.

13.2.2.Let now <t> have any number of variablesx,y,z, . By sin-gling out oneof the variables,say x,and treatingthe others,y y z, - , asconstants,we can view <t>(x, y, z, - - - ) as a one-variablefunction, of thevariable x. Hencewemay form Max<t>(x,y,z, ), Min </> (z,t/, 2, )as in 13.2.1.,of coursewith respectto this x.

But sincewe couldhave donethis equallywell for any one of the othervariablesy, z, it becomesnecessaryto indicatethat the operationsMax, Min wereperformed with respectto the variable x. We do this bywriting Max* </>(z, y, z, ), Min x <(z,y, z, ' ' * ) insteadof the incom-pleteexpressionsMax

<t>,Min </>. Thus we can now apply to the function

<t>(x> y, z>

' * ' ) anY ne of the operatorsMax*, Min z, Maxy, Min y, Max*,Min,, . They areall distinctand our notation isunambiguous.

This notation is even advantageousfor one variable functions, and wewill use it accordingly;i.e.we write Max* <(#),Min* <t>(x) instead of theMax (x),Min 4>(x) of 13.2.1.

Sometimesit will be convenient or even necessary to indicate thedomain S for a maximum or a minimum explicitly. E.g.when the func-tion <t>(x) is defined alsofor (some)x outsideof S,but it isdesiredto form themaximum or minimum within S only. In such a casewe write

Maxxin5 0(x), Minxin5 <t>(x)

insteadof Max* <(#),Min x <(z).Incertainothercasesit may besimplerto enumeratethe values of <f>(x)

say o,6, than to express<t>(x) as a function. We may then write1E.g.if <t>(x) ss x with all realnumbers asdomain, then neither Max 0(x)nor Min <f>(x)

exist.2Typicalexamples:Thefunctions 3C*(,n, , r n) of 11.2.3.(or of (e)in 13.1.2.),the

function 3C(ri,TZ) of 14.1.1.> > > > >

8 Typical examples:Thefunctions K( , 17 ), Max-> K( , 17 ), Min- K( , 17 ) in* *

0i ft,

17.4.,the functions Min Tl ^/ ^(n, T2) v Maxfi ^ 3C(n, r t )ij r in 17.5.2.The vari-

r^l^ ^ rt-lablesof all thesefunctions are or j or both, with respectto which subsequent maximaand minima are formed.

Another instance is discussedin 46.2.1.espec.footnote 1 on p. 384,where themathematical background of this subjectand its literature are considered.It seemsunnecessary to enter upon thesehere, sincethe aboveexamplesareentirely elementary.)))

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FUNCTIONALCALCULUS 91Max (a, &,-)>[Min (a, 6, )] instead of Max* <(z), [Min*))

13.2.3.Observethat while </>(x, y, z, - ) is a function of the variablesx, y, z, , Max* <t>(x, y, z, - - - ), Min* <f>(x, y, *,)arestillfunc-tions, but of the variables y, z, only. Purely typographically,x isstill presentin Max,</>(#, y, z, - ), Min* <t>(x, y, z, - - - ), but it is nolongera variable of thesefunctions. We say that the operationsMax*,Min* kill thevariable x which appearsas their index.2

SinceMax* <t>(x, y, z, ), Min* 4>(x, y, z, ) arestill functionsof the variablesy, z, ,3 we can go on and form the expressions

Maxy Max* <t>(x, y, z, ), Maxy Min* <f>(x, y, z, - ),Miny Max* <(z,y, z, - - ), Min y Min* <t>(x, y, z, - ),

We couldequallyform

Max* Maxy <j>(x, y, z, ), Max* Miny <t>(x, y, z, )

etc.;4 or use two other variablesthan x, y (if thereareany) ; or usemorevariablesthan two (if thereareany).

Infine, after having appliedas many operationsMaxor Minas therearevariablesof <t>(x, y, z, ) in any orderand combination,but preciselyonefor eachvariable x, y, z, - we obtain a function of no variablesat all, i.e.a constant.

13.3.Commutativity Questions13.3.1.Thediscussionsof 13.2.3.providethebasisfor viewing the Max*,

Min*, Maxy, Min tf , Max,, Min*, entirely as functional operations, eachone of which carriesa function into anotherfunction. 5 We have seenthatwe can apply severalof them successively. In this lattercaseit is primafacie relevant, in which orderthe successiveoperationsareapplied.

But is it really relevant? Precisely:Two operationsaresaid to commute

if, in caseof their successiveapplication(to the sameobject),the orderinwhich this is donedoesnot matter. Now we ask:Do the operationsMax*,Min*, Maxy, Min y, Max,,Min,, all commute with eachotheror not?

We shall answerthis question. Forthis purposewe needuseonly twovariables,sayx,y and then it is not necessarythat <f> be a function of furthervariablesbesidesx,y.6

1Of courseMax (a, b, - ) [Min (a, b, )] is simply the greatest [smallest]oneamong the numbers a, b, - .

1A well known operation in analysis which kills a variable x is the definite integral:f\\

4>(x)is a function of x, but / <j>(x)dx is a constant.8 We treated fy, z, as constant parameters in 13.2.2.But now that x has been

killed we releasethe variables y, z, .4Observethat if two or more operationsare applied, the innermost appliesfirst and

kills its variable; then the next one follows suit, etc6 With one variable less,sincetheseoperationskill onevariable each.8 Further variables of <, if any, may be treated as constants for the purpose of this

analysis.)))

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92 ZERO-SUMTWO-PERSONGAMES: THEORY

Sowe considera t Wo-variable function <t>(x, y). The significant ques-tions of commutativity arethen clearlythese:

Which of the threeequationswhich followaregenerally true:

(13:1) Max,Maxy 0(x,y) = Maxy Max,<t>(x f y),(13:2) Min, Miny <Kz, y) = Min y Min, <(x,y),(13:3) Max,Min y </>(x, y) = Miny Max,<t>(x, y).1

We shallseethat (13:1),(13:2)aretrue, while (13:3)is not; i.e.,any twoMaxor any two Min commute, while a Maxand a Min do not commute in

general. We shallalsoobtain a criterion which determinesin which specialcasesMax and Min commute.

This questionof commutativity of Max and Min will turn out to bedecisivefor the zero-sum two-persongame (cf. 14.4.2.and 17.6.).

13.3.2.Let us first consider(13:1).Itought to beintuitively clearthatMax,Maxy <f>(x, y) is themaximum of <(x,y) if we treatx,y togetheras onevariable;i.e.that for somesuitable x () , 2/0, </>(x , 2/o) = Maxx Maxy 0(x,y)and that for all x',y', <f>(x , y ) ^ <t>(x', y').

If a mathematical proof is neverthelesswanted,we give it here:Choosexosothat Maxy <f>(x, y) assumesits x-maximum at x = x , and then choosej/so that <t>(xQ, y) assumesits y-maximum at y = y Q. Then))

/o) = Maxy </>(x , y) = Max,Maxy <(x,y),

and for all x', y1

o, 2/0) = Maxy </>(x , y) ^ Maxy </>(x', y) ^ <t>(x', y').))

This completesthe proof.Now by interchanging x, y we seethat Maxy Max,<(x,y) is equally

the maximum of </>(x, y) if wetreat x, y asonevariable.Thus both sides of (13:1)have the same characteristicproperty, and

therefore they areequalto eachother. This proves(13:1).Literally the same arguments apply to Mm in placeof Max:we need

only use^ consistentlyin placeof ^ . Thisproves(13:2).This deviceof treating two variablesx, y as one,is occasionallyquite

convenient in itself. When we useit (as,e.g.,in 18.2.1.,with n,r 2, 3C(ri,r2)in placeof our present x, y, <(x,z/)), we shall write Max,,y <(x,y) andMin,.y <(x,2/).

13.3.3.At this point a graphicalillustration may be useful. Assumethat the domain of <f> for x, y is a finite set. Denote,for the sake of sim-plicity,the possiblevalues of x (in this domain) by 1, , t and thoseofy by 1, ,$. Then the values of </(x, y) correspondingto all x,y in thisdomain i.e.to all combinationsof x = 1, , /, y = 1, , $ canbe arranged in a rectangularscheme:We usea rectangleof t rowsand s

1Thecombination Min, Max,, requires no treatment of its own, sinceit obtains fromthe above Max,Min y by interchanging x, y.)))

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FUNCTIONALCALCULUS)) 93))

columns,using the number x = 1, , t to enumeratethe former andthe number y = 1, , s to enumeratethe latter. Intothefieldof inter-sectionof row x and column y to beknown briefly as the field x, y wewrite the value <t>(x t y) (Fig.11).Thisarrangement, known in mathematicsas a rectangular matrix, amounts to a completecharacterizationof the func-tion <t>(x, y). Thespecificvalues <(x,y) arethe matrix elements.))

1) 2) y) s)

1) 0(1,1)) $(1,2)) *>(!,y)) ^(l, ))

2) d(2,1)) 0(2,2)) <fr(2, y}) 0(2, ))

x) 4>(x,1)) <t>(x t 2)) 0(z,y}) 0(s,s])

-)

t) d(J, 1)) 4>(l 2)) <t>(t y}) 4>(L a))

Figure 11.Now MaXy <t>(x, y) is the maximum of <(#, y) in the row x.

Maxx))

is therefore the maximum of the row maxima.On the otherhand,

Max* <t>(x, y)

is the maximum of <f>(x, y) in the column y. Maxy Max* <(#,y) is thereforethe maximum of the column maxima. Our assertionsin 13.3.2.concerning(13:1)can now be stated thus: The maximum of the row maxima is thesameas the maximum of the column maxima;both arethe absolutemaxi-mum of <t>(x, y) in the matrix. In this form, at least,theseassertionsshouldbe intuitively obvious. The assertions concerning(13:2)obtain in thesameway if Min is put in placeof Max.

13.4.TheMixed Case. SaddlePoints

13.4.1.Let us now consider(13:3).Using the terminology of 13.3.3.the left-hand side of (13:3)is the maximum of the row minima and theright-handsideis theminimum of the column maxima.Thesetwo numbersareneitherabsolutemaxima, nor absoluteminima, and thereis no primafadeevidencewhy they shouldbegenerallyequal. Indeed,they arenot.Two functions for which they aredifferent aregiven in Figs.12,13.A)))

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94)) ZERO-SUMTWO-PERSON GAMES: THEORY))

function for which they areequal is given in Fig.14. (All thesefiguresshouldbe readin the senseof the explanationsof 13.3.3.and of Fig.11.)

Thesefigures as well as the entirequestionof commutativity of Maxand Min will play an essentialrolein the theory of zero-sumtwo-persongames. Indeed,it will be seenthat they representcertain games whichare typical for some important possibilitiesin that theory (cf. 18.1.2.).But for the moment we want to discussthem on their own account,without

any referenceto thoseapplications.))

t - s - 2)) t - a - 3))

1) 2)row)

minima)

1) 1) j) j)

2) -1) 1) -1)

column)1) 1)

maxima)

Maximum of row minima 1Minimum ofcolumn maxima 1

Figure 12.))

1) 2) 3)row

minima)

1) -1) 1) -1)2) 1) -1) -1)

3) -1) 1) -1)

column)1) 1) 1)

maxima)

Maximum of row minima 1Minimum of column maxima 1

Figure 13.))

t -*-2))

1) 2)row)

minima)

1) -2) 1) -2)2) -1) 2) -1)

column) -1) 2)maxima)

Maximum of row minima 1Minimum of column maxima 1

Figure 14.

13.4.2.Since(13:3)is neither generallytrue, nor generallyfalse, it isdesirableto discussthe relationshipof its two sides

(13:4) Max* Miny </>(#, y) y Miny Maxx </>(x, y),more fully. Figs.12-14,which illustrated the behavior of (13:3)to acertaindegree,give somecluesas to what this behavior is likely to be.Specifically:

(13:A) In all threefigures the left-hand sideof (13:3)(i.e.the firstexpressionof (13:4))is ^ the right-handsideof (13:3)(i.e.thesecondexpressionin (13:4)).)))

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FUNCTIONALCALCULUS 95

(13:B) In Fig.14 where (13:3)is true there existsa field in thematrix which contains simultaneously the minimum of its rowand the maximum of its column. (This happensto be the ele-ment 1 the left lower cornerfield of the matrix.) In theother figures 12,13,where (13:3)is not true, there existsnosuchfield.

It is appropriate to introduce a generalconceptwhich describesthebehavior of the field mentioned in (13:B).We define accordingly:

Let <f>(x, y) beany two-variable function. Then x , 3/0 is a saddlepointof <f> if at the sametime </>(x, y ) assumesits maximum at x = x and <(x, y)assumesits minimum at y = 3/0-

The reason for the use of the name saddlepoint is this:Imaginethematrix of all x, y elements(x = 1, ,,y = 1, a; cf. Fig.11)as an oreographicalmap, the height of the mountain over the field x, y

being the value of <t>(x, y) there. Then the definition of a saddle pointx , 2/0describesindeedessentiallya saddleor passat that point (i.e.over thefield xo, 2/0) ; the row XQ is the ridge of the mountain, and the column i/o isthe road (from valley to valley) which crossesthis ridge.

The formula (13:C*)in 13.5.2.also falls in with this interpretation.1

13.4.3.Figs.12,13show that a <f> may have no saddle points at all.On the otherhand it is conceivablethat < possessesseveralsaddlepoints.But all saddle points x , 2/0 i? they existat all must furnish the samevalue 0(x, 2/0)-

2 We denotethis value if it existsat all by Sax/v<(x,y),the saddlevalue of </>(x, y).8

We now formulate the theorems which generalizethe indicationsof(13:A), (13:B). We denotethem by (13:A*), (13:B*),and emphasizethat they arevalid for all functions <(x,y).(13:A*) Always

Max,Min y <(x,y) ^ Min y Max,#(x,y).(13:B*) We have

Max* Min y <(x,y) = Mmy Maxz </>(x, y)if and only if a saddlepoint x , t/o of <#> exists.

13.5.Proofso!the Main Facts13.5.1.We define first two sets A*, B+ for every function <(x, 2/)-

Miny <t>(x, 2/) is a function of x; let A+ bethe setof all thosex for which1All this is closelyconnectedwith although not preciselya specialcaseof; certain

more generalmathematical theoriesinvolving extremal problems, calculus of variations,etc. Cf.M.Morse:TheCritical Points of Functions and the Calculus of Variations inthe Large,Bull. Am. Math. Society,Jan .-Feb.1929,pp.38cont.,and What is Analysis inthe Large?,Am. Math. Monthly, Vol. XLIX, 1942,pp.358cont.

1This follows from (13:C*)in 13.5.2.Thereexists an equally simple direct proof:Considertwo saddlepoints a? , y o,say4,yj and 4',yi'. Then:

'*)-Max,*(*, y'o) fc *(*\", vl) Min, *(*'',y)))

i.e.:+(*;,yi) <^x'',j/'). Similarly *(*'',y'') +(*;,yj).Hence0(4,yi) -0(4',yi').1Clearlythe operation Sa/V0(x, y) kills both variablesx} y. Cf.13.2.3.)))

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96 ZERO-SUMTWO-PERSONGAMES: THEORY

this function assumesits maximum at x = XQ. Max,4>(x, y) is a functionof y ; letB+ bethe setof all thosey for which this function assumesits mini-mum at y = 2/0.

We now prove (13:A*),(13:B*).Proof of (13:A*) :ChooseXQ in A+ and y in B*. Then))

Max* Min y <l>(x, y) = Min y <(zo,y) ^ <(zo,2/0)

g Max,<f>(x, 2/0) = Min y Max* <t>(x, y),

i.e.:Max,Min y <f>(x, y) ^ Min y Max,<t>(x, y) as desired.Proof of the necessityof the existenceof a saddle point in (13:B*):

Assume that

Max,Miny <t>(x, y) = Miny Max,<t>(x, y).

ChooseXQ in A* and yo in 5*;then we have))

Max,<(z,2/0) = Min y Max,<t>(x, y)= Max,Miny <f>(x, y) = Min

y))y

Hencefor every xf

g Max,<l>(x, y<>)= Min))

i.e.<t>(x , i/o) ^ </>(x', 2/0) so </>(z, 2/o) assumes its maximum at x = x .And for every 2/'

y') ^ Min y ^(a:, y) = Max,</>(x, 2/0) ^ *(^o,2/o),))

i.e.0(x, 2/0) ^ *(^o,2/0 so *(^o,2/) assumesits minimum at y = 2/0.

ConsequentlytheseXD, 2/0 form a saddlepoint.Proof of the sufficiency of the existenceof a saddle point in (13:B*):

Let XQ, 2/0 be a saddlepoint. Then))

Max,Min y </>(z, y) ^ Miny </>(z , 2/) = *(^o,2/o),Min y Max,<f>(x, y) ^ Max,<t>(x, 2/0) =

</>(z , t/o),hence

Max,Miny <^(a:,y) ^ *(z, 2/0) ^ Min y Max,0(z,2/).

Combiningthis with (13:A*) gives

Max,Miny <t>(x, y) =<t>(x , 2/0) = Miny Max,<t>(x, y),

and hencethe desiredequation.13.5.2.The considerationsof 13.5.1.yield some further results which

are worth noting. We assume now the existenceof saddle points, i.e.the validity of the equationof (13:B*).

Forevery saddlepoint a; , yo))

(13:C*)0(x, 2/0) = Max,Miny <t>(x, y) = Miny Max,<t>(x, y).

Proof:This coincideswith the last equation of the sufficiency proof of(13:B*)in 13.5.1.)))

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FUNCTIONALCALCULUS 97

(13:D*) x , 2/0 is a saddlepoint if and only if x belongsto A* and yo

belongsto B*.1

Proof of sufficiency:Let x belongto A* and y* belongto B+. Then thenecessityproof of (13:B*)in 13.5.1.showsexactlythat this x , yo is a saddlepoint.

Proof of necessity:Let xo, 2/0 be a saddlepoint. We use (13:C*).Forevery x'

Min y 0(s',y) ^ Max,Min y <t>(x, y) = <t>(x Q, y<>)= Min y 4>(z<>,y),

i.e.Min y <(x, t/) ^ Min y <(x',i/) so Min y <(x,y) assumesits maximumat x = x . Hencex belongsto A+. Similarly for every y'

Max* <(>(x,i/O ^ Miny Max* <(x,y) = ^(x0| j/o) = Max* 0(x,2/0),

i.e.Maxx <(x,2/0) ^ Maxz <t>(x, y') so Max* 0(x,y) assumesits minimumat y = I/O- Hencey belongsto B*. This completesthe proof.

The theorems(13:C*); (13:D*)indicate,by the way, the limitationsof the analogy describedat the end of 13.4.2.;i.e.they show that our con-ceptof a saddle point is narrower than the everyday (oreographical)ideaof a saddle or a pass. Indeed,(13:C*)statesthat all saddles providedthat they exist areat the samealtitude. And (13:D*)states if wedepictthe setsA*, B+ as two intervals of numbers2 that all saddlestogetherarean areawhich has the form of a rectangularplateau.8

13.5.3.We concludethis sectionby proving the existenceof a saddlepoint for a specialkind of x, y and <(x,y). This specialcasewill be seento be of a not inconsiderablegenerality. Let a function ^(x, u) of twovariables x, u be given. We considerall functions /(x) of the variablewhich have values in the domain of u. Now we keepthe variable x butin placeof the variable u we use the function / itself.4 The expressionlK*/(z)) isdeterminedby x,/;hencewe may treat (x,/(ar))as a function ofthe variablesx, / and let it take the placeof <(x,y}.

We wish to prove that for thesex,/ and ^(x,/(x)) in placeof x, y

and <f>(x y y) a saddlepoint exists;i.e.that

(13:E) Maxx Min/ ^(x,/(x)) = Min/ Maxx f(x,/(x)).))

Proof:Forevery x choosea UQ with ^(x, u ) = Min u ^(x,w). This w

depends on x, hencewe can define a function / by w = /o(z). Thus^/(x, /o(z))= Min M ^(x,u). Consequently

Max*^(x,/o(x))= Max* Min u ^(x,u).1 Only under our hypothesis at the beginning of this section! Otherwise there exist

no saddlepoints at all.1If the x, y are positive integers, then this can certainly be brought about by two

appropriate permutations of their domain.3The general mathematical conceptsalluded to in footnote 1 on p.95are free from

theselimitations. They correspondpreciselyto the everyday ideaofa pass.4Thereaderis askedto visualize this: Although itself a function, / may perfectly well

be the variable of another function.)))

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98 ZERO-SUMTWO-PERSONGAMES:THEORY

A fortiori,

(13:F) Min/ Max, f(x,/(*)) ^ Max* Min u f(x,u).))

Now Min/ ^(z,f(x)) is the same thing as Min tt \\(/(x y u) since/ entersintothis expressiononly via its value at the one placex, i.e.f(x),for which wemay write u. SoMin/ \\l/(x, f(x)) = Min M ^(x,w) and consequently,))

(13:G) Max*Min/$(x,/(#)) = Maxx Min M ^(z,u).

(13:F),(13:G)togetherestablishthe validity of a ^ in (13:E).The ^in (13:E) holdsowing to (13:A*). Hencewe have = in (13:E),i.e.theproof is completed.

14.Strictly DeterminedGames14.1.Formulation of the Problem

14.1.1.We now proceedto the considerationof the zero-sum two-persongame. Again we beginby using the normalized form.

According to this the game consistsof two moves:Player 1 choosesanumber TI =!, , 0i,player2 choosesa number r2 = !, , #2, eachchoicebeing made in completeignorance of the other, and then players1and 2 get the amounts 3Ci(ri,r 2) and 3C2(Ti, T2), respectively.1

Sincethe game is zero-sum,we have, by 11.4.3Ci(Ti,TI) + JC(T,,TI) = 0.

We prefer to expressthis by writing

3Cl(Ti,T2) 35 3C(Ti,Tl), JCl(Ti,T2) = -JC^i,T2).We shall now attempt to understand how the obvious desiresof the

players1,2 will determinethe events,i.e.the choicesTI, r 2.It must again be remembered,of course,that TI, r 2 stand ultima analysi

not for a choice(in a move) but for the players'strategies;i.e.their entire\"theory\"or \"plan\" concerning the game.

Forthe moment we leave it at that. Subsequentlywe shall also go\"behind\"the TI, T2 and analyze the courseof a play.

14.1.2.The desiresof the players 1,2, are simpleenough. I wishesto make 5Ci(ri,T2) s= JC(TI,T2) a maximum, 2 wishesto make3C2(Ti, T2) s

#C(TI, T2) a maximum; i.e.1wants to maximize and 2 wants to minimize3C(n,Tt).

So the interestsof the two playersconcentrateon the sameobject:theone function 3C(Ti,T2). But their intentions are as is to be expectedin azero-sum two-persongame exactlyopposite:1wants to maximize, 2 wantsto minimize. Now the peculiardifficulty in all this is that neitherplayerhas full control of the objectof his endeavor of 3C(ri,T2) i.e.of bothits variables TI,T2. 1wants to maximize,but he controlsonly TI;2 wants tominimize, but he controlsonly T2: What is going to happen?

l Cf.(ll:D)in11.2.3.)))

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STRICTLYDETERMINEDGAMES)) 99))

Thedifficulty is that no particularchoiceof, say TI, needin itself make3C(ri,r 2) eithergreator small. Theinfluenceof r\\ on 3C(ri,r2) is, in general,no definite thing;it becomesthat only in conjunctionwith the choiceofthe other variable, in this caser2. (Cf. the correspondingdifficulty ineconomicsas discussedin 2.2.3.)

Observethat from the point of view of the player 1who choosesavariable, say n, the other variable can certainly not be consideredas achanceevent. The othervariable, in this caser2, is dependentupon thewill of the otherplayer,which must be regardedin thesamelight of \" ration-ality \" as one'sown. (Cf.alsothe end of 2.2.3.and 2.2.4.)

14.1.3.At this point it is convenient to make use of the graphicalrepresentationdevelopedin 13.3.3.We represent3C(ri,r2) by a rectangularmatrix:We form a rectangleof pi rows and 2 columns,using the numbern = 1, , #1to enumerate the former, and the number r 2 = 1, , 2

to enumeratethe latter;and into the fieldn,r 2 we write the matrix element3C(ri,r 2). (Cf.with Figure11in 13.3.3.The

</>, x,y, t, s therecorrespondto our OC,TI, r2, 01,2 (Figure15).)))

1) 2) 7>2)

1) 3C(1,1)) JC(1,2)) 3C(1 rj)) 3C(1.00)

2) 3C(2,1)) 3C(2,2)) 3C(2,T2)) 3C(2,0))

7-1) 3C(ri, 1)) 3C(n, 2)) 3C(T1,T)) 3C(n, 2))

0i) C(0i,1)) JC(0i, 2)) 3C(#i.n)) 3C(0i,0i))

Figure 15.It ought to be understoodthat the function 3C(ri,r 2) is subjectto no

restrictionswhatsoever;i.e.,we are free to chooseit absolutelyat will. 1

Indeed,any given function 5C(ri,r 2) definesa zero-sumtwo-persongamein the senseof (11:D)of 11.2.3.by simplydefining

3Ci(T!,T2) s JC(TI,r 2), 3C2(ri,r2) s -3C(ri,r2)

(cf. 14.1.1.).The desiresof the players 1,2, as describedabove in thelast section,can now be visualized as follows:Both players are solely))

1Thedomain, of course,is prescribed:It consistsof all pairsn, TI with n 1, ,;n = 1, , 0t. This is a finite set,soall Max and Min exist,cf.the end of 13.2.1.)))

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100 ZERO-SUMTWO-PERSONGAMES: THEORY

interestedin the value of the matrix elementJC(ri, r 2). Player 1tries tomaximize it, but he controlsonly the row, i.e.the numbern. Player 2triesto minimize it, but he controlsonly the column, i.e.the number r2.

We must now attempt to find a satisfactoryinterpretation for the out-comeof this peculiartug-of-war.1

14.2.The Minorant and the Majorant Gaifces

14.2.Insteadof attempting a directattackon the gameF itself forwhich we arenot yet prepared letus considertwo othergames,which arecloselyconnectedwith F and the discussionof which is immediatelyfeasible.

Thedifficulty in analyzing F is clearlythat the player 1,in choosingndoesnot know what choicer2 of the player 2 he is going to face and viceversa. Let us therefore compareF with othergameswhere this difficultydoesnot arise.

We define first a gameFi,which agreeswith F in every detailexceptthatplayer 1has to make his choiceof TI beforeplayer 2 makeshis choiceofT2, and that player 2 makeshis choicein full knowledgeof the value givenby player 1to TI (i.e.Tsmove is preliminary to 2'smove).2 In this gameFi player 1 is obviously at a disadvantageas comparedto his positionin the original gameF. We shall therefore call Fithe minor ant game of F.

We define similarly a secondgame F2 which again agreeswith F in everydetail exceptthat now player2 has to makehis choiceof r 2 beforeplayer 1makes his choiceof r\\ and that 1 makes his choicein full knowledgeof the value given by 2 to r 2 (i.e.2'smove is preliminary to 1'smove).3 Inthis game F2 the player 1is obviously at an advantage as comparedtohis positionin the gameF. We shall therefore call F2 the majorant gameof F.

The introductionof thesetwo gamesFI,F2 achievesthis:It ought tobe evident by common sense and we shall also establish it by an exactdiscussion that for FI,F2 the \"bestway of playing\" i.e.the conceptofrational behavior has a clearmeaning. On the otherhand, the game Fliesclearly\"between\"the two gamesFI,F2; e.g.from 1'spoint of view FIis always lessand F2 is always more advantageousthan F.4 Thus FI,F2

may be expectedto provide lower and upper bounds for the significantquantitiesconcerningF. We shall,of course,discussall this in an entirelypreciseform. A priori,these \"bounds\"could differ widely and leave aconsiderableuncertainty as to the understandingof F. Indeed,prima fadethis will seemto be the casefor many games.But we shall succeedinmanipulating this techniquein such a way by the introductionof certain

1Thepoint is, of course,that this is not a tug-of-war. Thetwo playershave oppositeinterests, but the means by which they have to promote them are not in opposition toeachother. On the contrary, these \" means \" i.e.the choicesof n, TJ are apparentlyindependent. This discrepancycharacterizesthe entire problem.1Thus Ti while extremely simple is no longer in the normalized form.

*Thus TI while extremely simple is no longer in the normalized form.4Ofcourse,to beprecisewe should say \"lessthan or equal to\" instead of \"less,\" and

\"more than or equal to\" instead of \"more.\)

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STRICTLYDETERMINEDGAMES 101further devices as to obtain in the end a precisetheory of F, which givescompleteanswersto all questions.

14.3.Discussionof the Auxiliary Games14.3.1.Let us first considerthe minorant game IV After player 1

has made his choiceTI the player 2makes his choiceT2 in full knowledgeof the value ofn. Since2'sdesireis to minimize JC(ri,T2), it is certainthathe will chooseT2 so as to make the value of 5C(ri,r2) a minimum for this TI.In otherwords:When 1choosesa particularvalue ofTIhe can alreadyforeseewith certaintywhat thevalue of3C(ri,r2) will be. Thiswill beMin TjX(TI,r 2).lThis is a function of TI alone. Now 1wishesto maximize 3C(ri,r 2) andsincehischoiceof TIisconducive to the value Min

Ta 3C(ri,r 2) which dependson TI only, and not at all on T2 so he will chooseTI so as to maximizeMin Tj JC(TI,T2). Thus the value of this quantity will finally be))

TfMinTj JC(TI,T2).2

Summingup:(14:A:a) The goodway (strategy)for 1to play the minorant game

Ti is to chooseTI, belongingto the setA, A beingthe setofthoseTI for which Min Tj 3C(n,T2) assumesits maximum valueMax

Tj Min,t OC(TI,T2).(14:A:b) The good way (strategy) for 2 to play is this:If 1has

chosena definite value of TI,S then T2 shouldbe chosenbelong-ing to the set Br^ BTi being the set of those T2 for which

3C(ri,T2) assumesits minimum value Min Tj 3C(n,T2).4

On the basisof this we can statefurther:

(14:A:c) If both players 1and 2 play the minorant game Fj well,i.e.if TI belongsto A and T2 belongsto BTi then the value ofJC(TI,T2) will beequal to

vi = MaxTj MinTj JC(TI,T2).1Observethat TZ may not be uniquely determined:For a given n the T2-function

3C(ri, T2) may assume its Tj-mmimum for severalvalues of T^ Thevalue of 3C(n, T2)will, however, be the samefor all theseT2,namely the uniquely defined minimum valueMinr t 3C(n,ri). (Cf.13.2.1.)

2For the samereasonas in footnote 1above,the value ofn may not beunique, but thevalue of Min

Tf 3C(ri, ri) is the same for all n in question, namely the uniquely-definedmaximum value

MaxTl Min Tl3C(n r2).8 2 is informed of the value of n when calledupon to make his choiceof r2, this is

the rule of PI. It follows from our conceptof a strategy (cf.4.1.2.and end of 11.1.1.)that at this point a rule must beprovided for 2'schoiceof T2for every value ofn, irre-spectiveof whether 1has played well or not, i.e.whether or not the value chosenbelongsto A.

4 In all, this n is treatedasa known parameter on which everything depends, includ-ing the set Brt from which n ought to be chosen.)))

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102 ZERO-SUMTWO-PERSONGAMES: THEORY

The truth of the above assertionis immediatelyestablishedin the mathe-matical senseby rememberingthe definitions of the setsA and BV and bysubstituting accordinglyin the assertion. We leave this exercisewhichis nothing but the classicaloperationof \" substituting the defining for thedefined\" to the reader. Moreover,the statement ought to be clearbycommon sense.

Theentirediscussionshouldmake it clearthat every play of the gameFi has a definite value for eachplayer. This value is the above Vi for theplayer1and therefore Vi for the player2.

An even more detailedidea of the significanceof Vi is obtained in thisway:

(14:A:d) Player 1can,by playing appropriately,securefor himselfa gain ^ Vi irrespectiveof what player 2 does. Player 2can,by playing appropriately,securefor himself a gain ^ Vi,

irrespectiveof what player 1does.(Proof:The former obtainsby any choiceof TI in A. The latterobtains

by any choiceof T^ in BT .* Again we leave the details to the reader;theyarealtogethertrivial.)

Theabove can be stated equivalently thus:

(14:A:e) Player 2 can,by playing appropriately,make it sure thatthe gain of player 1 is ^ Vi, i.e.prevent him from gaming> Vi irrespectiveof what player1does.Player1can,by play-ing appropriately,make it sure that the gain of player 2 isg Vi, i.e.prevent him from gaining > Vi irrespectiveofwhat player2 does.

14.3.2.We have carriedout the discussionof FI in rather profuse detailalthough the \" solution\"is a rather obvious one. That is, it is very likelythat anybodywith a clearvision of the situation will easilyreachthe sameconclusions \"unmathematically,\" just by exerciseof common sense.Neverthelesswe felt it necessaryto discussthis caseso minutely becauseit is a prototypeof severalothersto follow where the situation will be muchless open to \" unmathematical \" vision. Also, all essential elementsofcomplicationas well as the bases for overcoming them arereally presentin this very simplest case. By seeingtheir respectivepositions clearlyin this case,it will be possibleto visualize them in the subsequent,morecomplicated,ones. And it will be possible,in this way only, to judgepreciselyhow much can be achievedby every particularmeasure.

14.3.3.Let us now considerthe majorant game F2.F2 differs from Fi only in that the rolesof players 1and 2 areinter-

changed:Now player2 must makehis choicer 2 first and then the player 1makeshis choiceofr} in full knowledgeof the value ofr2.

1 Recall that n must be chosenwithout knowledge of r2, while r2 is chosen in full

knowledge of n.)))

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STRICTLYDETERMINEDGAMES 103But in sayingthat F2 arisesfrom FI by interchanging the players1and 2,

it must be rememberedthat theseplayers conservein the processtheirrespectivefunctions 3Ci(ri,r2), aC2(rj, r2), i.e.3C(ri,r 2), 3C(ri,r2). Thatis, 1still desiresto maximize and 2 still desiresto minimize X(TI,r2).

Thesebeingunderstood,we can leave the practicallyliteral repetitionof the considerationsof 14.3.1.to the reader. We confine ourselvestorestating the significant definitions, in the form in which they apply to F2.(14:B:a) Thegoodway (strategy) for 2 to play the majorant game

F2 is to chooser 2 belongingto the set JB, B beingthe setofthoser2 for which Max

Tf 3C(ri,r2) assumesits minimum valueMin Tj Max

Tj X(TI,r 2).(14:B:b) The good way (strategy) for 1 to play is this:If 2 has

chosena definite value of T2,* then TI shouldbechosenbelong-ing to the set A Ti, A T being the set of those r\\ for which

3C(ri,r2) assumesits maximum value MaxTi 3C(ri,r2).2

On the basisof this we can statefurther:

(14:B:c) If both players 1and 2 play the majorant game F2 well,i.e.if r2 belongsto B and TI belongsto A Tj then the value of5C(ri,T2) will be equal to

v2 = Min tt MaxTi 5C(ri,r 2).))

Theentirediscussionshouldmake it clearthat every play of the gameF2 has a definite value for eachplayer. This value is the above v2 for theplayer 1and therefore v 2 for the player2.

In orderto stressthe symmetry of the entirearrangement,we repeat,mutatis mutandis, the considerationswhich concluded14.3.1.They nowserve to give a more detailedidea of the significanceof v2.(14:B:d) Player 1can,by playing appropriately,securefor himself

a gain ^ v2, irrespectiveof what player 2 does. Player 2can, by playing appropriately, securefor himself a gaini v2, irrespectiveof what player 1does.

(Proof:Thelatterobtainsby any choiceof r2 in B. Theformer obtainsby any choiceof TI in A T Cf. with the proof, loc.cit.)

Theabove can again be stated equivalently thus :(14:B:e) Player 2 can, by playing appropriately, make it sure

that the gain of player1is g v2, i.e.prevent him from gaining1 1 is informed of the value ofT*when calledupon to make his choiceofn this is the

rule of ra (Cf.footnote 3 on p. 101).2In all this ra is treated asa known parameter on which everything depends,including

the set A T|from which n ought to be chosen.8 Remember that rj must be chosenwithout any knowledge of n, while n is chosen

with full knowledge of TI.)))

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104 ZERO-SUMTWO-PERSONGAMES: THEORY

> v2, irrespectiveof what player 1does.Player 1can,byplaying appropriately,make it sure that the gain of player 2is ^ v2, i.e.prevent him from gaining > v2, irrespectiveof what player2 does.

14.3.4.The discussionsof Fi and F2, as given in 14.3.1.and 14.3.3.,respectively,are in a relationshipof symmetry or duality to eachother;they obtain from eachother,as was pointedout previously (at the begin-ning of 14.3.3.)by interchanging the rolesof the players1and 2. In itselfneither game FI nor F2 is symmetricwith respectto this interchange;indeed,this is nothing but a restatementof the fact that the interchangeof theplayers1and 2 alsointerchangesthe two gamesFi and F2, and so modifiesboth. It is in harmony with this that the various statements which wemade in 14.3.1.and 14.3.3.concerningthe good strategiesof FI and F2,respectively i.e.(14:A:a),(l4:A:b),(14:B:a),(14:B:b),loc.cit. werenot symmetricwith respectto the players 1and 2 either. Again we see:An interchangeof the players1and 2 interchangesthe pertinentdefinitionsfor Fi and F2, and so modifiesboth.1

It is therefore very remarkablethat the characterizationof the valueof a play (vi for Fi,v2 for F2), as given at the end of 14.3.1.and 14.3.3.i.e.(14:A:c),(14:A:d),(14:A:e),(14:B:c),(14:B:d),(14:B:e),loc.cit.(exceptfor the formulae at the end of (14:A:c)and of (14:B:c))are fully

symmetric with respectto the players 1and 2. According to what wassaid above, this is the same thing as assertingthat thesecharacterizationsarestatedexactlythe sameway for Fi and F2.2 All this is,of course,equallyclearby immediateinspectionof the relevant passages.

Thus we have succeededin defining the value of a play in the samewayfor the gamesFI and F2, and symmetrically for the players 1 and 2:in(14:A:c),(14:A:d),(14:A:e),(14:B:c),(14:B:d)and (14:B:e)in 14.3.1.and in 14.3.3.,this in spite of the fundamental difference of the individualroleof eachplayer in these two games.From this we derive the hopethat the definition of the value of a play may be used in the same form forothergamesas well in particular for the gameF which, as we know,occupiesa middle position between FI and F2. This hope applies, ofcourse,only to the conceptof value itself, but not to the reasoningswhichleadto it;thosewerespecificto Fi and F2, indeeddifferent for Fi and forF2, and altogetherimpracticablefor F itself; i.e.,we expectfor the futuremorefrom (14:A:d),(14:A:e),(14:B:d),(14:B:e)than from (14:A:a),(14:A:b),(14:B:a),(14:B:b).

1Observethat the original game T was symmetric with respectto the two players1 and 2, if we let eachplayer take his function 3Ci(n, r a), 3Cs(n, r2) with him in an inter-change;i.e.the personalmoves of 1and 2 had both the samecharacterin P.

For a narrower concept of symmetry, where the functions 3Ci(n, T), JCs(ri, r) areheld fixed, cf.14.6.

2This point deservescareful consideration :Naturally these two characterizationsmust obtain from eachother by interchanging the rolesof the players 1 and 2. But inthis casethe statements coincidealsodirectly when no interchange of the playersis madeat all. This is due to their individual symmetry.)))

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STRICTLYDETERMINEDGAMES 105Theseare clearly only heuristic indications. Thus far we have not

even attempted the proof that a numerical value of a play can be definedin this manner for F. We shall now beginthe detaileddiscussionby whichthis gap will be filled. It will be seenthat at first definite and seriousdifficultiesseemto limit the applicabilityof this procedure,but that it will

be possibleto remove them by the introduction of a new device(Cf.14.7.1.and 17.1.-17.3.,respectively).

14.4.Conclusions

14.4.1.We have seenthat a perfectly plausibleinterpretation of thevalue of a play determinesthis quantity as

Vi = MaxT( Min Tj OC(ri,r2),v2 = Min Tj MaxTi 3C(ri,r2),

for the gamesFI, F2, respectively,as far as the player 1is concerned.1

Sincethe game FI is lessadvantageousfor 1than the game F2 in FIhe must makehis move prior to, and in full view of, his adversary,while inF2 the situation is reversed it is a reasonableconclusionthat the valueof Fi is lessthan, or equalto (i.e.certainly not greaterthan) the value of F2.Onemay argue whether this is a rigorous\" proof.\" Thequestionwhetherit is, is hard to decide,but at any ratea closeanalysisof the verbal argu-ment involved showsthat it is entirely parallelto the mathematical proofof the samepropositionwhich we alreadypossess.Indeed,the propositionin question,

Vi ^ V 2

coincideswith (13:A*) in 13.4.3.(The tf>, x, y there correspondto our3C,TI, T2.)

Insteadof ascribingVi, v2 as values to two gamesFI and F2 differentfrom F we may alternatively correlatethem with F itself, under suitableassumptionsconcerningthe \" intellect\"of the players1and 2.

Indeed,the rules of the game F prescribethat eachplayer must makehis choice(his personalmove) in ignoranceof the outcomeof the choiceof his adversary. It is neverthelessconceivablethat one of the players,say 2, \" finds out\" his adversary;i.e.,that he has somehowacquired theknowledgeas to what his adversary's strategy is.2 The basis for thisknowledgedoesnot concernus; it may (but neednot) beexperiencefrom

previous plays. At any ratewe assume that the player 2 possessesthisknowledge. It is possible,of course,that in this situation 1will changehis strategy; but again let us assume that, for any reasonwhatever, hedoesnot do it.8 Under theseassumptionswe may then say that player 2has \" found out\" his adversary.

1For player 2 the values are consequently vi, v.*In the game T which is in the normalized form the strategy is just the actual

choiceat the unique personalmove of the player. Remember how this normalized formwas derived from the original extensive form of the game; consequently it appearsthatthis choicecorrespondsequally to the strategy in the original game.

1For an interpretation of all theseassumptions! cf. 17.3.1.)))

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106 ZERO-SUMTWO-PERSONGAMES:THEORY

In this case,conditionsin F becomeexactlythe sameas if thegame wereFi,and henceall discussionsof 14.3.1.apply literally.

Similarly, we may visualize the oppositepossibility,that player 1has\" found out\" hisadversary. Then conditionsin F becomeexactlythe sameas if the gamewere F2; and henceall discussionsof 14.3.3.apply literally.

In the light of the above we can say:Thevalue of a play of the game F is a well-definedquantity if one of the

following two extremeassumptionsis made:Either that player 2 \" findsout\" his adversary,or that player 1 \" finds out\" his adversary. In thefirst casethe value of a play is Vi for 1,and vi for 2;in the secondcasethe value of a play is V2 for 1and V2 for 2.

14.4.2.This discussionshowsthat if the value of a play of F itselfwithout any further qualifications or modifications can be defined at all,then it must lie betweenthe values of Vi and v2. (We mean the valuesfor the player1.) I.e.if we write v for the hoped-forvalue of a play of Fitself (for player1),then theremust be

Vi 5jj V ^ V 2.Thelength of this interval, which is still available for v, is

A = v2 Vi ^ 0.At the same time A expressesthe advantage which is gained (in the

gameF) by \"finding out\" one'sadversaryinstead of being \" found out\"

by him.1Now the game may be suchthat it doesnot matter which player \" finds

out\" his opponent;i.e.,that the advantage involved is zero. Accordingto the above,this is the caseif and only if

A =or equivalently

Vi = V 2

Or, if we replaceVi, v2 by their definitions:Max

TjMinTj 3C(ri,r 2) = Min T2MaxTj JC(TI,r 2).))

If the game F possessestheseproperties,then we call it strictly determined.Thelast form of this conditioncallsfor comparisonwith (13:3)in 13.3.1.

and with the discussionsof 13.4.1.-13.5.2.(The0,x, y thereagain corre-spond to our 3C,7i, r2). Indeed,the statementof (13:B*)in 13.4.3.saysthat the game F is strictly determined if and only if a saddle point of3C(ri,T2) exists.

14.5.Analysis of Strict Determinateness

14.5.1.Let us assumethe game F to be strictly determined;i.e.that asaddlepoint of 3C(ri,rj) exists.

1Observethat this expressionfor the advantage in question appliesfor both players:Theadvantage for the player 1is v vi; for the player 2 it is (~vi) ( v2) and thesetwo expressionsare equal to eachother, i.e.to A.)))

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STRICTLY DETERMINEDGAMES 107

In this caseit is to be hoped consideringthe analysisof 14.4.2.thatit will be possibleto interpret the quantity

v = Vi = v2

as the value of a play of r (for the player 1). Recalling the definitions ofVi, v2 and the definition of the saddlevalue in 13.4.3.and using(13:C*)in13.5.2.,we seethat the above equationmay alsobe written as

v = MaxTi Min Tj 3C(7i,r 2) == Min Tj Max^OC(TI,r 2)= Sar/rj 3C(ri,T2).

By retracingthe stepsmadeat the endof 14.3.1.and at the endof 14.3.3.it is indeednot difficult to establishthat the above can be interpretedasthe value of a play of F (for player 1).

Specifically:(14:A:c),(14:A:d),(14:A:e),(14:B:c),(14:B:d),(14:B:e)of 14.3.1.and 14.3.3.where they apply to FI and F2 respectively,can nowbe obtained for F itself. We restatefirst the equivalent of (14:A:d)and(14:B:d):(14:C:d) Player 1can,by playing appropriately,securefor himself

a gain ^ v, irrespectiveof what player2 does.Player 2 can,by playing appropriately,securefor himself

a gain ^ v irrespectiveof what player 1does.In order to prove this, we form again the set A of (14:A:a)in 14.3.1.

and the setB of (14:B:a)in 14.3.3.Theseareactually the setsA+, B+ of13.5.1.(the <t> correspondsto our 5C). We repeat:(14:D:a) A is thesetof thoseTI for which Min

Ta 3C(ri,r 2) assumesitsmaximum value; i.e.for which

MinTi JC(ri,T2) = Max

TjMin rj 3C(n,r 2) = v.

(14:D:b) B is the setof those r 2 for which MaxTj 3C(ri,r 2) assumesits minimum value; i.e.for which

MaxTi 3C(ri,T2) = Min Tj MaxTi JC(TI,n) = v.

Now the demonstrationof (14:C:d)is easy:Let player 1chooseTI from A. Then irrespectiveof what player 2

does,i.e.for every r 2, we have X(TI,r 2) ^ Min Tj JC(TI,r 2) = v, i.e.,1'sgainis ^ v.

Let player 2 chooser 2 from B. Then,irrespectiveof what player 1does,i.e.for every n,we have JC(n,r 2) ^ Max

Tj 3C(riy T2) = v, i.e.1'sgainis ^ v and so 2*sgain is ^ v.

This completesthe proof.We pass now to the equivalent of (14:A:e)and (14:B:e).Indeed,

(14:C:d)as formulated above can beequivalently formulated thus:(14:C:e) Player 2 can,by playing appropriately,make it sure that

the gain of player1is : v, i.e.prevent him from gaining > virrespectiveof what player 1does.)))

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108 ZERO-SUMTWO-PERSONGAMES: THEORY

Player 1can,by playingappropriately,make it sure thaithe gain of player 2 is ^ v i.e.presenthim from gaining> v irrespectiveof what player 2 does.

(14:C:d)and (14:C:e)establishsatisfactorilyour interpretationof v as thevalue of a play of T for the player 1,and of v for the player2.

14.5.2.We considernow the equivalentsof (14:A:a),(14:A:b),(14:B:a),(14:B:b).

Owing to (14:C:d)in 14.5.1.it is reasonableto define a goodway for 1to play the gameF as onewhich guaranteeshim a gain which is greaterthan or equal to the value of a play for 1,irrespectiveof what 2 does;i.e.achoiceof TI for which X(TI, T2) ^ v for all r2. This may be equivalentlystatedas MinTj 3C(rj,r2) ^ v.

Now we have always MinTf 3C(ri,T2) ^ MaxTi Min Tj 3C(ri,r 2) = v.

Hencethe above conditionfor TI amounts to Min T 3C(ri,T2) = v, i.e.(by (14:D:a)in 14.5.1.)to r\\ beingin A.

Again, by (14:C:d)in 14.5.1.it is reasonableto define the goodway for2 to play the game T as one which guaranteeshim a gain which is greaterthan or equal to the value of a play for 2, irrespectiveof what 1 does;i.e.a choiceof r 2 for which 3C(ri,T2) ^ v for all TI. That is, JC(TI,T2) ^ vfor all TI. This may be equivalently stated as Max

Tj 5C(Ti,TZ) ^ v.Now we have always Max

Tj 3C(Ti, T2) ^ MinTj MaxT OC(TI, T2) = v.Hencethe above conditionsfor T2 amounts to MaxTi OC(TI,T2) = v, i.e.(by(14:D:b)in 14.5.1.)to T2 being in B.

Sowe have:(14:C:a) The good way (strategy) for 1 to play the game F is to

chooseany TI belongingto A, A beingthe setof (14:D:a)in

14.5.1.(14:C:b) The good way (strategy) for 2 to play the gameF is to

chooseany T2 belongingto B, B being the setof (14:D:b)in 14.5.1.1

Finally our definition of the good way of playing, as statedat thebeginningof this section,yields immediately the equivalent of (14:A:c)or (14:B:c):(14:C:c) If both players 1and 2 play the gameF well i.e.if r\\

belongsto A and T2 belongsto B then the value of 3C(n, T2)will beequal to the value of a play (for 1),i.e.to v.

We add the observationthat (13:D*)in 13.5.2.and the remarkconcerningthe setsA, B before(14:D:a),(14:D:b)in 14.5.1.togethergive this:(14:C:f) Both players1and 2 play the gameF well i.e.TI belongs

to A and T2 belongsto B if and only if TI, T2 is a saddlepointof 3C(T,,T2).

1Sincethis is the game F eachplayer must make his choice(of n or r t) withoutknowledge of the other player'schoice(of TI or n). Contrast this with (14:A:b)in14.3.1.for Ti and with (14:B:b)in 14.3.3.fo* Tf .)))

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STRICTLYDETERMINEDGAMES 109

14.6.TheInterchange of Players. Symmetry

14.6.(14:C:a)-(14:C:f)in 14.5.1.and 14.5.2.settleeverything as faras the strictly determined two-person games are concerned.In thisconnectionlet us remark that in 14.3.1.,14.3.3.for Ti, T2 we derived(14:A:d),(14:A:e),(14:B:d),(14:B:e)from (14:A:a),(14:A:b),(14:B:a),(14:B:b)while in 14.5.1.,14.5.2.for T itself we obtained (14:C:a),(14:C:b)from (14:C:d),(14:C:e).This is an advantage sincethe argu-ments of 14.3.1.,14.3.3.in favor of (14:A:a),(14:A:b),(14:B:a),(14:B:b)were of a much more heuristiccharacterthan those of 14.5.1.,14.5.2.infavor of (14:C:d),(14:C:e).

Our useof the function 3C(ri,r 2) = 3Ci(ri,r 2) impliesa certain asymmetryof the arrangement;the player 1is thereby given a specialrole. It oughtto be intuitively clear,however, that equivalent resultswould be obtainedif we gave this specialrole to the player 2 instead. Sinceinterchangingtheplayers1and 2 will play a certain rolelater,we shall neverthelessgive abrief mathematical discussionof this point also.

Interchangingthe players 1and 2 in the game T of which we neednot assumenow that it is strictly determined amounts to replacingthefunctions 5Ci(ri,r2), 3C2(ri, r 2) by 3C2(r2, TI),JCi(r2, ri).1 - 2 Itfollows,there-fore,that this interchangemeansreplacingthe function 3C(r i, r 2) by 3C(r2, r i).

Now the changeof sign has the effect of interchanging the operationsMaxand Min. Consequentlythe quantities

MaxTi

Min Tj OC(T],?2) = Vi,Min

rfMax

Ti 5C(ri,T2) = v2,

as defined in 14.4.1.becomenow

MaxTi Min Tj [-3C(>2, TI)]= -MinTi MaxT|JC(r2, TI)= Mint|MaxTi JC(TI,r 2) 8 = v2.

Min Tj MaxTi [ 3C(T2, TI)]= MaxTj Min Ti OC(r2, TI)= -MaxTi Min Tj JC(TI,T2) 8 = -VL

SoVi, v 2 become v2, Vi. 4 Hencethe value of

A = v 2 - Vi = (-Vi)- (-v2)1 This is no longer the operation of interchanging the players used in 14.3.4.There

we wereonly interested in the arrangement and the state of information at eachmove,and the players 1and 2 wereconsideredas taking their f unctions 3Ci(n, r2) and 3Cj(ri,TI)with them (cf.footnote 1 on p. 104). In this senser was symmetric, i.e.unaffected bythat interchange (id.).

At present we interchange the rolesof the players 1 and 2 completely, even in theirf unctions 3Ci(n, ra) and 3C2(Ti, r2).

*We had to interchange the variables n, rt sincer\\ representsthe choiceof player 1and T2the choiceofplayer 2. Consequently it isnow T which has the domain 1, , 0i.Thus it is again true for 3C*(r2,TI) as it was beforefor3C*(ri, TJ) that the variable beforethe comma has the domain 1, , ft and the variable after the comma, the domain1, , 02-

3This is a merechange of notations: Thevariables TI, r* arechanged around to rs, TI.4This is in harmony with footnote 1 on p. 105,as it should be.)))

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110 ZERO-SUMTWO-PERSONGAMES: THEORY

is unaffected,1 and if F is strictly determined,it remainsso,sincethis prop-erty is equivalent to A = 0. In this casev = Vi = v 2 becomes

V = Vi = ~ V2.

It is now easy to verify that all statements(14:C:a)-(14:C:f)in 14.5.1.,14.5.2.remain the samewhen the players1and 2 areinterchanged.

14.7.Non -strictly Determined Games

14.7.1.All this disposescompletelyof the strictly determinedgames,but of no others. Fora game I which is not strictly determinedwe haveA > i.e.in such a game it involves a positiveadvantage to \"find out\"one'sadversary. Hencethere is an essentialdifference between theresults,i.e.the values in FI and in F2, and therefore alsobetweenthe goodways of playing thesegames.Theconsiderationsof 14.3.1.,14.3.3.providetherefore no guidance for the treatmentof F. Thoseof 14.5.1.,14.5.2.do not apply either,sincethey make use of the existenceof saddle pointsof 3C(ri,r2) and of the validity of))

MaxTj Min Tj JC(71,r 2) = Min Tj MaxTi 3C(ri,r 2),))

i.e.of F beingstrictly determined.Thereis, of course,someplausibilityin the inequality at the beginningof 14.4.2.According to this, the value vof a play of F (for the player 1) if such a conceptcan be formed at all inthis generality,for which we have no evidenceas yet 2 is restrictedby

Vi g V g V2.

But this still leavesan interval of length A = v 2 Vi > open to v;and, besides,the entiresituation is conceptuallymost unsatisfactory.

One might be inclinedto give up altogether:Sincethere is a positiveadvantage in \"finding out\" one'sopponent in such a game F, it seemsplausibleto say that thereis no chanceto find a solution unlessonemakessomedefinite assumptionas to \"who finds out whom,\" and to what extent.3

We shall seein 17.that this is not so,and that in spite of A > a solu-tion can be found along the same lines as before.But we proposefirst,without attackingthat difficulty, to enumeratecertain gamesF with A > 0,and otherswith A = 0. The first which are not strictly determinedwill be dealt with briefly now; their detailedinvestigation will be under-taken in 17.1.The second which arestrictly determined will be ana-lyzed in considerabledetail.

14.7.2.Sincethere existfunctions 3C(ri, r 2) without saddle points (cf.13.4.1.,13.4.2.;the 4>(x, y) there,is our 3C(ri,r 2)) thereexist not strictlydeterminedgamesF. It is worth while to re-examinethoseexamplesi.e.

1This is in harmony with footnote 1 on p. 106,as it should be.2Cf.however, 17.8.1.*In plainer language:A > means that it is not possiblein this game for eachplayer

simultaneously to be clevererthan his opponent. Consequently it seemsdesirabletoknow just how clevereachparticular player is.)))

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STRICTLYDETERMINEDGAMES 111the functions describedby the matrices of Figs.12,13on p.94 in thelight of our presentapplication. That is, to describeexplicitlythe gamesto which they belong. (In eachcase,replace<t>(x, y) by our JC(TI,r a), r 2beingthe column number and n the row number in every matrix. Cf.alsoFig.15on p.99).

Fig.12:This is the game of \"MatchingPennies.\"Let for TI and forT2 1be \"heads\"and 2 be \"tails,\" then the matrix elementhas the value 1if n, r 2 \"match\" i.e.are equal to eachother and 1,if they do not.Soplayer 1\"matches\"player2:Hewins (oneunit) if they \"match\"andhe loses(oneunit), if they do not.

Fig.13:Thisis the game of \"Stone,Paper,Scissors.\"Let fornand forr 2 1be \"stone,\"2 be \"paper,\"and 3 be \"scissors.\"Thedistributionofelements1and 1over the matrix expressesthat \"paper\"defeats\"stone,\"\"scissors\"defeat \"paper,\"\"stone\"defeats \"scissors.\"1 Thus player 1wins (oneunit) if he defeatsplayer2, and he loses(oneunit) if he isdefeated.Otherwise(if both playersmake the samechoice)the gameis tied.

14.7.3.Thesetwo examplesshow the difficulties which we encounterin a not strictly determinedgame, in a particularly clearform; just becauseof their extremesimplicity the difficulty is perfectly isolatedhere,in vitro.The point is that in \"MatchingPennies\"and in \"Stone,Paper,Scissors,\"any way of playing i.e.any TI or any T2 is just as goodas any other:Thereis no intrinsic advantage or disadvantagein \"heads\"or in \"tails\"perse,nor in \"stone,\"\"paper\"or \"scissors\"perse. Theonly thing whichmatters is to guesscorrectlywhat the adversaryis going to do;but how arewe going to describethat without further assumptionsabout the players'\"intellects\"?2

Thereare,of course,more complicatedgames which are not strictlydeterminedand which areimportant from various more subtle, technicalviewpoints (cf. 18.,19.).But as far as the main difficulty is concerned,the simplegamesof \"MatchingPennies\"and of \"Stone,Paper,Scissors\"areperfectly characteristic.

14.8.Program of a DetailedAnalysis of Strict Determinateness

14.8.While the strictly determinedgames T for which our solutionis valid arethus a specialcaseonly, oneshouldnot underestimatethesizeof the territory they cover. The fact that we areusing the normalizedform for the game F may tempt to suchan underestimation:Itmakesthingslookmore elementarythan they really are. Onemust rememberthat theTI, T2 representstrategiesin the extensiveform of the game,which may beof a very complicatedstructure,as mentionedin 14.1.1.

In orderto understand the meaning of strict determinateness,it istherefore necessaryto investigate it in relation to the extensiveform of thegame. Thisbringsup questionsconcerningthe detailednature of the moves,

1\" Papercoversthe stone,scissorscut the paper,stonegrinds the scissors.\"2As mentioned before,weshall show in 17.1.that it canbedone.)))

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112 ZERO-SUMTWO-PERSONGAMES: THEORY

chanceor personal the stateof information of the players,etc.; i.e.wecometo the structural analysisbasedon the extensiveform, as mentionedin 12.1.1.

We areparticularly interestedin thosegamesin which eachplayerwhomakes a personal move is perfectly informed about the outcomeof thechoicesof all anteriormoves. Thesegames were already mentioned in6.4.1.and it was stated therethat they aregenerallyconsideredto beof aparticular rational character.We shall now establish this in a precisesense,by proving that all such gamesarestrictly determined. And thiswill be true not only when all moves arepersonal,but also when chancemoves too arepresent.

15.Gameswith PerfectInformation15.1.Statement of Purpose. Induction

15.1.1.We wish to investigate the zero-sumtwo-persongamessomewhatfurther, with the purpose of finding as wide a subclassamong them aspossible in which only strictly determined gamesoccur;i.e.where thequantities

vi = MaxTj MinTt 3C(ri,r2),v 2 = Min Tj Max

Ti JC(TI,r 2)

of 14.4.1.which turned out to be so important for the appraisal of thegame fulfill

Vi = V 2 = V.

We shall show that when perfect information prevailsin F i.e.whenpreliminarity is equivalent to anteriority (cf. 6.4.1.and the end of 14.8.)then F is strictly determined. We shall also discussthe conceptualsig-nificance of this result (cf. 15.8.).Indeed,we shallobtain this as a specialcaseof a more generalrule concerningvi, v2, (cf. 15.5.3.).

We begin our discussionsin even greatergenerality,by consideringaperfectly unrestrictedgeneraln-person gameT. The greatergeneralitywill be useful in a subsequentinstance.

15.1.2.Let T bea generaln-persongame,given in its extensiveform.We shallconsidercertainaspectsof F, first in our original pre-set-theoreticalterminology of 6.,7.,(cf. 15.1.),and then translateeverything into thepar-tition and set terminology of 9.,10.(cf. 15.2.,et sequ.).The readerwill

probablyobtain a full understandingwith the help of the first discussionalone;and the second,with its ratherformalistic machinery, is only under-taken for the sakeof absolute rigor,in orderto show that we arereallyproceedingstrictly on the basisof our axiomsof 10.1.1.

We considerthe sequenceof all moves in F:SfTli, 3T12, , 3TI,,. Letus fix our attentionon the first move, 9Tli, and the situation which existsat the moment of this move.

Sincenothing is anteriorto this move, nothing is preliminary to iteither;i.e.the characteristicsof this move depend on nothing, they areconstants. Thisappliesin thefirst placeto the fact, whether 9fRi isa chance)))

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GAMES WITH PERFECTINFORMATION 113move or apersonalmove;and in the lattercase,to which player9fTli belongs,i.e.to the value of fci = 0,1, , n respectively,in the senseof 6.2.1.And it appliesalso to the number of alternativesa\\ at 3fTCi and for a chancemove (i.e.when fci = 0)to the values of theprobabilitiespi(l), , p\\(ct\\).Theresult of the choiceat Sflli chanceor personal is a <TI = 1, , a\\.

Now a plausible step suggests itself for the mathematical analysisof the game T, which is entirely in the spirit of the method of \"completeinduction\"widely used in all branches of mathematics. It replaces,ifsuccessful,the analysisof T by the analysisof othergameswhich containone move lessthan T.1 This step consistsin choosinga *i = 1, , cm

and denotingby T9i a game which agreeswith F in every detail exceptthatthe move 9Tli is omitted,and instead the choice<n is dictated(by the rulesof the new game)the value ai = 9\\* T9i has, indeed,onemove lessthanF :Itsmoves are9TC 2, * * , 3TI,.3 And our \" inductive \" step will have beensuccessfulif we can derive the essentialcharacteristicsof F from thoseofall rfi , *i = 1, , i.

15.1.3.It must be noted, however, that the possibilitiesof formingF i aredependent upon a certain restrictionon F. Indeed,every playerwho makesa certainpersonalmove in the gameT9i must be fully informedabout the rulesof thisgame. Now this knowledgeconsistsof the knowledgeof the rulesof the original gameF plus the value of the dictatedchoiceatSfTli, i.e.91. HenceT9i can be formed out of F without modifying the ruleswhich govern the player's stateof information in F only if the outcomeof the choiceat SfTli, by virtue of the original rules F, is known to everyplayerat any personalmove of his 3TC2, , 9TI,;i.e.9fRi must be prelim-inary to all personalmoves 9dl 2, , 311,.We restatethis:(15:A) T9i can be formed without essentially modifying the

structureof F for that purpose only if F possessthe followingproperty:

(15:A:a) SfTli is preliminary to all personalmoves 9Tl2,* , 9Rr. 4

1I.e.have v I instead of v. Repeatedapplication of this \"inductive\" step iffeasibleat all will reducethe game Tto onewith steps;i.e.to oneof fixed, unalterableoutcome. And this means, ofcourse,a completesolution for r. (Cf.(15:C:a)in 15.6.1.)

*E.g.F is the game of Chess,and a\\ a particular opening move i.e.choiceat Dili of\"white,\" i.e.player 1.Then r^ is again Chess,but beginning with a move of the char-acterof the secondmove in ordinary Chess a \"black,\" player 2 and in the positioncreatedby the \"opening move\" a\\. This dictated \"opening move\" may, but neednot,be a conventional one (like E2-E4).

The same operation is exemplified by forms of Tournament Bridge where the\"umpires\" assign the players definite known and previously selected \"hands.\"(Thisis done,e.g.,in DuplicateBridge.)

In the first example, the dictated move 9TCi was originally personal (of \"white,\"player 1);in the secondexample it wasoriginally chance(the \"deal\.

In some games occasionally\"handicaps\" are used which amount to one or moresuch operations.*We should really use the indices1, , v 1 and indicate the dependenceon 9\\\\

e.g.by writing SJTlf1, , 3Hji,. But we prefer the simpler notation 3fRj, , STl*.

4This is the terminology of 6.3.; i.e.we use the specialform of dependencein thesenseof 7.2.1.Using the general description of 7.2.1.we must state (15:A:a)like this:For every personalmove 31k, * 2, , r, the set* contains the function <n.)))

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114 ZERO-SUMTWO-PERSONGAMES:THEORY

15.2.TheExact Condition (First Step)

16.2.1.We now translate 15.1.2.,15.1.3.into the partition and setterminology of 9.,10.,(cf. also the beginning of 15.1.2.).The notationsof10.1.will therefore be used.

diconsistsof the one set 12 ((10:1:f) in 10.1.1.),and it isa subpartitionof (&i ((10:1:a)in 10.1.1.);hence(Bi too consistsof the one set ft (the othersbeingempty) .** That is:

12 for preciselyone fc, say k = fci,for all fc ?^ fci.

This fci = 0,1, , n determinesthe characterof OTijit is the fci of 6.2.1.If fci = 1, , n i.e.if the move is personal then &i is alsoa subpar-tition of a)i(fci),((10:1:d)in 10.1.1.Thiswas only postulatedwithin 5i(fci),but Bi(ki)= 12). HenceSDi(fci) too consistsof the one set 12.3 And forfc 7* fc], the >i(fc) which is a partition in Bi(fc) = ((10:A:g)in 10.1.1.)must beempty.

Sowe have preciselyone AI of Cti, which is 12, and for fci = 1, , n

preciselyone Di in all SDi(fc),which is also 12;while for fci = therearenoDi in all >j(fc).

The move 3TZi consistsof the choiceof a Ci from Ci(fcj);by chanceif

fci = 0;by the player fci if fci = 1, , n. C\\ is automatically a subsetof the unique A\\( = 12) in the former case,and of the unique DI(= 12)in the latter. The number of theseCiis c*i(cf.9.1.5.,particularly footnote 2on p.70); and sincethe AI or DI in questionis fixed, this a\\ is a definiteconstant, ai is the number of alternatives at 'JTli, the a\\ of 6.2.1.and 15.1.2.

TheseCi correspondto the a\\ = 1, , a\\ of 15.1.2.,and we denotethem accordinglyby Ci(l), , Ci(ai).4 Now (10:1:h)in 10.1.1.showsas is readilyverified that Ct 2 is also the setof the Ci(l), , Ci(ai),i.e.equal to d.

Sofar our analysishas been perfectly general, valid for m\\ (and to acertain extentfor ffil 2) of any game F. The readershould translate thesepropertiesinto everyday terminology in the senseof 8.4.2.and 10.4.2.

We pass now to iy. This shouldobtain from F by dictatingthe moveSflli as describedin 15.1.2.by putting <j\\

= 9\\. At the same time themoves of the game arerestrictedto 9fTl2,

' , STC,. This meansthat the

1 This (Bi is an exceptionfrom (8:B:a)in 8.3.1.;cf.the remark concerning this (8:B:a)in footnote 1 on p.63,and alsofootnote 4 on p. 69.

8 Proof:ft belongsto Q,\\, which is a subpartition of (Bij hence ft is a subset of anelement of (Bi. This element is necessarilyequal to ft. All other elements of (Bi arethereforedisjunct from (cf.8.3.1.),i.e.empty.s

Cli, S)i(fci)unlike i (cf.above) must fulfill both (8:B:a),(8:B:b)in 8.3.1.;henceboth have no further elementsbesidesft.

4They representthe alternatives Cti(l), , Cli(ai) of 6.2.and 9.1.4.,9.1.5.)))

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GAMES WITH PERFECTINFORMATION 115elementw which representsthe actual play can no longervary over allft, but is restrictedto Ci(ffi). And the partitionsenumeratedin 9.2.1.arerestrictedto thosewith * = 2, , v,

1 (and K- v + 1for Ct).

15.2.2.We now cometo the equivalent of the restrictionof 15.1.3.The possibilityof carrying out the changesformulated at the end of

15.2.1.is dependentupon a certain restrictionon F.As indicated,we wish to restricttheplay i.e.v within Ci(fri). There-

fore all thosesetswhich figured in the descriptionof F and which weresubsetsof Q, must be madeover into subsetsof C}(9\\) and the partitionsinto partitions within Ci(ffi) (or within subsets of Ci(frj)). Howis this tobe done?

Thepartitions which makeup the descriptionsof F (cf.9.2.1.)fall intotwo classes:thosewhich representobjectivefacts the Ctc, the (B = (S(0),J3,(l), , BK(n)) and the C<(fc),k = 0, 1, , n and thosewhich

representonly the player'sstateof information, 2 the ><(&),k = 1, ,n.We assume,of courseK ^ 2 (cf. the end of 15.2.1.).

In the first class of partitions we needonly replaceeachelementbyits intersectionwith Ci(frj). Thus (B* is modified by replacingits ele-ments B,(0),B,(l), , BK(n) by C,(*,)n B.(0),Ci(*i)n B.(l), ,Ci(fri) n BK(ri). In Ct< even this is not necessary:It is a subpartition of&2 (sinceK *t 2, cf. 10.4.1.),i.e.of the system of pairwisedisjunct sets(Ci(l), , Ci(i))(cf. 15.2.1.);hencewe keeponly thoseelementsof a,which aresubsets of Ci(fri), i.e.that part of ft* which lies in Ci(fri). TheQ*(k) shouldbe treatedlike (B, but we prefer to postponethis discussion.

In the secondclassof partitions i.e.for the )*(&) we cannot do any-thing like it. Replacingthe elementsof 3)(fc) by their intersectionswith

Ci(fri) would involve a modification of a player'sstateof information3 andshould therefore be avoided. The only permissibleprocedurewould bethat which was feasiblein the caseof CtK:replacementof >(&) by thatpart of itself which liesin Ci(fri). But this is applicableonly if 33<(fc) like& before is a subpartitionof ($2 (for K ^ 2). Sowe must postulatethis.

Now e(fc) takescareof itself:It is a subpartition of ^> K(k) ((10:1:c)in 10.1.1.),henceof dz (by the above assumption);and so we can replaceit by that part of itself which liesin Ci(fri).

So we see:The necessaryrestrictionof F is that every D^fc) (withK ^ 2) must be a subpartition of ($2. Recallnow the interpretation of8.4.2.and of (10:A:d*),(10:A:g*)in 10.1.2.They give to this restrictionthe meaning that every player at a personalmove 9n2, , 9TC r is fully

1We do not wish to change the enumeration to K = 1, , v 1,cf. footnote 3on p. 113.

2a representsthe umpire's state of information, but this is an objectivefact: theevents up to that moment have determined the courseof the play preciselyto thatextent (cf.9.1.2.).

3Namely, giving him additional information.)))

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116 ZERO-SUMTWO-PERSONGAMES:THEORY

informed about the stateof things after the move SfTli (i.e.beforethe move9Hlj) expressedby &2. (Cf. also the discussionbefore (10:B)in 10.4.2.)That is, 3Tli must bepreliminary to all moves Sfll 2, , 3TI,.

Thus we have again obtained the condition (15:A:a)of 15.1.3.Weleave to the readerthe simpleverification that the gameT9i fulfills therequirementsof 10.1.1.

15.3.TheExact Condition (Entire Induction)

15.3.1.As indicatedat the end of 15.1.2.,we wish to obtain the char-acteristicsof T from thoseof all F^, fri = 1, , i, sincethis if suc-cessful would be a typicalstep of a \"completeinduction.\"

Forthe moment, however, the only classof gamesfor which we possessany kind of (mathematical) characteristicsconsistsof the zero-sumtwo-persongames:for thesewe have the quantitiesVi, v 2 (cf.15.1.1.).Let ustherefore assumethat F is a zero-sum two-persongame.

Now we shallseethat the Vi, v2 of F can indeedbe expressedwith thehelp of thoseof the r

fj , 9i = 1, , i (cf. 15.1.2.).This circumstancemakes it desirable to push the \"induction\"further, to its conclusion:i.e.,to form in the same way I\\,s, F^.^,^, , r it s 9,.1 Thepoint is that the number of steps in thesegames decreasessuccessivelyfrom v (for F),v - 1(for F^),over v - 2, v - 3, to (for rfi ,fi ,);i.e.F*

jf f9, is a \"vacuous\" game (like the one mentioned in the

footnote 2 on p.76). Thereareno moves; the player k gets the fixedamount $*(fri, -,#).

This is the terminology of 15.1.2.,15.1.3.,i.e.of 6.,7. In that of15.2.1.,15.2.2.i.e.of 9.,10.we would say that ft (for F) is graduallyrestrictedto a Ci(fri) ofa2 (for rf|),a C2(fri, fr 2) of ft 8 (for F^,^),a C8(fri, fr 2, 9*)of a4 (for rf

iff

f ,f|), etc.,etc.,and finally to a C,(*i,fr 2, , 9,)of a,+i(forrt |(f t 9,). And this last set has a unique element((10:1:g)in 10.1.1.),say it. Hencethe outcomeof the game r

f 9, is fixed:Theplayerk

getsthe fixed amount $*(*).Consequently,the nature of the gameF^,^ ,9y is trivially clear;

it is clearwhat this game'svalue is for every player. Therefore the processwhich leads from the F^ to F if established canbe used to work back-wards from T9l .9t 9, to F,ir ,a

9V _ t to r

f|ffi 9,_t etc.,etc.,toF,i,,i to T9l and finally to F.

But this is feasibleonly if we areable to form all gamesof the sequencerV T'i'*i> *X.it. $ ,

' ' '>rfi ,f t 9V, i.e.if the final condition of 15.1.3.

or 15.2.2.is fulfilled for all thesegames.This requirementmay again beformulated for any generaln-persongameF;so we return now to thoseF.

16.3.2.The requirementthen is, in the terminology of 15.1.2.,15.1.3.(i.e.of 6.,7.) that 3Tli must be preliminary to all 9fR 2, SflZa, , 9ffl,; that

1, , on; 1 -1, , at where a> *af (*i); *i -1, , ai where*t); etc.,etc.)))

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GAMES WITH PERFECTINFORMATION 1173TC2must be preliminary to all STls, 3TZ 4, , 3fTC,;etc.,etc.;i.e.that pre-liminarity must coincidewith anteriority.

In the terminology of 15.2.1.,15.2.2.i.e.of 9.,10.of coursethe sameis obtained:All >K(k),K ^ 2 must be subpartitionsof a2; all SD(fc),* ^ 3must be subpartitionsof Ct$, etc.,etc.;i.e.all 5)(fc) must besubpartitionsof a\\ if K ^ X. 1 SinceOt< is a subpartitionof Q,\\ in any case(cf.10.4.1.),itsuffices to require that all >(&) be subpartitionsof a. However a, is asubpartition of >(/0 within &K(k) ((10:1:d) in 10.1.1.);consequentlyourrequirementis equivalent to saying that )K(k) is that part of &K which liesinB(fe).2 By (10:B)in 10.4.2.this means preciselythat preliminarity andanteriority coincidein F.

By all theseconsiderationswe have establishedthis:(15:B) In orderto beable to form the entiresequenceof games(15:1) F, ly, rf itf

f> F^,,^,' * '>r j, t *9

ofv, v - 1,v -2, ,

moves respectively,it is necessaryand sufficientthat in thegameF preliminarity and anteriority shouldcoincide,i.e.that per-fect information shouldprevail. (Cf.6.4.1.and the endof 14.8.)

If F is a zero-sum two-persongame,then this permits theelucidationof F by going through the sequence(15:1)back-wards from the trivial game F^,^, ... , r to the significantgame F performing eachstep with the helpof the devicewhichleadsfrom the T9i to F as will beshown in 15.6.2.

15.4.Exact Discussionof the Inductive Step15.4.1.We now proceedto carry out the announcedstep from the T

9g9

to F, the \"inductive step.\" F needtherefore fullfill only the final conditionof 15.1.3.or 15.2.2.,but it must be a zero-sumtwo-persongame.

Hencewecan form all F , <r i = 1, , a \\ ,andthey alsoarezero-sumtwo-persongames.We denotethe two players'strategiesin F by Sj, , S?1

and 2'2, , 2?;and the \"mathematical expectation\"of the outcomeofthe play for the two players,if the strategiesSJi,2j areused,by

3Ci(n,rt) sOC(TI,TI), 3Ci(T,,TI) s -3C(r!,T2)

(cf. 11.2.3.and 14.1.1.).We denotethe correspondingquantities in F,,by z;t/1> , S^r and s;i/2, , zj^1

, and if the strategiesS^1

,

S^/2*areused,by

flCfj/lOlVj/lj Te^/l) s30^(^/1,T^/j), ^Crj/aCTcj/l, T^/j) S JC^^/i,T^/|).1We stated this abovefor X = 2, 3, ; for X = 1 it is automatically true:every

partition is a subpartition of Oti sinceCti consistsof the oneset ((10:1:f)in 10.1.1.).1For the motivation if one is wanted cf.the argument of footnote 3 on p.63.1From now on we write <n, **, , <r* instead of 9\\, ft, , S9 becauseno mis-

understandings will bepossible.)))

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118 ZERO-SUMTWO-PERSONGAMES: THEORY

We form the vi, v2 of 14.4.1.for T and for F^ denotingthem in the lattercaseby v^/i, v,i/2. So

Vi = MaxT|MinT|3C(ri,7 2),v 2 = Min

TfMaxTi3C(n, r 2),))

and))

v, /2))

Our aim is to expressthe vi, v2 in terms of the vr /i, v^/j.Thefcj of 15.1.2.,15.2.1.which determinethe characterof the move 3Ej

will play an essentialrole. Sincen = 2,its possiblevalues arek\\ = 0,1,2.We must considerthesethreealternatives separately.

15.4.2.Considerfirst the casek\\ = 0;i.e.let 3Tli bea chancemove. Theprobabilitiesof its alternatives <TI =!, , i arethe pi(l), , PI(I)mentionedin 15.1.2.(pi(<n) isthepi(Ci)of (10:A:h)in 10.1.1.withCi = Ci(<7i)in 15.2.1.).

Now a strategy of player 1in T, S^, consistsobviously in specifyinga

strategy of player 1 in F^, S^// for every value of the chancevariable

ai = 1, , i,1i.e.,the 2Icorrespondto the aggregates21/7, , Z^1//

for all possiblecombinationsTI/I, , r ai /i.Similarlya strategy of player2 in F, 2J consistsin specifyinga strategy

of player2 in F^, 2^/2 for every value of the chancevariable <TI = 1, ,

ij i.e.the 22 correspondto the aggregates2V/ 2, ' * ' , ^a\"1^ for all possi-

ble combinationsTI/Z, , raj /2.

Now the \" mathematical expectations\"of the outcomesin F and in Fff|

areconnectedby the obvious formula

i

JC(TI,T2) = Jr t-l

Thereforeour formula for vi gives

Vi = MaxTi Min Tj3C(n,?2)

= Maxvi.....',/,Min v,)) r,-l

The<ri-term of the sum V on the extremeright-hand side))

1This is clearintuitively. The readermay verify it from the formal istic point ofview by applying the definitions of 11.1.1.and (11:A)in 11.1.3.to the situation describedin 15.2.1.)))

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GAMES WITH PERFECTINFORMATION 119containsonly the two variablesr^/i, T,i/2. Thus the variable pairs

TI/I,Ti/ 2; ; Tai /i, rttj /2

occurseparately,in the separateo-i-terms

Hencein forming the Min Vj Ta ,, we can minimize each<ri-term sep-arately, and in forming the MaxVi r<x /v we can again maximize each<ri-term separately. Accordingly, our expressionbecomes

<*t <*i

pi(cri)MaxT<Vi

Min rvt JCri (Tr|/i, rV2) = pi(n)vri /i.

Thus we have shown

(15:2) Vi = piOrOv^/i.

If the positionsof Max and Min are interchanged,then literally thesameargument yields

(15:3) v 2 = pi(cri)v<rj/2.

15.4.3.The caseof fci = 1 comesnext,and in this casewe shallhaveto makeuseof the resultof 13.5.3.Consideringthe highly formal characterof this result,it seemsdesirableto bring it somewhatnearerto the reader'simagination by showing that it is the formal statementof an intuitivelyplausiblefact concerninggames.This will also make it clearerwhy thisresult must play a roleat this particularjuncture.

Theinterpretationwhich wearenow going to give to the resultof 13.5.3.is based on our considerationsof 14.2.-14.5.particularly those of 14.5.1.,14.5.2.and for this reasonwe couldnot proposeit in 13.5.3.

Forthis purposewe shall considera zero-sumtwo-persongameF in itsnormalized form (cf.14.1.1.)and alsoits minorant and majorant gamesFi,F2 (cf.14.2.).

If we decidedto treatthe normalized form of F as if it were an extensiveform, and introduced strategiesetc. with the aim of reachinga (new)normalized form by the procedureof 11.2.2.,11.2.3.then nothing wouldhappen, as describedin 11.3.and particularly in footnote 1on p.84. Thesituationis different, however, for the majorant and minorant gamesFI,Fa;thesearenot given in the normalized form, as mentionedin footnotes 2 and 3on p.100.Consequentlyit is appropriateand necessaryto bringthem intotheir normalized forms which we are yet to find by the procedureof11.2.2.,11.2.3.)))

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120 ZERO-SUMTWO-PERSONGAMES:THEORY

Sincecompletesolutionsof FI,F2 were found in 14.3.1.,14.3.3.,it is tobeexpectedthat they will turn out to be strictly determined.1

It sufficesto considerFI (cf. the beginning of 14.3.4.),and this we now

proceedto do.We use the notations TI, r 2, 3C(n,r 2) and vi, v2 for F and we denotethe

correspondingconceptsfor FI by r(, r2, 3C'(r i> r i) and vi, v2.A strategy of player 1 in FI consists in specifying a (fixed) value

TI(=1, , 0i) while a strategyof plajrer2 in FI consistsin specifyingavalue of r2(= 1, , 2) dependingon TIfor every value of TI(=!,-,0i).2 Soit is a function of TI:TI = 32(ri).

Thus T'J is TI, while TZ correspondsto the functions 32, andSC'^i,rj) to3C(ri,32(ri)). Accordingly))

Tj Min 3|3C(Ti,32(rj)),v a

= Min3f

MaxT|3C(Ti,32(ri)).

Hencethe assertion that FI is strictly determined,i.e.the validity ofvi = vi coincidespreciselywith (13:E)in 13.5.3.; thereweneedonly replacethe X, U, /(*), ifr(x, f(x))by our n,T2, 32(r,),JC(TI,32(ri)).

This equivalenceof the result of 13.5.3.to the strictly determinedcharacterof Fi makes intelligible why 13.5.3.will play an essentialrolein the discussionwhich follows. FI is a very simpleexampleof a gameinwhich perfectinformation prevails, and thesearethe gameswhich aretheultimate goal of our presentdiscussions(cf. the end of 15.3.2.).And thefirst move in Fi is preciselyof the kind which is coming up for discussionnow:It is a personalmove of player1, i.e.ki = 1.

15.5.Exact Discussionof the Inductive Step(Continuation)

15.6.1.Considernow the casefci = 1;i.e.let 3Tli be a personalmoveof the player1.

A strategy of player1in F, 2{iconsistsobviously in specifyinga (fixed)

value <rj(= 1, , on) and a (fixed)strategyof player1 in F,,Zj1/^

1 8; i.e.the Z5> correspondto the pairs<rj, T^/I.

1This is merely a heuristic argument, sincethe principles on which the \"solutions\"of 14.3.1.,14.3.3.arebasedarenot entirely the sameasthoseby which wedisposedof thestrictly determined casein 14.5.1.,14.5.2.,although the former principles werea steppingstoneto the latter. It is true that the argument could bemade pretty convincing by an\"unmathematical,\" purely verbal, amplification. We prefer to settle the mattermathematically, the reasonsbeing the sameasgiven in a similar situation in 14.3.2.

*This is clearintuitively. The readermay verify it from the formalistic point ofview, by reformulating the definition of r } as given in 14.2.in the partition and setterminology, and then applying the definitions of 11.1.1.and (11:A)in 11.1.3.

Theessentialfact is, at any rate, that in Ti the personalmove of player 1 is pre-liminary to that of player 2.

1Cf.footnote 1 on p.118or footnote 2 above.)))

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GAMES WITH PERFECTINFORMATION 121A strategy of player2 in F, 2^, on the otherhand, consistsin specifying

a strategy of player 2 in IV, 2^'/2f

, for every value of the variable <rj=

1, , ai.1 Sor^/2isa function of <rJ:r^ /2 = 3s(<7j);i.e.theS5 correspondto the functions 32 and clearly))

Therefore our formula for Vi gives:vi = Max,;.TrJ/1 Min^OC^r,;,!,32(or))

= Maxr<r?/i Max,jMin^JC^j/i,32(<rJ)).

Now

Max,,;Min^JC^Ovj/i,3s(<rJ)) = Max,;Min^SC,;^,;/!,r,;/i)owing to (13:G)in 13.5.3.;therewe needonly replacethex,u, f(x), \\l/(x, u)by our erj, r,;/2, 32(cr?),3C,;(T,j/i,rff j/2).2 Consequently))

vi = MaxT<rJ/l

Maxff ;MinTr,/t JC^r,;/!,T,;/t )= Max^jMax

T<r0/iMinv/t JCr j(r rj/i, rcj/t)

= Max,;vr j/i

And our formula for v2 gives:8

v2 = Min 3j Max^o/^ajCraJ/i,= Min 3j Max,;Max

rr;/1JC,;(rrj/Now

Min 3l Max^j Maxrr!/IJC^j/i,3t (aJ))= Max,jMin 3j Max

r<r;/l ^jKj/i,32(<r5))

= Maxaj MinTff;/1 Max^o/.X^^j/i,Trj/,)

owing to (13:E)and (13:G)in 13.5.3.;therewe needonly replacethe x,u,f(x),\\fr(x, u) by our a\\, r,;/2, 32(<r?), Maxv/iacr j(rrj/i,r^/).4 Consequently

v2 =))

Summingup (and writing <n insteadof crj) :1Cf.footnote 1 on p. 118or footnote 2 on p. 120.1

TrJ/i must be treated in this caseas a constant.This stepis of coursea rather trivial one, cf.the argument loc.cit.

8 In contrast to 15.4.2.,there is now an essentialdifference between the treatments ofvi and vj.

4rrj/i is killed in this caseby the operation Maxr a0/1.This stepis not trivial. It makes use of (13:E)in 13.5.3.,i.e.of the essentialresult

of that paragraph, as statedin 15.4.3.)))

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122 ZERO-SUMTWO-PERSONGAMES:THEORY

(15:4) vi = Max,,v^/i,(15:5) v2 = Max^ v, i/2 .))

16.6.2.Considerfinally the casefci = 2;i.e.let SfTli be a personalmove ofplayer2.

Interchangingplayers1and 2carriesthis into the precedingcase(fci = 1).As discussedin 14.6.,this interchangereplacesVi, v2 by v2, Vi

and henceequallyv^/i, v, i/2 by v^/j, v^/i. Substitutingthesechangesinto the above formulae (15:4),(15:5),it becomesclearthat theseformulaemust bemodified only by replacingMax in them by Min. Sowe have:

(15:6) vi = Min^ v, t/ i,(15:7) v2 = Min<r i vri /i.

16.6.3.We may sum up the formulae (16:2)-(15:7)of 15.4.2.,15.5.1.,15.5.2.,as follows:

For all functions /(<TI) of the variable <TI(= 1, , i) define threeoperationsMjj, fci = 0,1,2 as follows:

for fci = 0,(15:8)

for))

for fci = 2.Then

v* = Mk\\\\9i/k for fc = 1,2.

We wish to emphasizesome simple facts concerningtheseoperationsM\".\\.

First,M*\\ kills the variable a\\ ; i.e.M9\\f(ai) no longerdependson <n.Forfci = 1,2 i.e.for Max^, Min^ this was pointedout in 13.2.3.Forfci = it is obvious;and this operationis, by the way, analogousto theintegral used as an illustration in footnote 2 on p.91.

Second,Mk

\\ dependsexplicitlyon the gameF. This is evident sincefci occursin it and <r\\ has the range 1, , a\\. But a further dependenceis due to the use of the pi(l), , PI(I),in the caseof fci = 0.

Third, the dependenceof v* on v ff /k is the same for fc = 1,2 for eachvalue of fci.

We concludeby observingthat it would have beeneasy to make these

formulae involving the average ] PI(^I)/(^I)for a chancemove, the

maximum for a personalmove of the first player, and the minimum forone of his opponent plausible by a purely verbal (unmathematical)argument. Itseemedneverthelessnecessaryto give an exactmathematicaltreatmentin orderto do full justiceto the precisepositionof Vi and of v2.A purely verbal argument attempting this would unavoidably becomesoinvolved if not obscure as to beof little value.)))

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GAMES WITH PERFECTINFORMATION 123

16.6.TheResult in the Caseof PerfectInformation

15.6.1.We return now to the situation describedat the end of 15.3.2.and make all the hypothesesmentioned there;i.e.we assumethat perfectinformation prevailsin the game F and alsothat it is a zero-sum two-persongame. Theschemeindicatedloc.cit.,togetherwith the formula (15:8)of15.5.3.which takescareof the \"inductive\"step,enableus to determinetheessentialpropertiesof F.

We prove first without going any further into details that such a Fis always strictly determined.We do this by \" completeinduction \" with

respectto the length v of the game (cf.15.1.2.).This consistsof provingtwo things :(15:C:a) That this is true for all gamesof minimum length;i.e.for

v = 0.(15:C:b) That if it is true for all gamesof length v 1,for a given

v = 1,2, , then it is also true for all gamesof length v.

Proof of (15:C:a):If the length v is zero,then thegamehas no moves atall;it consistsof paying fixedamounts to the players1,2, say theamountsw, w.1 Hence0i= 2 = 1 so TI = TZ = 1,3C(n,r 2) = w,2 and so

vi = v2 = w;

i.e.F is strictly determined,and its v = w.3

Proof of (15:C:b):Let F be of length v. Then every F,t is of lengthv 1;henceby assumptionevery F^ is strictly determined.Thereforev9i/i ss v^/2. Now the formula (15:8)of 15.5.3.shows4 that Vi = v t .HenceF is alsostrictly determinedand the proof is completed.

15.6.2.We shallnow go more into detailand determinetheVi = V2 = vof F explicitly. Forthis wedonot even needthe above resultof 15.6.1.

We form, as at the end of 15.3.2.,the sequenceof games(15:9) r, rv r,rv - ,r,it,......,/of the respectivelengths

v, v - 1,v - 2, , 0.Denotethe Vi, v2 of thesegamesby))

1Cf. the game in footnote 2 on p. 76 or F^,^.....i v in 15.3.1.In the partition and

set terminology: For v = (10:1:f), (10:1:g) in 10.1.1.show that 12has only one ele-ment, say TT:ft = (TT). Sow = $I(TT), w = 5i(ir) play the role indicated above.

8 I.e.eachplayer has only onestrategy, which consistsof doing nothing.*This is, of course,rather obvious. Theessentialstepis U5:C:b).4I.e.the fact mentioned at the end of 15.5.3.,that the formula is the samefor A; = 1,2

for eachvalue of k\\.5Cf.footnote 3 on p. 117.)))

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124 ZERO-SUMTWO-PERSONGAMES: THEORY

Let us apply (15:8)of 15.5.3.for the \" inductive \" step describedatthe end qf 15.3.2.;i.e.letus replacethe <TI,T, T^of 15.5.3.by<r,r,t ^_if

T9i ,K_itt ,Kfor each * =!,,v. The ki of 15.5.3.then refers

to the first move of T0i OK j i.e.to the move 311*in F. It is thereforeconvenient to denoteit by fc(*i, * * ' , ow). (Cf. 7.2.1.)Accordinglywe form the operationMk*

K

(ffl '\"-1',replacingthe Mk

\\of 15.5.3.In this

way we obtain

(15:10) v., w * = M*,f\" '' ' '^ v, t ,K/ * for fc = 1,2.Considernow the last elementof the sequence(15:9),thegame F^ v

This falls under the discussionof (15:C:a)in 15.6.1.;it has no moves at all.Denoteits unique play1 by if =

ff(<ri, , <? v). Henceits fixed w 2 isequal to (Fi(((ri, , <r,)). Sowe have:

(15:11) v^ %/i = v^ ,,/t= ffi(ir(<ri, , <r,)).

Now apply (15:10)with K = v to (15:11)and then to the result,with

K =s v lt- ,2,1successively. In this manner

(15:12)v, = v2 = v = Mk9 \\M

k^ M**9* \"'*Si(*(ffi, , O).obtains.

This provesoncemore that T is strictly determined,and also gives anexplicitformula for its value.

15.7.Application to Chess15.7.1.The allusionsof 6.4.1.and the assertionsof 14.8.concerning

thosezero-sumtwo-persongames in which preliminarity and anterioritycoincide i.e.where perfect information prevails arenow established.Wereferredthereto the generalopinion that thesegamesareof a particularlyrational character,and wehave now given this vague view a precisemeaningby showingthat the gamesIn questionarestrictly determined.And wehave also shown a fact much less founded on any \" generalopinion\"that this is also true when the gamecontainschancemoves.

Examplesof gameswith perfectinformation were alreadygiven in 6.4.1.:Chess(without chancemoves) and Backgammon(with chancemoves).Thus we have establishedfor all thesegames the existenceof a definitevalue (of a play) and of definite beststrategies.But we have establishedtheirexistenceonly in the abstract,while our methodfor their constructionis in most casestoo lengthy for effective use.3

In this connectionit is worth while to considerChess in a little moredetail.

1Cf.the remarks concerning r^ ^ in 15.3.1.1Cf.(15:C:a)in 15.6.1.,particularly footnote 1 on p. 123.8 This is due mainly to the enormous value of v. For Chess,cf.the pertinent part of

footnote 3on p.59. (Thev* there is our K, cf.the end of 7.2.3.))))

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GAMES WITH PERFECTINFORMATION 125Theoutcomeof a play in Chess i.e.every value of the functions SF* of

6.2.2.or 9.2.4.is restrictedto thenumbers1,0, I.1 Thus the functions9* of 11.2.2.have the samevalues, and sincethereareno chancemoves inChess,the sameis true for the function OC*of 11.2.3.2 In what follows weshallusethe function 3C= 3Ciof 14.1.1.

Sincerehas only the values, 1,0, 1,the number

(15:13) v = MaxTi Mm Tj3C(n,r 2) = MinTj MaxTj3C(r,,r2)

has necessarilyoneof thesevalues

v- 1,0,-1.We leave to the readerthe simplediscussionthat (15:13)meansthis:(15:D:a) If v = 1then player1(\" white \") possessesa strategy with

which he \"wins,\" irrespectiveof what player 2 (\"black\does.

(15:D:b) If v = then both players possessa strategy with whicheach one can \"tie\" (and possibly \"win\,") irrespectiveofwhat the otherplayerdoes.

(15:D:c) If v = 1then player 2 (\"black\")possessesa strategywith which he \"wins,\" irrespectiveof what player1(\"white\does.8

15.7.2.This showsthat if the theory of Chesswere really fully knowntherewould be nothing left to play. Thetheory would show which of thethreepossibilities(15:D:a),(15:D:b),(15:D:c)actually holds,and accord-ingly the play would be decidedbeforeit starts:Thedecisionwould be incase(15:D:a)for \"white,\" in case(15:D:b)for a \"tie,\" in case(15:D:c)for \"black.\"

But our proof, which guaranteesthe validity of one (and only one)of thesethreealternatives,gives no practicallyusablemethodto determinethe true one. This relative, human difficulty necessitatesthe use of thoseincomplete,heuristicmethodsof playing, which constitute \"good\"Chess;and without it therewould be no elementof \"struggle\"and \"surprise\"inthat game.

1This is the simplest way to interpret a \"win,\" \"tie,\" or \"loss\" of a play by theplayer k.

*Every value of 9* is one of ff *;every value of 3C* in the absenceofchancemoves isone of 9*> ^ l c- cit. If there were chancemoves, then the value of 3C*would be theprobability of a \"win\" minus that of a \"loss,\" i.e.a number which may lie anywherebetween 1 and 1.

8 When there are chancemoves, then 3C(ri, r2) is the excessprobability of a \"win\"over a \"loss,\" cf. footnote 2 above. The players try to maximize or to minimize thisnumber, and the sharp trichotomy of (15:D:a)(15:D:c)abovedoesnot, in general,obtain.

Although Backgammon isa game in which completeinformation prevails, and whichcontains chancemoves, it is not a goodexamplefor the abovepossibility; Backgammonis played for varying payments, and not for simple \"win,\" \"tie\" or \"loss,\" i.e.thevalues of the $*arenot restrictedto the numbers 1,0, 1.)))

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126 ZERO-SUMTWO-PERSONGAMES: THEORY

15.8.TheAlternative, Verbal Discussion

16.8.1.We concludethis chapterby an alternative, simpler,lessformaisticapproachto our main result, that all zero-sumtwo-persongames,iwhich perfectinformation prevails,arestrictly determined.

It canbe questionedwhether the argumentation which follows is realla proof;i.e.,we prefer to formulate it as a plausibilityargument by whichvalue can be ascribedto eachplayof any gameF of the above type, but thi

is still opento criticism.It is not necessaryto show in detailhow thoscriticismscan be invalidated,sincewe obtain the samevalue v of a play cT as in 15.4.-15.6.,and there we gave an absolutely rigorous procusing preciselydefined concepts.The value of the presentplausibilitargument is that it is easierto grasp and that it may berepeatedfor othegames,in which perfect information prevails,which arenot subjectto th

zero-sumtwo-personrestriction.The point which we wish to bring onis that the samecriticismsapply in the generalcasetoo,and that they cano longerbe invalidated there. Indeed,the solution there will be foun

(even in gameswhere perfect information prevails)along entirely differenlines. This will make clearerthe nature of the difference between th

zero-sum two-person caseand the generalcase. That will be ratheimportant for the justification of the fundamentally different methodwhich will* have to be used for the treatment of the generalcas(cf.24.).

16.8.2.Considera zero-sumtwo-persongameF in which perfectinformation prevails. We use the notations of 15.6.2.in all respects:For thSTCi,3TC2, , 311,;the<n, cr 2, , a,]the fci, fc 2(<n), , k,(<n, **, , <r,_0the probabilities;the operatorsMh

.\\, Af*;(<ri> , , M*?'1'9'.....''-''jth

sequence(15:9)ofgamesderivedfrom F;and the function &I(*(<TI, - - ,0-,))We proceedto discuss the gameF by startingwith the last move 3M

and then goingbackwardfrom therethrough the moves 9fTC,_i, 3TC,_2,Assumefirst that thechoices<TI,o-2, , <r -\\ (of themoves 9Tli, 3fE 2, '3fll,_i) have already beenmade,and that the choice<r, (of the move 9TC,)inow to be made.

If 9TC, is a chancemove, i.e.if k,(<n, 0-2, , <r,_i) = 0,then v v wil

have the values 1,2, , a,,(<ri, , v v-\\) with therespectiveprobabilitiesp,(l),p*(2), ,p9(cL 9((r\\, , o-,_i)). Sothemathematicalexpectation of the final payment (for the player 1) 9ri(*(o-i, , <r,_i, <7,)) i))

*-If 3TI, is a personalmove of players 1or2, i.e.if k 9 (<r\\, , <r v-\\) =

or 2,then that player canbe expectedto maximize or to minimize$i(*(<n,* * * , <r,-i,<r*)) by his choiceof <r,; i.e.the outcomeMax,,2Fi(lf(<ri, , <r,-i,<r,)) or Min^ $i(ff(<ri, , <r,_i, <r,)), respectively is to be expected.)))

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GAMES WITH PERFECTINFORMATION 127

I.e.,the outcome to beexpectedfor the play after the -choices<TI,v-\\ have beenmade is at any rate))

Assume next that only the choices<n, , ar_2 (of the moves, 9TC,_i)have beenmadeand that the choice<r,_i (of the move9K,_i)

is now to be made.Sincea definite choiceof <r,_i entails, as we have seen,the outcome

Afjj(r|.....*'~l)

&i(*(<ri, . , 0v)) which is a function of <n, , a,-ionly, sincethe operationMjv (ffl .....'\"** kills cr, we can proceedas above.We needonly replaceF; <n, ,*,;Mk.*'1.....'\"-l} friOKn, , or,)) byv- 1;en, , a,_i;Mj;:;('>.....\"-'> Atf'/'.....\"-> *,(*(, , a,)).Consequentlythe outcometo be expectedfor the play after the choices<n,

* * , cr,_2 have beenmade is))

Similarlythe outcometo be expectedfor the play after the choices<n, , <r,_s have beenmade is))

Finally, the outcometo be expectedfor the play outright before ithas begun is))

And this is preciselythe v of (15:12)in 15.6.2.1

15.8.3.The objectionagainst the procedureof 15.8.2.is that thisapproach to the \"value\" of a play of T presupposes\"rational\"behaviorof all players;i.e.player 1'sstrategy is based upon the assumptionthatplayer2'sstrategy is optimal and vice-versa.

Specifically:Assume fc,_i(<n, , <r,_2) = 1,k,(*i, , <r,_i) = 2.Then player 1,whose personalmove is 9H v_i chooseshis <r,_i in the convic-tion that player 2, whosepersonalmove is 9fR, chooseshis <r, \"rationally.\"Indeed,this is his soleexcusefor assumingthat his choiceof <r v-\\ entailsthe outcomeMin,,SiOr(en, ,O),i.e.M^'1.....'\"-'^i^^i, ,*,)),of the play. (Cf. the discussionof 3TC,_i in 15.8.2.)

1 In imagining the application of this procedureto any specificgame it must beremem-beredthat we assume the length v of T to be fixed. If v is actually variable and it isso in most games (cf. footnote 3 on p. 58) then we must first make it constant, bythe deviceof adding \"dummy moves\" to T as describedat the end of 7.2.3.It is onlyafter this has beendone that the aboveregression through 3TC,,9M*-i, , 9Ri becomesfeasible.

For practical construction this procedureis of courseno better than that of 15.4.-15.6.

Possibly some very simple games, like Tit-tat-toe,could be effectively treated ineither manner.)))

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128 ZERO-SUMTWO-PERSONGAMES: THEORY

Now in the secondpart of 4.1.2.we cameto the conclusionthat thehypothesisof \" rationality \" in othersmust be avoided. Theargumentationof 15.8.2.did not meetthis requirement.

Itis possibleto arguethat in a zero-sumtwo-persongamethe rationalityof the opponentcan be assumed,becausethe irrationality of his opponentcan never harm a player. Indeed,sincethereareonly two players andsincethe sum is zero,every losswhich the opponent irrationally inflictsupon himself, necessarilycausesan equal gain to the otherplayer.1 As it

stands, this argument is far from complete,but it couldbe elaboratedcon-siderably. However,we do not needto be concernedwith its stringency:We have the proof of 15.4.-15.6.which is not open to thesecriticisms.2

But the above discussionis probably neverthelesssignificant for anessentialaspectof this matter. We shall seehow it affects the modifiedconditionsin the more generalcase not subjectto the zero-sumtwo-personrestriction referredto at the end of 15.8.1.

16.Linearity and Convexity16.1.GeometricalBackground

16.1.1.The task which confronts us next is that of finding a solutionwhich comprisesall zero-sumtwo-persongames, i.e.which meetsthedifficulties of the non-strictlydeterminedcase. We shallsucceedin doingthis with the help of the same ideaswith which we masteredthe strictlydeterminedcase:It will appear that they can be extendedso as to cover allzero-sumtwo-persongames. In orderto do this we shall have to makeuse of certainpossibilitiesof probability theory (cf. 17.1.,17.2.).Andit will benecessaryto use somemathematicaldeviceswhich arenot quitethe usual ones. Our analysisof 13.providesone part of the tools;for theremainderit will be most convenient to fall back on the mathematico-geometricaltheory of linearity and convexity. Two theoremson convexbodies3 will beparticularlysignificant.

Forthesereasonswe arenow going to discuss to the extentto whichthey areneeded the conceptsof linearity and convexity.

16.1.2.It is not necessaryfor us to analyze in a fundamental way thenotion of n-dimensionallinear (Euclidean)space. All we needto say isthat this spaceis describedby n numerical coordinates.Accordingly wedefine for eachn = 1,2, , the n-dimensionallinear spaceLn as thesetof all n-upletsof realnumbers [x\\, - - ,xn }. Thesen-upletscan alsobelookeduponas functions i of thevariable i,with the domain (1, , n)

1 This is not necessarilytrue if the sum is not constantly zero,or if there are morethan two players. For details cf.20.1.,24.2.2.,58.3.

Cf. in this respectparticularly (14:D:a),(14:D:b),(14:C:d),(14:C:e)in 14.5.1.and (14:C:a),(H:C:b)in 14.5.2.

1Cf. T. Bonessenand W. Fenchel:Theorieder konvexen Korper, in ErgebnissederMathematik und ihrer Grenzgebiete,Vol. III/l,Berlin 1934. Further investigations inH.Weyl: Elementare Theoriederkonvexen Polyeder. Commentarii Mathematici JGEelve-

tici, Vol. VII, 1935,pp.290-306.)))

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LINEARITY AND CONVEXITY 129in the senseof 13.1.2.,13.1.3.1 We shall in conformity with generalusage call ian indexand not a variable;but this doesnot alterthe natureof the case. In particularwe have

{xi, , xn }= jtfi, , y n ]

if and only if x = yi for all i = 1, , n (cf.the end of 13.1.3.).Onecouldeven take the view that Ln is thesimplestpossiblespaceof (numerical)functions, where the domain is a fixed finite set the set(1, , n).2

We shall alsocall thesen-uplets or functions of Ln pointsor vectorsofLn and write

(16:1) \"? = {xi,- - ,x*\\.The Xi for the specifici = 1, , n the values of the function x arethe componentsof the vector x.

16.1.3.We mention although this is not essentialfor our further workthat Ln is not an abstract Euclideanspace,but one in which a frame of

reference (systemof coordinates)has already beenchosen.3 This is dueto the possibilityof specifyingthe origin and the coordinatevectorsof Ln

numerically (cf. below) but we do not proposeto dwell upon this aspectof the matter.

Thezerovector or origin of Ln is

0*={0, - - ,0}.Then coordinatevectorsof Ln arethe

7= {0, ,1, ,0)= {!,, ,6n,l j = 1, , n,

wherefor i = j,4-5for i ? j.

After thesepreliminarieswe can now describethe fundamental oper-ationsand propertiesof vectorsin Ln.

16.2.Vector Operations16.2.1.The main operations involving vectors are those of scalar

>

multiplication, i.e.the multiplication of a vector x by a number t, and of1I.e.the rMiplets [x\\, , xn \\ are not merely sets in the senseof 8.2.1.The

effective enumeration of the x, by means of the index i = 1, , n is just as essentialasthe aggregateof their values. Cf.the similar situation in footnote 4on p.69.

1Much in modern analysis tends to corroboratethis attitude.8 This at leastis the orthodox geometrical standpoint.4Thus the zero vector has all components 0, while the coordinatevectors have all

components but one that onecomponent being 1,and its index j for thej-th coordinatevector.

6 in is the \"symbol of Kroneckerand Weierstrass,\" which is quite useful in manyrespects.)))

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130 ZERO-SUMTWO-PERSONGAMES: THEORY

vector addition, i.e.additionof two vectors. Thetwo operationsaredefinedby the correspondingoperations,i.e.multiplication and addition, on thecomponentsof the vector in question. Moreprecisely:Scalarmultiplication:t{x\\,

- - , xn }= [txi, , tx n}.

Vector addition:{zi, , xn } + [yi, - , y n ] = [xi+ t/i, , xn + y n \\.

The algebraof theseoperationsis so simpleand obvious that we foregoits discussion.We note,however, that they permit the expressionof any

vector x = [x\\,- - , xn ] with the help of its componentsand the coordi-

natevectorsof Ln))

Someimportant subsetsof Ln :(16:A:a) Considera (linear,inhomogeneous)equation

n

(16:2:a) a >x>= li-i

(ai, , an, b areconstants). We excludeai = = a =

sincein that casethere would be no equation at all. All

points(vectors)x = {zi, , xn \\which fulfill thisequation,

form a hyperplane.2

(16:A:b) Given a hyperplanen

(16:2:a) X a'*'= b>-i

it definestwo parts of Ln. It cuts Ln into thesetwo parts:n

(16:2:b) J) OM > 6,t-iand

(16:2:c) ^ a lxi < b.t-i

Thesearethe two half-spacesproducedby the hyperplane.>

n1 Thex,arenumbers, and hencethey actmx } 8 'asscalarmultipliers. 2^ isa vector

y-isummation.

*For n 3, i.e.in ordinary (3-dimensional Euclidean) space,these are just theordinary (2-dimensional) planes. In our generalcasethey arethe ((n 1)-dimensional)analogues;hencethe name.)))

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LINEARITY AND CONVEXITY)) 131))

Observethat if we replacea i, , an , b by ai, , a n , 6, thenthe hyperplane(16:2:a)remains unaffected, but the two half-spaces(16:2:b),(16:2:c)areinterchanged.Hencewe may always assumea half spacetobe given in the form (16:2:b).

> >

(16:A:c) Given two points(vectors)x , y andaJ^ Owithl t ^ 0;then the center of gravity of x, y with the respectiveweights

I, 1 t in the senseof mechanics is / x + (1- t) y.

Theequations>

X = {Xl, , Xn ), y = (2/1,' , 2/n),

*7+ (1-07= Itei+ (1- Ol/i, - - , ten + (1- t)y n ]

shouldmakethis amply clear.A subset,C,of Ln which containsall centersof gravity of all its points

i.e.which containswith x, y all x + (1 f) y , Q t 1 is convex.Thereaderwill note that for n = 2,

3 i.e.in the ordinary planeor spacethis is the customary conceptof con-vexity. Indeed,the set of all pointst x + (1 y , ^ t ^ 1 ispreciselythe linear (straight) interval connect-

ing the points x and y , the interval

[ x , y]. And so a convex set is one

which, with any two of its points x ,x

interval i* i

> Figure 16.y , alsocontainstheir interval [ x , y ].Figure16showsthe conditionsfor n = 2, i.e.in the plane.16.2.2.Clearly the intersectionof any number of convex setsis again

> >

convex. Henceif any number of points (vectors)x ', , x p is given,thereexistsa smallestconvex set containing them all:the intersectionof

all convexsetswhich contain x ', , x p. Thisis the convexset.spanned>

by x',,x p. It is again useful to visualize the casen = 2 (plane).Cf. Fig.17,where p = 6. It is easy to verify that this setconsistsof allpoints (vectors)))

(16:2:d))) (/ *''forally 0, , tp ^ with)) /, =1.))

Proof:The points (16:2:d)form a set containing all x ',x 'is such a point:put tj = 1and all otherti 0.)))

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132 ZERO-SUMTWO-PERSONGAMES: THEORY

> * >

The points(16:2:d)form a convexset:If x = 2}J/ * ; and y =

pthen *z + (1- y =

t*v x 'with w/ = **,-+(1- O*/-y-i

Any convex set,D, containing a;',,z p containsalso all \"pointsof (16:2:d):We prove this by induction for all p = 1,2, .

Proof:Forp = 1it is obvious;sincethen ti = 1and so x ' is the onlypoint of (16:2:d).

P-IAssume that it is true for p 1. Considerp itself. If 2) <; = then

jj = = <p_1 = 0,the point of (16:2:d)is x p and thus belongsto D. If))

Shaded area: Convex spanned by 7-,....,?Figure 17.))

p-i))P-I p-l p

2 *, >0,then put * = 5) *,,sol-J^JJ^-S^5\" 8 ^. HenceO<^1.y-i y-i y-i y-i

P-IPut 5, = <7/< for j = 1, , p 1. So 2) Sj? = 1. Hence,by our

y-iP-I _

assumptionfor p -1,% 8jX > is in D. D is convex, hencey-i))

y-i))

is also in D;but this vector is equal to))

which thus belongsto Z>.))

5;y-i)))

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LINEARITY AND CONVEXITY)) 133))

Theproof is therefore completed.The h, , tp of (16:2:d)may themselvesbe viewed as the com-

ponents of a vector t = {t\\, , tp ] inLp. It is therefore appropriateto give a name to the setto which they arerestricted,defined by

ti ^ 0, - - , tp 0,))and))

r,-txit))

- 1.))

Zt-axis))

Figure 18.

n ?))

Figure 19.))

Figure 20. Figure 21.We shall denoteit by Sp. It is also convenient to give a name to the setwhich is describedby the first line of conditionsabove alone,i.e.by t\\ ^ 0,

* , tp ^ 0. We shalldenoteit by Pp. Both setsSp, PP areconvex.Let us picture the casesp = 2 (plane)and p = 3 (space). P2 is the

positivequadrant, the areabetweenthe positivex\\ and x* axes(Figure18).P8 is the positiveoctant, the spacebetweenthe positivex\\, x* and x8 axes,i.e.betweenthe planequadrants limitedby the pairs x\\, #2; x\\, x8 ; x2, x8 ofthese(Fig.19). /S2 isa linear interval crossingP2 (Figure18).58 isa planetriangle,likewisecrossingP8 (Figure19).It is useful to draw Si,Sisep-)))

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134 ZERO-SUMTWO-PERSONGAMES: THEORY

arately,without the Pa, P8 (or even the La,L8) into which they arenaturallyimmersed(Figures20,21).We have indicatedon thesefigures thosedis-tanceswhich areproportionalto Xi, Xa or x\\, x2, x8, respectively.

(We re-emphasize:Thedistancesmarkedx\\, x2, x8 in Figures20, 21arenot the coordinatesx\\, xa, x8 themselves.Theselie in La or L8 outsideofSa or S8, and therefore cannot be pictured in Szor83} but they areeasilyseento be proportionalto thosecoordinates.)

16.2.3.Another important notion is the length of a vector. Thelengthof x = {xi, , xn } is))

1*1= A' t-i))

Thedistanceof two points (vectors)is the length of their difference:))

Thus the length of x is the distancefrom the origin .l

16.3.TheTheoremof the Supporting Hyperplanes

16.3.We shall now establishan important generalproperty of convexsets:(16:B) Let p vectors x l , - - - , x p be given. Then a vector y

>

either belongs to the convex C spanned by x',,x p (cf.(16:A:c)in 16.2.1.),or thereexistsa hyperplanewhich contains

y (cf. (16:2:a)in 16.2.1.)such that all of C is containedinone half-spaceproduced by that hyperplane (say (16:2:b)in16.2.1.;cf.(16:A:b)id.).

This is true even if the convexspannedby x ', , x p is replacedbyany convex set. In this form it is a fundamental tool in the moderntheoryof convex sets.

A picturein the casen = 2 (plane)follows:Figure22 uses the convexset C of Figure17 (which is spannedby a finite number of points, as inthe assertionabove),while Figure23 showsa generalconvex setC.2

Beforeproving (16:B),we observethat the secondalternative clearlyexcludesthe first, since y belongsto the hyperplane,hencenot to the halfspace. (I.e.it fulfills (16:2:a)and not (16:2:b)in (16:A:b)above.)

We now give the proof:Proof:Assumethat y doesnot belongto C. Thenconsidera point of

C which liesas nearto y as possible, i.e.for which1TheEuclidean Pythagorean meaning of thesenotions is immediate.*For the readerwho is familiar with topology, we add: To be exact,this sentence

should be qualified the statement is meant for closedconvex sets. This guaranteesthe existenceof the minimum that we use in the proof that follows. Regarding these)))

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LINEARITY AND CONVEXITY

|7-7l2 = ( -ViY))

135))

assumesits minimum value.))

Thehyperplane))

t-l))

Thehalf space))

Figure 22.))

Thehyperplane))

Thehalf space))

Figure 23.

Considerany otherpoints u of C. Thenfor every t with ^ t ^ 1,tu + (I t) z also belongsto the convex C. By virtue of the minimum

property of z (cf.above)this necessitates))

i.e.))

i.e.))

- y)+t(u -))

By elementaryalgebrathis means

2 <*)) ^ 0.))t-l)))

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136 ZERO-SUMTWO-PERSONGAMES: THEORY

Sofor t > (but of courset ^ 1)even

2 5) (ft ~ Vi)(Ui - ft) + t (Ui - z<)*t * 0.t-i t-i

n

If t convergesto0,then the left-hand sideconvergesto 2 J) (ft ~~&

Hence

(16:3) (*- y,)(t*< - ft) ^ 0.

As w, yi = (wt- Zi) + (ft ~ 2/.),this means))

ft).))

-i

2.))

t'-l)) t-1))

Now 2; 7^ y (as z belongsto C, but y doesnot);hence\\

z y |2 > 0.Sothe left-hand sideabove is >0. I.e.))

(16:4))) (ft - t/)2/.))-i))

Put a< = y, then ai))

Figure 24.))

= = an = is excludedby z j* y (cf.n

above). Put alsob = a^,. Thus))

(16:2*)))

defines a hyperplane, to which y clearlybe-1 ;s. Next))

<z, >))(16:2:b*)))

is a half spaceproducedby this hyperplane,and (16:4)statespreciselythat u belongsto this half space.

Sinceu wasan arbitrary elementof C this completesthe proof.This algebraicproof canalsobe statedin the geometricallanguage.Letus do this for the casen = 2 (plane)first. Thesituationis pictured

in Figure24:z isapointofCwhich isasnearto thegiven point y aspossible;i.e.for which thedistanceof y and z , |z y\\ assumesits minimum value.)))

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LINEARITY AND CONVEXITY 137_* > _+

Sincey , 2 arefixed,and w isa variable point (ofC),therefore (16:3)defineda hyperplaneand one of the half spacesproducedby it. And it is easyto verify that z belongsto this hyperplane,and that it consistsof those

points u for which the angle formed by the threepoints is a right-angle

(i.e.for which the vectors z y and u z areorthogonal).Thismeans,))

Interval (x ,u J

Part of the interval nearer to y than *))

^Hyperplane of (1 3)

Figure 25.))

Hyperplane of (16:4)

Hyperplane of (16:3)))

Figure 26.n

indeed,that JJ (zi yi)(ui z>) = 0. Clearly all of C must lie on this-ihyperplane,or on that side of it which is away from y . If any point u

of C did lie on the y side,then somepointsof the interval [ z , u ] would be

nearerto y than z is. (Cf.Figure25. Thecomputation on pp.135-136>

properly interpreted showspreciselythis.) SinceC contains z and u ,

andsoall of [ z , u],this would contradictthe statement that z is as near

to y as possiblein C.Now our passagefrom (16:3)to (16:4)amounts to a parallelshift of

this hyperplanefrom z to y (parallel,becausethe coefficientsa =y)))

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138 ZERO-SUMTWO-PERSONGAMES: THEORY

of Uiy i = 1, , n areunaltered). Now y lies on the hyperplane,andall of C in onehalf-spaceproducedby it (Figure26).

Thecasen = 3 (space)couldbe visualized in a similar way.It is even possibleto accountfor a generaln in this geometricalmanner.

If the readercan persuadehimself that he possessesn-dimensional\"geo-metrical intuition \" he may acceptthe above as a proof which is equallyvalid in n dimensions. It is even possibleto avoid this by arguing asfollows:Whatever n, the entireproof deals with only threepoints at once,namely y , z , u . Now it is always possibleto lay a (2-dimensional)planethrough three given points. If we consideronly the situation in thisplane,then Figures24-26and the associatedargument can beusedwithout

any re-interpretation.Bethis as it may, the purely algebraicproof given above is absolutely

rigorousat any rate. We gave the geometricalanalogiesmainly in thehope that they may facilitate the understandingof the algebraicoperationsperformedin that proof.

16.4.TheTheoremof the Alternative for Matrices16.4.1.The theorem (16:B)of 16.3.permitsan inference which will be

fundamental for our subsequentwork.We start by consideringa rectangularmatrix in the senseof 13.3.3.

with n rowsand m columns,and thematrix elementa(t,j). (Cf.Figure11in 13.3.3.The <, z, y, t, s there correspondto our a, i,j, n, m.) I.e.a(t,j) is a perfectly arbitrary function of the two variablesi = 1, , n\\

j 1, , m. Nextwe form certainvectorsin Ln :Foreachj'= 1, ,

m the vector x 1 = {x\\, , x'n ] with x{ = a(i,j) and for eachI = 1,, n the coordinatevector 5 l = {Bli}. (Cf. for the latter the end of

16.1.3.;we have replacedthe j there by our 1.) Let us now apply the

theorem(16:B)of 16.3.for p = n + m to thesen + m vectors x',,x m , d',,6 n. (They replacethe x',,x p loc.cit.) We put

7=7.>

Theconvex Cspannedby x',,x m, 6',,6 n may contain 0.If this is the case,then we can concludefrom (16:2:d)in 16.2.2.that

*/7;>-+i>7<=K;- 1 f-1with

(16:5) *! ^ 0, , tm ^ 0,81 ^ 0, - , sn ^ 0.(16:6) t,: + Sl =

1.)))

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LINEARITY AND CONVEXITY 139<!,* , Jm, i, * ' , sn replacethe t\\ t , tp (loc.cit.).In termsof thecomponentsthis means

m n

2) *x*> j) + 2}*/** = o.y=i j-i

Thesecondterm on the left-hand sideis equal to s,so we write

(16:7) a(i,j)ti=-,.y-i

m

If we had % tj = 0,then ti = = tm = 0,henceby (16:7)sl = =y-i

m

sn = 0, thus contradicting(16:6).Hence2) */ > - We replace(16:7)y-i

by its corollary

(16:8) aftJH-SO.))

2^ ^ for j = 1, , m. Thenwe have 2}*/ = 1j-i y-i

(16:5)gives x\\ ^ 0, , xm ^ 0. Hence

(16:9) x = jxi, , xm \\ belongsto Sm

and (16:8)givesm

U6:10) 2 *(*'/)*/ for f = 1, , n.))

Consider,on the otherhand, the possibilitythat C doesnot contain .Then the theorem (16:B)of 16.3.permits us to infer the existenceof a

hyperplane which contains y (cf. (16:2:a)in 16.2.1.),such that all of Cis containedin one half-spaceproducedby that hyperplane(cf.(16:2:b)in16.2.1.).Denotethis hyperplaneby

n

2}a tXi = 6.t-i

Since belongsto it, therefore 6 = 0. Sothe half spacein questionisn

(16:11) o,^> 0.)))

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140 ZERO-SUMTWO-PERSONGAMES: THEORY

x',*,x m, d ',*,6 n belongto this half space. Statingthis for- n

5 ', (16:11)becomes ot$,j >0,i.e.ai >0. Sowe have-i(16:12) ai > 0, , an >0.

Statingit for x J , (16:11)becomesn

(16:13) J) a(t,j)a<>0.i-i

/nn

2) 0* for i = 1, , n. Then we have ti\\ = 1and-i t-i(16:12)gives wi >0, , w n >0. Hence

(16:14) w = {wi, , w n \\ belongsto Sn .And (16:13)gives

m

(16:15) a(it j)Wi ^ for j = 1, , m.% - i

Summingup (16:9),(16:10),(16:14),(16:15),we may state:(16:C) Let a rectangularmatrix with n rows and m columns be

given. Denoteits matrix elementby a(i,j),i = 1, , n;

j = 1, , m. Then there existseither a vector x =[xi, - - - , xm ] inSm with

m

(16:16:a) a(t,j>,^0 for i = 1, , n,/-iora vectorw = {101, , w n ] in <S n with

n

(16:16:b) a(i,j)w< > for j = 1, , m.-iWe observefurther:Thetwo alternatives(16:16:a),(16:16:b)excludeeachother.Proof:Assumeboth (16:16:a)and (16:16:b).Multiplyeach(16:16:a)by

n m

Wi and sum over i = 1, ,n;this gives % a(i,j)w&j^ 0. Multiply)))

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LINEARITY AND CONVEXITY 141each(16:16:b)by x, and sum over j = 1, , w; this gives))

t-iy-iThus we have a contradiction.

16.4.2.We replacethe matrix a(i,j) by its negative transposedmatrix;i.e.we denotethe columns (andnot, as before,the rows)by i = 1, , nand the rows (and not, as before, the columns)by j = 1, , m. Andwe let the matrix elementbe a(t, j) (and not, as beforea(i,j)). (Thusn, m too areinterchanged.)

We restatenow the final resultsof 16.4.1.as appliedto this new matrix.But in formulating them, we let x ' = {x\\, , x'm \\ play the rolewhich

w = [wi, , w n ] had, and w' = {w(, - - , w'n \\the rolewhich x =*

[x\\y- , m } had. And we announce the result in terms of the original

matrix.Then we have:

(16:D) Let a rectangularmatrix with n rows and m columnsbegiven. Denoteits matrix elementby a(i,j),i = 1, , n;

j = 1, , m. Then there existseither a vector x' =(x'j, , x'J in Sm with

m

(16:17:a) a(i,j)x'; < for t = 1,- , n,y-i

or a vector w' = {w(, - - - , u^) in Sn with

n

(16:17:b) & JX ^0 f r))i-lAnd the two alternatives excludeeachother.

16.4.3.We now combinethe resultsof 16.4.1.and 16.4.2.They implythat we must have (16:17:a),or (16:16:b),or (16:16:a)and (16:17:b)simultaneously;and also that thesethreepossibilitiesexcludeeachother.

Usingthe samematrix a(t,j) but writing x , w , x\\ w 'for the vectors

x', w y x , w' in 16.4.1.,16.4.2.we obtain this:1 > and not only < 0. Indeed, would necessitatex\\ = xm * which

m

it impossible since J) *, 1.)))

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142 ZERO-SUMTWO-PERSONGAMES:THEORY

(16:E) Thereexistseither a vector x = \\x\\,- - - , zm j in Sm with

m

(16:18:a) a(t,;>,< for t = 1, , n,>-i

or a vector w = [wi, - , w n j in Sn with

n

(16:18:b) a(i,j)wi > for j = 1, , m,;= i

ortwovectorsx'= [x(,- - ,x'm \\ inSm and w f = \\w' lt- ,u/n j

in & with))

for i = 1, , n,

(16:18:c) '\"'2) a(i,j)^ ^0 for j = 1, , m.t-i

Thethreealternatives (16:18:a),(16:8:b),(16:8:c)excludeeachother.

By combining (16:18:a)and (16:18:c)on onehand and (16:18:b)and(16:18:c)on the other,we getthis simplerbut weakerstatement.1 '2

(16:F) Thereexistseithera vector x = [xi, - - - , xm \\in Sm with

m

(16:19:a) % a(i,j)xf ^0 for i = 1, , n,y-i

or a vector w = {wi, - - , wn \\in Sn with

n

(16:19:b) J) a(ij)w, 0 for j = 1, , m.i-i16.4.4.Considernow a skew symmetric matrix a(i, j), i.e.one which

coincideswith its negative transposedin the senseof 16.4.2.; i.e.n = m and

ofej) = -aQ', for i,j = 1, , n.

1Thetwo alternatives (16:19:a),(16:19:b)do not excludeeachother:Their conjunc-tion is precisely(16:18:c).

8 This result could alsohave beenobtained directly from the final result of 16.4.1.:(16:19:a)is precisely (16:16:a)there, and (16:19:b)is a weakened form of (16:16:b)there. We gave the abovemore detaileddiscussion becauseit gives a better insightinto the entire situation.)))

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MIXED STRATEGIES.THESOLUTION 143

Then the conditions(16:19:a)and (16:19:b)in 16.4.3.expressthe samething:Indeed,(16:19:b)is

n

2) a(i,j)w< 0;

this may be written

n n

n

We needonly write jf, i for i,j l so that this becomes a(i,j)Wj ^ 0,and

._ ^ ^ n

then x for 10,l so that we have V a(i,j)xjg 0. And this is precisely))

Therefore we can replacethe disjunctionof (16:19:a)and (16:19:b)by eitherone of them, say by (16:19:b).So we obtain:

(16:G) If the matrix a(i,j) is skew-symmetric(and therefore n = m

cf. above), then there existsa vector w = {wi, , w n } in

Sn with))

2) a(i,j)wi 0 for j = 1, , n.t-i

17.MixedStrategies.TheSolutionfor All Games17.1.Discussionof Two Elementary Examples

17.1.1.Inorderto overcomethe difficultiesin the non-strictlydeterminedcase which we observedparticularly in 14.7.it is best to reconsiderthesimplestexamplesof this phenomenon. Thesearethe gamesof MatchingPenniesand of Stone, Paper,Scissors(cf. 14.7.2.,14.7.3.).Since anempirical,common-senseattitude with respectto the \"problems\"of thesegamesexists,we may hope to get a cluefor the solutionof non-strictlydetermined(zero-sumtwo-person)gamesby observingand analyzing theseattitudes.

It was pointed out that, e.g.in Matching Pennies,no particular wayof playing i.e.neither playing \"heads\"nor playing \"tails\" is anybetterthan the other,and all that matters is to find out the opponent'sinten-tions. This seemsto block the way to a solution,sincethe rules of the

gamein questionexplicitlybar eachplayer from the knowledgeabout theopponent'sactions,at the moment when he has to make his choice.But

1 Observethat now, with n = m this is only a change in notation !)))

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144 ZEROSUM TWO-PERSONGAMES: THEORY

the above observationdoesnot correspondfully to the realitiesof the case:In playing Matching Penniesagainst an at leastmoderately intelligentopponent,the playerwill not attempt to find out the opponent'sintentionsbut will concentrateon avoiding having his own intentionsfound out, byplaying irregularly \"heads\"and \" tails\" in successivegames.Sincewewish to describethestrategy in one play indeedwe must discussthe coursein oneplay and not that of a sequenceof successiveplays it is preferableto expressthis as follows:Theplayer'sstrategy consistsneither of playing\"tails\"nor of playing \"heads,\"but of playing \"tails\"with the probabilityof i and \"heads\"with the probabilityof .

17.1.2.One might imagine that in orderto play MatchingPenniesin arational way the player will before his choicein each play decidebysome 50:50chancedevice whether to play \"heads\"or \"tails.\"1 Thepoint is that this procedureprotectshim from loss.Indeed,whateverstrategy the opponent follows, the player's expectationfor the outcomeof the play will be zero.2 This is true in particular if with certainty theopponentplays \"tails,\"and also if with certainty he plays \"heads\";andalso,finally, if he like the player himself may play both \"heads\"and\"tails,\"with certainprobabilities.3

Thus, if we permit a player in MatchingPenniesto use a \"statistical\"strategy, i.e.to \"mix\" the possibleways of playing with certain proba-bilities(chosenby him), then he can protecthimself againstloss.Indeed,we specifiedabove such a statistical strategy with which he cannot lose,irrespectiveof what hisopponentdoes.Thesameis true for the opponent,i.e.the opponentcan use a statistical strategy which prevents the playerfrom winning, irrespectiveof what the playerdoes.4

Thereaderwill observethe greatsimilarity of this with the discussionsof14.5.5 In the spirit of thosediscussionsit seemslegitimate to considerzero as the value of a play of MatchingPenniesand the 50:50 statisticalmixture of \"heads\"and \"tails\"as a goodstrategy.

The situation in Paper,Stone,Scissorsis entirely similar. Commonsensewill tell that the goodway of playing is to play all threealternativeswith the probabilitiesof|each.6 Thevalue of a play as well as the inter-

1E.g.he could throw a die of coursewithout letting the opponent seethe resultand then play \"tails\" if the number of spotsshowing is even,and \"heads\" if that num-ber is odd.

1I.e.his probability of winning equals his probability of losing, becauseunder theseconditions the probability of matching as well as that of not matching will be J, what-ever the opponent'sconduct.

3Say p, I p. For the player himself we used the probabilities J, J.4All this, of course,in the statistical sense:that the player cannot lose,means that

his probability of losing is his probability of winning. That he cannot win, meansthat the former is 2 to the latter. Actually eachplay will bewon or lost,sinceMatchingPenniesknows no ties.

We mean specifically (14:C:d),(14:C:e)in 14.5.1.A chancedevicecould be introduced as before. Thediementioned in footnote 1,

above,would be a possibleone. E.g.the player could decide\"stone\" if 1 or 2 spotsshow, \"paper\" is 3 or 4 spotsshow, \"scissors\" if 5 or 6 show.)))

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MIXEDSTRATEGIES.THESOLUTION 145

pretation of the above strategy as a goodonecan be motivated as before,again in the senseof the quotation there,1

17.2.Generalization of This Viewpoint

17.2.1.It is plausibleto try to extendthe results found for MatchingPenniesand Stone,Paper,Scissorsto all zero-sumtwo-persongames.

We use the normalized form, the possiblechoicesof the two playersbeingn = 1, , Pi and r 2 = 1, , 2, and the outcomefor player13C(ri,r2), as formerly. We makeno assumptionof strictdeterminateness.

Let us now try to repeatthe procedurewhich was successfulin 17.1.; i.e.letus again visualize playerswhose\"theory\"of the gameconsistsnot inthe choiceof definite strategiesbut rather in the choiceof severalstrategieswith definite probabilities.2 Thus player 1 will not choosea numberTI = 1, , pi i.e.the correspondingstrategy 2^ but 0i numbers

i, , ifljthe probabilitiesof thesestrategiesSi, , 2J?, respec-

tively. Equally player 2 will not choosea number TJ = 1, , 2 i.e.the correspondingstrategy S but /9 2 numbers iji, , ifo s the proba-bilitiesof thesestrategies2'2, * ' *

> 2|,respectively. Sincetheseprob-abilitiesbelongto disjointbut exhaustive alternatives,the numbers T|, *?r,aresubjectto the conditions

(17:l:a) all ^ fe 0, {T| = 1;ri-i

(17:1:b) all^^O, ^ = 1.r,-l

and to no others.We form the vectors = {1, , &J and ty

={771, , J?0 t}.

>

Then the above conditionsstatethat { must belongto Sp^ and rj to S0tin the senseof 16.2.2.

In this setup a player doesnot, as previously, choosehis strategy, buthe playsall possiblestrategiesand choosesonly the probabilitieswith whichhe is going to play them respectively. This generalization meetsthe majordifficulty of the not strictly determinedcaseto a certain point:We haveseenthat the characteristicof that casewas that it constituteda definitedisadvantage3 for each player to have his intentions found out by his

1In Stone,Paper,Scissorsthere existsa tie, but no lossstill means that the probabilityof losing is ^ the probability of winning, and no gain means the reverse. Cf.footnote4 on p. 144.

8 That theseprobabilities werethe same for all strategies (i,1or J, |,i in the exam-plesof the last paragraph) was, of courseaccidental. It is to be expectedthat thisequality was due to the symmetric way in which the various alternatives appearedinthose games. We proceednow on the assumption that the \"appearance of probabilitiesin formulating a strategy was the essential thing, while the particular values -wereaccidental.

'TheA > Oof14.7.1.)))

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146 ZERO-SUMTWO-PERSONGAMES: THEORY

opponent. Thus oneimportant consideration1for a player in sucha gameis to protecthimselfagainsthaving his intentionsfound out by his opponent.Playing severaldifferent strategiesat random, so that only their probabili-ties aredetermined,is a very effective way to achieve a degreeof suchprotection:By this devicethe opponentcannot possiblyfind out what theplayer's strategy is going to be,sincethe playerdoesnot know it himself.2

Ignoranceis obviously a very goodsafeguardagainstdisclosinginformationdirectlyor indirectly.

17.2.2.It may now seemthat we have incidentally restrictedtheplayer's freedom of action. It may happen, after all, that he wishestoplay one definite strategy to theexclusionof all others;or that, while desir-ing to use certainstrategieswith certainprobabilities,he wants to excludeabsolutelythe remaining ones.3 We emphasizethat thesepossibilitiesareperfectly within the scopeof our scheme.A player who doesnot wish toplay certainstrategiesat all will simplychoosefor them the probabilitieszero. A player who wishesto play one strategy to the exclusionof allothers will choosefor this strategy the probability 1 and for all otherstrategiesthe probabilityzero.

Thus if player 1wishesto play the strategy 2\\i only, he will choosefor

the coordinatevector 6 Ti (cf. 16.1.3.).Similarly for player2, the strategy2 and the vectors T? and 6 T.

>

In view of all theseconsiderationswe call a vector of S$i or a vector

ry of S/3t a statistical or mixed strategy of player 1or 2, respectively. The>

coordinatevectors 5 Ti or 6 T* correspond,as we saw,to the originalstrategies

TI or r 2 i.e.2^i or S^ of player 1or 2, respectively. We call them strictor pure strategies.

17.3.Justification of the ProcedureAs Applied to an Individual Play

17.3.1.At this stagethe readermay have becomeuneasyand perceivea contradictionbetweentwo viewpoints which we have stressedas equallyvital throughout our discussions.On the one hand we have always insistedthat our theory is a static one (cf.4.8.2.),and that we analyze the course

1But not necessarilythe only one.2If the opponent has enough statistical experienceabout the player's\"

style,\" or if

he is very shrewd in rationalizing his expectedbehavior, he may discoverthe probabilitiesfrequencies of the various strategies. (We neednot discusswhether and how this

may happen. Cf. the argument of 17.3.1.)But by the very conceptof probabilityand randomness nobody under any conditions can foreseewhat will actually happen in

any particular case. (Exception must be made for such probabilities as may vanish;cf.below.)

8 In this casehe clearly increasesthe danger of having his strategy found out by theopponent. But it may be that the strategy or strategies in question have such intrinsicadvantages over the others as to make this worth while. This happens, e.g.in anextremeform for the \"good\" strategiesof the strictly determined case(cf.14.5.,particu-larly (14:C:a),(14:C:b)in 14.5.2.).)))

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MIXEDSTRATEGIES.THESOLUTION 147

of one play and not that of a sequenceof successiveplays (cf. 17.1.).Buton the otherhand we have placedconsiderationsconcerningthe dangerofone'sstrategy being found out by the opponentinto an absolutelycentralposition (cf. 14.4.,14.7.1.and again the last part of 17.2.).How canthe strategy of a player particularly one who plays a random mixture ofseveraldifferent strategies be found out if not by continuedobservation!We have luledout that this observationshouldextendover many plays.Thus it would seemnecessaryto carry it out in a singleplay. Now evenif the rulesof the game shouldbe suchas to makethis possible i.e.if theylead to long and repetitiousplays the observationwould be effected onlygradually and successivelyin the courseof the play. Itwould not be avail-able at the beginning.And the whole thing would be tied up with variousdynamical considerations, while we insistedon a statictheory! Besides,the rulesof the gamemay not even give suchopportunitiesfor observation;1

they certainly do not in our original examplesof Matching Pennies,andStone,Paper,Scissors.Theseconflictsand contradictionsoccurboth in thediscussionsof 14. where we used no probabilitiesin connectionwith thechoiceof a strategy and in our presentdiscussionsof 17.where probabilitieswill be used.

Howarethey to be solved?17.3.2.Our answeris this:To beginwith, the ultimate proof of the resultsobtainedin 14.and 17.

i.e.the discussionsof 14.5.and of 17.8.do not contain any of thesecon-flicting elements.So we could answer that our final proofs are correcteven though the heuristicprocedureswhich lead to them arequestionable.

But even theseprocedurescan be justified. We make no concessions:Our viewpoint is staticand we areanalyzing only a singleplay. We aretrying to find a satisfactory theory, at 'thisstagefor the zero-sumtwo-persongame. Consequentlywe arenot arguing deductivelyfrom the firmbasisof an existingtheory which has already stood all reasonabletestsbut we aresearchingfor sucha theory.2 Now in doingthis, it is perfectlylegitimatefor us touse the conventional tools of logics,and in particularthat of the indirect proof. This consistsin imagining that we have a satis-factory theory of a certaindesiredtype,3 trying to picturethe consequencesof this imaginary intellectualsituation, and then in drawing conclusionsfrom this as to what thehypotheticaltheory must belike in detail. If this

processis appliedsuccessfully,it may narrow the possibilitiesfor the hypo-theticaltheory of the type in question to such an extent that only one

1I.e.\"gradual,\" \"successive\" observations of the behavior of the opponent withinone play.1Our method is, of course,the empirical one:We are trying to understand, formalizeand generalizethose features of the simplest gameswhich impress us as typical. This is,after all, the standard method of all scienceswith an empirical basis.

3This is full cognizanceof the fact that we do not (yet) possessone, and that wecannot imagine (yet) what it would be like, if we had one.

All this is in its own domain no worsethan any other indirect proof in any part ofscience(e.g.the perabsurdum proofs in mathematics and in physics).)))

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148 ZERO-SUMTWO-PERSONGAMES: THEORY

possibilityis left, i.e.that the theory is determined,discoveredby thisdevice.1 Of course,it can happen that the applicationis even more \" suc-cessful,\"and that it narrows the possibilitiesdown to nothing i.e.that itdemonstratesthat a consistenttheory of the desiredkind is inconceivable.2

17.3.3.Let us now imagine that thereexistsa completetheory of thezero-sumtwo-persongamewhich tells a player what to do, and which isabsolutelyconvincing. If the playersknewsucha theory then eachplayerwould have to assumethat his strategy has been \" found out\" by his oppo-nent. Theopponentknowsthe theory, and he knowsthat a playerwouldbeunwisenot to follow it.8 Thus the hypothesisof the existenceof a satis-factory theory legitimatizesour investigation of the situation when a play-er'sstrategy is \" found out\" by his opponent. And a satisfactory theory 4

can existonly if we areable to harmonize the two extremesFI and F2,strategiesof player1\"found out\" or of player2 \"found out.\"

Forthe original treatment free from probability(i.e.with purestrate-gies) the extentto which this can be donewas determinedin 14.5.

We sawthat the strictlydeterminedcaseis the one where thereexistsatheory satisfactoryon that basis. We arenow trying to push further, byusing probabilities(i.e.with mixed strategies). The same device whichweusedin 14.5.when therewereno probabilitieswill do again, theanalysisof \"finding out\" the strategy of the other player.

It will turn out that this time the hypotheticaltheory can be determinedcompletelyand in all cases(not merely for the strictly determinedcasecf. 17.5.1.,17.6.).

After the theory is found we must justify it independentlyby a directargument. 6 This was done for the strictly determinedcasein 14.5.,andwe shalldo it for the presentcompletetheory in 17.8.

1 Thereareseveral important examplesof this performance in physics. Thesucces-sive approachesto Specialand to General Relativity or to Wave Mechanicsmay beviewed as such. Cf.A. D'Abro:The Declineof Mechanism in Modern Physics, NewYork 1939.

1This too occursin physics. TheN. Bohr-Heisenberganalysis of \"quantities whichare not simultaneously observable\" in Quantum Mechanicspermits this interpretation.Cf.N. Bohr:Atomic Theory and the Descriptionof Nature, Cambridge 1934and P.A.M.Dirac:ThePrinciples ofQuantum Mechanics,London 1931,Chap.I.

8 Why it would beunwise not to follow it is none of our concernat present;we haveassumed that the theory is absolutely convincing.

That this is not impossible will appearfrom our final result. We shall find a theorywhich is satisfactory ;neverthelessit implies that the player'sstrategy is found out by hisopponent. But the theory gives him the directions which permit him to adjust himselfso that this causesno loss. (Cf.the theorem of 17.6.and the discussion of our completesolution in 17.8.)

4I.e.a theory using our present devicesonly. Of coursewe do not pretend to beableto make \"absolute\" statements. If our present requirements should turn out to beunfulfillable we should have to look for another basisfor a theory. We have actuallydone this onceby passing from 14.(with pure strategies)to 17.(with mixed strategies).*Theindirect argument, as outlined above,gives only necessaryconditions. Henceit may establish absurdity (perabsurdum proof), or narrow down the possibilities to one;but in the latter caseit is still necessaryto show that the one remaining possibility issatisfactory.)))

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MIXEDSTRATEGIES.THESOLUTION 149

17.4.The Minorant and the Majorant Games(For Mixed Strategies)17.4.1.Our present picture is then that player 1choosesan arbitrary

element from S$ and that player2 choosesan arbitrary elementi? fromsf,

Thus if player 1wishesto play the strategy 2^ only, he will choosefor

the coordinatevector d Ti (cf. 16.1.3.); similarly for player2, the strategy

ZS and the vectors rj and 6 T.We imagine again that player 1makes his choiceof in ignoranceof

player2'schoiceof y and viceversa.The meaning is, of course,that when thesechoiceshave beenmade

player 1will actually use (every) TI = 1, , ft\\ with the probabilitiesTI and the player2 will use (every) TI = 1, , 0iwith the probabilities

iyv Sincetheir choicesare independent, the mathematical expectationof the outcomeis

- > 0i 0t

(17:2) K( , n ) = 3C(n,n)^,.n-lr.-lIn otherwords,we have replacedthe original gameT by a new oneof

essentiallythe same structure,but with the following formal differencei:Thenumbersn,r 2 the choicesof the players arereplacedby the vectors

, q . Thefunction 5C(ri,r 2) the outcome,or rather the \" mathematical>

expectation\" of the outcomeof a play is replacedby K( , 17 ). All theseconsiderationsdemonstrate the identity of structureof our presentviewof T with that of 14.1.2.,the soledifference beingthe replacementof

TI, T2, 3C(n,TJ) by , t\\ , K( , t\\ ), describedabove. This isomorphismsuggeststhe applicationof the samedeviceswhich we usedon the originalT, the comparisonwith the majorant and minorant games FI and F* asdescribedin 14.2.,14.3.1.,14.3.3.

17.4.2.Thus in TI player1chooseshis first and player2 chooseshis ij

afterwardsin full knowledgeof the chosenby his opponent. In Tj theorderof their choicesis reversed. So the discussionof 14.3.1.applied

literally. Player 1,choosinga certain , may expectthat player 2 will

choosehis /j , soas to minimize K( , 17 ) ; i.e.playerTschoiceof leadsto>

the value Min~* K( , 17 ). This is a function of alone;henceplayer 1shouldchoosehis so as to maximize Min-+K( , ij ). Thus the value of

a play of TI is (for player 1))))

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150 ZERO-SUMTWO-PERSONGAMES:THEORY

vj = Max-Min-> K(T,7).Similarly the value of a play of F2 (for player 1)turns out to be

v'2 = Min-> Max-* K( , rj ).*? f

(Theapparent assumptionof rational behavior of the opponentdoes notreally matter,sincethe justifications (14:A:a)-(14:A:e),(14:B:a)-(14:B:e)of 14.3.1.and 14.3.3.again apply literally.)

As in 14.4.1.we can argue that the obvious fact that FI is lessfavorablefor player 1than F2 constitutesa proof of))

vi))

and that if this is questioned,a rigorousproof is containedin (13:A*) in

13.4.3.Thex,y, therecorrespondto our , 17 , K.l If it shouldhappenthat))

v'i))

then the considerationsof 14.5.apply literally. Thearguments (14:C:a)-(14:C:f),(14:D:a),(14:D:b)loc.cit.,determinethe conceptof a \" good\"{and i? and fix the \"value\" of a play of (for the player 1)at

v' = v{ = v'2.2All this happensby (13:B*)in 13.4.3.if and only if a saddle point of K

>

exists. (Thex,y, </> therecorrespondto our , rj , K.)17.5.GeneralStrict Determinateness

17.5.1.We have replacedthe YI, v 2 of (14:A:c)and (14:B:c)by ourpresentv{, v'2, and the above discussionshowsthat the lattercan performthe functions of the former. But we are just as much dependent uponv[ = v2 as we were then upon v l = v2. It is natural to ask, therefore,whether thereis any gain in this substitution.

Evidently this is the caseif, as and when there is a betterprospectof having vj = v'2 (for any given F) than of having vi = v2. We calledFstrictly determined when Vi = v2; it now seemspreferableto make a dis-tinction and to designateF for Vi = v2 as speciallystrictly determined,andfor vi = v2 as generally strictly determined. This nomenclature is justifiedonly providedwe can show that the former impliesthe latter.

> ->1Although , 17 are vectors,i.e.sequencesof real numbers (1, ,$ and

'nit ' ' ' 1 10) it is perfectly admissible to view eachasa single variable in the maxima andminima which we are now forming. Their domains are, of course,the sets8ft , Sft

which we introduced in 17.2.8 For an exhaustive repetition of the arguments in question cf. 17.8.)))

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MIXEDSTRATEGIES.THESOLUTION 151Thisimplication isplausibleby commonsense:Our introduction of mixed

strategieshas increasedthe player'sability to defendhimself againsthavinghis strategy found out;so it may beexpectedthat vj, v2 actually liebetweenVi, v2. Forthis reasonone may even assertthat

(17:3) V! ^ vl g v'2 g v2.(This inequality secures,of course,the implication just mentioned.)

Toexcludeall possibilityof doubt we shall give a rigorousproof of(17:3).It is convenient to prove this as a corollary of another lemma.

17.5.2.First we prove this lemma:

(17:A) Forevery { in S^-* - ft *

Min-K( f , r, ) = Min- 3C(n,))

= Min Tj JC(n,r^l

Forevery 77 in Spt

> 0i ft

Max-K( , 77 ) = Max-n-lr.-l

/5i

= MaxTi ^ JC(TI,r 2)r/Tj.r z -l

Proof:We prove the first formula only ; the proof of the secondis exactlythe same,only interchanging Max and Min as well as ^ and ^ .

> > ,Considerationof the specialvector r?

= 6 Ti (cf. 16.1.3.and the end of17.2.)gives

0! ft 01 ft ^1

Min- rcd^rOfc^g S^i>^)Mv;=Z X(T1>r ')k'-

rj-lTt-l T!-!T2=l r, =l

Sincethis is true for all r'2, so01 ft 01

(17:4:a)Min-^ JC(n,njfc^g Minr; ^ 3C(n,r;)fv1))

On the otherhand, for all r 2

ft))

ac(n,))

Given any 77 in S08, multiply this by r\\ T and sum over r 2 = 1,a

Since ^ yTa= 1,therefore

r.-l)))

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152 ZERO-SUMTWO-PERSONGAMES: THEORY

ti f t fi3C(r,,rt) r, Min,, JC(nf r,){r ,T!-!r,-l fi-1

results. Sincethis is true for all i; , so

0i 0t fti

(17:4:b)Min- 3C(n,r.)^*, Min r, % OC(n,r) v))

(17:4:a),(17:4:b)yieldtogetherthe desiredrelation.If we combinethe above formulae with the definition of v'^ v'2 in 17.4.,

then we obtain

0i

(17:5:a) vi = Max-MinTt K(rl9 T,) V

(17:5:b) v = Min-MaxT ?))

Theseformulae have a simpleverbal interpretation:In computing v't weneedonly to give player 1the protectionagainsthaving his strategy found

out which liesin theuseof (insteadof TI); player2 might as wellproceedin the oldway and user 2 (andnot 17 ). In computingvj the rolesareinter-changed. This is plausibleby common-sense:v{ belongsto the gameFI(cf. 17.4.and 14.2.);there player 2 choosesafter player 1 and is fullyinformed about thechoiceof player1, hencehe needsno protectionagainsthaving his strategy found out by player 1. Forvi which belongsto thegameTj (cf.id.)the rolesareinterchanged.

Now the value of v( becomes if we restrictthe variability of in

the Max of the above formula. Let us restrictit to the vectors = d r'\\

(rj = !,-,/>!,cf. 16.1.3.and the end of 17.2.).Since0i

X fri, ri)*r/i= 3e(r'lf ri),TI-I

this replacesour expressionby

Max/ Min Tj$(/!,TZ) = VL

Sowe have shown thatvi ^ v;.

Similarly(cf.the remarkat thebeginningof the proof of our lemma above)

restrictionof 77 to the rj= 8 T establishes

v vj.)))

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MIXEDSTRATEGIES.THESOLUTION 163Togetherwith v{ ^ v (cf.17.4.),theseinequalitiesprove

(17:3) Vl g v; ^ vi g v f ,

as desired.17.6.Proof of the Main Theorem

17.6.We have establishedthat generalstrictdeterminateness(v( = vj)holds in all casesof specialstrict determinateness(vi = v2) us is to beexpected.That it holds in some further casesas well i.e.that we canhave Vi = v'2 but not Vi = v2 is clearfrom our discussionsof MatchingPenniesand Stone,Paper,Scissors.1 Thus we may say, in the senseof17.5.1.that the passagefrom specialto generalstrictdeterminatenessdoesconstitutean advance. But for all we know at this moment this advancemay not cover the entireground which shouldbe controlled;it couldhap-pen that certaingamesF arenot even generallystrictly determined, i.e.we have not yet excludedthe possibility

vl < v2.If this possibilityshould occur,then all that was said in 14.7.1.wouldapply again and to an increasedextent:finding out one'sopponent'sstrategywould constitutea definite advantage

A' = v2 - vj > 0,and it would be difficult to seehow a theory of the game shouldbe con-structedwithout someadditionalhypothesesas to \"who finds out whosestrategy.\"

The decisivefact is, therefore, that it can be shown that this neverhappens. Forall gamesF

v; = v2i.e.

(17:6) Max-Min-K(7,\"7 ) = Min-Max-K(7,7),or equivalently (againuse (13:B*)in 13.4.3.thex,y, <t> therecorrespondingto our , q , K):A saddlepoint of K( ( , rj ) exists.

This is a generaltheoremvalid for all functions K( { , ?? ) of the form

(17:2) K(7,7) = |W(TI, r,)r>i.Ti-lT,-!Thecoefficients3C(ri,r t ) areabsolutelyunrestricted;they form, as described

* *in 14.1.3.a perfectly arbitrary matrix. The variables $ , 17 are really

1In both games YI 1,vi 1 (cf.14.7.2.,14.7.3.),while the discussion of 17.1.can be interpreted as establishing v{ vj 0.)))

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154 ZERO-SUMTWO-PERSONGAMES:THEORY

sequencesof realnumbers {i, , ^ and rji, , <]^\\their domains

>

being the setsS^, S0t (cf. footnote 1on p.150).The functions K( , 1? )of the form (17:2)arecalledbilinear forms.

With the help of the resultsof 16.4.3.the proof is easy.1 This is it:We apply (16:19:a),(16:19:b)in 16.4.3.replacingthe i,j,n, m, a(iy j)

thereby our TI, r 2, 0i, 2,3C(ri,r 2) and the vectorsw , x thereby our , ry .If (16:19:b)holds,then we have a in Spi with

0i

2j3C(Ti,TI){TI ^ for r 2 = 1, , |8,,n-i

i.e.with

0iMin Tj J JC(n,r 2) T ^ 0.))

Therefore the formula (17:5:a)of 17.5.2.gives

vl ^ 0.If (16:19:a)holds,then we have an rj in Sp2

with

0*

2) 3C(ri,rz)^ ^ for n = 1, , 0,,r,-l

1This theorem occurredand was proved first in the original publication of one of theauthors on the theory of games:/.von Neumann: \"Zur Theorieder Gesellschaftsspiele,\"Math. Annalen, Vol. 100(1928),pp.295-320.

A slightly more generalform of this Min-Max problem arisesin another question ofmathematical economicsin connection with the equations of production :J. von Neumann: \"tlber ein okonomisches Gleichungssystem und eine Verall-gemeinerung desBrouwer'schen Fixpunktsatzes,\" Ergebnisseeines Math. Kolloquiums,Vol. 8 (1937),pp.73-83.

It seemsworth remarking that two widely different problems related to mathe-matical economicsalthough discussedby entirely different methods leadto the samemathematical problem, and at that to oneof a rather uncommon type:The \" Min-Maxtype.\" Theremay be some deeperformal connections here, as well as in some otherdirections, mentioned in the secondpaper. Thesubject should be clarified further.

Theproof of our theorem, given in the first paper, made a rather involved use ofsometopology and of functional calculus. Thesecondpapercontained a different proof,which was fully topologicaland connectedthe theorem with an important deviceofthat discipline:the so-called\" Fixed Point Theorem\" of L.E.J.Brouwer. This aspectwas further clarified and the proof simplified by S. Kakutani: \"A Generalization ofrtrouwer's Fixed Point Theorem,\" Duke Math. Journal, Vol. 8 (1941),pp.457-459.

All theseproofsaredefinitely non-elementary. Thefirst elementary one was givenby J.Ville in the collectionby E.Boreland collaborators,\"Trait6 du CalculdesProb-abilit& et de sesApplications,\" Vol. IV, 2:\"Applications aux Jeux de Hasard,\" Paris(1938),Note by J.Ville: \"Sur la TheorieG6ne>aledesJeux ou intervient THabilete* desJoueurs,\" pp.105-113.

The proof which we are going to give carriesthe elementarization initiated by/.Ville further, and seemsto be particularly simple. Thekey to the procedureis, ofcourse,the connection with the theory of convexity in 16.and particularly with theresults of 16.4.3.)))

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MIXEDSTRATEGIES.THESOLUTION 155

i.e.with

ft*

MaxT 3e(n,r 2)i7 T ^ 0.))

Therefore the formula (17:5:b)of 17.5.2.gives

v'2 ^ 0.Sowe see:Either vi ^ or v'2 ^ 0,i.e.

(17:7) Never vi < < v'2.Now choosean arbitrary number w and replacethe function 3C(ri,r 2)

byJC(n, r 2) - w.1 - _> 0i 0* > >

This replacesK( , T;) by K( , T;) w ^ ] TI TJ V that is as , 17

r, =l r,= l

01 02 _ _lie in S^,fi^ andso TI

= ^ 77^= 1 by K( , TJ ) w. ConsequentlyT!-! r,-l

vj, v 2 arereplacedby vi w, v'2 w.2 Therefore applicationof (17:7)tothesevi w, v'2 w gives

(17:8) Never vi < w < v'2.Now w was perfectly arbitrary. Hencefor vi < v'2 it would be possible

to choosew with vi < w < v'2 thus contradicting(17:8).So vi < v'2 isimpossible,and we have proved that vi = v 2 as desired.This completesthe proof.

17.7.Comparison of the Treatments by Pure and by Mixed Strategies17.7.1.Beforegoing further let us oncemore considerthe meaning of

the result of

Theessentialfeature of this is that we have always v{ = v2 but not alwaysVi = v 2, i.e.always generalstrict determinateness,but not always specialstrict determinateness(cf. the beginning of 17.6.).

Or, to expressit mathematically :We have always

(17:9) Max-Min-K(7,7) = Min- Max-K(7,7),, 6 v n

1I.e.the game Tis replacedby a new onewhich is played in preciselythe sameway asT exceptthat at the end player 1gets less(and player 2gets more) by the fixed amount wthan in T.

2This is immediately clear if we remember the interpretation of the precedingfootnote.)))

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156 ZERO-SUMTWO-PERSONGAMES: THEORY

i.e.(17:10)Max-Min- SC^rj) -

0i ft*

Min-+Max->V Tn * ^ ^

Using(17:A)we may even write for this

(17:11)Max^Minrt ^ 3C(n,r2) ri=

Min^ Max,t % 3C(n,T2)i?v

But we do not alwayshave

(17:12) MaxT|Min^^COn,r2) = Min Ti MaxTj3C(ri,r2).

Let us compare(17:9)and (17:12):(17:9)is always true and (17:12)is not. Yet thedifferencebetweentheseismerelythat of , ij , Kandn,n,3C. Why doesthe substitution of the former for the latterconvert theuntrue assertion(17:12)into the true assertion(17:9)?

Thereasonis that the5C(ri,r a) of (17:12)is a perfectly arbitrary func-

tion of its variablesn, r2 (cf. 14.1.3.),while the K( , rj ) of (17:9)is an> >

extremelyspecialfunction of its variables , ij i.e.of the i, , &Vi;

' * * t *?/v namely a bilinearform. (Cf.the first part of 17.6.)Thusthe absolutegeneralityof JC(TI,r 2) rendersany proof of (17:12)impossible,while the specialbilinearform nature of K( , rj ) providesthebasisforthe proof of (17:9),as given in 17.6.l

17.7.2.While this is plausibleit may seemparadoxicalthat K( , q)shouldbemorespecialthan 3C(ri,r2), although the former obtained fromthe latterby a processwhich bore all the marks of a generalization:Weobtained it by the replacementof our original strictconceptof a purestrategy by the mixedstrategies,as describedin 17.2.;i.e.by the replace-ment of n,r2 by , 17 .

But a closerinspectiondispelsthis paradox.K( , i\\ ) isa very specialfunction when comparedwith 3C(ri,r2) ; but its variableshave an enormously

1 That the K( , 17 ) is a bilinear form is due to our useof the \" mathematical expecta-tion \" wherever probabilities intervene. It seemssignificant that the linearity of thisconceptis connectedwith the existenceof a solution, in the sensein which we found one.Mathematically this opensup a rather interesting perspective:Onemight investigatewhich other concepts,in placeof \"mathematical expectation,1'would not interfere withour solution, i.e.with the result of 17.6.for zero-sum two-person games.

Theconceptof \"mathematical expectation\" is clearly a fundamental one in manyways. Its significance from the point of view of the theory of utility was brought forthparticularly in 3.7.1.)))

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MIXEDSTRATEGIES.THESOLUTION 157

widerdomain than the previous variablesn, r 2. IndeedT\\ had the finite

set(1, , pi) for its domain, while varies over the setS^,which isa

(Pi l)-dimensionalsurface in the 0i-dimensionallinear spaceS^ (cf.the>

end of 16.2.2.and 17.2.).Similarly for r 2 and rj .lThereareactually among the in S

fti specialpoints which correspondto the various TI in (1, , 0i). Given such a r\\ we can form (as in

16.1.3.and at the end of 17.2.)the coordinatevector = 6 Ti, expressing

the choiceof the strategy S^ to the exclusionof all others. We can corre-latespecialrj in S^ with the r 2 in (1, , fit) in thesameway:Given such

>

a T2 we can form the coordinatevector rj= 6 r , expressingthe choiceof the

strategy 22 to the exclusionof all others.Now clearly:))

SCW, T'I) *,**,I /-I

, r 2).2

Thus the function K( , T;) contains,in spite of its specialcharacter,theentirefunction 3C(ri,r 2) and it is therefore really the more generalconceptof the two, as it ought to be. It is actually more than 3C(ri,r 2) sincenot all

, 77 areof the specialform 6 r , 5 %, not all mixedstrategiesarepure.*Onecouldsay that K( , 17 ) is the extensionof 3C(ri,r 2) from the narrower

domain ofthe ri,r2 i.e.ofthe5 Ti, 6 r to the widerdomain ofthe , rj i.e.

to all of Sp, Spt from the purestrategiesto the mixedstrategies.Thefact

that K( , TJ ) isa bilinearform expressesmerely that thisextensioniscarriedout by linear interpolation. That it is this processwhich must be used, isof coursedue to the linear characterof \" mathematical expectation/*4

1 Observethat = | i, , t l with the components TI, n = !, , 0i, alsocontains r\\ ;but there is a fundamental difference. In 3C(n,TJ),r\\ itself isa variable. In

K( , 17 ), is a variable, while n is, so to say, a variable within the variable. isactually a function of TI (cf.the end of 16.1.2.)and this function assuch is the variable

of K( , q ). Similarly for r2and T;.>

Or, in terms of n, r2:3C(n, r*) is a function of n, TJ while K( , 17 ) is a functionof functions ofn,*2 (in the mathematical terminology :a functional).

2The meaning of this formula is apparent if we considerwhat choiceof strategies> >

d ri, 5 r represent.8 I.e.severalstrategiesmay beused effectively with positive probabilities.4Thefundamental connection between the conceptof numerical utility and the linear

\" mathematical expectation\" was pointed out at the end of 3.7.1.)))

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158 ZERO-SUMTWO-PERSONGAMES: THEORY

17.7.3,Reverting to (17:9)-(17:12),we seenow that we canexpressthetruth of (17:9)-(17-11)and the untruth of (17:12)as follows:

(17:9),(17:10)expressthat eachplayeris fully protectedagainsthavinghis strategy found out by his opponent if he can use the mixedstrategies

, ij insteadof the pure strategiesn,r 2. (17:11)statesthat this remainstrue if the player who finds out his opponent'sstrategy uses the T\\, r2

while only the player whosestrategy is being found out enjoys the pro-

tectionof the , r; . Thefalsity of (17:12), finally, showsthat bothplayersand particularly the playerwhosestrategy happensto be found out may

> >

not forego with impunity the protectionof the {, rj .17.8.Analysis of GeneralStrict Determinateness

17.8.1.We shall now reformulate the contentsof 14.5.as mentionedat the end of 17.4.with particular considerationof the fact establishedin 17.6.that every zero-sumtwo-persongame F is generally strictly deter-mined. Owing to this result we may define:

v' = Max-Min-K(7,7)= Min-Max-K(7,7)= Sa-|-K(7,7).

(Cf.also (13:C*)in 13.5.2.and the end of 13.4.3.)Let us form two setsA, B subsetsof S^, S/^,respectively in analogy

to the definition of the sets A, B in (14:D:a),(14:D:b)of 14.5.1.Thesearethe setsA+, B+ of 13.5.1.(the </> correspondingto our K). We define:

(17:B:a) A is the set of those (in Sp) for which Min-K( , y )*>

assumesits maximum value, i.e.for which

Min-K(7,7) = Max-Min-K(7,7) = V.* { *

(17:B:b) B is the set of those 77 (in Spt) for which Max-K( , 7)assumesits minimum value, i.e.for which

Max-^K(7,7) = Min-Max-K(7,7)= V.

It is now possibleto repeatthe argumentation of 14.5.In doingthis we shallusethehomologousenumeration for the assertions

(14:C:a)-(14:C:f)asinl4.5.1

1 (a)-(f)will therefore appearin an order different from the natural one. This wasequally true in 14.5.,sincethe enumeration there was basedupon that of 14.3.1.,14.3.3.,and the argumentation in those paragraphs followed a somewhat different route.)))

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MIXEDSTRATEGIES.THESOLUTION 159

We observefirst:

(17:C:d) Player 1can,by playing appropriately,securefor himselfa gain ^ v', irrespectiveof what player2 does.

Player 2 can, by playing appropriately,securefor himselfa gain ^ v', irrespectiveof what player 1does.

Proof:Let player 1choose{ from A ; then irrespectiveof what player2

does,i.e.for every T; wehave K( , y ) ^ Min->K({ , rj ) = v'. Letplayeri?

2 chooserj from B. Then irrespectiveof what player1does,i.e.for every

, we have K( , 77 ) g Max->K( { , rj ) = v'. Thiscompletesthe proof.

Second,(17:C:d)is clearlyequivalent to this:(17:C:e) Player 2 can,by playing appropriately,make it sure that

the gain of player 1 is ^ v', i.e.prevent him from gaining> v', irrespectiveof what player 1does.

Player 1can,by playing appropriately,make it sure thatthe gain of player 2 is ^ v', i.e.prevent him from gaining> v', irrespectiveof what player2 does.

17.8.2.Third, we may now assert on the basisof (17:C:d)and (17:C:e)and of the considerationsin the proof of (17:C:d)that:

(17:C:a) The good way (combination of strategies)for 1 to play

the game F is to chooseany belongingto A, A beingthe setof (17:B:a)above.

(17:C:b) The good way (combinationof strategies)for 2 to play

the gameF is to chooseany rj belongingto 5, B beingthe setof (17:B:b)above.

Fourth, combination of the assertions of (17:C:d)or equally wellof thoseof (17:C:e)gives:

(17:C:c) If both players 1 and 2 play the game F well i.e.if {belongsto A and ij belongsto B then the value of K( , rj )will beequal to the value of a play (for 1), i.e.to v'.

We add the observationthat (13:D*)in 13.5.2.and the remarkconcerningthe setsA, B before(17:B:a),(17:B:b)above give togetherthis:

(17:C:f) Bothplayers1and 2 play the game F well i.e.{ belongs

to A and t; belongsto B if and only if , rj is a saddlepoint

of K(7,7)-!)))

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160)) ZERO-SUMTWO-PERSONGAMES: THEORY))

All this shouldmake it amply clearthat v' may indeedbe interpretedas the value of a play of F (for 1),and that A, B contain the goodways ofplaying T for 1,2,respectively. Thereis nothing heuristicor uncertainabout the entire argumentation (17:C:a)-(17:C:f).We have made noextrahypothesesabout the \" intelligence\" of the players, about \"whohas found out whosestrategy\" etc. Nor areour results for one playerbased upon any belief in the rational conduct of the other, a point theimportanceof which we have repeatedlystressed.(Cf. the end of 4.1.2.;also 15.8.3.)

17.9.Further Characteristicsof GoodStrategies17.9.1.The last results (17:C:c)and (17:C:f)in 17.8.2.give also a

simpleexplicitcharacterizationof the elementsof our present solution,i.e.of the number v' and of the vector setsA and B.

By (17:C:c)loc.cit.,A, B determinev';hencewe needonly study A,B,and we shall do this by meansof (17:C:f)id.

Accordingto that criterion, belongsto A, and t\\ belongsto B if and

only if , i? is a saddlepoint of K( , rj ). This meansthat

Max-K(7',7Min-K(T,7'

We make this explicitby using the expression(17:2)of 17.4.1.and

17.6.for K( , 17 ), and the expressionsof the lemma (17:A)of 17.5.2.for

Max-*,K( ', if) and Min-K( , y ') Then our equationsbecome:I' V))

MaxT))

r.-lr.-l))ac(rlf))

Consideringthat ^ TI=

r,-! '))

Tj 1))

[Maxr;

[-Minr; { ^))

XTt 1))

Min T'

, we can alsowrite for these

* _. ^^ fin t \\ It ^s

r,-l))

r,-1 rt-1

Now on the left-hand side of theseequationsthe v tjTfhave coefficients

which are all ^ O.1 The TI, T7Tfthemselvesare also ^ 0. Hencethese

1Observehow the Max and Min occurthere!)))

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MIXED STRATEGIES.THESOLUTION 181equationshold only when all termsof their left hand sidesvanish separately.I.e.when for eachn = 1,. , $\\ for which the coefficientis not zero, wehave r = 0;and for eachr* = 1, , fa for which the coefficientis notzero, we have T/TI

= 0.Summing up;

(17:D) belongsto A and rj belongsto B if and only if thesearetrue:

0tFor eachn = 1, , 0i,for which JC(TI,TZ)rjTl does

r,-lnot assumeits maximum (in n) we have TI

= 0.0i

Foreachr 2 = 1, , ft, for which OC(ri, T2) ri doesn-i

not assumeits minimum (in r 2) we have r;Tj= 0.

It is easy to formulate theseprinciplesverbally. They expressthis: If

, 77 aregoodmixed strategies,then excludesall strategiesT\\ which arenot optimal (for player1)against t\\ , and rj excludesall strategiesr 2 which

are not optimal (for player 2) against ; i.e. , rj are as was to beexpectedoptimal againsteachother.

17.9.2.Another remarkwhich may be madeat this point is this:

(17:E) The game is speciallystrictly determinedif and only if thereexistsfor eachplayera goodstrategy which is a pure strategy.

In view of our past discussions,and particularly of the processof gen-eralization by which we passed from pure strategiesto mixed strategies,this assertionmay be intuitively convincing. But we shall also supply amathematical proof, which is equally simple.This is it :

We sawin the last part of 17.5.2.that both Vi and v\\ obtain by applyingfe -*

Max->to MinTj 2) & (TI> T*)^*only with different domainsfor :The set

* \\-iof all 5 Ti (TI = 1, , 00for YI, and all of8^ for vi; i.e.the purestrategiesin the first case,and the mixed ones in the second.HenceVi = v\\, i.e.the two maxima areequalif and only if the maximum of the seconddomainis assumed(at leastonce)within the first domain. This meansby (17:D)above that (at least)one pure strategymust belongto A, i.e.be a goodone.I.e.

(17:F:a) Vi = vi if and only if there exists for player 1 a goodstrategy which is a pure strategy.)))

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162 ZERO-SUMTWO-PERSONGAMES: THEORY

Similarly:

(17:F:b) V2 = v if and only if there existsfor player 2 a goodstrategy which is a pure strategy.

Now vj = v'2 v' and strict determinatenessmeans Vi = V2 = v',i.e.vi = v; and v2 = v. So(17:F:a),(17:F:b)give together(17:E).

17.10.Mistakesand Their Consequences.Permanent Optimality

17.10.1.Our past discussionshave made clearwhat a good mixedstrategy is. Let us now say a few wordsabout the othermixedstrategies.We want to expressthe distancefrom \" goodness\"for thosestrategies(i.e.vectors {, rj ) which arenot good;and to obtainsomepictureof the conse-quencesof a mistake i.e.of the useof a strategy which is not good.How-ever, weshallnot attempt to exhaustthissubject,which hasmany intriguingramifications.

Forany in S0i and any ?? in S02we form thenumerical functions

(17:13:a) a(7)= V - Min-K(T,7),))

(17:13:b) 0()= Max-K(, ) - V.

By the lemma (17:A) of 17.5.2.equally

-> *(17:13**) ( O V - Min Tj % 3C(n,T,)$TI>

Tt-l- *

(17:13:b*) ft( *) = Max,,V jc(n,r^- v;.r,-l

Thedefinition

v' = Max-Min-K(7,7)= Min-Max-K(7,7)guaranteesthat always

(7) ^ 0, 0(7)^ 0.

And now (17:B:a),(17:B:b)and (17:C:a),(17:C:b)in 17.8.imply that 7is goodif and only if a( ) =0,and ij is goodif and only if 0( 77 ) =0.

> > >

Thusa( ^)jft(r))areconvenient numerical measuresfor the general{, 77

expressingtheir distancefrom goodness.The explicitverbal formulation>

of what a( ),0( t\\ ) are,makesthis interpretationeven more plausible:Theformulae (17:13:a),(17:13:b)or (17:13:a*),(17:13:b*)above make clear)))

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MIXED STRATEGIES.THESOLUTION 163how much of a lossthe playerrisks relative to the value of a play for him 1

by using this particular strategy. We mean here \"risk\" in the senseof the worst that can happenunder the given conditions.2

>It must be understood,however, that a( ( ),0( i? ) donot disclosewhichstrategyof the opponentwill inflict this (maximum) lossupon the playerwho

>

is using { or rj . It is, in particular,not at all certainthat if the opponentusessomeparticulargoodstrategy,i.e.an 17 in 5or a o in A, this in itself

>

impliesthe maximum lossin question. If a (not good) or rj isusedby the

player,then the maximum losswill occurfor those 17'or {'of the opponent,

for which))

K(, ') = Min 7 K(, ),

K(7,7) = Max K(7,7),))

i.e.if 17' isoptimal againstthe given , or 'optimal againstthe given 17 .

>

And we have never ascertainedwhether any fixed 17 o or { can beoptimal^

againstall or 77 .17.10.2.Let us therefore call an TJ

' or a 'which isoptimal againstall

7 or 7 i.e.which fulfills (17:14:a),or (17:14:b)in 17.10.1.for all 7,7~permanently optimal. Any permanently optimal 17

' or ' is necessarilygood;this shouldbe clearconceptuallyand an exactproof is easy.8 But

1I.e.we mean by lossthe value of the play minus the actual outcome:v' K( , q )> > >

for player 1 and (-v') - (~K( , 77 )) = K( , 17 ) - v' for player 2.'2Indeed,using the previous footnote and (17:13:a),(17:13:b)

(7) -V - Min-.K(T,7) = Max- (v' - K(7,7)1,1? 1

3(7) = Max-K(7,7) -V = Max- |K(7,7) -v'|.I.e.eachis a maximum loss.

8 Proof:It suffices to show this for rj'; the proof for ' is analogous.

Let 77 'bepermanently optimal. Choosea *which isoptimal against 17',i.e.with

K(T',V)= Max-K(7,7)By definition

K(7*.7) = Min-K(7*.7).If

>

Thus |*, 17' is a saddlepoint of K( , 17 ) and therefore V belongs to B i.e.it isgoodby (17:C:f)in 17.8.2.)))

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164 ZERO-SUMTWO-PERSONGAMES: THEORY

the question remains:Are all good strategiesalso permanently optimal?And even:Do any permanently optimal strategiesexist?

In generalthe answeris no. Thus in Matching Penniesor in Stone,Paper,Scissors,the only goodstrategy (for player1 as wellas for player2)is

^

f =rj

= (i,i)or {i,|,i}, respectively.1 If player1playeddifferentlye.g.always \" heads\"2 or always \" stone\"2 then he would loseif the oppo-nent counteredby playing \" tails\"3 or \"paper.\"3 But then the opponent'sstrategy is not good i.e.{,?}or (i,i, },respectively either. If theopponent played the good strategy, then the player's mistake would notmatter.4

We shallgetanother exampleof this in a more subtleand complicatedway in connectionwith Pokerand the necessityof \"

bluffing,\" in 19.2and19.10.3.

All this may be summedup by saying that while our good strategiesareperfectfrom the defensivepoint of view, they will (in general)not getthe maximum out of the opponent's(possible)mistakes, i.e.they arenotcalculatedfor the offensive.

It should be remembered,however, that our deductionsof 17.8.areneverthelesscogent;i.e.a theory of the offensive, in this sense,is notpossiblewithout essentiallynew ideas. The readerwho is reluctant toacceptthis, ought to visualize the situation in MatchingPenniesor inStone,Paper,Scissorsoncemore;the extremesimplicity of these twogamesmakesthe decisivepointsparticularlyclear.

Another caveat against overemphasizingthis point is:A great dealgoes,in common parlance,under the name of \"offensive,\"which is not atall \"offensive\"in the above sense, i.e.which is fully coveredby our pres-ent theory. This holdsfor all gamesin which perfectinformation prevails,as will beseenin 17.10.3.6 Also suchtypically\"aggressive\"operations(andwhich arenecessitatedby imperfect information) as \"bluffing\" in Poker.6

17.10.3.We concludeby remarkingthat thereis an important classof(zero-sumtwo-person) games in which permanently optimal strategiesexist. Thesearethe gamesin which perfect information prevails,whichwe analyzed in 15.and particularly in 15.3.2.,15.6.,15.7.Indeed,a smallmodification of the proof of specialstrictdeterminatenessof thesegames,asgiven loc.cit.,would suffice to establishthis assertiontoo. It would givepermanently optimal purestrategies.But we do not enterupon thesecon-siderationshere.

1 Cf.17.1.Any otherprobabilities would leadto losseswhen \" found out.\" Cf.below.

'Thisis = * ' (1,0) or (1,0, 0),respectively.*This is i?

- S\" - (0, 1)or {0,1,0),respectively.4I.e.the bad strategy of \" heads\" (or \"stone 1') can bedefeatedonly by \"tails\" (or

\"paper\,") which is just as bad in itself.1Thus Chessand Backgammon are included.The preceding discussion appliesrather to the failure to \"bluff.\" Cf. 19.2.and

19.10.3.)))

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MIXED STRATEGIES.THESOLUTION 165Sincethe gamesin which perfect information prevailsarealways spe-

cially strictly determined(cf.above),onemay suspecta more fundamentalconnectionbetweenspeciallystrictly determinedgamesand those in which

permanently optimal strategiesexist(for both players). We do not intendto discussthesethings hereany further, but mention the following factswhich arerelevant in this connection:

(17:G:a) Itcan beshownthat if permanentlyoptimal strategiesexist(for both players) then the game must be speciallystrictlydetermined.

(17:G:b) It can be shown that the converseof (17:G:a)is not true.(17:G:c) Certain refinements of the conceptof specialstrictdeter-

minatenessseemto bear a closerrelationshipto the existenceof permanently optimal strategies.

17.11.TheInterchange of Players. Symmetry

17.11.1.Let us considerthe roleof symmetry, or more generally theeffects of interchanging the players 1 and 2 in the game F. This will

naturally be a continuation of the analysisof 14.6.As was pointed out there,this interchangeof the playersreplacesthe

function OC(T!,r 2) by -3C(r2, n). The formula (17:2)of 17.4.1.and 17.G.>

showsthat the effectof this for K( {, TJ ) is to replaceit by K( r; , {). Inthe terminology of 16.4.2.,we replacethe matrix (of5C(ri,r 2) cf. 14.1.3.)byits negative transposedmatrix.

Thus the perfectanalogy of the considerationsin 14.continues;againwe have the same formal results as there, provided that we replaceTI, r 2, 3C(ri,T2) by , rj , K( f , T;). (Cf. the previous occurrenceof thisin 17.4.and 17.8.)

We saw in 14.6.that this replacementof 3C(ri,r 2) by 3C(r2, n) carriesVi, v2 into v2, VL A literal repetitionof thoseconsiderationsshows

now that the correspondingreplacementof K( , 77 ) by K( 17 , )carriesvj, v 2 into v'2, vj. Summing up:Interchangingthe players1,2,carriesVi, v2, v'lt v'2 into v2, Vi, v'2, v't.

The result of 14.6.establishedfor (special)strictdeterminatenesswasthat v = Vi = v2 is carriedinto v = Vi = v2. In the absenceofthat property no such refinement of the assertionwas possible.

At the presentwe know that we always have generalstrictdeter-minateness,so that v' = v'i = v 2. Consequently this is carriedinto))

Verbally the contentof this resultisclear:Sincewesucceededin defininga satisfactoryconceptof the value of a play of F (for the player1),v', it isonly reasonablethat this quantity should changeits sign when the rolesof the playersareinterchanged.

17.11.2.We can also staterigorously when the game F is symmetric.This is the casewhen the two players 1 and 2 have preciselythe same)))

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166 ZEROSUM TWO-PERSONGAMES: THEORY

rolein it, i.e.if the gameF is identical with that gamewhich obtainsfrom it by interchanging the two players1,2. According to what was saidabove,this meansthat

JC(n,TI) = -JC(r,TI),or equivalently that

K(T,7)= -x(7,7).This property of the matrix 3C(ri,r 2) or of the bilinearform K( { , rj ) wasintroducedin 16.4.4.and calledskew-symmetry. 1'2

In this caseVi, v2 must coincidewith v2, Vi; henceVi = v2,and sinceVi ^ v2, so Vi ^ 0. But v' must coincidewith v';thereforewe can even assertthat

v' = O.8

Sowe see:Thevalue of eachplay of a symmetricalgame is zero.It shouldbe noted that the value v' of eachplay of a game F couldbe

zero without F being symmetric. A gamein which v' = will be calledfair.

Theexamplesof 14.7.2.,14.7.3.illustrate this:Stone,Paper,Scissorsissymmetric (and hencefair); Matching Penniesis fair (cf. 17.1.)without

beingsymmetric.4

>1 For a matrix 3C(n, r2) or for the corresponding bilinear form K( , 77 ) symmetry

is defined by

3C(n, Tj) = X(r 2, TI),or equivalently by))

It is remarkable that symmetry of the game F is equivalent to skew-symmetry, and not tosymmetry, of its matrix or bilinear form.

1Thus skew-symmetry means that a reflection of the matrix schemeof Fig. 15 in14.1.3.on its main diagonal (consisting of the fields (1,1),(2, 2),etc.)carriesit into itsown negative. (Symmetry, in the senseof the precedingfootnote, would mean that itcarriesit into itself.)

Now the matrix schemeof Fig. 15is rectangular; it has 2columns and /3i rows. Inthe caseunder consideration its shapemust be unaltered by this reflection. Henceitmust be quadratic, i.e.0i = #2. This is so,however, automatically, sincethe players1,2 are assumed to have the same role in r.

3This is, of course,due to our knowing that v\\ = v2. Without this i.e.withoutthe general theorem (16:F)of 16.4.3.we should assertfor the vj, v2only the samewhichwe obtained abovefor the Vi, v2:vj = vj and sincevj ^ v^, sov\\ ^ 0.

4The players 1 and 2 have different rolesin Matching Pennies:1 tries to match,and 2 tries to avoid matching. Ofcourse,onehas a feeling that this difference is inessen-tial and that the fairness of Matching Penniesis due to this inessentiality of the assyme-try. This could beelaboratedupon, but we do not wish to do this on this occasion. Abetter exampleof fairness without symmetry would begiven by a game which is grosslyunsymmetric, but in which the advantages and disadvantages of each player are sojudiciously adjusted that a fair game i.e.value v' = results.

A not altogether successfulattempt at such a game is the ordinary way of \"Rolling

Dice.\" In this game player 1 the \"player

\" rolls two dice,eachof which bear thenumbers 1, , 6. Thus at eachroll any total 2, ,12may result. Thesetotals)))

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MIXEDSTRATEGIES.THESOLUTION 167

In a symmetricalgame the setsA, S of (17:B:a),(17:B:b)in 17.8.areobviously identical. SinceA = B wemay put =

T\\ in the final criterion(17:D)of 17.9.We restateit for this case:

(17:H) In a symmetrical game, belongsto A if and only if this is(*i

true:Foreachr 2 = 1, 0* for which ^ 3C(ri,T2) ri doesr,-l

not assumeits minimum (in r2 ) we have T = 0.Usingthe terminology of the concludingremarkof 17.9.,we seethat the

above conditionexpressesthis:{ is optimal against itself.17.11.3.The results of 17.11.1.,17.11.2.that in every symmetrical

game v' = can be combinedwith (17:C:d)in 17.8.Then we obtainthis:(17:1) In a symmetrical game eachplayercan,by playing appropri-

ately, avoid loss1 irrespectiveof what the opponentdoes.We can statethis mathematically as follows:If the matrix 3C(n, r 2) is skew-symmetric,then thereexistsa vector))

n)) with))

for)) r2 =))

This couldalso have been obtained directly, becauseit coincideswith

the last result (16:G)in 16.4.4.To seethis it suffices to introduce thereour presentnotations:Replacethe i,j,a(i,j) thereby our r\\ t r 2, X(n, r 2)

and the w thereby our {.have the following probabilities:))

Total .....)2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)Chanceout of 36) 1) 2) 3) 4) 5) 6) 5) 4) 3) 2) 1)

Probability) A) A) A) A) A) A) A) A) A) A) A))

Therule is that if the \"player

\" rolls 7 or 11(''natural\") he wins. If he rolls 2, 3, or 12he loses. If he rolls anything else(4, 5, 6, or 8, 9, 10)then he rolls again until he rollseither a repeatof the original one (in which casehe wins), or a 7 (in which casehe loses).Player2 (the \"house\") has no influence on the play.

In spite of the great differences of the rules as they affect players 1 and 2 (the\"player\" and the \"house\") their chancesarenearly equal:A simple computation, whichwe do not detail, shows that the \"player\" has 244chancesagainst 251for the \"house,\"out ofa total of495;i.e.the value ofa play played for a unit stake is))

244- 251495))

---1.414%.))

Thus the approximation to fairness is reasonably good,and it may bequestioned whethermore was intended.

1 I.e.securehimself a gain * 0.)))

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168 ZERO-SUMTWO-PERSONGAMES: THEORY

It is even possibleto base our entiretheory on this fact, i.e.to derivethe theoremof 17.6.from the above result. In otherwords:Thegeneralstrictdeterminatenessof all F can bederivedfrom that oneof thesymmetricones. Theproof has a certaininterestof its own, but we shallnot discussit heresincethe derivation of 17.6.is more direct.

Thepossibilityof protectingoneselfagainst loss (in a symmetricgame)existsonly due to our useof themixedstrategies , ij (cf.the endof 17.7.).If theplayersarerestrictedto purestrategiesTI,r2 then the dangerof havingone'sstrategy found out, and consequentlyof sustaining losses,exists.To seethis it suffices to recallwhat we found concerningStone,Paper,Scissors(cf. 14.7.and 17.1.1.).We shall recognizethe samefact in con-nectionwith Pokerand the necessityof \"bluffing\" in 19.2.1.)))

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CHAPTER IV

ZERO-SUMTWO-PERSONGAMES: EXAMPLES

18.SomeElementaryGames18.1.TheSimplest Games

18.1.1.We have concludedour generaldiscussionof the zero-sum two-persongame. We shall now proceedto examine specificexamplesof suchgames.Theseexampleswill exhibit betterthan any generalabstractdiscussionscould, the true significance of the various componentsof ourtheory. They will show, in particular, how someformal steps which aredictatedby our theory permit a direct common-senseinterpretation. Itwill appear that we have herea rigorousformalization of the main aspectsof such \"practical\"and \"psychological\"phenomenaas thoseto bemen-tioned in 19.2.,19.10.and 19.16.1

18.1.2.Thesizeof the numbers0i, 2 i.e.the number of alternativesconfronting the two players in the normalized form of the game gives areasonablefirst estimatefor the degreeof complicationof the gameF.Thecasethat either,or both, of thesenumbersis 1may bedisregarded:Thiswould mean that the playerin questionhas no choiceat all by which he caninfluence the game.2 Therefore the simplest games of the class whichinterestsus arethose with

(18:1) 0i = fa = 2.We saw in 14.7.that MatchingPenniesis such a game;its matrix schemewas given in Figure12in 13.4.1.Another instanceof sucha game isFigure14,id.))

1) 2)

1) 3C(1,1)) 3C(1,2))

2) 3C(2,1)) 3C(2,2)))

Figure 27.

Letus now considerthe most generalgame falling under (18:1),i.e.underFigure27. Thisapplies,e.g.,to MatchingPenniesif the various waysof matching do not necessarilyrepresentthe same gain (ora gain at all),

1 We stressthis becauseof the widely held opinion that thesethings are congenitallyunfit for rigorous (mathematical) treatment.

1Thus the game would really be one of oneperson;but then, of course,no longer ofzerosum. Cf.12,2.

169)))

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170 ZERO-SUMTWO-PERSONGAMES: EXAMPLES

nor the various ways of not matching the sameloss (ora lossat all).1 We

proposeto discussfor this casethe resultsof 17.8.,the value of the gameFand the setsof goodstrategies4,5. Theseconceptshave beenestablishedby the generalexistentialproof of 17.8.(basedon the theorem of 17.6.);but we wish to obtain them again by explicitcomputation in this specialcase,and therebygain somefurther insight into their functioning and theirpossibilities.18.1.3.Thereare certaintrivial adjustments which can be made ona gamegiven by Figure27,and which simplify an exhaustive discussionconsiderably.

First it is quite arbitrary which of the two choicesof player1 wedenoteby TI = 1 and by TI = 2;we may interchangethese, i.e.the two rowsofthe matrix.

Second,it is equallyarbitrary which of the two choicesof player 2 wedenoteby TI = 1and by r2 = 2;we may interchangethese, i.e.the twocolumns of the matrix.

Finally, it is alsoarbitrary which of the two playerswe call 1 and which2;we may interchangethese, i.e.replace5C(ri,r 2) by 3C(ri,r 2) (cf.14.6.and 17.11.).This amounts to interchanging the rows and the columnsof the matrix, and changing the signof its elementsbesides.

Putting everything together,we have here 2X2X2= 8 possibleadjustments,all of which describeessentiallythe samegame.

18.2.DetailedQuantitative Discussionof TheseGames

18.2.1.We proceednow to the discussionproper. This will consistin the considerationof severalalternative possibilities,the \"Cases\"to beenumeratedbelow.

TheseCasesaredistinguishedby the various possibilitieswhich existfor the positionsof those fields of the matrix where X(TI, r 2) assumesitsmaximum and its minimum for both variables n, r 2 together. Theirdelimitations may first appear to be arbitrary; but the fact that theylead to a quickcataloguingof all possibilitiesjustifies them ex post.

Consider accordingly MaxVTj 3C(ri, r 2) and Min VTj JC(ri, r 2). Eachone of thesevalues will be assumedat leastonceand might beassumedmorethan once;2 but this doesnot concernus at this juncture. We beginnowwith the definition of the various Cases:

18.2.2.Case(A): It is possible to choosea field where MaxVTj isassumedand one where Min

Ti ,Tj is assumed,so that the two areneitherinthe same row nor in the samecolumn.

By interchanging TI = 1,2 as well as r 2 = 1,2 we can make the first-mentioned field (of MaxTi,Tj) to be (1,1). The second-mentionedfield

1 Comparison of Figs.12and 27shows that in Matching Pennies3C(1,1) -3C(2,2) - 1(gain on matching) ;3C(1,2) -3C(2,1)- -1(losson not matching).2In Matching Pennies (cf. footnote 1 above) the Max

Tj ,ri is 1 and is assumed at(1,1)and (2,2),while the Min V rf is -1and is assumed at (1,2) and (2, 1).)))

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SOME ELEMENTARYGAMES 171

(of Min ri .r)) must then be (2,2). Consequentlywe have

(18:2)))

Therefore (1,2) is a saddlepoint.1Thus the game is strictly determinedin this caseajid

(18:3) v' = v = 3C(1,2).18.2.3.Case(B):It is impossibleto make the choicesas prescribed

above:Choosethe two fields in question(of Max

Ti ,Tj and Min ri ,Tj); then theyarein the samerow or in the samecolumn. If the formershouldbethe case,

then interchangethe players 1,2, so that thesetwo fields are at any ratein the samecolumn.2

By interchanging TI = 1,2 as well as r 2 = 1,2 if necessary,we can againmake the first-mentioned field (of MaxVT2) to be (1,1). Sothe column in

questionis r 2 = 1. Thesecond-mentionedfield (of Min ri ,Tj) must then be(2,I).3 Consequentlywe have:))

Actually 3C(1,1) = JC(1,2) or JC(2,2) = 3C(2,1)areexcludedbecausefor the Max

Tj ,rj and Min T ,Tj fields they'would permit the alternative choicesof (1,2), (2, 1)or (1,1),(2, 2), thus bringing about Case(A).4

Sowe can strengthen(18:4)to

(18:5)

We must now makea further disjunction:18.2.4.Case(Bi):

(18:6) 3C(1,2) OC(2,2)Then (18:5)can be strengthenedto

(18:7) 3C(1,1)>3C(1,2) ^ OC(2,2) >3C(2,1).Therefore (1,2) is again a saddlepoint.

Thus the game is strictlydeterminedin this casetoo;and again

(18:8) V = v =3C(1,2).1 Recall13.4.2.Observethat we had to take (1,2) and not (2,1).2This interchange of the two players changes the sign of every matrix element (cf.

above),hence it interchanges MaxTi ,T2

and MinTi>Tj

. But they will nevertheless be in

the same column.3Tobeprecise:It might alsobe (1,1). But then 3C(n, r,) has the sameMax

Tl ,Tsand

Min r .T , and soit isa constant. Thenwe can use (2, 1)alsofor Minrj>Tj

.<3C(1,1) -3C(2,2) and JC(1,2) -3C(2,1)are perfectly possible,as the example of

Matching Pennies shows. Cf.footnote 1on p. 170and footnote 1on p. 172.)))

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172 ZERO-SUMTWO-PERSONGAMES: EXAMPLES

18.2.6.Case(B2):(18:9) 3C(1,2) <3C(2,2)Then (18:5)can be strengthenedto

(18:10) 3C(1,1) ^ 3C(2,2) >3C(l,2) JC(2,I).1Thegame is not strictly determined.2

It iseasyhowever, to find goodstrategies,i.e.a { in A and an rj in B,bysatisfying the characteristiccondition(17:D)of 17.9.We can doeven more:

-> 2 -We can choose77 so that ^ 3C(ri,TJ)T;TIis thesamefor all r\\ and so that

r,-l2

2}3C(fi,T2) ri is the samefor all r*. Forthis purposewe need:))

-U*'11; (30(1,l){i+3C(2,1){,=5C(1,2){i+3C(2,2)fa.

This means))

) - 3C(2,1) -3C(1,2):OC(1,1)-JC(2,1).

We must satisfy theseratios,subjectto the permanentrequirements

fa 0, fa ^ fa + fa = 1i?i S 0, r/2 ^ r?i + 172

= 1This is possiblebecausethe prescribedratios (i.e.the right-hand sides in

(18:12))arepositiveby (18:10).We have

, _ 3C(2,2) - 3C(2,1)_))_ _\" 3C(1,1)+ 5C(2,2) -3C(1,2) - 3C(2,1)'> ^_X(l, 1) -30(1,2)_** 3C(1,1)+ 3C(2,2)-3C(1,2) - JC(2,1)*

and further _ OC(2,2) -3C(1,2)_171 3C(1,1)+ 3C(2,2) - 3C(1,2) - 3C(2,1)'

__ge(i,i)-ae(2,i)_772 5C(1,1)+ 3C(2,2) - 3C(1,2)- JC(2,1)'

We can even showthat these , 17 areunique,i.e.that A, Bpossessno otherelements.

1This is actually the casefor Matching Pennies. Cf.footnotes i on p. 170and 4on p. 171.

*Clearlyvi - MaxTi Min f|5C(n,rt) -3C(1,2),v, - MmTj MaxTl3C(r,,r t) -JC(2,2),80V t < Vj.)))

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SOME ELEMENTARY GAMES 173

Proof: If either or rj were somethingelsethan we found above,then

77 or respectivelymust have a component0,owing to the characteristiccondition (17:D)of 17.9.But then rj or { would differ from the abovevalues sincein theseboth componentsarepositive. Sowe see:If either

> >

or 77 differs from the above values, then both do. And then both musthave a component0. Forboth the othercomponentis then 1,i.e.botharecoordinatevectors.1 Hencethe saddlepoint of K( , T;) which they repre-sent is really oneof 3C(ri,r 2), cf. (17:E) in 17.9.Thus the game would bestrictly determined;but we know that it is not in this case.

This completesthe proof.All four expressionsin (18:11)arenow seento have the samevalue,

namely3C(1,1)3C(2,2) - 3C(1,2)3C(2,1)

JC(1,1)+ 3C(2,2)-JC(1,2)-X(2,1)and by (17:5:a),(17:5:b)in 17.5.2.this is the value of v'. Sowe have

na-nn v' - 3C(1,1)3C(2,2)-3C(1,2)OC(2,1)^10.10, v x( ^ +^ 2) _ ^ 2) _ ^ l y

18.3.Qualitative Characterizations

18.3.1.The formal results in 18.2.can be summarizedin various wayswhich make their meaning clearer.We beginwith this criterion:

Thefields (1,1),(2,2)form one diagonal of thematrix schemeof Fig.27,the fields (1,2), (2,1)form the otherdiagonal.

We say that two setsof numbersE and F areseparatedeitherif everyelementof E is greaterthan every elementof F, or if every elementof Eis smallerthan every elementof F.

Considernow the Cases(A), (Si),(B2) of 18.2.In the first two casesthe game is strictly determinedand the elementson onediagonalof thematrix arenot separatedfrom thoseon the other.2 In the last casethegameis not strictly determined,and the elementson one diagonalof thematrix areseparatedfrom thoseon the other.8

Thus separationof the diagonalsis necessaryand sufficient for thegamenot being strictly determined.This criterionwas obtained subject tothe usemade in 18.2.of the adjustmentsof 18.1.3.But the threeprocessesof adjustment describedin 18.1.3.affect neitherstrictdeterminateiiessnor separationof the diagonals.4 Henceour first criterionis always valid.We restateit:

Ml,0|or (0,1).2 Case(A):3C(1,1) OC(1,2) ^ 3C(2,2) by (18:2).Case(Bi):JC(1,1)>JC(1,2)

^ 3C(2,2) by (18:7).8 Case(Bi):3C(1,1) JC(2,2) >3C(1,2) OC(2,1)by (18:10).4Thefirst is evident sincetheseareonly changesin notation, inessential for the game.

Thesecondis immediately verified.)))

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174 ZERO-SUMTWO-PERSONGAMES:EXAMPLES

(18:A) Thegame isnot strictlydeterminedif and only if the elementson one diagonal of the matrix are separatedfrom those on theother.

18.3.2.In case(#2), i.e.when the game is not strictlydetermined,both

the (unique){of A and the (unique) iy of B which we found, have both com-ponents 5** 0. This, as well as the statement of uniqueness,is unaffectedby adjustmentsdescribedin 18.1.3.1 Sowe have:(18:B) If the game is not strictly determined,then thereexistsonly

> -one goodstrategy { (i.e.in A) and only one goodstrategy 77

(i.e.in 5),and both have both their componentspositive.I.e.both playersmust really resortto mixedstrategies.According to (18:B)no componentof or rj ( in A, vj in B) is zero.

Hencethe criterion of 17.9.shows that the argument which preceded(18:11)which was then sufficient without being necessary is nownecessary(and sufficient). Hence(18:11)must be satisfied,and thereforeall of its consequencesaretrue. This appliesin particular to the values$1, {2, i?i, 772 given after (18:11),and to the value of v' given in (18:13).All theseformulae thus applywhenever the game is not strictlydetermined.

18.3.3.We now formulate another criterion:In a generalmatrix 3C(ri, r 2) cf. Fig.15on p.99 (we allow for a

moment any 0i, 2) we say that a row (sayT() or a column (sayr'2) majorizesanother row (sayr\") or column (sayr'2'),respectively,if this is true for theircorrespondingelements without exception.I.e.if 3C(rJ, r 2) ^ 3C(r7, r 2)for all T2, or if 3C(n,T{) ^ 3C(rj,/,')for all r*.

This concepthas a simplemeaning:It means that the choiceof r(is at leastas goodfor player 1as that of r\" or that the choiceof r( is atmost as goodfor player 2 as that of r'2' and that this is so in both casesirrespectiveof what the opponentdoes.2

Let us now return to our present problem (/?i = 2 = 2). Consideragain the Cases(A), (J5t), (/?2) of 18.2.In the first two casesa row or acolumn majorizesthe other.8 In the last caseneither is true.4

Thus the fact that a row or a column majorizesthe other isnecessaryandsufficient for T beingstrictly determined.Like our first criterionthis issubjectto the usemadein 18.2.of the adjustmentsmade in 18.1.3.And, asthere,thoseprocessesof adjustment affect neither strict determinatenessnor majorization of rows or columns. Henceour present criteriontoo isalways valid. We restateit:(18:C) Thegame r is strictly determinedif and only if a row or a

column majorizesthe other.1Thesetoo areimmediately verified.'Thisis, of course,an exceptionaloccurrence:In general the relative merits of two

alternative choiceswill dependon what the opponent does.'Case(A): Column 1 majorizes column 2, by (18:2)Case(Bi):Row 1 majorizes

row 2 by (18:7).4Case(J?t):(18:10)excludesall four possibilities, as is easily verified.)))

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SOME ELEMENTARYGAMES)) 175))

18.3.4.That the conditionof (18:C)issufficientfor strictdeterminatenessis not surprising:Itmeansthat for one of the two playersone of his possiblechoicesisunderall conditionsat leastasgoodasthe other (cf.above). Thushe knowswhat to do and his opponentknowswhat to expect,which is likelyto imply strict determinateness.

Of coursetheseconsiderationsimply a speculationon the rationalityof the behavior of the other player, from which our original discussionisfree. The remarks at the beginning and at the end of 15.8.apply to acertain extentto this, much simpler,situation.

What really matters in this result (18:C)however is that the necessityof the condition is also established;i.e.that nothing more subtle thanoutright majorization of rows or columns can causestrict determinateness.

It shouldbe rememberedthat we areconsideringthe simplestpossiblecase:0i = 2 = 2. We shall seein 18.5.how conditionsgetmore involvedin all respectswhen pi, fa increase.18.4.Discussionof SomeSpecificGames. (GeneralizedForms of Matching Pennies)

18.4.1.The following are someapplicationsof the resultsof 18.2.and18,3.

(a) MatchingPenniesin its ordinary form, where the JC matrix of Figure27 is given by Figure12on p.94. We know that this game has the value

v' =and the (unique) goodstrategies

7=7={*,*}(Cf.17.1.The formulae of 18.2.will, of course,give this immediately.)

18.4.2.(b) MatchingPennies,where matching on headsgives a doublepremium. Thus the matrix of Figure27 differs from that of Figure12bythe doublingof its (1,1)element:))

1) 2)

1)(\\) -1)

2) -1) 1))

Figure 28o.The diagonalsare separated(1and 2 are > than 1),hencethe goodstrate-giesareunique and mixed (cf. (18:A), (18:B)).By using the pertinentformulae of case(#2) in 18.2.5.,we obtain the value

and the goodstrategies))

Itwill beobservedthat the premium put on matching headshasincreasedthe value of a play for player 1 who triesto match. It alsocauseshim to)))

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176)) ZERO-SUMTWO-PERSONGAMES: EXAMPLES))

chooseheadslessfrequently, sincethe premium makesthis choiceplausibleand therefore dangerous.Thedirectthreatof extralossby beingmatchedon heads influences player 2 in the same way. This verbal argument hassomeplausibilitybut is certainly not stringent. Our formulae which yieldedthis result, however, werestringent.

18.4.3.(c) MatchingPennies,where matching on headsgives a doublepremium but failing to match on a choice(by player 1) of heads gives atriple penalty. Thus the matrix of Figure27 is modified as follows:))

1) 2)

1) 2) -3)2) -1) 1))

Figure 286.

Thediagonalsareseparated(1and 2, are> than 1, 3),hencethe goodstrategiesareuniqueand mixed(cf. as before). Theformulae usedbeforegive the value))

v = -))

and the goodstrategies))

We leave it to the readerto formulate a verbal interpretationof thisresult,in the same senseas before.The constructionof otherexamplesof this type is easy along the linesindicated.

18.4.4.(d) We saw in 18.1.2.that thesevariants of MatchingPenniesare,in a way, the simplestforms of zero-sumtwo-persongames.By thiscircumstancethey acquirea certaingeneralsignificance,which is furthercorroboratedby the results of 18.2.and 18.3.:indeedwe found therethatthis class of gamesexhibits in their simplest forms the conditionsunderwhich strictly and not-strictly determinedcasesalternate.As a furtheraddendum in the samespirit we point out that the relatednessof thesegamesto Matching Penniesstressesonly one particular aspect. Othergameswhich appear in an entirely different material garb may, in reality,well belongto this class. We shallgive an exampleof this:

Thegameto beconsideredis an episodefrom theAdventures of SherlockHolmes.1-2

1Conan Doyle:The Adventures of SherlockHolmes,New York, 1938,pp.550-551.1Thesituation in question is of courseagain to beappraisedas a paradigm of many

possibleconflicts in practicallife. It was expounded as such by 0.Morgenstern: Wirt-schaftsprognose,Vienna, 1928,p.98.

Theauthor doesnot maintain, however, somepessimistic views expressedid.or in\" Vollkommene Voraussicht und wirtschaftliches Gleichgewicht,\" Zeitschrift fur Nation-alokonomie, Vol. 6, 1934.

Accordingly our solution also answers doubts in the samevein expressedby K.Menger: Neuere Fortschritte in den exacten Wissenschaften, \"Einige neuere Fort-schritte in der exactenBehandlung Socialwissenschaftlicher Probleme,\" Vienna, 1936,pp.117and 131.)))

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SOME ELEMENTARY GAMES)) 177))

SherlockHolmesdesiresto proceedfrom Londonto Dover and henceto the Continent in orderto escapefrom ProfessorMoriarty who pursueshim. Having boarded the train he observes,as the train pulls out, theappearanceof Professor Moriarty on the platform. SherlockHolmestakes it for granted and in this he is assumedto be fully justified thathis adversary,who has seenhim, might securea specialtrain and overtakehim. SherlockHolmesis faced with the alternative of going to Dover orof leaving the train at Canterbury, the only intermediatestation. Hisadversary whoseintelligenceis assumedtobe fully adequateto visualizethesepossibilities has the samechoice.Bothopponentsmust choosetheplaceof their detrainmentin ignoranceof theother'scorrespondingdecision.If, as a result of thesemeasures,they should find themselves,in fine, onthe sameplatform, SherlockHolmesmay with certainty expectto be killedby Moriarty. If SherlockHolmesreachesDover unharmed he can makegoodhis escape.

What arethe goodstrategies,particularlyfor SherlockHolriies? Thisgamehas obviously a certainsimilarity to Matching Pennies,ProfessorMoriarty being the one who desiresto match. Let him therefore beplayer1,and SherlockHolmesbeplayer2. Denotethechoiceto proceedtoDoverby 1and the choiceto quit at the intermediatestationby 2. (Thisappliesto both r\\ and T2.)

Let us now considerthe X matrix of Figure27. The fields (1,1) and(2,2)correspondto ProfessorMoriartycatchingSherlockHolmes,which itis reasonableto describeby a very high value of the correspondingmatrixelement, say 100.The field (2, 1) signifiesthat SherlockHolmessuc-cessfullyescapedto Dover,while Moriartystoppedat Canterbury. This isMoriarty'sdefeat as far as the presentactionis concerned,and*shouldbedescribedby a big negative value of the matrix element in the orderofmagnitudebut smallerthan the positivevalue mentionedabove say, 50.The field (1,2) signifies that SherlockHolmesescapesMoriarty at theintermediatestation, but fails to reachthe Continent. This is best viewedas a tie,and assignedthe matrix element0.

The5Cmatrix is given by Figure29:))

1) 2)

1) 100)

2) -50) 100))

Figure 29.

As in (b), (c)above,the diagonalsareseparated(100is > than 0, 50);hencethe good strategiesare again unique and mixed.The formulaeusedbeforegive the value (for Moriarty)))

40)))

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178 ZERO-SUMTWO-PERSONGAMES: EXAMPLES

and the goodstrategies( for Moriarty, rj for SherlockHolmes):

7 = {*,!}, 7 = {*,D.Thus Moriarty should go to Dover with a probabilityof 60%,while

SherlockHolmesshouldstop at the intermediatestation with a probabilityof60%, the remaining40% beingleft in eachcasefor the otheralternative. l

18.5.Discussionof SomeSlightly More Complicated Games

18.6.1.Thegeneralsolutionof the zero-sum two-persongame which weobtained in 17.8.brings certain alternatives and conceptsparticularlyinto the foreground:Thepresenceor absenceof strict determinateness,thevalue v' of a play, and the setsA, S of goodstrategies.Forall theseweobtainedvery simpleexplicitcharacterizationsand determinationsin 18.2.Thesebecameeven more striking in the reformulation of those results in18.3.

This simplicitymay even lead to some misunderstandings. Indeed,theresultsof 18.2.,18.3.were obtainedby explicitcomputationsof the mostelementary sort. The combinatorial criteriaof (18:A), (18:C)in 18.3.for strict determinatenesswere at leastin their final form alsoconsider-ably more straightforward than anything we have experiencedbefore.Thismay give occasionto doubtswhether the somewhat involved consider-ations of 17.8.(and the correspondingconsiderationsof 14.5.in the caseof strict determinateness)were necessary, particularly since they arebasedon the mathematical theorem of 17.6.which necessitatesour analysisof linearity and convexity in 16. If all this couldbe replacedby discussionsin thestyleof 18.2.,18.3.then our modeof discussionof 16.and 17.would beentirely unjustified.2

This is not so. As pointedout at the end of 18.3.,the greatsimplicityof the proceduresand resultsof 18.2.and 18.3.is due to the fact that theyapplyonly to the simplesttype of zero-sumtwo-persongames:the MatchingPenniesclassof games,characterizedby ft\\ = 2 = 2. For the generalcasethe more abstract machinery of 16.and 17.seemssofar indispensable.

1The narrative of Conan Doyle excusably disregards mixed strategies and statesinstead the actual developments. According to these Sherlock Holmes gets out at theintermediate station and triumphantly watches Moriarty 's specialtrain going on toDover. Conan Doyle'ssolution is the best possibleunder his limitations (to purestrategies),insofar ashe attributes to eachopponent the coursewhich we found to be themore probable one (i.e.he replaces60% probability by certainty). It is, however,somewhat misleading that this procedureleadsto Sherlock Holmes'scompletevictory,whereas, as we saw above,the odds(i.e.the value of a play) are definitely in favor of

Moriarty. (Our result for {, 17 yields that Sherlock Holmesis as goodas 48%deadwhen his train pulls out from Victoria Station. Comparein this connection the sugges-tion in Morgenstern, loc.cit.,p.98,that the whole trip is unnecessary becausethe losercouldbe determined beforethe start.)

'Of courseit would not lack rigor, but it would be an unnecessary use of heavymathematical machinery on an elementary problem.)))

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SOME ELEMENTARYGAMES)) 179))

It may help to seethesethings in their right proportionsif we show bysome exampleshow the assertionsof 18.2.,18.3.fail for greatervaluesof 18.

18.5.2.It will actually suffice to considergames with 0i= fa = 3.In fact they will be somewhat relatedto MatchingPennies, more generalonly by introduction of a third alternative.

Thusboth playerswill have the alternative choices1,2, 3 (i.e.the valuesfor TI, r 2). Thereaderwill best think of the choice1in terms of choosing\"heads,\"the choice2 of choosing\"tails\"and the choice3 as somethinglike\"callingoff.\" Player 1again tries to match. If eitherplayer \"callsoff,\"then it will not matter whether the otherplayerchooses\"heads\"or \"tails,\"

the only thing of importanceis whether he choosesone of thesetwo atall or whether he \"callsoff\" too. Consequentlythe matrix has now theappearanceof Figure30:))

\\) 1) 2) 3)

1) 1) -1) 7)

2) -1) 1) y)

3) a) a) ft))

Figure 30.

The four first elements i.e.the first two elementsof the first two rowsarethe familiar pattern of MatchingPennies(cf.Fig.12).Thetwo fieldswith a areoperative when player 1 \"callsoff\" and player2 doesnot. Thetwo elementswith 7 areoperative in the oppositecase. Theelementwith

|8refers to the casewhere both players\" call off.\" By assigningappropriatevalues (positive,negative or zero) we can put a premium or a penaltyon any one of theseoccurrences,or make it indifferent.

We shallobtain all the exampleswe needat this junctureby specializingthis scheme, i.e.by choosingthe above a, 0,y appropriately.

18.5.3.Our purposeis to show that none of the results (18:A),(18:B),(18:C)of 18.3.is generally true.

Ad (18:A):This criterionof strict determinatenessis clearly tied to thespecialcase$\\ = fa = 2:For greatervalues of 0i, 2 the two diagonalsdonot even exhaustthe matrix rectangle,and therefore the occurrenceon thediagonalalonecannot be characteristicas before.

Ad (18:B):We shall give an exampleof a gamewhich is not strictlydetermined,but where neverthelessthereexistsa goodstrategy which ispure for one player (but of coursenot for theother). Thisexamplehas thefurther peculiarity that one of the playershas severalgoodstrategies,whilethe otherhas only one.)))

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180 ZERO-SUMTWO-PERSONGAMES: EXAMPLES

We choosein the game of Figure30 , ft, y as follows:))

\\) 1) 2) 3)

1) 1) -1)2) -1) 1)

3) a) a) -a))

Figure 31.a >0,6 > 0. Thereaderwill determinefor himself which combinationsof\" calling off\" areat a premium or arepenalizedin the previously indicatedsense.

This is a completediscussionof the game,usingthe criteriaof 17.8.For = {i,i,0}always K({ , rj ) = 0,i.e.with this strategy player 1

cannot lose.Hencev' jg 0. For t\\= 6 8 = {0,0,1}always K( , rj )

^ O;1i.e.with this strategy player2 cannot lose.Hencev' g 0. Thuswehave

v' =

Consequently is a goodstrategyif and only if always K( { , rj ) ^ andTJ

is a goodstrategy if and only if always K( , rj ) ^ O.2 Theformer is easilyseento betrue if and only if

*i = {2 = i {3= 0,and the latterif and only if

5))

172))

2(a))

Thus the setA of all goodstrategies containspreciselyone element,and this is not a pure strategy. ThesetBof all goodstrategies77 , on theotherhand, contains infinitely many strategies,and one of them is pure:namely rj

= d 8 = {0,0,1}.ThesetsA, B can be visualized by makinguse of the graphicalrepre-

sentationof Figure21(cf.Figures32,33):Ad (18:C):We shall give an exampleof a gamewhich is strictly deter-

mined but in which no two rowsand equallyno two columnsmajorizeeachother. We shall actually do somewhatmore.

18.5.4.Allow for a moment any 0i,/32. Thesignificanceof themajoriza-tion of rowsor of columnsby eachotherwasconsideredat the end of 18.3.It was seentomean that oneof the players had a simpledirectmotive

1 It is actually equal to 3 .*We leaveto the readerthe simple verbal interpretation of thesestatements.)))

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SOME ELEMENTARYGAMES)) 181))

for neglectingone of his possiblechoicesin favor of another, and thisnarrowedthe possibilitiesin a way which couldbeultimately connectedwith strict determinateness.

Specifically:If the row T\" is majorized by the row r\\ i.e.if3(T\"> TS) ^ 3CM, T2) for all r2 then player 1 need never consider thechoiceTit sinceT\\ is at leastas goodfor him in every contingency. And:If the column r 2

' majorizesthe column/2 i.e.if JC(TI,r'2') ^ 3C(n,r2) for alln then player2 neednever considerthe choicer'2',sincer'2 isat leastasgoodfor him in every contingency. (Cf. loc.cit.,particularly footnote 2 onp.174. Theseareof courseonly heuristicconsiderations,cf. footnote 1,p.182.)))

Figure 33.Now we may use an even more generalset-up:If the row r\", i.e.

the player1'spurestrategycorrespondingto r\" ismajorizedby an average

of all rows T\\ j T\" i.e.by a mixed strategy with the componentT\" = then it is still plausibleto assumethat player 1neednever con-

sider the choiceof r\", sincethe other T\\ areat leastasgoodfor him in everycontingency. Themathematical expressionof this situation is this:

0i

it T2) ^ y 3C(ri,r 2) T for all T2))

n)) 0.))

The correspondingsituation for player 2 arisesif the columnrj' i.e.player 2;s pure strategy correspondingto r2

' majorizesan average of all>

columnsr'2 ^ //, i.e.a mixedstrategy ?? with the component t\\ r\" = 0.

Themathematical expressionof this situation is this:0*

(n, r2;) Y JC(n,rj)^ for all n))

in 8^)) = 0.))

Theconclusionsarethe analoguesof the above.)))

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182)) ZERO-SUMTWO-PERSONGAMES: EXAMPLES))

Thus a gamein which (18:14:a)or (18:14:b)occurs,permits of animmediateand plausiblenarrowing of the possiblechoicesfor oneof theplayers.1

18.5.5.We arenow going toshow that the applicabilityof (18:14:a),(18:14:b)is very limited:We shall specifya strictly determinedgameinwhich neither(18:14:a)nor (18:14:b)is ever valid.

Let us therefore return to the classofgamesof Figure30. (0i = /32 = 3).We choose < a < 1, = 0,y = -a:))

1) 2) 3)

1) 1) -1) a)

2) 1) i) a)

3) a) a)

Figure 34.

The readerwill determinefor himself which combinationsof \" calling off\"

areat a premium or arepenalizedin the previously indicatedsense.This is a discussionof the game:The element(3, 3) is clearly a saddle point, so the game is strictly

determinedandv = v' = 0.

It is not difficult to seenow (with the aid of the method used in 18.5.3.),that the setA of all goodstrategies as wellas the setB of all goodstrate-

gies 17 , containspreciselyone element:the pure strategy d 8 = {0,0,1).On the otherhand, the readerwill experiencelittle trouble in verifying

that neither(18:14:a)nor (18:14:b)is ever valid here,i.e.that in Figure34no row is majorizedby any averageof the two other rows, and that nocolumn majorizesany average of the othertwo columns.

18.6.Chanceand Imperfect Information

18.6.1.The examplesdiscussedin the precedingparagraphs make itclearthat the roleof chance more precisely,of probability in a gameisnot necessarilythe obviousone,that which is directlyprovidedfor in therules of the game. Thegamesdescribedin Figures27 and 30 have rules

1This is ofcoursea thoroughly heuristic argument. We do not needit, sincewe havethe completediscussionsof 14.5.and of 17.8.But one might suspectthat it can beusedto replaceor at leastto simplify those discussions.Theexamplewhich we aregoing togive in the text seemsto dispelany such hope.

Thereis another coursewhich might produceresults:If (18:14:a)or (18:14:b)holds,then a combination of it with 17.8.can beusedto gain information about the setsofgoodstrategies,A and B. We donot proposeto take up this subjecthere.)))

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SOME ELEMENTARYGAMES 183which do not provide for chance;the moves arepersonalwithout excep-tion.l Neverthelesswefound that mostof them arenot strictlydetermined,i.e.that their good strategiesaremixed strategiesinvolving the explicituse of probabilities.

On the other hand, our analysisof thosegamesin which perfectinforma-tion prevailsshowedthat thesearealways strictly determined, i.e.thatthey have goodstrategieswhich arepure strategies,involving no probabili-ties at all. (Cf. 15.)

Thus from the point of view of the players'behavior i.e.of the strate-giesto be used the important circumstanceis whether thegame is strictlydeterminedor not, and not at all whether it containsany chancemoves.

The results of 15.on games in which perfect information prevailsindicatethat thereexistsa closeconnectionbetweenstrictdeterminatenessand the ruleswhich govern the players'stateof information. To establishthis point quite clearly, and in particular to show that the presenceofchancemoves is quite irrelevant, we shall now showthis:In every (zero-sum two-person)game any chancemove can be replacedby a combinationof personalmoves, so that the strategicalpossibilitiesof the game remainexactlythe same. Itwill be necessaryto allowfor rulesinvolving imperfectinformation of the players,but this is just what we want to demonstrate:That imperfect information comprises(among other things) all possibleconsequencesof explicitchancemoves.2

18.6.2.Let us consider,accordingly,a (zero-sumtwo-person)game F,and in it a chancemove 3TC*.3 Enumerate the alternatives as usual byaK

= 1, , aK and assumethat their probabilitiesp(Kl), , p(?** are

all equal to l/ex<.4 Now replace3TI, by two personal moves 2fTl^, 9TC\".

1Thereduction of all gamesto the normalized form shows even more:It proves thatevery game is equivalent to one without chancemoves, sincethe normalized form con-tains only personalmoves.

2 A direct way of removing chancemoves existsofcourseafter the introduction of the(pure) strategiesand the umpire's choice,as describedin 11.1.Indeed as the laststep in bringing a game into its normalized form we eliminated the remaining chancemoves by the explicit introduction of expectationvalues in 11.2.3.

But we now proposeto eliminate the chancemoves without upsetting the structureof the game so radically. We shall replaceeachchancemove individually by personalmoves (by two moves, as will beseen),so that their respectiverolesin determining theplayers'strategies will always remain differentiated and individually appraisable.Thisdetailedtreatment is likely to give a clearerideaof the structural questions involved thanthe summary procedurementioned above.

3For our present purposes it is irrelevant whether the characteristicsof 9Tl dependupon the previous courseof the play or not.

4This is no real lossof generality. To seethis, assume that the probabilities in

question have arbitrary rational values, say r\\/t, , ra /t (r\\, , ra and /integers). (Herein liesan actual restriction but an arbitrary small one sinceanyprobabilities canbeapproximated by rational onesto any desireddegree.)

Now modify the chancemove 9fR so that it has r\\ + - -f rajc

t alternatives

(instead of a,),designatedby </K 1, , t (instead of <r 1,- , a);and sothateachof the first n values of Jn has the same effecton the play as <r = 1,eachof thenext rj values of a'K the same as *K 2, etc.,etc. Then giving all *'K 1, , t thesame probability l/l,has the same effectas giving <r 1, , aK the original prob-abilities Ti/t, ' ' , Tat.)))

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184)) ZERO-SUMTWO-PERSONGAMES: EXAMPLES))

Wl( and 9(Tl\" arepersonalmoves of players1and 2 respectively. Both havea,alternatives;we denotethe correspondingchoicesby o-^

= 1,- , aK

and <r\" = 1, , aK. It is immaterial in which orderthesemoves aremade,but we prescribethat both moves must bemade without any infor-mation concerningthe outcomeof any moves (includingthe othermove3fl, 9TC\-") We define a function S(<r', <r\") by this matrix scheme.(Cf.Figure35. Thematrix elementis 5(a',a\-")1)- Theinfluence of 3ffl^, 9TC\"

i.e.of the corresponding(personal)choicesa, <r\" on the outcomeof thegameis the same as that of STl* would have beenwith the corresponding(chance)choicea* = 6(<r, <r\.") We denotethis new game by F*. Weclaim that the strategicalpossibilitiesof F* arethe sameas thoseof F.))

a*))

a))

-2))

-1))

Figure 35.

18.6.3.Indeed:Let player1use in F* a given mixedstrategy of F with

the further specificationconcerningthe move 9Tl^2 to chooseall a'K = 1,-,,with the same probability1/a*. Thenthe gameF* with this

strategy of player 1 will befrom player2'spoint of view the sameas F.This is so becauseany choiceof his at 9Tl^ (i.e.any cr\" = 1, , a)producesthe same result as the original chancemove 9R,: One look atFigure35 will show that the a\" = cr\" column of that matrix containseverynumber <r = 6(a',<r\")

=!,,,precisely once, i.e.that 5(er', <r\will assumeevery value!,,(owing to player1'sstrategy) with thesameprobability!/<,just as 9H, would have done. So from player Tspoint of view, F* is at leastas goodas F.

The sameargument with players 1 and 2 interchangedhencewith

the rows of the matrix in Figure35 playing the roleswhich the columnhad above showsthat from player2'spoint of view too F* is at leastasgoodas F.))

1 Arithmetically))<r' a\" -f 1 for a' ^ <r'*'-*\"+ 1 + a* for <r' < <r'))

Hence5(<r',a\") is always oneof the numbers 1, , ot.131li is his personalmove, sohis strategy must provide for it in r*. Therewas no

needfor this in r, since9R, was a chancemove.)))

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SOME ELEMENTARYGAMES 185Sincethe viewpoints of the two playersareopposite,this means that

F* and F areequivalent.1

18.7.Interpretation of This Result

18.7.1.Repeatedapplicationto all chancemoves of F, of the operationdescribedin 18.6.2.,18.6.3.,will remove them all, thus proving the finalcontention of 18.6.1.Themeaning of this resultmay beeven betterunder-stood if we illustrate it by somepracticalinstancesof this manipulation.

(A) Considerthe following quite elementary \"gameof chance.\"Thetwo playersdecide,by a 50%-50%chancedevice,who pays the otheroneunit. Theapplicationof the device of 18.6.2.,18.6.3.transforms this game,which consistsof preciselyone chancemove, into one of two personalmoves.A lookat the matrix of Figure35 for <*<

= 2 with the3(</,<r\") values 1,2replacedby the actual payments 1, 1 shows that it coincideswith

Figure12.Remembering14.7.2.,14.7.3.we seethat this means what isplain enough directly that this is the game of MatchingPennies.

I.e.:MatchingPenniesis the natural deviceto producethe probabilitiesi> i by personalmoves and imperfect information. (Recall17.1.!)

(B) Modify (A) so as to allow for a \"tie\":Thetwo playersdecideby a33%,33%,33^%chancedevicewho pays the otherone unit, or whethernobody pays anything at all. Apply again the device of 18.6.2.,18.6.3.Now the matrix of Figure35 with aK

= 3 with the &(&',a\") values 1,2,3replacedby the actual payments0,1, 1 coincideswith Figure13.By14.7.2.,14.7.3.we seethat this is the game of Stone,Paper,Scissors.

I.e.,Stone,Paper,Scissorsis the natural devicetoproducethe proba-bilities i,i,i by personalmoves and incompleteinformation. (Recall17.1.!)

18.7.2.(C) Thed(<r', <r\") of Figure35 can be replacedby another func-tion, and even the domainsa'K = !, , aand<r\"= !, , a,by otherdomains<r'K

= 1, , a'K and o-\" = 1, , a\",providedthat the follow-ing remainstrue:Every column of the matrix of Figure35 containseachnumber 1, , ac the same number of times,2 and every row containseachnumber !,-,the same number of times.8 Indeed,the con-siderationsof 18.6.2.made use of thesetwo propertiesof S(<7<,a\") (and of:,*'.')only.

It is not difficult to seethat the precautionof \"cutting\" the deckbeforedealingcards falls into this category. When one of the 52 cardshas to bechosenby a chancemove, with probability^, this is usually achievedby\"mixing\" the deck. This is meant to bea chancemove, but if the playerwho mixesis dishonest,it may turn out to bea \"personal\"move of his.

1We leaveit to the readerto casttheseconsiderations into the preciseformalism of 11.and 17.2.,17.8.:This presentsabsolutely no difficulties, but it is somewhat lengthy.Theaboveverbal arguments convey the essentialreasonof the phenomenon under con-sideration in a clearerand simpler way wehope.

*Hence<*i/ times; consequently a'K must bea multiple ofo.1Hencea\"/ctimes; consequently a\" must be a multiple of a,.)))

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186 ZERO-SUMTWO-PERSONGAMES: EXAMPLES

As a protectionagainst this, the otherplayeris permitted to point out theplacein the mixeddeck,from which the card in questionis to betaken,by\"cutting\" the deckat that point. This combination of two^moves evenif they arepersonal is equivalent to the originally intendedchancemove.Thelack of information is, of course,the necessaryconditionfor the effec-tivenessof this device.

Herea,= 52, a'K= 52!= the number of possiblearrangementsof the

deck,a\" = 52 the number of ways of \" cutting.\" We leave it to the readertofill in the detailsand to choosethe 5(<r^, a\") for this set-up.1

19.Pokerand Bluffing

19.1.Descriptionof Poker

19.1.1.It has beenstressedrepeatedly that the casefti = ft 2 = 2 asdiscussedin 18.3.and more specifically in 18.4.,comprisesonly the verysimplestzero-sumtwo-persongames. We then gave in 18.5.someinstancesof the complicationswhich can arisein the generalzero-sum two-persongame,but the understanding of the implicationsof our generalresult(i.e.of 17.8.)will probablygain more by the detaileddiscussionof a specialgameof the more complicatedtype. This is even more desirablebecausefor the gameswith 0i= $2 = 2 the choicesof the r\\, r 2, called(pure)strate-gies,scarcelydeservethis name:just calling them \"moves\"would havebeen lessexaggerated.Indeed,in these extremely simple gamestherecouldbehardly any differencebetweenthe extensiveand the normalized form ;and sotheidentity of moves and strategies,a characteristicof the normalizedform, is inescapablein thesegames. We shallnow considera game in theextensiveform in which the playerhas severalmoves, so that the passagetothe normalized form and to strategiesis no longer a vacuous operation.

19.1.2.Thegameof which we give an exactdiscussionis Poker.2 How-ever,actualPokeris really a much too complicatedsubjectfor an exhaustivediscussionand so we shall have to subjectit to somesimplifying modifica-

1We assumed that the mixing is used to produceonly one card. If whole \" hands\"aredealt,\"cutting\" is not an absolutesafeguard. A dishonest mixer can producecorre-lations within the deckwhich one \"cut\" cannot destroy, and the knowledge of which

gives this mixer an illegitimate advantage.1The general considerations concerning Poker and the mathematical discussionsofthe variants referred to in the paragraphs which follow, were carried out by J. vonNeumann in 1926-28,but not published before. (Cf.a generalreferencein \" Zur Theorieder Gesellschaftsspiele,\"Math. Ann., Vol. 100[1928]).This appliesin particular to thesymmetric variant of 19.4.-19.10.,the variants (A), (B)of 19.11.-19.13.,and to the entireinterpretation of \"Bluffing\" which dominates all thesediscussions. Theunsymmetricvariant (C)of 19.14.-19.16.was consideredin 1942for the purposesof this publication.

The work of E. Boreland /. Ville, referred to in footnote 1 on p. 154,alsocon-tains considerations on Poker (Vol. IV, 2:\"Applications aux Jeux de Hasard,\" Chap.V: \"Le jeu de Poker\.") They are very instructive, but mainly evaluations of prob-abilities appliedto Poker in a more or lessheuristic way, without a systematic use of anyunderlying general theory of games.

A definite strategicalphaseof Poker (\"La Relance\"- \"TheOverbid\") isanalyzedon pp.91-97loc.cit. This may be regarded alsoas a simplified variant of Poker,)))

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POKERAND BLUFFING 187

tions, someof which are,indeed,quite radical.1 It seemsto us, neverthe-less,that the basicideaof Pokerand its decisivepropertieswill be conservedin our simplified form. Therefore it will be possibleto basegeneralcon-clusionsand interpretationson the results which we aregoing to obtainby the applicationof the theory previously established.

To begin with, Pokeris actually played by any number of persons,2but sincewe arenow in the discussionof zero-sumtwo-persongames,weshallsetthe number of playersat two.

The game of Pokerbeginsby dealingto eachplayer 5 cards out of adeck.8 The possiblecombinationsof 5 which he may get in this waythereare2,598,960of them 4 arecalled\"hands\"and arrangedin a linearorder,i.e.thereis an exhaustive rule defining which hand is the strongestof all, which is the second,third, strongest down to the weakest.5Pokeris played in many variants which fall into two classes:\"Stud\"and\"Draw\"games. In a Stud game the player's hand is dealt to him in itsentirety at the very beginning,and he has to keepit unchangedthroughoutthe entireplay. In \"Draw\"gamestherearevarious ways for a player toexchangeall or part of his hand, and in somevariants he may gethis handin several successivestagesin the courseof the play. Sincewe wish todiscussthe simplestpossibleform, we shall examine the Stud game only.

In this casethere is no point in discussingthe hands as hands, i.e.ascombinationsof cards. Denoting the total number of hands by SS = 2,598,960for a full deck,as we saw we might as well say that each

comparableto the two which we considerin 19.4.-19.10.and 19.14-19.16.It is actuallycloselyrelated to the latter.

Thereaderwho wishes to comparethese two variants, may find the following indica-tions helpful:

(I)Our bids a, 6 correspondto 1 + a, 1 loc.cit.(II)Thedifference between our variant of 19.4.-19.10.and that in loc.cit. is this:

If player 1begins with a \"low\" bid, then our variant provides for a comparison of hands,while that in loc.cit.makes him losethe amount of the \"low\" bid unconditionally. I.e.we treated this initial \"low\" bid as \"seeing\" cf. the discussion at the beginning of19.14.,particularly footnote 1 on p. 211while in loc.cit. it is treated as \"passing.\"We believethat our treatment is a better approximation to the phase in question in realPoker;and in particular that it is neededfor a proper analysis and interpretation of\"Bluffing.\" For technical details cf.footnote 1 on p.219.

1 Cf.however 19.11.and the end of 19.16.2The \"optimum\" in a sensewhich we do not undertake to interpret is supposed

to be 4 or 5.3This is occasionallya full deckof 52cards,but for smaller numbers of participants

only parts of it usually 32or 28 are used. Sometimes one or two extra cardswith

specialfunctions, \"jokers,\" areadded.4This holds for a full deck. Thereaderwho is conversant with the elements of com-

binatorics will note that this is the number of \"combinations without repetitions of5 out of 52\":))

/52\\ ^52.51-50.49.48U/ * 1.2-3.4.5= 2>598>960-))

& This description involves the well known technical terms \"Royal Flush,\" \"StraightFlush,\" \"Four of a Kind,\" \"Full House,\" etc. Thereis no needfor us to discussthemhere.)))

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188 ZERO-SUMTWO-PERSONGAMES: EXAMPLES

player draws a number s = 1, , S instead. The ideais that s = <S

correspondsto the strongestpossible hand, a = S 1 to the secondstrongest,etc.,and finally s = 1to the weakest. Sincea \" square deal\"amounts to assumingthat all possiblehands aredealt with the sameprob-ability, we must interpretthe drawing of the above number s as a chancemove, eachone of the possiblevalues s = 1, , S having the sameprobability 1/8.Thus the game begins with two chancemoves:Thedrawing of the number s for player 1and for player 2,1 which we denoteby 81and $2.

19.1.3.The next phase of the generalgame of Pokerconsistsof themaking of \"Bids\"by the players. Theideais that after one of the playershas madea bid, which involves a smalleror greateramount of money, hisopponent has the choiceof \" Passing,\"\"Seeing,\"or \"Overbidding.\"Passing means that he is willing to pay, without further argument, theamount of hislast precedingbid (which isnecessarilylower than the presentbid). In this caseit is irrelevant what hands the two playershold. Thehands arenot disclosedat all. \"Seeing\"means that the bid is accepted:the hands will be comparedand the playerwith the strongerhand receivesthe amount of the presentbid. \"Seeing\"terminatesthe play. \"Overbid-ding\" means that the opponentcountersthe presentbid by a higher one,in which the rolesof the playersarereversedand the previous bidder hasthe choiceof Passing,Seeingor Overbidding,etc.2

19.2.Bluffing

19.2.1.Thepoint in all this is that a player with a stronghand is likelyto make high bids and numerous overbids sincehe has goodreason toexpectthat he will win. Consequentlya player who has madea high bid,or overbid, may be assumedby his opponent a posteriori!to have astronghand. Thismay provide the opponentwith a motive for \" Passing.\"However,sincein the caseof \"Passing\"the handsarenot compared,evena playerwith a weakhand may occasionallyobtaina gain againsta strongeropponent by creatingthe (false) impressionof strength by a high bid, orby overbid, thus conceivably inducinghis opponentto pass.

This maneuver is known as \"Bluffing.\" It is unquestionablyprac-ticedby all experiencedplayers. Whether the above is its realmotivationmay be doubted;actually a secondinterpretationis conceivable.That isif a playeris known to bid high only when his hand is strong,his opponentislikelyto passin suchcases. The playerwill, therefore, not be ableto collecton high bids, or on numerous overbids,in just thosecaseswhere his actualstrength gives him the opportunity. Henceit is desirablefor him to create

1In actual Poker the secondplayer draws from a deckfrom which the first player'shand has already beenremoved. We disregard this as we disregard someother minorcomplications of Poker.

2This schemeis usually complicatedby the necessityof making unconditional pay-ments, the \"ante,\" at the start, in somevariants for the first bidder, in others for allthose who wish to participate, again in others extra payments are required for the privi-legeof drawing, etc. We disregard all this.)))

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POKERAND BLUFFING 189uncertainty in his opponent'smind as to this correlation,i.e.to make itknown that he doesoccasionallybid high on a weakhand.

To sum up:Of the two possiblemotives for Bluffing, the first is thedesireto give a (false)impressionof strength in (real)weakness;thesecondis the desireto give a (false) impressionof weaknessin (real)strength.Both areinstancesof inverted signaling(cf. 6.4.3.),i.e.of misleadingtheopponent. It shouldbe observedhowever that the first- type of Bluffingis most successfulwhen it \"succeeds,\"i.e.when the opponent actually\"passes,\"sincethis securesthe desired gain; while the secondis mostsuccessfulwhen it \"fails,\" i.e.when the opponent \"sees,\"sincethis will

convey to him thedesiredconfusing information.119.2.2.Thepossibilityof such indirectlymotivated henceapparently

irrational bids has also another consequence.Suchbids arenecessarilyrisky,and therefore it can conceivably be worth while to makethem riskierby appropriatecountermeasures, thus restrictingtheiruseby the oppon-ent. But such countermeasuresareipsofacto also indirectly motivatedmoves.

We have expounded these heuristic considerationsat such lengthbecauseour exacttheory makes a disentanglementof all these mixedmotives possible.It will beseenin 19.10.and in 19.15.3.,19.16.2.how thephenomena which surround Bluffing can be understood quantitatively,and how the motives areconnectedwith the main strategicfeaturesof thegame,like possessionof the initiative, etc.

19.3.Description of Poker(Continued)

19.3.1.Let us now return to the technicalrules of Poker. In ordertoavoid endlessoverbiddingthe numberof bids is usuallylimited.2 Inorderto avoid unrealisticallyhigh bids with hardly foreseeableirrational effectsupon the opponent there are also maxima for each bid and overbid.It is also customaryto prohibit too smalloverbids;we shall subsequentlyindicatewhat appears to bea goodreasonfor this (cf.the end of 19.13.).We shall expresstheserestrictionson the size of bids and overbidsin thesimplestpossibleform:We shall assumethat two numbersa, b

a > b >1 At this point we might beaccusedoncemore of disregarding our previously stated

guiding principle; the abovediscussion obviously assumes a seriesof plays (so thatstatistical observation of the opponent'shabits is possible)and it has a definitely\"dynamical\" character. And yet we have repeatedlyprofessedthat our considerationsmust beapplicableto oneisolatedplay and alsothat they arestrictly statical.

We refer the readerto 17.3.,where this apparent contradiction has beencarefullyexamined. Thoseconsiderations are fully valid in this casetoo, and should justify our

procedure. We shall add now only that our inconsistency the use of many plays andof a dynamical terminology is a merely verbal one. In this way we wereableto makeour discussionsmore succinctand more akin to the way that thesethings aretalked aboutin everyday language. But in 17.3.it waselaboratedhow all thesequestionable picturescan bereplacedby the strictly static problem of finding a goodstrategy.

1This is the stop rule of 7.2.3.)))

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190 ZERO-SUMTWO-PERSONGAMES: EXAMPLES

aregiven ab inilio,and that for every bid thereareonly two possibilities:the bid may be \"high,\" in which caseit is a;or \"low,\" in which caseit is6. By varying the ratio a/b which isclearlythe only thing that matterswe can makethegamerisky when a/b is much greaterthan 1,or relativelysafe when a/b is only a little greaterthan 1.

The limitation of the number of bidsand overbidswill now be usedfor asimplification of the entirescheme.In the actual play one of the playersbeginswith the initial bid;after that the playersalternate.

The advantage or disadvantage contained in the possessionof theinitiative by one player but concurrentwith the necessityof acting first!constitutesan interestingproblemin itself. We shall discussan (unsym-metric)form of Pokerwhere this plays a rolein 19.14.,19.15.But wewish at first to avoid beingsaddled^yvith this problemtoo. In otherwords,we wish to avoid for the moment all deviations from symmetry, so as toobtain the otheressentialfeatures of Pokerin their purest and simplestform. We shall therefore assumethat the two playersboth make initialbids, eachone ignorant of the other'schoice.Only after both have madethis bid is eachoneinformed of what the otherdid, i.e.whether his bid was\"high \"or\"low.\"

19.3.2.We simplify further by giving to the playersonly the choiceof\"Passing\"or \"Seeing,\"i.e.by excluding\"Overbidding.\"Indeed,\"Over-bidding\"is only a more elaborateand intensive expressionof the tendencywhich is alreadycontainedin a high initial bid. Sincewe wish to do thingsas simplyas possible,we shall avoid providing severalchannelsfor the sametendency. (Cf.however (C)in 19.11.and 19.14.,19.15.).

Accordingly we prescribe these conditions:Consider the momentwhen both playersareinformed of eachother'sbids. If it then developsthat both bid \"high\" or that both bid \"low,\" then the handsare comparedand the playerwith the strongerhand receivesthe amount a or b respectivelyfrom his opponent. If their hands areequal,no payment is made. If onthe otherhand onebids \"high\" and onebids \"low,\" then the player with

the low bid has the choiceof \"Passing\"or \"Seeing.\"\"Passing\"meansthat he pays to the opponent the amount of the low bid (without anyconsiderationof their hands). \"Seeing\"means that he changesoverfrom his \"low\" bid to the \"high\" one,and the situation is treatedas ifthey both had bid \"high\" in the first place.

19.4.Exact Formulation of the Rules

19.4.We can now sum up the precedingdescriptionof our simplifiedPokerby giving an exactstatement of the rulesagreedupon:

First,by a chancemove eachplayer obtains his \"hand,\" a numbers = 1, S,eachoneof thesenumbershaving the sameprobability1/5.We denotethe hands of players1,2, by 81,82respectively.

After this eachplayer will, by a personalmove, chooseeithera or b,the \"high\" or \"low\" bid. Eachplayer makes his choice(bid) informed)))

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POKERAND BLUFFING 191about his own hand, but not about his opponent'shand or choice(bid).Lastly, eachplayer is informed about the other'schoicebut not about hishand. (Eachstill knowshis own hand and choice.) If it turns out thatonebids \"high\" and the other \"low,\" then the latterhas the choiceof''Seeing\"or \"Passing.\"

This is the play. When it is concludedthe payments are made asfollows:If both playersbid \"high,\" or if one bids \"high,\" and the other

bids \"low\" but subsequently\"Sees,\"then for i = $2 player 1 obtains<

afrom player 2 the amount respectively. If both playersbid \"low,\"

a> b

then for si= s2 player 1obtainsfrom player2 the amount respectively.< -bIf one player bids \"high,\" and the other bids \"low\" and subsequently\"Passes,\"then the \"high bidder\"being^ player 1obtains from player 2

the amount , .o

19.6.Description of the Strategies19.6.1.A (pure)strategy in this game consistsclearlyof the following

specifications:To statefor every \" hand \" s = 1, , S whether a \"high\"or a \"low\" bid will be made,and in the lattercasethe further statementwhether, if this \"low\" bid runs into a \"high\" bid of theopponent,the playerwill \"See\"or \"Pass.\"It is simplerto describethis by a numerical indexi.= 1,2,3;it = I meaning a \"high\" bid; i,= 2 meaning a \"low\" bidwith subsequent\"Seeing\"(if the occasionarises);t. = 3 meaning a \"low\"bid with subsequent\"Passing\"(if the occasionarises).Thus the strategyis a specificationof such an indexi,for every s = 1, , S, i.e.of thesequencet'i, , is.

This appliesto both players1and 2. Accordingly we shall denotetheabove strategy by 2i(t'i, is) or 22(ji, , js).

Thus eachplayer has the samenumber of strategies:as many as therearesequencesi\\ } , is, i.e.precisely3s. With the notationsof 11.2.2.

0i= 182 = = 3s.1For the sakeof absolute formal correctnessthis should still be arranged according

to the patterns of 6. and 7. in Chapter II. Thus the two first-mentioned chancemoves(the dealing of hands) should becalledmoves 1and 2; the two subsequent personalnjoves(the bids), moves 3 and 4;and the final personal move (\"Passing\" or \"Seeing\,") move 5.

In the caseof move 5, both the player whose personalmove it is, and the numberof alternatives, dependon the previous courseof the play asdescribedin 7.1.2.and 9.1.5.(If both playersbid \"high\" or both bid \"low,\" then the number of alternatives is 1,andit doesnot matter to which player we ascribethis vacuous personalmove. If one bids\"high\" and the other bids \"low,\" then the personalmove is the \"low\" bidder's).

A consistent use of the notations loc.cit. would alsonecessitatewriting <n, <n for

i, I;<TI,<rifor the \"high\" or \"low\" bid;**for \"Passing\" or \"Seeing.\"We leaveit to the readerto iron out all thesedifferences.)))

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192)) ZERO-SUMTWO-PERSONGAMES: EXAMPLES))

If we wanted to adhererigorously to the notations of loc.cit.,we shouldnow enumeratethe sequencesfi, , is with a r\\ = 1, , ft and thendenotethe (pure)strategiesof the players1,2 by S^, S^. But we preferto continue with our presentnotations.

We must now expressthe payment which player1receivesif the strate-gies2i(t\"i, , is), 2j(ji, , js) areused by the two players. Thisis the matrix element3C(i\\, , t'alji, * * , js).1

If the players have actually the \"hands\" i, $2 then the paymentreceivedby player 1can be expressedin this way (usingthe rules statedabove):It is 90n (t-j(iit j,)where sgn($i s2) is the signof s\\ s2,2 andwhere the threefunctions

+(i,j), JB (t, j), -(*,j) f, j = 1,2, 3.can berepresentedby the followingmatrix schemes:3))

^) 1) 2) 3)

1) a) a) &)

2) a) 6) &)

3) -6) 6) 6))

\\ 3)

1) 2) 3)

1) &)

2)

3) -6)

\\j)1) 2) 3)

1) a) a) b)

2) a) -6) -b)

3) -6) -b) -&))

Figure 36. Figure 37. Figure 38.Now i, s2 originate from chancemoves, as describedabove. Hence

81

9 a i, , ja) 2

19.5.2.We now pass to the (mixed)strategiesin the senseof 17.2.Theseare the vectors , rj belongingto Sp. Consideringthe notations

1Theentire sequencet'i, , is is the row index, and the'entire sequencejit ,ja is the column index. In our original notations the strategies were Si,Sj and thematrix element 3C(n, r2).

-I- >1I.e. for si 8t respectively. It expressesin an arithmetical form which hand is

stronger.8 Thereaderwill do well to comparethesematrix schemeswith our verbal statements

of the rules, and to verify their appropriateness.Another circumstance which is worth observing is that the symmetry of the game

correspondsto the identities

M*.3) m --(/0. o(i,j) -JBo(j,4Thereadermay verify))

as a consequenceof the relations at the end of footnote 3 above. I.e.3C(tif , ia\\ji, - ,j5)

is skew-symmetric, expressing oncemore the,symmetry of the game.)))

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POKERAND BLUFFING 193which we arenow using, we must indexthe componentsof thesevectorsalso in the new way: We must write fy i , ^ ig insteadof v i?v

We express(17:2)of 17.4.1.,which evaluatesthe expectationvalue ofplayer 1'sgain))

, if) =))

Thereis an advantage in interchanging the two S and writing

K( {, 77 ) = gj2} 2^ Snri.c,-.,)(*'.,,j'Ofc,.....^.....vi.t i...../!.....Js

If we now put))

(19:1)*ii i^d 'excluding ,))

(19:2) <rj, =/IP J'sexcluding /.

f))

then the above equationbecomes

(19:3) K(7,7) =))

i,i tj

It is worth while to expoundthe meaning of (19:1)-(19:3)verbally.(19:1)showsthat pji is the probabilitythat player 1,using the mixed

strategy , will choosei when his \"hand\" is a\\\\ (19:2)showsthat a-J is the

probabilitythat player2,usingthe mixedstrategy y , will choosej when his\"hand\"is S2.1 Now it is intuitively clearthat the expectationvalue

> >K( , rj ) dependson theseprobabilitiesp{,<r$ only, and not on the underly-ing probabilities fct <a, ^ ^ themselves.2 The formula (19:3)can

1We know from 194. that i or;- 1means a \"high\" bid, t - 2, 3 a \"low\" bid with

(the intention of) a subsequent \"Seeing\" or \"Passing\" respectively.2This means that two different mixtures of (pure) strategiesmay in actual effectbe

the same thing.Letus illustrate this by a simple example. Put S 2,i.e.let there beonly a \"high

\"

and a \"low\" hand. Consideri - 2, 3 asonething, i.e.let there beonly a \"high\" and a)))

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194 ZERO-SUMTWO-PERSONGAMES:EXAMPLES

easily be seento be correctin this directway: It suffices to rememberthe meaning of the JB w <ti-.t)(i,j) and the interpretationof the pji, <rj.

19.5.3.It is clear,both from the meaning of the pji, o-Ji and from theirformal definition (19:1),(19:2),that they fulfill the .conditions

3(19:4) all P; ^ 0, p\\>

= 1t-i3

(19:5) allcr}.0, irj. = 1

On the other hand, any pji, aj which fulfill theseconditionscan be obtained

from suitable , 77 by (19:1),(19:2).This is clearmathematically, 1 andintuitively as well. Any suchsystemof pji, <rj isoneof probabilitieswhichdefine a possiblemodus procedendi, so they must correspondto somemixedstrategy.

(19:4),(19:5)make it opportuneto form the 3-dimensionalvectors

P'1 = (Pi1, P?SPa 1}, *'*= {*if , <r'22, <rj|.Then (19:4),(19:5)statepreciselythat all p*i, a**belongto 53.

This shows how much of a simplification the introduction of thesevectorsis: (or 77 ) was a vector in 50,i.e.dependingon ft 1 = 3s 1

* *numerical constants;the p '* (or the a ) are5 vectorsin 58, i.e.eachonedepends on 2 numerical constants;hencethey amount togetherto 25numerical constants. And 3s 1 is much greaterthan 25,even for moder-ate5.2

\"low\" bid. Then there arefour possible(pure) strategies,to which we shall give names*\"Bold\": Bid \"high\" on every hand.\"Cautious\": Bid \"low\" on every hand.\"Normal\": Bid \"high\" on a \"high\" hand, \"low\" on a \"low\" hand.\"Bluff\": Bid \"high\" on a \"low\" hand, \"low\" on a \"high\" hand.Then a 50-50mixture of \"Bold\" and \"Cautious\" is in effect the same thing as a

50-50mixture of \"Normal\" and \"Bluff\": both mean that the player will accordingto chance bid 50-50\"high\" or \"low\" on any hand.Nevertheless theseare,in our presentnotations, two different \"mixed\" strategies,

i.e.vectors .This means, of course,that our notations, which were perfectly suited to the general

case,areredundant for many particular games. This is a frequent occurrencein mathe-matical discussions with generalaims.

Therewas no reasonto take accountof this redundance as long aswe were workingout the general theory. But we shall remove it now for the particular game underconsideration.

1Put e.g.fy is - P{t ... p* , i,,t ,a - <7}v ... <r, s and verify the (17:1:a),(17:1:b)of 17.2.1.as consequencesof the above(19:4),(19:5).

8Actually Sis about 2J millions (cf.footnote 4 on p.187);so3s 1 and 25areboth

great, but the former is quite exorbitantly the greater.)))

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POKERAND BLUFFING 195

19.6.Statement of the Problem

19.6.Sincewe aredealingwith a symmetric game,we canuse the char-\"\"\"*

acterizationof the good (mixed)strategies i.e.of the in A given in

(17:H)of 17.11.2.It stated this: must be optimal against itself, i.e.> > >

Min- K( , ij ) must be assumedfor rj= .

Now wesawin 19.5.that K({, t\\ ) dependsactually on the p % a . Sowe may write for it, K(7s ,7V1, 75). Then (19:3)in 19.5.2.states(we rearrangethe S somewhat)

(19:6)K(7', ' ,7V\\ , O =55

* *And the characteristicof the p l , , p 8 of a goodstrategy is that

Min^ 7,K(7l, ' ' ' ,7V1, - ,7s)

is assumedat a l = p 1y

- - , a s = p s. The explicitconditionsfor thiscanbefound in essentiallythe sameway as in the similar problemof 17.9.1.;we will give a brief alternative discussion.

The Min-* -> of (19:6)amounts to a minimum with respecttoff l , .. .,O 8

> >

each crl, , a s separately. Consider therefore such a a . It is

restrictedonly by the requirementto belongto $3, i.e.by3

all o}i SO, X *'f = 1-y-i

(19:6)is a linearexpressionin thesethreecomponents V, orj*, cr|. Henceit assumesits minimum with respectto a '*therewhereall thosecomponents<rj

which do not have the smallestpossiblecoefficient (with respectto j,cf. below),vanish.

Thecoefficientof <r* is

^ V .Cn(.r.,)(i,J)P? to bedenotedby g7^.,,f

Thus (19:6)becomes

(19:7) K(7S , 75|7s , 7)))

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196 ZERO-SUMTWO-PERSONGAMES: EXAMPLES

>

And the conditionfor the minimum (with respectto a ) is this:(19:8) Foreachpair s2, j,for which 7}doesnot assumeits minimum

(in j *), we have crjt= 0.

Hencethecharacteristicofa goodstrategy minimization at a '= pl , ,))

(19:A) p *, , p a describea goodstrategy, i.e.a in J!,if andonly if this is true:

Foreachpair s2, j for which 7j doesnot assumeits minimum

(in j l), we have p;* = 0.We finally statetheexplicitexpressionsfor the 7<,of courseby usingthe

matrix schemesof Figures36-38.They are))

(19:9:a)7*1' =*))

2 (ap\\i + ap'2 --i

(19:9:b)))

(19:9:c)))

19.7.Passagefrom the Discreteto the Continuous Problem

19.7.1.Thecriterion(19:A) of 19.6.,togetherwith the formulae (19:7),(19:9:a),(19:9:b),(19:9:c),can be used to determineall goodstrategies.2

This discussionis of a rathertiresomecombinatorialcharacter,involvingthe analysisof a numberof alternatives. Theresults which areobtained

1We mean in j and not in s,j\\2This determination hasbeencarriedout by oneofus and will bepublished elsewhere.)))

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POKERAND BLUFFING)) 197))

arequalitatively similarto those'whichwe shallderive belowundersome-what modified assumptions,exceptfor certaindifferences in very delicatedetail which may becalledthe \"fine structure\"of the strategy. We shallsay more about this in 19.12.

Forthe moment we arechieflyinterestedin themain featuresof thesolu-tion and not in the questionof \"fine structure.\" We beginby turning ourattention to the \" granular \" structureof the sequenceof possiblehands8 = 1,- ' - , S.

If we try to picturethe strengthof all possible\"hands\"on a scalefrom0% to 100%,or ratherof fractions from to 1,then the weakestpossiblehand, 1,will correspondto 0,and the strongestpossiblehand, 5,to 1.Hencethe \"hand\" s(=1, , S) shouldbe placedat z = ~ __ 1 on this

scale. I.e.we have this correspondence:))

Oldscale:) S SB) 1) 2) 3) S -1) S)

1) 2) S -2)I)S -

1) S -I) S -

1)

Figure 39.

Thus the values of z fill the interval

(19:10) ^ z g 1

very densely,1but they form neverthelessa discretesequence.This is the\"granular\"structurereferred to above. We will now replaceit by acontinuousone.

I.e.we assumethat the chancemove which choosesthe hand s i.e.zmay produceany z of the interval (19:10).We assumethat theprobabilityof any part of (19:10)is the length of that part, i.e.that z is equidistributedover (19:10).2 We denotethe \"hands\"of the two players 1,2 by *i, za

respectively.19.7.2.This changeentails that we replacethe vectors

>

p , a -i (si, ,,= 1, , S) by vectors p , <r (0 ^ *i, z* ^ 1);butthey are,of course,still probabilityvectors of the samenature as before,i.e.belonging to S*. In consequence,the components (probabilities)p'1, *',* (*i> s2 = 1, , S;i,j = 1,2,3) give way to the componentsPjs<rj(0 *,,z2 ^ l;i,j= 1,2,3).Similarlythe y',* (in (19:9:a),(19:9:b),(19:9:c)of 19.6.)become75*.

We now rewrite the expressionsfor K and the 7}in the formulae (19:7),(19:9:a),(19:9:b),(19:9:c)in 19.6.Clearlyall sums))

1It will beremembered (cf.footnote 4on p. 187)that.Sis about 2J millions.1This is the so-calledgeometrical probability.)))

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198 ZERO-SUMTWO-PERSONGAMES: EXAMPLES

s s1y

must bereplacedby integrals))

/:))sums

1-1))

by integralsr _ r +

dzi,))/*./O))

while isolatedterms behinda factor 1//Smay be neglected.1'2 Thesebeingunderstood,the formulae for K and the 7}(i.e.7*) become:

(19:7*) K =

(19:9:a*)y{> = [\" (-opf- op}))

(19:9:b*)7J.= * (-op[.- bp;>- fepj.)<fci + (opj.+ bp't >

(19:9:c*)yl*

And thecharacterization(19:A) of 19.6.goesover into this:

(19:B) The 7 (0 ^ 2 g 1) (they all belongto 89) describea goodstrategy if and only if this is true:

Foreach z, j for which 7* doesnot assume its minimum

(in ,;'a)> we have pj = O.4

Specificallywe mean the middle terms bp\\* and 6pJ in (19:9:a)and (19:9:c).1Theseterms correspondto i = *2, in our present set-upto z\\ j, and sincethei, 2sarecontinuous variables, the probability of their (fortuitous) coincidenceis indeed0.

Mathematically one may describethese operations by saying that we are nowcarrying out the limiting processS > oo.

8 We mean in ,; and not in z, j\\4Theformulae (19:7*),(19:9:a*),(19:9:6*),(19:9:c*)and this criterion couldalsohave

> *beenderived directly by discussing this \"continuous\" arrangement, with the p fi, p f i

in placeof the , 17 from the start. We preferredthe lengthier and more explicit proce-dure followed in 19.4.-19.7.in order to make the rigor and the completenessof our proce-dure apparent. The readerwill find it a good exerciseto carry out the shorter directdiscussion, mentioned above.

It would be tempting to build up a theory of games, into which such continuousparametersenter, systematically and directly; i.e.in sufficient generality for applicationslike the present one,and without the necessityfor a limiting processfrom discretegames.

An interesting stepin this direction was taken by /.Ville in the work referredto infootnote 1on p. 154:pp.110-113loc.cit. Thecontinuity assumptions madethereseem,however, to be too restrictive for many applications, in particular for the present one.)))

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POKERAND BLUFFING 199

19.8.Mathematical Determination of the Solution

19.8.1.We now proceedto the determination of the goodstrategiesp ',i.e.of the solution of the implicit condition(19:B)of 19.7.Assume first that p|> ever happens.1 For such a z necessarily

Min,75 = y\\ hencey\\ ^ y*2 i.e.7*2

-y\\ ^ 0.

Substituting(19:9:a*),(19:9:b*)into this gives

(19:11) (a - 6) ( fydz, - f*Pjcfei)+ 26 p^dz,g 0.Now letz be the upper limit of thesez with p\\ > O.2 Then (19:11)holdsby continuity for z = Z Q too. As fy > does not occurfor Zi >2 by

hypothesis so the / p dz\\ term in (19:11)is now 0. Sowe may write itJ*o

with + insteadof , and (19:11)becomes:))

(a - 6) P pjufei + 26 P pjufei ^ 0.o J *))

But p|iis always ^ and sometimes> 0, by hypothesis;hencethe firstterm is > O.8'4 The secondterm is clearly g:0. So we have deriveda contradiction.I.e.we have shown

(19:12) pj = O.6

19.8.2.Having eliminatedj = 2 we now analyze the relationshipofj = I and j = 3. Sincep|= so p\\ + pj = 1i.e.:(19:13) PJ = 1- pj,and consequently(19:14) ^ pj g 1.

Now there may existin the interval ^ z g 1subintervalsin whichalways p* = or always p\\ = I.6 A z which is not insideany interval of

1I.e.that the good strategy under consideration provides for j 2, i.e.\"low\"

bidding with (the intention of) subsequent \"Seeing,\" under certain conditions.1I.e.the greatest 2for which pj > occursarbitrarily near to z. (But we do not

require pj > for all z < ZQ.) This 2 existscertainly if the zwith pj > exist.1Ofcoursea b > 0.4 It doesnot seemnecessaryto go into the detailed fine points of the theory of integra-

tion, measure, etc. We assume that our functions aresmooth enough so that a positivefunction has a positive integral etc. An exacttreatment could begiven with easeif wemade use of the pertinent mathematical theoriesmentioned above.

8 The reader-should reformulate this verbally: We excluded \"low\" bids with (theintention of) subsequent \"Seeing\" by analysing conditions for the (hypothetical) upperlimit of the hands for which this would bedone;and showed that near there, at least,an

outright \"high\" bid would bepreferable.This is, of course,conditioned by our simplification which forbids \"overbidding.\"

6I.e.where the strategy directs the player to bid always \"high,\" or where it directshim to bid always \"low\" (with subsequent \"Passing\.)))

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200 ZERO-SUMTWO-PERSONGAMES: EXAMPLES

eitherkind i.e.arbitrarily near to which both pj' 5* and pf ^ 1 occurwill be called intermediate. Sincep\\' ^ or p\\' 7* 1 (i.e.p|'/^ 0) implyMin, 75' = 7*' or y\\

f

respectively,therefore we see:Both y[' g ys' and

7*' ^ 7a' occurarbitrarily near to an intermediate2. Hencefor sucha z,y\\ = 7s by continuity, 1i.e.

7'8 - 7!= 0.Substituting(19:9:a*),(19:9:c*)and recalling(19:12),(19:13),gives

(a + 6) (Z

p^dz,- (a - 6) flp^dz,+ 26 f

l(1- P\\^dz l =

JQ Js Jti.e.(19:15) (a + 6) p^dz,- Vi^i + 26(1- *) = 0.))

Considernext two intermediate', z\". Apply (19:15)to z = z' ande = z\" and subtract. Then

2(a+ b)

obtains,i.e.

(19:16)))

Verbally:Betweentwo intermediatez',z\" the average of p{is -r ,

Soneither p* s= nor p\\ = 1can hold throughout the interval

z' g z ^ z\"

sincethat would yieldthe average or 1. Hencethis interval must contain(at least)a further intermediatez, i.e.betweenany two intermediateplacestherelies(at least)a third one. Iterationof this resultshowsthat betweentwo intermediatez', z\" the further intermediatez lie everywhere dense.Hencethe z', z\" for which (19:16)holdslie everywhere densebetweenz',z\" .But then (19:16)must hold for all 5', z\" between2',z\", by continuity. 2

This leavesno alternative but that pi = r r everywhere betweenz' z\" *a -r o

1The7*aredefined by integrals (19:9:a*),(19:9:b*),(19:9:c*),hencethey arecertainlycontinuous.

1Theintegral in (19:16)is certainly continuous.*Clearly isolated exceptionscovering a z area i.e.of total probability zero(e.g.a

finite number of fixed z's) could be permitted. They alter no integrals. An exactmathematical treatment would be easybut doesnot seemto becalledfor in this context

(cf.footnote 4 on p. 199).So it seemssimplest to assume pf pr in 2' ^ z <: 2\"a -powithout any exceptions.

This ought to bekept in mind when appraising the formulae of the next pageswhichdealwith the interval 2' ^ z ^ 2\" on one hand, and with the intervals : z < I1 andI\" < z' g 1on the other; i.e.which count the points 2',I\" to the first-mentioned interval.This is, of course,irrelevant: two fixed isolated points z' and z\" in this case couldbe disposedof in any way (cf.above).

The readermust observe,however, that while there is no significant difference)))

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POKERAND BLUFFING 20119.8.3.Now if intermediatez existat all, then thereexistsa smallest

one and a largestone;choosez',2\" as these. We have

(19:17) p\\= ^pj throughout ti z z\".

If no intermediatez exist,then we must have p\\ = (for all z) or p\\ ss 1(for all z). It iseasyto seethat neither is a solution.1 Thusintermediatezdo existand with them z',z\" existand (19:17)is valid.

19.8.4.Theleft hand sideof (19:15)is y\\-

y\\ for all z\\ hencefor z = 1))

(sincepi i = isexcluded).Bycontinuity y\\ y\\ > 0,i.e.7i < 7!remainstrueeven when z is merely near enough to 1. Hencepj = 0,i.e.p* = 1forthesez. Thus (19:17)necessitatesz\" < 1. Now no intermediatez existsin 2\" ^ z g 1;hencewe have p\\ = or p\\ = 1throughout this interval.Our precedingresult excludesthe former. Hence(19:18) pi s 1 throughout z\" ^ z 1.

19.8.5.Considerfinally the lower end of (19:17),z'. If z' > then wehave an interval ^ z g z'. This interval contains no intermediatez;hencewehave p\\ = or pi = 1throughout z '. Thefirst derivativeof 7S -7ii i-e-of the left sideof (19:15),is clearly 2(a+ 6)pi-26. Hencein ^ z < z' this derivative is 2(a+ 6) - 26 = 26 < if pi as

there,2(a+ 6) 1- 26 = 2a > if p\\ = 1there,i.e.7*3-

y\\ is monotonedecreasingor increasingrespectively, throughout ^ z < zf. Sinceitsvalue is at the upperend (theintermediatepoint z')>we have 73 7* >or < respectively, i.e.y\\ < y\\ or 75 < y\\ respectively, throughout

^ z < '. The former necessitatesp\\ = 0,pi ss 1 the latterpi = in

^ z < z'\\ but the hypotheseswith which we startedwerepi as orpi B 1respectively,there. Sothereis u contradictionin eachcase.

Consequently(19:19) z' = 0.

19.8.6.And now we determinez\" by expressingthe validity of (19:15)for the intermediatez = z' = 0. Then (19:15)becomes

-(a+ 6) f1pi 1+ 26 =))

between a < and a ^ when the z'a themselves are compared,this is not so for the y'rThus we saw that y' > y\\ implies p\\

- 0, while y\\ ^ 71 nas no such consequence.(Cf.alsothe discussion of Fig. 41and of Figs. 47, 48.)

1I.e.bidding \"low\" (with subsequent \"Pass\") under all conditions is not a goodstrategy; nor is bidding \"high\" under all conditions.

Mathematical proof: For p\\ m 0:Compute 7? - -6,71 - & hence 7? < y\\

contradicting p\\- 1 ^ 0. For pj m 1:Compute 7? - <*, 7i - & hence 7? < ?!

contradicting pj 1 ^ 0.)))

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202 ZERO-SUMTWO-PERSONGAMES: EXAMPLES

But (19:17),(19:18),(19:19)give))

r^-r.-J + u-n-iJo a + b))

+ 6Sowe have

26))

+ 6o

2/, = t _ 26 = g-6+ 6 + 6 + 6'

i.e.(19:20) 2\" = \"--

Combining(19:17),(19:18),(19:19),(19:20)gives:

6 for g z g ^=-))(19:21) ,1 + b))

= 1 for))

Togetherwith (19:12),(19:13)this characterizesthe strategycompletely.

19.9.DetailedAnalysis of the Solution

19.9.1.Theresultsof 19.8.ascertainthat thereexistsone and only onegoodstrategy in the form of Pokerunder consideration.1 It is describedby (19:21),(19:12),(19:13)in 19.8.We shallgive a graphicalpictureof thisstrategy which will make it easierto discussit verbally in what follows.(Cf. Figure 40. The actual proportions of this figure correspond toa/6~3.)

Theline plots the curve p =p\\. Thus the height of above

the line p = is the probabilityof a \"high\" bid:p\\; the height of the linep = 1above is the probabilityof a \"low\" bid (necessarilywith sub-sequent\"pass\:")p\\

= 1 p\\.

19.9.2.The formulae (19:9:a*),(19:9:b*),(19:9:c*)of 19.7.permit usnow to computethe coefficients7*. We give the graphicalrepresentationsinstead of the formulae, leaving the elementaryverification to the reader.(Cf.Figure41. Theactual proportionsarethoseof Figure40,i.e.a/6~3cf. there.) The line plots the curve 7 = 7*;the line plots thecurve 7 = 7j; the line plotsthecurve 7 = y\\. Thefigure showsthat

1We have actually shown only that nothing elsebut the strategy determined in 19.8.can begood. That this strategy is indeedgood,couldbeconcludedfrom the establishedexistenceof (at least)a good strategy, although our passageto the \"continuous\" casemay there createsomedoubt. But we shall verify below that the strategy in question isgood,i.e.that it fulfills (19:B)of 19.7.)))

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POKERAND BLUFFING

and (i.e.y\\ and 75)coincidein g z g a ~))

203

and that and))

(i.e.y\\ and 7$) coincidein ^ z ^ 1. All threecurves aremade))

a -6a

Figure 40.))

Figure 41.))

of two linear pieceseach,joining at z =))a -

6)) Theactual values of the))

7* at the criticalpoints)) 0, , 1 are given in the figure.1))

1Thesimple computational verification of theseresults is left to the reader.)))

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204 ZERO-SUMTWO-PERSONGAMES: EXAMPLES

19.9.3.Comparisonof Figures40 and 41shows that our strategy is

indeedgood,i.e.that it fulfills (19:B)of 19.7.Indeed:In ^ z ^ ^where both p{ 5^0,pj 5* both y{ and yl arethe lowestcurves,i.e.equal

to Min, y'j. In fl ~\" < z ^ 1 where only p{ 5* there only 7? is the

lowestcurve, i.e.equal to Min, y]. (Thebehavior of y\\ doesnot matter,sincealways pj = 0.)

We can alsocomputeK from (19:7*)in 19.7.,thevalue of a play. K =is easilyobtained;and this is the value to be expectedsincethe gameissymmetric.

19.10.Interpretation of the Solution

19.10.1.The results of 19.8.,19.9.,although mathematically complete,call for a certainamount of verbal comment and interpretation,which wenow proceedto give.

Firstthe pictureof the goodstrategy, as given in Figure40, indicatesthat for a sufficiently strong hand p{ = 1;i.e.that the player shouldthen

bid \"high,\" and nothing else. This is the casefor hands z > For

weakerhands,however, p\\=

^r~r> p\\= 1 pf = . , ; soboth p{,pj 7* 0;

i.e.theplayershouldthen bid irregularly \"high\" and \"low\" (with specified

probabilities).This is the casefor hands z g The \"high\" bids

(in this case)shouldbe rarerthan the \"low\" ones,indeed-|= -=- and a > 6.PI o

This last formula showstoothat the last kind of \"high\" bids becomeincreasinglyrare if the costof a \"high\" bid (relative to a \"low\" one)increases.

Now these\"high\" bids on \"weak\"hands made irregularly, governedby (specified)probabilitiesonly, and gettingrarerwhen thecostof \"high\"

bidding is increasedinvite an obvious interpretation:These are the\"Bluffs\" of ordinaryPoker.

Due to the extremesimplifications which we applied to Pokerfor thepurposeof this discussion,\"Bluffing\" comesup in a very rudimentary formonly; but the symptoms are neverthelessunmistakable:The player is

advised to bid always \"high\" on a stronghand f z > J and to bid

mostly \"low\" (with the probabilityjpr-j)on a \"low\" one (z < ?-ZjMbut with occasional,irregularly distributed \"Bluffs\" (with the probability)))

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POKERAND BLUFFING 205

19.10.2.Second,theconditionsin thezoneof \"Bluffing,\" z a,a

throw somelight on anothermatter too, the consequencesof deviatingfrom thegoodstrategy, \"permanentoptimality,\" \"defensive,\"\"offensive,\"as discussedin 17.10.1.,17.10.2.

Assumethat player 2 deviatesfrom the goodstrategy, i.e.usesproba-bilities (Ty

which may differ from the pj obtainedabove. Assume,further-more,that player1still usesthosep*, i.e.the goodstrategy. Thenwe canusefor the 7'of (19:9a*),(19:9:b*),(19:9:c*)in 19.7.,the graphicalrepre-sentationof Figure41,and expressthe outcomeof the play for player1by (19:7*)in 19.7.(19:22)

Consequentlyplayer2's<r* areoptimal againstplayer1'sp* if the analogueof thecondition(19:8)in 19.6.is fulfilled:

(19:C) Foreachpair z, j for which y] doesnot assumeits minimum

(in j *) we have a* = 0.I.e.(19:C)is necessaryand sufficientfor <7* beingjust as goodagainstpj as p*

itself, that is, giving a K = 0. Otherwisea] is worse, that is, giving aK >0. In other words:

(19:D) A mistake, i.e.a strategy <r* which deviates from the goodstrategy pj will causeno losseswhen the opponentsticks to thegoodstrategy if and only if the a* fulfill (19:C)above.

Now one glanceat Figure41 sufficesto makeit clearthat (19:C)means

a|= ff\\= for z > a \"\"

but merely a\\ = for z ^ a~^^ I.e.:(19:C)ct a

prescribes\"high\" bidding,and nothing else,for stronghands( z > J;it forbids \"low\" bidding with subsequent \"Seeing\"for all hands, but itfoils to prescribethe probability ratio of \"high\" bidding and of \"low\"

bidding (with subsequent \"Passing\")for weak hands, i.e.in the zone of

\"Bluffing\19.10.3.Thus any deviation from the good strategy which involves

more than just incorrect\"Bluffing,\" leadsto immediatelosses.It sufficesfor the opponent to stick to the good strategy. Incorrect\"Bluffing\"

causesno lossesagainst an opponent playing the good strategy; but the

1We mean in j,and not in *,j!a b1Actually even ?* would be permitted at the one placez But this

isolatedvalue of * has probability and so it can be disregarded.Of.footnote 3 onp. 200.)))

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206 ZERO-SUMTWO-PERSONGAMES: EXAMPLES

opponent could inflict lossesby deviating appropriately from the goodstrategy. I.e.the importanceof \"Bluffing\" lies not in the actual play,playedagainsta goodplayer,but in theprotectionwhich it providesagainstthe opponent'spotential deviations from the good strategy. This is in

agreementwith the remarksmadeat the end of 19.2.,particularly with thesecondinterpretationwhich we proposedtherefor \"

Bluffing.\"1 Indeed,the

elementof uncertainty createdby \"Bluffing\" is just that type of constrainton the opponent'sstrategy to which we referredthere,and which was ana-lyzed at the end of 19.2.

Our results on \"bluffing\" fit in also with the conclusionsof 17.10.2.We seethat the unique goodstrategy of this variant of Pokeris not per-manently optimal;henceno permanently optimal strategy existsthere.(Cf. the first remarksin 17.10.2.,particularlyfootnote 3 on p.163.)And

\"Bluffing\" is a defensive measurein the sensediscussedin the secondhalfof 17.10.2.

19.10.4.Thirdand last,letus takea lookat the offensivestepsindicatedloc.cit.,i.e.the deviationsfrom goodstrategy by which a playercan profitfrom his opponent'sfailure to \"Bluff\" correctly.

We reversetheroles:Let player1 \" Bluff \" incorrectly,i.e.usep* differentfrom thoseof Figure40. Sinceonly incorrect\"Bluffing\" is involved, westill assume

p*2 == for all za-b))Pf = 1I

Pi =/))

r n ^for all z >))

Sowe areinterestedonly in the consequencesof

(19:23) p\\ % ^-~ for some z = Z Q <^~'2Theleft hand sideof (19:15)in 19.8.is stilla valid expressionfor y\\ 7*!.

Considernow a z < z . Then ^ in (19:23)leaves / p^dziunaffected, butJo

., increasesn _ , , ., decreases,, , ~, , , ., f ,+* *e\\it , / pi^azi henceit . the left hand side of (19:15),i.e.decreasesJ z increases7s 7i- Sincey\\ y\\ would be without the change(19:23)(cf. Figure41),so it will now be ^ 0. I.e.y\\ ^ y\\. Considernext a z in

/ ^ a - 6ZQ < Z ^ a

1 All this holds for the form of Poker now under consideration. For further view-points cf.19.16.

2 We need this really for more than one z, cf. footnote 3 on p. 200. The simplestassumption is that theseinequalities hold in a small neighborhood of the ZD in question.

It would beeasyto treat this matter rigorously in the senseof footnote 4 on p. 199and of footnote 3on p.200. We refrain from doing it for the reasonstated there.)))

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POKERAND BLUFFING 207

Then ^ in (19:23)\"1\"easesf* p Zl wn iie jt leaves f'p{feiunaffected;

increaseshenceit , the left hand sideof (19:15),i.e.7} y\\. Sincey\\ y\\

would be without the change(19:23)(cf.Fig.41),so it will now be ^ 0.I.e.7f ^ 75. Summing up:(19:E) Thechange(19:23)with ^ causes

7s % y\\ for z < z ,

7'3 % 7l for z, < z ~^-Hencethe opponentcan gain, i.e.decreasethe K of (19:22),t?y using &*

which differ from the presentpj:Forz < z by increasing* at the expensei

offf

\\, i.e.by . -erffrom the value of pf, jj-ito the extremevalue ,<rj increasing ' a + b 1

And for ZQ < z ^ ---by increasing *at the expenseof ** i.e.by , .a <r|^

<r\\ decreasinga\\ from the value of pf, . to the extremevalue n In otherwords:

(19:F) If the opponent\" Bluffs\" too ... for a certainhand ZQ, thenlittlehe can be punishedby the following deviationsfrom the good))

strategy:\" Bluffing\" for handsweakerthan z^ and \"Bluff-

ing\" , for handsstrongerthan ZQ.lessI.e.by imitating his mistake for hands which arestronger

than ZQ and by doing the oppositefor weakerones.Thesearethe precisedetailsof how correct\" Bluffing\" protectsagainst

too much or too little \"Bluffing\" of the opponent, and its immediateconsequences.Reflections in this directioncouldbecarriedeven beyondthis point, but we do not proposeto pursuethis subjectany further.

19.11.MoreGeneralForms of Poker

19.11.While the discussionswhich we have now concludedthrow agooddeal of light on thestrategicalstructureand the possibilitiesof Poker,they succeededonly due to our far reachingsimplification of the rules ofthe game. Thesesimplifications were formulated and imposed in 19.1.,19.3.and 19.7.For a realunderstanding of the game we should nowmake an effort to remove them.

By this we do not mean that all the fanciful complicationsof the gamewhich we have eliminated (cf. 19.1.)must necessarilybe reinstated,1

1Nor do we wish, yet to consideranything but a two-persongame!)))

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208 ZERO-SUMTWO-PERSONGAMES: EXAMPLES

but somesimpleand important features of the game were equallylost andtheir reconsiderationwould be of greatadvantage. We mean in particular:

(A) The \" hands\" shouldbe discrete,and not continuous. (Cf.19.7.)(B) Thereshouldbe more than two possibleways to bid. (Cf.19.3.)(C) Thereshouldbe more than oneopportunityfor eachplayer to bid,

and alternating bids,insteadof simultaneousones,shouldalsobeconsidered.(Cf. 19.3.)

The problemof meeting thesedesiderata(A), (B),(C)simultaneouslyand finding the goodstrategies is unsolved. Therefore we must besatis-fied for the moment to add (A), (B),(C)separately.

The completesolutionsfor (A) and for (B) areknown, while for (C)only a very limited amount of progresshas beenmade. It would leadtoo far to give all thesemathematical deductionsin detail, but we shallreport briefly the resultsconcerning(A), (B),(C).

19.12.DiscreteHands

19.12.1.Considerfirst (A). I.e.let us return to the discretescaleofhands s = 1, , S as introduced at the end of 19.1.2.,and used in19.4-19.7.In this casethe solution is in many ways similar to that ofFigure40. Generallypj = and thereexistsa certain5 such that pf = 1for s >s,while p\\ j 0,1 for s <s. Also, if we changeto the z scale(cf.

so __ } a ^Fig.39), then -~ r is very nearly l Sowe have a zoneof \"

Bluffing\"

o 1 dand above it a zone of \"high\" bids, just as in Fig.40.

But the p\\ for s <s,i.e.in the zone of \"Bluffing,\" arenot at all equal

to or near to the , , of Fig.40.2 They oscillatearound this value by

amountswhich dependon certainarithmetical peculiaritiesof Sbut do nottend to disappear for S > oo. The averagesof the p\\ however, tend to

: r-3 In otherwords:a + b

The good strategy of the discretegameis very much like the goodstrategy of the continuous game:this is true for all details as far as thedivision into two zones (of \"Bluffing\" and of \"high\" bids) is concerned;alsofor the positionsand sizesof thesezones,and for theeventsin the zoneof \"high\" bids.But in the zone of \"Bluffing\" it appliesonly to averagestatements(concerningseveral hands of approximatelyequal strength).The preciseproceduresfor individual hands may differ widely from those

* Precisely:%-^-l-^^ for S -* .o 1 fl

b2I.e.not PI -* 7 for S > < whatever the variability of .

1 b*Actually (pf + P?*1) - T for most a < .)))

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POKERAND BLUFFING 209

given in Figure40, and dependon arithmetical peculiaritiesof 8 and S(with respectto a/6).119.12.2.Thus the strategywhich correspondsmorepreciselyto Figure40

i.e.where p[ m for all a < s is not good, and it differs quite

considerablyfrom the goodone. Neverthelessit canbe shown that themaximal loss which can be incurredby playing this \"average\"strategyis not great. Moreprecisely,it tends to for S >.*

So we see:In the discretegame the correctway of \"Bluffing

\" has avery complicated\"fine structure,\"which howeversecuresonly an extremelysmall advantage to the playerwho usesit.

This phenomenon is possiblytypical, and recursin much more compli-catedrealgames.Itshowshow extremelycareful one must be in assertingor expectingcontinuity in this theory.3 But the practicalimportance-i.e.the gainsand lossescaused seemsto be small,and the whole thingis probablyterra incognita to even the most experiencedplayers.

19.13.m possibleBids

19.13.1.Consider,second,(B):I.e.letuskeepthe handscontinuous,butpermitbiddingin more than two ways. I.e.we replacethe two bids

a >6(>0)by a greaternumber, say m, ordered:

ai > a2 > > am_i > am (>0).In this casetoo the solution bearsa certainsimilarity to that of Figure40.4Thereexistsa certainz 6 such that for z > ZQ the playershouldmakethehighestbid, and nothing else,while for z < 2 he shouldmake irregularlyvarious bids (always including the highest bid a, but also others),with

specifiedprobabilities. Which bids he must make and with what proba-

1Thus in the equivalent of Figure 40,the left part of the figure will not bea straight

line (P \" TT in S 2 ^ j, but one which oscillatesviolently around this

average.1It is actually of the order 1/S. Remember that in real PokerSisabout 2}millions.

(Cf.footnote 4 on p. 187.)1Recallin this connection the remarks made in the secondpart of footnote 4 on p.198.4 It has actually beendetermined only under the further restriction of the rules which

forbids \"Seeing\" a higher bid. I.e.eachplayer is expectedto make his final, highest bidat once,and to \"Pass\" (and acceptthe consequences)if the opponent'sbid should turnout higher than his.

a - 61Analogue of the z in Figure 40.)))

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210 ZERO-SUMTWO-PERSONGAMES: EXAMPLES

bilities,is determinedby the value of z.1 Sowe have a zone of \"Bluffing\"and above it a zone of \"high\" bids actually of the highestbid and nothingelse just as in Figure40. But the \"Bluffing\" in its own zone z g Z Q

has a much more complicatedand varying structure than in Figure40.We shall not go into a detailed analysis of this structure,although

it offers somequite interestingaspects.We shall,however, mention one ofits peculiarities.19.13.2.Lettwo values

a > b >be given, and use them as highestand lowestbids:

di = dj am == b.

Now letm > and choosethe remaining bidsa*, - , am-iso that theyfill the interval

(19:24) 6 g x ^awith unlimited increasingdensity. (Cf. the two examplesto begiven infootnote 2 below.) If the goodstrategy describedabove now tends to alimit i.e.to an asymptoticstrategy for m oo then one couldinterpretthis as a goodstrategy for the game in which only upper and lower boundsaresetfor the bids (a and 6),and the bids can beanything in between(i.e.in (19:24)).I.e.the requirement of a minimum interval betweenbidsmentionedat thebeginning of 19.3.is removed.

Now this is not the case. E.g.we can interpolatethe a2, , am_ibetweenai = a and am = b both in arithmetic and in geometricsequence.2

In both casesan asymptoticstrategyobtainsfor m > but the two strate-giesdiffer in many essentialdetails.

If we considerthe game in which all bids (19:24)arepermitted,as onein its own right, then a directdetermination of its goodstrategiesispossible.

1 If the bids which he must make are ai, ap , afl , , an (l < p < q < < n),then it can be shown that their probabilities must be111 If 1,1, , 1, ,_,..., I c --

j

---h +ca\\ cap caq can V at ap a,,

respectively. I.e.if a certain bid is to be made at all, then its probability must beinversely proportional to the cost.

Which Op, ac, am actually occurfor a given z is determined by a more com-plicatedcriterion, which we shall not discusshere.

Observethat the c abovewas neededonly to make the sum of all probabilitiesequal to 1.Thereadermay verify for himself that the probabilities in Figure 40havethe abovevalues.

1Thefirst one is defined by

a, --T ((m - p)a + (p - 1)6) for p - 1,2, , m - 1,mm i

the secondone is defined by))

apm~am~*b*~l for p 1,2, , m 1,m.)))

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POKERAND BLUFFING 211It turns out that both strategiesmentionedabove aregood,togetherwith

many others.Thischowsto what complicationsthe abandonmentof a minimum inter-

val betweenbidscan lead:a goodstrategy of the limiting casecannot be anapproximationfor the goodstrategiesof all nearbycaseswith a finite num-berof bids.The concludingremarksof 19.12.arethus re-emphasized.

19.14.Alternate Bidding

19.14.1.Third and last, consider(C):Theonly progressso far made inthis directionis that we can replacethesimultaneousbidsof the two playersby two successiveones;i.e.by an arrangementin which player 1bids firstand player2 bids afterwards.

Thus the rulesstated in 19.4.aremodified as follows:Firsteachplayerobtains,by a chancemove, his hand 8 = 1, , S,

eachoneof thesenumbershaving the same probabilityl/S. We denotethe hands of players1,2,by Si, 2 respectively.

After this1player 1will, by a personalmove, chooseeithera or 6, the\"high\" or the \"low\" bid.2 Hedoes this informed about his own handbut not about the opponent'shand. If his bid is \"low,\" then the play isconcluded. If his bid is \"high,\" then player 2 will, by a personalmove,chooseeithera or 6, the \"high

\" or the \"low \" bid.3 Hedoesthis informedabout his own hand, and about the opponent'schoice,but not his hand.

This is the play. When it is concluded,the payments are made as>

follows:If player 1 bids \"low,\" then for si = s2 player1obtainsfrom player<b >

2 the amount respectively. If both playersbid \"high,\" then for Si = s2-6 <

aplayer 1obtains from player 2 the amount respectively. If player 1

abids \"high\" and player 2 bids \"low,\" then player1obtainsfrom player2the amount b.4

19.14.2.The discussionof the pure and mixed strategiescan now becarriedout, essentiallyas we did for our original variant of Pokerin 19.5.

We give the main lines of this discussionin a way which will be per-fectly clearfor the readerwho rememberstheprocedureof 19.4.-19.7.

A pure strategy in this game consistsclearlyof the followingspecifica-tions:to statefor every hand s = 1, , S whether a \"high\" or a \"low\"bid will be made. It is simpler to describethis by a numerical indexi.= 1,2;t.= 1meaning a \"high\" bid,i.= 2 meaning a \"low\" bid. Thus

1We continue from here on as if player 2 had already made the \"low\" bid, and thiswere player 1'sturn to \"See\"or to \"Overbid.\" We disregard \"Passing\" at this stage.

1I.e.\"Overbid\" or \"See,\" cf. footnote 1 above.8 I.e.\"See\"or \"Pass.\" Observethe shift of meaning sincefootnote 2 above.4 In interpreting theserules, recallthe abovefootnotes. From the formalist ic point

of view, footnote 1 on p.191should berecalled,mutatis mutandis.)))

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212)) ZERO-SUMTWO-PERSONGAMES:EXAMPLES))

the strategy is a specificationof such an indext,for every s = 1, , Si.e.of a sequencet'i, , is.

This applies to both players 1and 2;accordinglywe shall denotetheabovestrategy by 2i(ii, , ia) or 2*(jiy , ja). Thus eachplayerhas the samenumber of strategies,as many as there are sequencesi, , <*; i.e.precisely25. With the notationsof 11.2.2.

0i- A - ft = 2*.

(Butthe gameis not symmetrical!)We must now expressthe paymentwhich player1receivesif the strate-

gies2i(ti, , is), *0*i, * , ja) areusedby the two players. This isthe matrix elementOC(t'i, , ia\\ji 9 , ja). If the players haveactually hands*i, Sjthen thepayment receivedby player1can beexpressedin this way (using the rules statedabove):It is JB^nc^-*,)^,j f ) wheresgn(si *) is the signof 81 s and where the threefunctions))

canberepresentedby the following matrix schemes:))

t \\)

1) 2)

1) a) 6)

2) b) b))

\\j)1) 2)

1) 6)

2)

\\jt \\)

1) 2)

1) a) &)

2) -b) -6))

Figure 42. Figure 43. Figure 44,

Now Si,Sjoriginatefrom chancemoves, as describedabove. Hence:S))

, ja))) -1))

19.14.3.We now passto themixedstrategiesin thesenseof 17.2.These>

arevectors , 17 belongingto Sa. We must indexthe componentsof thesevectors like the (pure)strategies:we must write ^.....<f> ^.....<f

insteadOf fe,, 1?r,.

We express(17:2)of 17.4.1.which evaluatestheexpectationof player1'sgain))

.....))'tis,

j'i,)))

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t. ,ta excluding t))

POKERAND BLUFFING 213Thereisan advantage in interchanging the two Z and writing

> i VAK( , ? ) = -

2 2,pj

If we now put

(^25) P',= ......V

(19:26) < =J.' i ja excluding ;j

then the aboveequationbecomes l

(19:27) K(7,7)= gi

19.14.4.All this is preciselyas in 19.5.2.As there,(19:25)showsthat

p* is the probabilitythat player1,usingthe mixedstrategy will chooseiwhen his hand is s\\. (19:26)showsthat a',*is the probabilitythat player2,using the mixedstrategy 77 will choosej when his hand is s2. It is again

>

clearintuitively that the expectationvalue K( , 17 ) dependsupon theseprobabilitiesonly, and not on the underlying probabilities{ .....s, 17^,.____ ,athemselves.(19:27)expressesthis and could have easily been deriveddirectly,on this basis.

It is alsoclear,both from the meaning of the p',<rj and from their formaldefinitions (19:25),(19:26),that they fulfill the conditions:

2

(19:28) all p}. V p}i = 1-i2

(19:29) alUji V <rj= 1,

;-iand that any p's<r} which fulfill theseconditionscan be obtained from

suitable T> V by (19:25),(19:26).(Cf. the correspondingstep in 19.5.?.particularly footnote 1on p.194.)It is therefore opportuneto form the2-dimensionalvectors

P *!- {pfsPi'K * *' = (*i'> ^J1)-

Then(19:28),(19:29)statepreciselythat all 7\\ ^ ' belongto S.)))

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214 ZERO-SUMTWO-PERSONGAMES:EXAMPLES

Thus (or 17 ) was a vector in 8$ i.e.dependingon ft 1 = 2s 1

constants;the p * (or <r ')areS vectorsin Sti.e.eachone dependson onenumerical constant,hencethey amount togetherto S numerical constants.Sowe have reduced2s 1to S. (Cf.the end of 19.5.3.)

19.14.5.We now rewrite (19:27)as in 19.6.

(19:30) K(7s ' ' , 7V\\ , 7s) =^ V 7;-*;',

with the coefficients

IS'

i.e.usingthe matrix schemesof Figures42-44,

,-! S

(19:31:a)7l =g

,-i a

- * I V V M)

Sincethe game is no longer symmetric,we need also the correspondingformulae in which the rolesof the two playersare interchanged.Thisis:

(19:32) K( p ', , p V S ' ' ' , O =g

with the coefficients

H* .81S * 02o

i.e.usingthe matrix schemesof Figures42-44,i-l 8

(19:33:a)J.=g))

(19:33:b) - g ^^+ MO +))

Thecriteriafor goodstrategiesarenow essentiallyrepetitionsof those in

19.6.I.e.due to the asymmetry of the variant now under considerationour presentcriterion will be obtained from the generalcriterion (17:D))))

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POKER AND BLUFFING 215of 17.9.in the sameway as that of 19.6.couldbe obtainedfrom the sym-metrical criterion at the end of 17.11.2.I.e.:

(19:G) The7l, , 7*and the 7l, , 7* they all belongto /Si describegoodstrategieefifand only if this istrue:

Foreach t, j, for which 7* doesnot assumeits minimum(in j l) we have crj

= 0. Foreach i, i for which 6J does notassumeits maximum (in i *) we have pj = 0.

19.14.6.Now we replacethe discretehands *i, *i by continuous ones,in the senseof 19.7.(Cf. in particular Figure 39 there.) As describedin

19.7.this replacesthe vectors p \\ <r (sit t = 1, , S) by vectors

p *, <r *(0 *i,2s 1),which arestill probabilityvectorsofthe samenatureas before,i.e.belongingto Sj. Sothe componentspji, <r*- makeplacefor thecomponentspji, <rjt. Similarly the 5*-,7j become5*,7j. Thesumsin ourformulae (19:30),(19:31:a),(19:31:b),and (19:32),(19:33:a),(19:33:b)goover into integrals,just as in (19:7*),(19:9:a*),(19:9:b*),(19:9:c*)in 19.7.Sowe obtain:))

(19:30*) K

(19:31*) y\\> = /''(-ap|.- bp'.Odz,+ f'(apj.+ 6p{>)(bt|^0 / *t

(19:31:b*) T!= f\"0>til ~ bP'/o))

and

(19:32*)))

(19:33:a*)',.(19:33:b*)<; = f\" (hrjt + bff' t >)dzt + f

1

(-bo*?- fo;)<bi.y '*t

Our criterion for goodstrategiesis now equally transformed. (This is the

sametransition as that from the discretecriterion of 19.6.to the continuouscriterion of 19.7.)We obtain

(19:H) The p f i and the <r' (0 g * Ip zi g 1)they all belongto Si-

describegoodstrategiesif and only if this is true:Foreach*s, j for which 751 doesnot assumeits minimum

(in j 2) we have <r* = 0. Foreach* lf i for which $Ji doesnotassumeits maximum (in i 2) we have pj = 0.

*We mean in j (t) and not in **,,; (t i, i)!s We mean in j (t) and not in zs,; (ci> j) !)))

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216)) ZERO-SUMTWO-PERSONGAMES: EXAMPLES))

19.15.Mathematical Descriptionof All Solutions

19.15.1.The determinationof the goodstrategiesp 'and a *,i.e.of thesolutionsof the implicit conditionstated at the end of 19.14.,can becarriedout completely. Themathematfcal methodswhich achieve this aresimilarto thosewith which we determinedin 19.8.the goodstrategiesof our originalvariant of Poker, i.e.the solutionsof the implicit conditionstatedat theend of 19.7.

We shallnot give themathematical discussionhere,but we shalldescribethe goodstrategiesp * and a * which it produces.

Thereexistsoneand only one goodstrategy p *while thegoodstrategies<r form an extensivefamily. (Cf.Figures45-46.Theactual proportionsof thesefigures correspondto a/6~ 3.)))

Figure 45.)) Figure 46.))

(o - b)b\"

a(a + 36)a + 2a6- 6*

a(a + 36)-plot the curvesp =

p\\ anda = a\\ respectively. Thus the))Thelines-height of above the line p = (a = 0) is the probabilityof a \"high\"bid, p\\ (or}); the height of the line p = 1(<r

= 1)above isthe probability

of a \"low\" bid, pj = 1- pj fa = 1- <rj). The irregularpart of thev = <r* curve (in Figure46) in the interval u ^ z ^ v representsthe multiplicity of the goodstrategiesa z: Indeed,this part of the o- = a\\curve is subjectto the following (necessaryand sufficient) conditions:))

I a\\dzv - *<> J..))

baba))

when

when))

u))

U < 2 < V.))

Verbally:Betweenu and v the average of <rj is 6/a,and on any right endof this interval the average of a\\ is j 6/a.)))

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POKERAND BLUFFING)) 217))

Thus both p * and a * exhibitthreedifferent types of behavior on thesethreeintervals:1

First:0g z < u. Second:u<> z v. Third:v <z ^ 1. Thelengthsof thesethreeintervals areu, v u, 1 v, and the somewhatcomplicatedexpressionsfor M, v can bebest rememberedwith the help of theseeasilyverified ratios:

tt:l t; = a b:a+ b

v u:l t; = a:b.19.15.2.The formulae (19:31:a*),(19:31:b*)and (19:33:a*),(19:33:b*)

of 19.14.6.permitus now to computethe coefficients?J,3J. We give (asin19.9.in Figure41)the graphicalrepresentations,instead of the formulae,leaving the elementaryverification to the reader. Foridentification of thep *, <r *asgoodstrategiesonly the differences, t\\ 6J,y\\ y\\ matter:Indeed,thecriterionat the endof 19.14.can beformulated as stating that wheneverthis 'differenceis > then p = or <r| = respectively,and that whenever))

Figure 47. Figure 48.tg a = 2a, tg ft -26, tg y - 2(a - b)

this differenceis < then pj = or <rj= respectively. We give therefore

the graphs of thesedifferences. (Cf. Figures47, 48. Theactual propor-tions arethoseof Figures45, 46;i.e.a/6~3, cf. there.)

The line plots the curve y = y\\ yj; the lineplots the curve 5 =

&{ b\\. Theirregularpart of the d = $5 - d'2 curve(in Figure 48) in the interval u ^ z ^ v correspondsto the similarlyirregularpart of the <r = o-J curve (in Figure46) in the same interval,i.e.it alsorepresentsthemultiplicity of the goodstrategiesa '. Therestric-tion to which that part of the o- = a\\ curve is subjected(cf.the discussionafter Figure46) means that this part of the 6 =

b\\ 5J curve must liewithin the shadedtriangle /////////(cf. Figure48).19.15.3.Comparisonof Figure45 with Figure47, and of Figure46with Figure48 shows that our strategiesare indeed good,i.e.that theyfulfill (19:H).We leave it to the readerto verify this, in analogy with thecomparisonof Figure40 and Figure41in 19.9.

1Concerning the endpoints of theseintervals, etc.,cf.footnote 3on p. 200.)))

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218 ZERO-SUMTWO-PERSONGAMES: EXAMPLES

Thevalue of K can alsobe obtainedfrom (19:30*)or (19:32*)in 19.14.6.Theresult is:))

Thus player 1 has a positive expectationvalue for the play, i.e.an advan-tage2 which is plausiblyimputableto his possessingthe initiative.

19.16.Interpretation of the Solutions. Conclusions

19.16.1.The resultsof 19.15.shouldnow be discussedin the samewayas thoseof 19.8.,19.9.were in 19.10.We do not wish to do this at full

length, but just to makea few remarkson this subject.We seethat instead of the two zonesof Figure40 threezonesappear

in Figures45, 46. Thehighestone (farthest to the right) correspondsto\"high\" bids, and nothing else,in all thesefigures (i.e.for both players).Thebehavior of the otherzones,however, is not so uniform.

Forplayer2 (Figure46) the middlezone describesthat kind of \" Bluff-ing\" which we had on the lowestzone in Figure40, irregular \"high\" and\"low\" bids on the samehand, But the probabilities,while not entirelyarbitrary, arenot uniquely determinedas in Figure40.3 And thereexistsa lowestzone (in Figure46) where player 2 must always bid \"low,\" i.e.where his hand is too weakfor that mixedconduct.

Furthermore,in player2'smiddlezone the 7* show the sameindifferenceas in Figure41 y\\

- y{ = there, both in Figure41and in Figure47so the motives for his conductin this zone areas indirectas thosediscussedin the last part of 19.10.Indeed,these\"high\" bidsaremore of a defenseagainst \"Bluffing,\" than \"Bluffing\" proper. Sincethis bid of player 2concludesthe play, thereis indeedno motive for the latter,while thereis aneed to put a rein on the opponent's\"Bluffing\" by occasional\"high\"bids, by \"Seeing\"him.

Forplayer1(Figure45) the situationis different. Hemust bid \"high,\"

and nothing else,in the lowestzone;and bid \"low,\" and nothing else,inthe middle zone. These\"high\" bids on the very weakesthands whilethe bid on the medium hands is \"low\" areaggressive\"Bluffing\" in its

1 For numerical orientation: If a/6 = 3, which is the ratio on which all our figures arebased,then u J, v J and K = -

1For a/6 ~ 3 this is about 6/9 (cf. footnote 1 above),i.e.about 11per cent, of the\"low \"bid.

1Cf.the discussion after Figure 46. Indeed,it is evenpossibleto meet those require-ments with o\\ and 1 only; e.g.aj in the lower-fraction and a\\ * 1 in the

6upper -fraction of the middle interval.

The existenceof such a solution (i.e.never o\\ ^ 0,1, by Figure 45never p[ y* 0, 1either) means, of course,that this variant is strictly determined. But a discussion onthat basis(i.e.with pure strategies)will not disclosesolutions like the oneactually drawnin Figure 46.)))

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POKERAND BLUFFING 219purest form. The d* arenot at all indifferent in this zone of \"Bluffing\"

(i.e.the lowestzone): t>\\ 6J > there in Figure48. i.e.any failure to\"Bluff\" under theseconditionsleadsto instant losses.

19.16.2.Summing up:Our new variant of Pokerdistinguishestwovarietiesof \"Bluffing\": the purely aggressiveonepracticedby the playerwho has the initiative; and a defensive one \"Seeing\"irregularly, evenwith a medium hand, the opponent who is suspectedof \"Bluffing\"

practicedby the player who bids last. Our original variant where theinitiative was split betweenthe two players becausethey bid simultane-ously containeda procedurewhich we can now recognizeas a mixture ofthesetwo things.1

All this gives valuable heuristic hints how realPoker with longersequencesof (alternating)bids and overbids ought to be approached.Themathematical problemis difficult, but probablynot beyondthereachofthetechniquesthat areavailable. Itwill beconsideredin otherpublications.

1The variant of E. Borel, referredto in footnote 2 on p. 186,is treated loc.cit. in away which bearsa certain resemblanceto our procedure. Using our terminology, thecourseof E.Borel can be describedas follows:

The Max-Min (Max for player 1,Min for player 2) is determined both for pure andfor mixed strategies. The two are identical, i.e.this variant is strictly determined.The good strategieswhich are obtained in this way are rather similar to those of ourFigure 46. Accordingly the characteristicsof \"Bluffing\" do not appearas clearlyas inour Figures 40and 45. Of.the analogous considerations in the text above.)))

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CHAPTER V

ZERO-SUMTHREE-PERSONGAMES

20.Preliminary Survey20.1.GeneralViewpoints

20.1.1.The theory of the zero-sum two-person game having beencompleted,we takethe nextstep in the senseof 12.4.:We shall establishthe theory of the zero-sumthree-persongame. This will bring entirelynew viewpoints into play. The types of gamesdiscussedthus far havehad also their own characteristicproblems.We saw that the zero-sumone-persongame wascharacterizedby the emergenceof a maximum problemand the zero-sumtwo-persongameby the clearcut oppositionof interestwhich couldno longerbe describedas a maximum problem.And just asthe transition from the one-personto the zero-sum two-person gameremoved the pure maximum characterof the problem,so the passagefromthe zero-sumtwo-persongame to the zero-sumthree-persongameobliteratesthe pure oppositionof interest.

20.1.2.Indeed,it is apparent that the relationshipsbetweentwo playersin a zero-sumthree-persongame can be manifold. In a zero-sumtwo-persongameanything one player wins is necessarilylost by the otherandvice versa, so there is always an absolute antagonism of interests.In azero-sumthree-persongame a particular move of a player which, for thesakeof simplicity,we assumeto be clearlyadvantageousto him may bedisadvantageousto both other players,but it may also be advantageousto oneand (afortiori) disadvantageousto theotheropponent.1 Thus someplayersmay occasionallyexperiencea parallelismof interestsand it may beimaginedthat a more elaboratetheory will have to decideeven whetherthis parallelismis total,or partial, etc. On the otherhand, oppositionofinterestmust also existin the game(it is zero-sum) and so the theorywill have to disentanglethe complicatedsituationswhich may ensue.

It may happen, in particular,that a playerhas a choiceamong variouspolicies:That he can adjust his conductso as to get into parallelismofinterest with another player, or the opposite;that he can choosewith

which of theothertwo playershe wishesto establishsucha parallelism,and(possibly)to what extent.

1All this, of course,is subjectto all the complications and difficulties which we havealready recognizedand overcomein the zero-sum two-person game:whether a particularmove is advantageous or disadvantageous to a certain player may not dependon thatmove alone, but alsoon what other playersdo. However, we are trying first to isolatethe new difficulties and to analyze them in their purest form. Afterward we shall dis-cussthe interrelation with the old difficulties.

220)))

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PRELIMINARY SURVEY 22120.1.3.As soon as there is a possibilityof choosingwith whom to

establishparallelinterests,this becomesa caseof choosingan ally. Whenalliancesareformed, it is to beexpectedthat somekind of a mutual under-standing between the two players involved will be necessary.One canalso state it this way: A parallelismof interestsmakes a cooperationdesirable,and therefore will probablylead to an agreementbetween theplayers involved. An oppositionof interests,on the otherhand, requirespresumablyno more than that a player who has electedthis alternativeactindependentlyin his own interest.

Of all this there can be no vestige in the zero-sumtwo-persongame.Betweentwo players,where neither can win except(precisely)the other'sloss,agreementsor understandingsarepointless.1 This shouldbeclearbycommon sense. If a formal corroboration (proof) be needed,one canfind it in our ability to completethe theory of the zero-sumtwo-persongame without ever mentioning agreementsor understandings betweenplayers.

20.2.Coalitions

20.2.1.We have thus recognizeda qualitatively different feature of thezero-sumthree-persongame (as against the zero-sumtwo-persongame).Whether it is the only one is a questionwhich canbedecidedonly later.If we succeedin completingthe theory of the zero-sumthree-persongamewithout bringing in any further new concepts,then we can claim to haveestablishedthis uniqueness.Thiswill be thecaseessentiallywhen we reach23.1.Forthe moment we simply observethat this is a new major elementin the situation,and we proposeto discussit fully before takingup anythingelse.

Thus we wish to concentrateon the alternativesfor acting in cooperationwith, or in oppositionto, others,among which a playercan choose.I.e.wewant to analyze the possibilityof coalitions the questionbetweenwhichplayers,and against which player,coalitionswill form.2

1 This is,ofcourse,different in a general two-person game (i.e.onewith variable sum) :there the two playersmay conceivably cooperateto producea greater gain. Thus thereis a certain similarity between the general two-person game and the zero-sum three-persongame.

We shall seein Chap.XI, particularly in 56.2.2.,that there is a generalconnectionbehind this: the general n-person game is closelyrelated to the zero-sum n + I-persongame.1The following seemsworth noting: coalitions occurfirst in a zero-sum game whenthe number of participants in the game reachesthree. In a two-person game there arenot enough players to go around: a coalition absorbsat least two players, and then

nobody is left to oppose. But while the three-persongame of itself implies coalitions,the scarcity of players is still such as to circumscribe thesecoalitions in a definite way: acoalition must consistof preciselytwo players and bedirectedagainst preciselyone (theremaining) player.

If there are four or more players, then the situation becomesconsiderably moreinvolved, severalcoalitions may form, and thesemay merge or opposeeachother, etc.Someinstances of this appearat the end of 36.1.2.,et seq.,the end of 37.1.2.,et seq.;another alliedphenomenon at the end of 38.3.2.)))

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222 ZERO-SUMTHREE-PERSONGAMES

Consequentlyit is desirableto form an exampleof a zero-sumthree-persongame in which this aspectis foremost and all othersaresuppressed;i.e.,a game in which the coalitionsarethe only thing that matters,and theonly conceivableaim of all players.1

20.2.2.At this point we may mention also the following circumstance:A playercan at best choosebetweentwo possiblecoalitions,sincetherearetwo otherplayerseitherof whom he may try to induceto cooperatewith him

againstthe third. We shallhave to elucidateby thestudy of the zero-sumthree-persongame just how this choiceoperates,and whether any particularplayerhas such a choiceat all. If, however, a playerhas only one possi-bility of forming a coalition (in whatever way we shall in fine interpret thisoperation)then it is not quite clearin what sensethereis a coalition at all:moves forced upon a player in a unique way by the necessitiesof the rulesof the game aremore in the nature of a (onesided)strategythan of a (cooper-ative) coalition. Of coursethese considerationsare rather vague anduncertain at the presentstageof our analysis. We bringthem up neverthe-less,becausethesedistinctionswill turn out to be decisive.

It may also seemuncertain,at this stageat least,how the possiblechoicesof coalitionswhich confront one playerarerelatedto thoseopen toanother;indeed,whether the existenceof severalalternatives for one playerimpliesthe samefor another.

21.TheSimpleMajority Game of ThreePersons21.1.Descriptionof the Game

21.1.We now formulate the examplementionedabove:a simplezero-sum three-persongame in which the possibilitiesof understandings i.e.coalitions betweenthe playersarethe only considerationswhich matter.

This is the game in question:Each player, by a personalmove, choosesthe number of one of the

two other players.2 Each one makes his choiceuninformed about thechoicesof the two otherplayers.

After this the payments will be made as follows:If two players havechoseneachother'snumbers we say that they form a couple* Clearly

1This is methodically the same deviceas our consideration of Matching Pennies inthe theory of the zero-sum two-person game. We had recognizedin 14.7.1.that thedecisivenew feature of the zero-sum two-person game was the difficulty of decidingwhich player \"finds out\" his opponent. Matching Pennieswas the game in which this\"

finding out\" dominated the picture completely, where this mattered and nothing else.2 Player 1chooses2or 3, player 2 chooses1or 3, player 3 chooses1or 2.*It will be seenthat the formation of a coupleis in the interest of the players who

createit. Accordingly our discussion of understandings and coalitions in the paragraphswhich follow will show that the players combine into a coalition in order to be abletoform a couple. Thedifference between the conceptsofa coupleand a coalition neverthe-lessshould not beoverlooked:A coupleis a formal conceptwhich figures in the setof rulesof the game which we define now; a coalition is a notion belonging to the theory concern-ing this game (and, as will beseen,many other games).)))

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THESIMPLEMAJORITY GAME 223

therewill be preciselyone couple,or none at all.1- 2 If thereis preciselyonecouple,then the two playerswho belongto it getone-half unit each,whilethe third (excluded)player correspondinglylosesoneunit. If thereis nocouple,then no one getsanything. 8

The readerwill have no difficulty in recognizingthe actual socialproc-essesfor which this game is a highly schematizedmodel.We shall call itthe simplemajority game (of three players).

21.2.Analysis of the Game. Necessityof \"Understandings\"

21.2.1.Let us try to understand the situation which existswhen thegame is played.

To begin with, it is clearthat thereis absolutelynothing for a playerto do in this game but to look for a partner, i.e.for another playerwhois preparedto form a couplewith him. The game is so simpleand abso-lutely devoidof any other strategicpossibilitiesthat therejust isno occasionfor any other reasonedprocedure.Sinceeachplayer makes his personalmove in ignoranceof those of the others, no collaborationof the playerscan be establishedduring the courseof the play. Two playerswho wish tocollaboratemust get togetheron this subjectbefore the play, i.e.outsidethe game. The player who (in making his personalmove) lives up to hisagreement(by choosingthe partner'snumber) must possessthe convictionthat the partner too will do likewise. As long as we areconcernedonlywith the rulesof the game, as stated above, we arein no positionto judgewhat the basis for such a conviction may be. In other words what, if

anything, enforces the \"sanctity\"of such agreements?Theremay begameswhich themselves by virtue of the rulesof the game as defined in6.1.and 10.1.provide the mechanism for agreementsand for their enforce-ment.4 But we cannot base our considerationson this possibility,sinceagame neednot provide this mechanism;the simplemajority game describedabove certainly does not. Thus there seemsto be no escapefrom thenecessityof consideringagreementsconcludedoutsidethe game. If we donot allow for them, then it is hard to seewhat, if anything, will govern theconductof a player in a simplemajority game. Or, to put this in a some-what different form :

1 I.e.there cannot be simultaneously two different couples. Indeed,two couplesmust have one player on common (sincethere are only three players), and the numberchosenby this player must be that of the other player in both couples, i.e.the twocouplesare identical.

2 It may happen that no couplesexist:e.g.,if 1chooses2, 2chooses3,and 3chooses1.3 For the sakeof absolute formal correctnessthis should still be arranged according

to the patterns of 6.and 7. in Chap.II. We leave this to the reader,as in the analogoussituation discussedin footnote 1 on p.191.

4 By providing personalmoves of one player, about which only one other player isinformed and which contain (possibly conditional) statements of the first player'sfuture

policy; and by prescribing for him to adhere subsequently to these statements, or byproviding (in the functions which determine the outcome of a game) penalties for thenon-adherence.)))

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224 ZERO-SUMTHREE-PERSONGAMES

We aretrying to establisha theory of the rational conductof the partici-pants in a given game. In our considerationof the simplemajority gamewe have reachedthe point beyondwhich it is difficult to go in formulatingsuch a theory without auxiliary conceptssuch as \" agreements/'\"under-standings/1 etc. On a lateroccasionwe proposeto investigatewhat theo-reticalstructures arerequired in orderto eliminate theseconcepts.Forthis purposethe entiretheory of this bookwill berequiredasa foundation,and the investigation will proceedalong the linesindicatedin ChapterXII,and particularly in 66. At any rate,at presentour positionis tooweakandour theory not sufficiently advancedto permit this \"self-denial.\"We shalltherefore, in the discussionswhich follow,makeuseof the possibilityof theestablishmentof coalitionsoutsidethe game;this will includethe hypothesisthat they arerespectedby the contractingparties.

21.2*2.Theseagreementshave a certain amount of similarity with\" conventions \" in somegameslikeBridge with the fundamental difference,however, that those affected only one \" organization \" (i.e.one player splitinto two \"persons\")while we arenow confronted with the relationshipoftwo players. At this point the readermay rereadwith advantage ourdiscussionof \"conventions\"and relatedtopicsin the last part of 6.4.2.and6.4.3.,especiallyfootnote 2 on p.53.

21.2.3.If our theory wereappliedas a statisticalanalysisof a long seriesof playsof the samegame and not as theanalysisof one isolatedplay analternative interpretation would suggestitself. We should then viewagreementsand all forms of cooperationas establishingthemselvesbyrepetitionin such a long seriesof plays.

It would not be impossibleto derivea mechanismof enforcement fromtheplayer'sdesireto maintain his recordand to be ableto rely on the recordof his partner. However,we prefer to view our theory as applyingto anindividual play. But theseconsiderations,nevertheless,possessa certainsignificance in a virtual sense. The situation is similar to the one whichwe encounteredin the analysisof the (mixed)strategiesof a zero-sumtwo-person game. The readershould apply the discussionsof 17.3.mutatismutandis to the presentsituation.

21.3.Analysis of the Game:Coalitions. TheRoleof Symmetry

21.3.Onceit is concededthat agreementsmay existbetweenthe playersin the simplemajority game,the path is clear. This game offers to playerswho collaboratean infallible opportunity to win and the gamedoesnotoffer to anybodyopportunitiesfor rational actionof any otherkind. Therulesaresoelementarythat this point ought to be fully convincing.

Again the gameis wholly symmetricwith respectto the threeplayers.That is trueas far as the rulesof the gameareconcerned:they do not offertoany playerany possibilitywhich is not equallyopen to any otherplayer.What theplayersdo within thesepossibilitiesis,of course,anothermatter.Theirconductmay beunsymmetric;indeed,sinceunderstandings,i.e.coali-)))

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FURTHEREXAMPLES 225

tions, are sure to arise,it will of necessitybe unsymmetric. Amongthe three playersthereis room for only one coalition (of two players)andone player will necessarilybeleft out. Itisquite instructive to observehowthe rulesof the game areabsolutelyfair (in this case,symmetric),but theconductof the playerswill necessarilynot be.1*2

Thus the only significant strategicfeature of this game is the possibilityof coalitionsbetweentwo players.8 And sincethe rules of the gameareperfectly symmetrical,all three possiblecoalitions4 must be consideredon the same footing. If a coalition is formed, then the rulesof the gameprovide that the two alliesgetone unit from the third (excluded)playereachone getting one-half unit.

Which of thesethreepossiblecoalitions will form, is beyondthe scopeof the theory, at leastat the presentstageof its development. (Cf.the endof 4.3.2.)We can say only that it would be irrational if no coalitionswereformed at all, but as to which particular coalitionwill beformed must dependon conditionswhich we have not yet attempted to analyze.

22.Further Examples22.1.Unsymmetric Distribution. Necessityof Compensations

22.1.1.The remarksof the precedingparagraphsexhaust, at leastforthe time being, the subject of the simplemajority game. We must nowbeginto remove, one by one, the extremely specializingassumptionswhichcharacterizedthis game:its very specialnature was essential for us inorderto observethe role of coalitions in a pureand isolatedform in vitro

1We saw in 17.11.2.that no such thing occursin the zero-sum two-person games.There,if the rules of the game are symmetric, both players get the same amount (i.e.the value of the game is zero),and both have the samegoodstrategies. I.e.there is noreason to expecta difference in their conduct or in the results which they ultimatelyobtain.

It is on emergenceof coalitions when more than two players arepresent and ofthe \"squeeze\" which they produce among the players, that the peculiar situationdescribedabovearises. (In our presentcaseof three players the \"squeeze\" is due to thefact that eachcoalition can consistof only two players, i.e.lessthan the total number ofplayers but more than one-half of it. It would beerroneous, however, to assume that nosuch \"squeeze\" obtains for a greater number of players.)

*This is,of course,a very essentialfeature of the most familiar forms of socialorgani-zations. It is also an argument which occursagain and again in the criticism directedagainst these institutions, most of all against the hypothetical orderbasedupon \"Jowserfaire.\" It is the argument that even an absolute, formal fairness symmetry of the rulesof the game doesnot guarantee that the use of theserules by the participants will befair and symmetrical. Indeed,this \"doesnot guarantee\" is an understatement: it is tobe expectedthat any exhaustive theory of rational behavior will show that the partici-pants aredriven to form coalitions in unsymmetric arrangements.

Tothe extent to which an exacttheory of thesecoalitions is developed,a real under-

standing of this classicalcriticism is achieved. It seemsworth emphasizing that this

characteristically \"social\" phenomenon occursonly in the case of three or moreparticipants.

5Such a coalition is in this game, ofcourse,simply an agreement to chooseeachother'snumbers, so as to form a couplein the senseof the rules. This situation was foreseenalready at the beginning of 4.3.2.

4 Between players 1,2;1,3;2,3.)))

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226 ZERO-SUMTHREE-PERSONGAMES

but now this step is completed.We must begin to adjust our ideastomore generalsituations.

22.1.2.Thespecializationwhich we proposeto remove first is this:Inthe simplemajority game any coalition can getone unit from the opponent;the rules of the game provide that this unit must be divided evenlyamong the partners. Let us now considera game in which eachcoalitionoffers the same total return, but where the rulesof the gameprovide for adifferent distribution. Forthesakeof simplicityletthis be the caseonly inthe coalition ofplayers1and2,where player1,say,isfavored by an amount e.The rulesof the modified game aretherefore as follows:

The moves are the same as in the simplemajority game describedin21.1.The definition of a coupleis the sametoo. If the couple1,2forms,then player 1getsthe amount + l, player2 getsthe amount e, andplayer 3 loses one unit. If any othercoupleforms (i.e.1,3or 2,3)thenthe two playerswhich belongto it get one-half unit eachwhile the third

(excluded)playerlosesone unit.What will happen in this game?To beginwith, it isstillcharacterizedby the possibilityof threecoalitions

correspondingto the three possible couples which may arisein it.Primafacie it may seemthat player1has an advantage,sinceat leastin hiscouplewith player2 he getsmore by than in the original, simplemajoritygame.

However,this advantage is quite illusory. If player 1would reallyinsist on gettingthe extrae in the couplewith player 2, then this wouldhave the followingconsequence:Thecouple1,3would never form, becausethecouple1,2is more desirablefrom 1'spoint of view; the couple1,2wouldnever form, becausethe couple2,3is more desirablefrom 2'spoint of view;but the couple2,3is entirely unobstructed,sinceit canbe brought aboutby a coalition of 2,3who then needpay no attention to 1and his specialdesires.Thus the couple2,3and no otherwill form ; and player 1will notgeti + enor even one-half unit, but he will certainly be the excludedplayerand loseoneunit.

Soany attempt of player1to keephis privilegedpositionin the couple1,2is bound to lead to disaster for him. The best he can do is to takesteps which make the couple1,2just as attractive for 2 as the competingcouple2,3. That is to say, he actswisely if, in caseof the formation ofa couplewith 2,he returns the extra to his partner. It shouldbe notedthat he cannot keepany fraction of e;i.e.,if he shouldtry to keepan extraamount e'for himself,2 then theabove argumentscouldberepeatedliterallywith 'in placeof e.3

1 It seemsnatural to assume < e < J.2 We mean of course <'<.8 So the motives for player 1'sultimate disaster the certain formation of couple

2,3 would be weaker, but the disasterthe same and just as certain as before. Cf. inthis connection footnote 1on p. 228.)))

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FURTHEREXAMPLES 227

22.1.3.One could try some other variations of the original, simple,majority game,still always maintaining that the total value ofeachcoalitionis oneunit. E.g.we couldconsiderrules where player 1 getsthe amounti + in eachcouple1,2,1,3;while players2 and 3 split even in the couple2,3. In this caseneither 2 nor 3 would careto cooperatewith 1if 1shouldtry to keephis extra or any fraction thereof. Henceany such attemptof player1would again leadwith certainty to a coalition of 2,3againsthimand to a lossof one unit.

Another possibilitywould be that two playersarefavored in all coupleswith the third:e.g.in the couples1,3and 2,3,players1and 2 respectivelygeti + while 3 getsonly i ; and in the couple1,2both get one-halfunit each. In this caseboth players1and 2 would loseinterestin a coalitionwith eachother,and player3 will becomethe desirablepartner for eachofthem. One must expectthat this will lead to a competitive biddingfor hiscooperation.Thismust ultimately leadto a refund to player3 of the extraadvantage e. Only this will bring the couple1,2back into the field ofcompetitionand therebyrestoreequilibrium.

22.1.4.We leave to the reader the considerationof further variants,where all threeplayersfare differently in all threecouples. Furthermorewe shall not push the above analysisfurther, although this couldbe doneand would even be desirablein order to answer someplausibleobjections.We aresatisfiedwith having establishedsomekind of a generalplausibilityfor our present approach which can be summarized as follows:It seemsthat what a playercan get in a definite coalition dependsnot only on whatthe rules of the game provide for that eventuality, but also on the other(competing)possibilitiesof coalitionsfor himself and for his partner. Sincethe rules of the game are absoluteand inviolable, this means that undercertainconditionscompensationsmust be paid among coalition partners;i.e.that a player must have to pay a well-defined price to a prospectivecoalition partner. The amount of the compensationswill dependon whatotheralternatives areopento eachof the players.

Our examplesabove have servedasa first illustration for theseprinciples.This being understood,we shall now takeup the subjectde novo and inmore generality,and handle it in a more precisemanner. l

22.2.Coalitions of Different Strength. Discussion

22.2.1.In accordancewith the above we now take a far reaching steptowardsgenerality. We considera game in which this is the case:

If players1,2cooperate,then they can get the amount c, and no more,from player3;if players1,3cooperate,they can get the amount 6, and nomore, from player2;if players2,3cooperate,they can get the amount a,andno more, from player1.

1This is why we need not analyze any further the heuristic arguments of this para-graph the discussion of the next paragraphs takes careof everything.

All thesepossibilities wereanticipated at the beginning of4.3.2.and in 4.3.3.)))

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228 ZERO-SUMTHREE-PERSONGAMES

We make no assumptionswhatsoever concerningfurther particularsabout the rulesof this game. Sowe neednot describeby what steps ofwhat orderof complication the above amounts aresecured.Nor do westatehow theseamounts aredividedbetweenthe partners,whether and howeitherpartner can influence ormodify this distribution,etc.

We shall neverthelessbe able to discussthis gamecompletely. Butit will be necessaryto rememberthat a coalition is probably connectedwith compensationspassingbetween the partners. The argument is asfollows:

22.2.2.Considerthe situation of player1. Hecan entertwo alternativecoalitions:with player 2 or with player 3. Assume that he attempts toretain an amount x under all conditions.In this caseplayer 2 cannotcount upon obtaining more than the amount c x in a coalition with

player 1. Similarly player 3 cannot count on getting more than theamount 6 x in a coalition with player 1. Now if the sum of theseupperbounds i.e.the amount (c x) + (b x) is less than what players 2and 3 can get by combining with eachotherin a coalition, then we maysafely assumethat player 1will find no partner.1 A coalition of 2 and 3can obtain the amount a. Sowesee:If player1desiresto getan amount xunder all conditions,then he is disqualified from any possibilityof findinga partner if his x fulfills

(c - x) + (b - x) < a.I.e.the desireto getx is unrealisticand absurdunless

(c - x) + (b - x) ^ a.This inequality may be written equivalently as.-a+ b + c** 2We restatethis :(22:l:a) Player 1 cannot reasonablymaintain a claim to get under

all conditionsmore than the amount a = ~2i

Thesameconsiderationsmay be repeatedfor players2 and 3, and theygive:(22:l:b) Player 2 cannot reasonablymaintain a claim to get under

all conditionsmore than the amount ft =i

(22:l:c) Player 3 cannot reasonablymaintain a claim to get under

all conditionsmore than the amount 7 = 5&1We assume,of course,that a player is not indifferent to any possibleprofit, however

small. This was implicit in our discussion of the zero-sum two-person game aswell.The traditional ideaof the \"homo oeconomicus,\" to the extent to which it is clearly

conceivedat all, alsocontains this assumption.)))

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FURTHEREXAMPLES 229

22.2.3.Now the criteria(22:l:a)-(22:l:c)were only necessaryones,andone could imagine a priori that further considerationscould further lowertheir upper bounds, a, 0,y or lead to some otherrestrictionsof whatthe players can aim for. This is not so,as the following simple con-siderationshows.

One verifies immediately that

a + = c, a + 7 = 6, + 7 = <*

In other words:If the players1,2,3do not aim at more than permittedby(22:1:a),(22:1:b), (22:1:c),i.e.than a, 0,y respectively,then any twoplayers who combinecan actually obtain these amounts in a coalition.Thus theseclaimsarefully justified. Of courseonly two players the twowho form a coalition can actually obtain their \"justified\"dues. Thethird player,who is excludedfrom the coalition, will not geta, ft, y respec-tively, but a, 6, c instead.1

22.3.An Inequality. Formulae

22.3.1.At this point an obvious question presents itself:Any player1,2,3can get the amount a, 0,y respectivelyif he succeedsin enteringacoalition;if he doesnot succeed,he getsinstead only a, b, c. Thismakessenseonly if a,0,y aregreaterthan the corresponding a, 6, c,sinceotherwisea player might not want to enter a coalition at all, butmight find it more advantageous to play for himself. So the questioniswhether the threedifferences

p a _ (__ a) = a + a,g = 0-(-&) =0 + 6,r = y ~ (-<0= y + c,

areall ^ 0.It is immediately seenthat they areall equal to eachother. Indeed:

a + b + cP = 9 = r =----

We denotethis quantity by A/2. Then our questionis whether))

This inequality can be demonstratedas follows:22.3.2.A coalition of the players 1,2can obtain (from player 3) the

amount c and no more. If player 1 playsalone, then he can prevent players2,3from reducinghim to a result worsethan a sinceeven a coalition ofplayers2,3can obtain (from player 1) the amount +a and no more;i.e.player 1 can get the amount a for himself without any outside help.Similarly,player 2 can get the amount 6 for himself without any outsidehelp. Consequentlythe two players1,2betweenthem can get the amount

1Theseare indeed the amounts which a coalition of the other players can wrest from

players 1,2,3respectively. Thecoalition cannot take more.)))

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230)) ZERO-SUMTHREE-PERSONGAMES))

(a -f 6) even if they fail to cooperatewith eachother. Sincethe maxi-mum they can obtain togetherunder any conditions is c, this impliesc ^ -a- b i.e.A = a + 6 +c0.

22.3.3.This proof suggeststhe followingremarks:First:We have based our argument on player 1. Owing to the sym-

metry of the result A=a+6+c^Owith respectto the threeplayers,the sameinequality would have obtained if we had analyzedthe situationof player2 or player3. This indicatesthat thereexistsa certainsymmetryin the roleof the threeplayers.

Second:A = meansc = a 6 or just as well a = a, and the twocorrespondingpairs of equationswhich obtain by the cyclicpermutationof the threeplayers. So in this caseno coalition has a raisond'&tre:Anytwo playerscan obtain, without cooperating,the sameamount which theycanproducein perfectcooperation(e.g.for players1and 2 this amount is

a b = c). Also, after all is said and done,eachplayer who succeedsin joining a coalition getsno more than he couldget for himself withoutoutsidehelp (e.g.for player 1this amount is a = a).

If, on the otherhand, A > then every playerhas a definite interestinjoining a coalition. The advantage contained in this is the same for allthreeplayers:A/2.

Herewe have again an indicationof the symmetry of certainaspectsof the situation for all players:A/2 is the inducementto seeka coalition;itis the samefor all players.

22.3.4.Our result canbeexpressedby the following table:))

Player) i) 2) 3)

Value of a play)

With coalition) a) ft) 7)

Without coalition) a) -b) c))

Figure 49.))

If we put))

., L, 1A a 1

Aa-26+ c6'= -6+ -A = /J-g A

g,

/ . 1A

1A a + b - 2cc'=-c+ A = T-gA=

g,

then we havea'+ V + c'= 0,

and we can expressthe abovetable equivalently in the followingmanner:

(22:A) A play has for the players1,2,3thebasicvaluesa',6',c'respec-tively. (Thisis a possiblevaluation, sincethe sum of these)))

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THEGENERAL CASE 231values is zero,cf. above). Theplay will, however, certainly beattended by the formation of a coalition. Thosetwo playerswho form it get(beyondtheir basicvalues) a premium of A/6 andthe excludedplayersustainsa lossof A/3.

Thus the inducement to form a coalition is A/2 for eachplayer,and always A/2 ^ 0.

23.TheGeneralCase23.1.Exhaustive Discussion.Inessentialand Essential Games

23.1.1.We can now remove all restrictions.Let T be a perfectly arbitrary zero-sumthree-persongame. A simple

considerationsufficesto bring it within the reachof the analysisof 22.2.,22.3.We argue as follows:

If two players,say 1and 2, decideto cooperatecompletely postponingtemporarily,for a latersettlement,the questionof distribution,i.e.of thecompensationsto be paid betweenpartners then F becomesa zero-sumtwo-persongame. The two playersin this new game are:the coalition 1,2(which is now a compositeplayer consistingof two \"natural persons\,and the player3. Viewed in this manner T falls under the theory of thezero-sumtwo-persongame of Chapter III. Each play of this gamehas awell defined value (we mean the v' defined in 17.4.2.).Let us denoteby cthe value of a play for the coalition 1,2(which in our presentinterpretationis one of the players).

Similarly we can assumean absolutecoalition betweenplayers1,3andview T as a zero-sum two-persongame betweenthis coalition and the player2. We then denoteby b the value of a play for the coalition 1,3.

Finally we can assumean absolutecoalition betweenplayers2,3,andview F as a zero-sum two-persongamebetween this coalition and theplayer 1. We then denoteby a the value of a play for the coalition 2,3.

It ought to be understoodthat we do not yet! assumethat any suchcoalition will necessarilyarise. The quantitiesa, 6, c aremerely computa-tionally defined;we have formed them on the basis of the main (mathe-matical) theorem of 17.6.(Forexplicitexpressionsof a,6, c cf. below.)

23.1.2.Now it is clearthat the zero-sumthree-persongame F fallsentirely within the domain of validity of 22.2.,22.3.:a coalition of theplayers1,2or 1,3or 2,3can obtain (from the excludedplayers3 or 2 or 1)the amounts c, 6, a respectively,and no more. Consequentlyall resultsof22.2.,22.3.hold,in particularthe one formulated at the end which describesevery player'ssituation with and without a coalition.

23.1.3.Theseresults show that the zero-sumthree-persongamefallsinto two quantitatively different categories,correspondingto the possi-bilitiesA = and A > 0. Indeed:

A = 0:We have seenthat in this casecoalitionshave no raisond'etre,and eachplayer canget the sameamount for himself, by playing a \"lonehand\"againstall others,as he couldobtainby any coalition. Inthis case,)))

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232 ZERO-SUMTHREE-PERSONGAMES

and in this casealone,it is possibleto assumea uniquevalue of eachplayfor eachplayer, the sum of thesevalues beingzero. Thesearethe basicvalues a',&', c'mentionedat the end of 22.3.In this casethe formulae of22.3.show that a!= a = -a, V = ft = -b, c1 = y = -c. We shallcall a game in this case,in which it is inessentialto considercoalitions,aninessentialgame.

A >0:In this casethereis a definite inducementto form coalitions,asdiscussedat the end of 22.3.Thereis no needto repeatthe descriptiongiven there;we only mention that now a > a'> a, ft > b' > 6,7 >c'> c. We shall call a game in this case,in which coalitionsareessential,an essentialgame.

Our above classification,inessentialand essential,applies at presentonly to zero-sum three-persongames.But weshallseesubsequentlythat itcan be extendedto all games and that it is a differentiation of centralimportance.

23.2.CompleteFormulae

23.2.Beforewe analyze this result any further, letus makea few purelymathematical remarksabout the quantitiesa, 6, c and the a,0,7,a',6',c',A basedupon them in termsof which our solution was expressed.

Assume the zero-sumthree-persongameF in the normalized form of11.2.3.Therethe players 1,2,3choosethe variablesTI, r2, r 3 respectively(eachoneuninformed about the two otherchoices)and get the amounts3Ci(ri,T2, r 3),3C2(ri,T2, r3),3C3(n, T2, T3) respectively. Of course(thegame iszero-sum):

3Ci(n,T2, T8) + JC2(ri,T2, T3) + 3C3(n, r 2, r 3) s 0.Thedomainsof the variablesare:

T! = 1,2, - - - , 0,,T, = 1,2, , 02,T3 = 1,2, , 3.

Now in the two-persongame which arisesbetweenan absolutecoalition ofplayers1,2,and the player3,we have the followingsituation:

The compositeplayer 1,2has the variablesn, r 2; the other player 3has the variable r8. Theformer getsthe amount

3Cl(Tl,T2, T3) +3C2(n, T2, T3) 3 -3Cs(Ti,T2, T3),the latterthe negative of this amount.

A mixedstrategy of the compositeplayer 1,2is a vector of S^,the

componentsof which we may denoteby {fiiT|.1 Thus the of S

fti pt arecharacterizedby

*V r, 0, 2 *v,= 1-*!/*

1Thenumber of pairs n, TJ is, of course,0ifa.)))

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DISCUSSIONOF AN OBJECTION 233

A mixed strategyof the player3 is a vector 17 of Sp the componentsof

which we denoteby jv The i\\ of Sft arecharacterizedby

Hr, 0, *,= 1.*\\

The bilinear form K( , r? ) of (17:2)in 17.4.1.is therefore

K( , 1? ) = 2) iJCi(n,T2, T3) + JC(n,T2, T,)|TI r^))

and finally> . >

c = Max-* Min-> K( , 77 ) = Min- Max-*K( , rj ).f n n {

Theexpressionsfor 6, a obtain from this by cyclical permutations of theplayers1,2,3in all detailsof this representation.

We repeatthe formulaeexpressinga, /3, 7,a',6',cf and A:

A = a + b + c necessarily ^ 0,-a+ b + c , -2a+ b + c))

_ a + 6 c , a + 6 2c

and we have))

A ^ 0,a'+ b' + c'= 0,))

,'+,))

24.Discussionof an Objection24.1.TheCaseof PerfectInformation and Its Significance

24.1.1.We have obtaineda solution for the zero-sumthree-persongamewhich accountsfor all possibilitiesand which indicatesthe directionthat thesearchfor the solutionsof the n-persongame must take:the analysisof allpossiblecoalitions,and the competitive relationshipwhich they bear toeachother, which shoulddeterminethe compensationsthat playerswhowant to form a coalition will pay to eachother.

We have noticedalreadythat this will be a much more difficult problemforn ^ 4 playersthan it wasfor n = 3 (cf.footnote 2, p.221).)))

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234 ZERO-SUMTHREE-PERSONGAMES

Before we attackthis question, it is well to pause for a moment toreconsiderour position. In the discussionswhich follow we shall put themain stresson the formation of coalitionsand the compensationsbetweenthe participants in thosecoalitions,using the theory of the zero-sumtwo-persongame to determinethe values of the ultimate coalitionswhich opposeeachotherafter all playershave \"takensides\"(cf. 25.1.1.,25.2.).But isthis aspectof the matter really as universal as we proposeto claim?

We have adducedalreadysomestrong positive argumentsfor it, in ourdiscussionof the zero-sumthree-persongame. Our ability to build thetheory of the n-persongame(for all ri) on this foundation will, in fine, bethe decisivepositive argument. But there is a negative argument anobjection to be considered,which arisesin connectionwith those gameswhere perfectinformation prevails.

The objectionwhich we shall now discussapplies only to the abovementionedspecialcategoryof games. Thus it would not, if found valid,provideus with an alternative theory that appliesto all games. But sincewe claim a generalvalidity for our proposedstand, we must invalidate allobjections,even thosewhich applyonly to somespecialcase.1

24.1.2.Gameswith perfect information have alreadybeendiscussedin15.We saw there that they have important peculiaritiesand that theirnature can beunderstoodfully only when they areconsideredin the extensiveform and not merely in the normalized one on which our discussionchieflyrelied(cf. also 14.8.).

The analysisof 15.began by consideringn-persongames (for all n),but in its laterparts we had to narrow it to the zero-sumtwo-persongame.At the end, in particular,wefound a verbal method of discussingit (cf. 15.8.)which had some remarkablefeatures:First,while not entirely free fromobjections,it seemedworth considering.Second,the argumentation usedwasratherdifferent from that by which we had resolvedthe generalcaseofthe zero-sumtwo-persongame and while applicableonly to this specialcase,it was more straight forward than the otherargumentation.Third,it led for the zero-sumtwo-persongameswith perfect information tothe sameresult as our generaltheory.

Now onemight betempted to use this argumentation for n ^ 3 playerstoo;indeeda superficial inspectionof the pertinent paragraph15.8.2.doesnot immediatelydiscloseany reasonwhy it shouldbe restricted(asthere)ton = 2 players (cf.,however, 15.8.3.).But this proceduremakesno men-tion of coalitionsor understandingsbetweenplayers,etc.;so if it is usablefor n = 3 players,then our presentapproachis opento grave doubts.2 We

1In other words :in claiming general validity for a theory onenecessarilyassumestheburden of proof against all objectors.

2Onemight hopeto evadethis issue,by expecting to find A = for all zero-sum three-person games with perfectinformation. This would make coalitions unnecessary. Cf.the end of 23.1.

Just asgameswith perfectinformation avoidedthe difficulties of the theory ofzero-sum two-persongamesby being strictly determined (cf.15.6.1.),they would now avoidthose of the zero-Bum three-persongameeby being inessential.)))

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DISCUSSIONOF AN OBJECTION 235

proposeto show therefore why the procedureof 15.8.is inconclusive whenthe number of playersis threeor more.

To do this, let us repeatsomecharacteristicstepsof the argumentationin question(cf. 15.8.2.,the notations of which we are alsousing).24.2.DetailedDiscussion.Necessityof Compensations betweenThreeor MorePlayers

24.2.1.Consider accordinglya game F in which perfect informationprevails. Let SfTCi, 3fE 2, , 9(11,be its moves, <TI,(72, , <r, the choicesconnectedwith these moves, T(<TI, , 0-,) the play characterizedbythesechoices,and 3v(7r(oi, , <r,)) the outcome of this play for the playerj(= 1,2, - - ,n).

Assume that the moves 3TCi, 3Tl2, , 3TC,-ihave alreadybeen made,the outcome of their choicesbeing<TI,<r2, * , cr,_i and considerthe lastmove 3TC,and its <r,. If this is a chancemove i.e.ft, (en, , <r,~\\) = 0,then the various possiblevaluesa, = 1,2, , a,(cri, , a,-i)have theprobabilitiesp (1), pv(2), , p,(a,(o-i, , cr,_i)),respectively. If thisisa personalmove of player A: i.e.fc,(ai, ,<r,-i)= k = 1,2, , n,then player k will chooseaf so as to make SFjb(ir(ori, , 0v-A| tr,)) a maxi-mum. Denotethis <r, by <r,(<n, , *,_i). Thus one can argue that thevalue of the play is already known (for eachplayerj = 1, n) afterthe moves Sflfcj, 3Tl2,

* * * , 2(TC,-i(and before 9ftl,!), i.e.as a function of0*1,0% , <r,-ialone. Indeed:by the above it is))

*-!for fc,(<7i,

* , <r,_i) = 0,

where <r, = <r,(<n, , <r,-i)maximizesS^Or^i, , <r,-i,<r,)) for

Consequentlywe can treat the game T as if it consistedof the moves9Tli, 9fn 2, , 3Tl,_i only (without 3TC,).

By this devicewe have removed the last move 9TC,. Repeatingit, wecansimilarly remove successivelythe moves 3fTC,_i,9fH,_a, , 9Ha , 9Tli andfinally obtain a definite value of the play (for eachplayerj = 1,2, n).

24.2.2.Fora criticalappraisalof this procedureconsiderthe last twosteps 9Tl,_i, 9fll, and assumethat they arepersonalmoves of two different

This, however, is not the case. To seethat, it suffices to modify the rules ofthe

simple majority game (cf.21.1.)as follows: Let the players 1,2,3make their personalmoves (i.e.the choicesof n, TJ, r* respectively,cf.loc.cit.)in this order,eachonebeinginformed about the anterior moves. It is easy to verify that the values c, 6, a of thothree coalitions 1,2,1,3,2,3are the sameas before

c-6-a-l,A-a+6+c-3>0.A detaileddiscussion of this game, with particular respectto the considerationsof

21.2.,would be of a certain interest, but we do not proposeto continue this subjectfurther at present.)))

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236 ZERO-SUMTHREE-PERSONGAMES

players, say 1,2respectively. In this situation we have assumed thatplayer 2 will certainly choosecr, so as to maximize $2(0-1, , o-,_i,<ry).This gives a a> = 0-,(<ri, , <r,_i). Now we have also assumed thatplayer1,in choosing<r,_i can rely on this;i.e.that he may safely replacethe$I(<TI, , ov-i, 0-,),(which is what he will really obtain),by$1(0-1, , (rr_i, OV(<TI, , cr,_i)) and maximize this latterquantity.1But canherely on this assumption?

To beginwith, <r y(cri, , <r,,_i) may not even be uniquely determined:$2(^1, , cr,_i, (T,) may assumeits maximum (for given <TI, , o>_i)at severalplacesov In the zero-sumtwo-persongame this was irrelevant:there$1= #2,hencetwo o> which give thesamevalue to $F 2, alsogive thesamevalue to 9Fi.

2 But even in the zero-sumthree-persongame,$2 doesnotdetermine$1,due to the existenceof the third player and his $3!So it

happensherefor the first time that a difference which is unimportant foroneplayer may besignificant for another player. This was impossibleinthe zero-sumtwo-persongame,where eachplayerwon (precisely)what theotherlost.

What then must player 1 expectif two v v areof the sameimportanceforplayer2,but not for player1? Onemust expectthat he will try to induceplayer 2 to choosethe a, which is more favorable to him. Hecouldofferto pay to player 2 any amount up to the difference this makesfor him.

This beingconceded,one must envisagethat player 1may even try toinduceplayer2to choosea av which doesnot maximizeS^i, * ' * , ov-i, ov).As long as this changecausesplayer2 lessof a lossthan it causesplayer 1a gain,3 player 1can compensateplayer 2 for his loss, and possiblyevengive up to him somepart of his profit.

24.2.3.But if player1can offer this to player2, then he must alsocounton similar offerscoming from player3 to player2. I.e.thereis no certaintyat all that player 2 will, by his choiceof o-,,maximize 2(0-1, , <r,_i, cr v).In comparingtwo av one must considerwhether player 2Js loss is over-compensatedby playerTs or player3'sgain, sincethis couldlead to under-standingsand compensations.I.e.onemust analyze whether a coalition 1,2or 2,3would gain by any modification of av.

24.2.4.Thisbringsthe coalitionsbackinto the picture. A closeranalysiswould lead us to the considerationsand results of 22.2.,22.3.,23.in everydetail. But it doesnot seemnecessaryto carry this out herein completedetail:after all,this is just a specialcase,and the discussionof 22.2.,22.3.,23.wasof absolutelygeneralvalidity (for the zero-sumthree-persongame)

1Sincethis is a function of <TI, , a>_2, <r,_i only, ofwhich <TI, , o>_2areknownat 9Rr-iy and <r,,_i is controlled by player 1,he isableto maximize it.

Hecannot in any sensemaximize $1(0-1, , 0>-i,ov) sincethat alsodependson <r,

which he neither knows nor controls.2 Indeed,we refrained in 15.8.2.from mentioning $2at all: instead of maximizing $2,

we talked of minimizing $1.Therewas no needeven to introduce 9(a\\ t , r-i)and everything was describedby Max and Min operationson JFi.

3I.e.when it happens at the expenseof player 3.)))

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DISCUSSIONOF AN OBJECTION 237

providedthat theconsiderationof understandingsand compensations,i.e.ofcoalitions,is permitted.

We wanted to show that the weaknessof the argument of 15.8.2.,alreadyrecognizedin 15.8.3.,becomesdestructiveexactly when we go beyond thezero-sumtwo-persongames,and that it leadspreciselyto the mechanismofcoalitionsetc.foreseen in the earlierparagraphs of this chapter. Thisshouldbeclearfrom the aboveanalysis,and sowe can return to our originalmethodin dealingwith zero-sumthree-persongames, i.e.claim full validityfor the resultsof 22.2.,22.3.,23.)))

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CHAPTER VI

FORMULATIONOF THE GENERAL THEORY:ZERO-SUMn-PERSONGAMES

25.TheCharacteristicFunction25.1.Motivation and Definition

25.1.1.We now turn to thezero-sumn-persongame for generaln. Theexperiencegained in ChapterV concerningthe casen = 3 suggeststhatthe possibilitiesof coalitionsbetweenplayerswill play a decisiverolein thegeneraltheory which we are developing. It is therefore important toevolve a mathematical tool which expressesthese \" possibilities\"in aquantitative way.

Sincewe have an exactconceptof \" value\" (of a play) for the zero-sumtwo-persongame,we can also attribute a \" value\" to any given group ofplayers,providedthat it is opposedby the coalitionof all the otherplayers.We shall give theseratherheuristicindicationsan exactmeaning in whatfollows. The important thing is, at any rate,that we shall thus reachamathematical concepton which one can try to basea generaltheory andthat the attempt will, in fine, prove successful.

Letus now statethe exactmathematicaldefinitions which carry out thisprogram.25.1.2.Supposethen that we have a gameF of n playerswho, for thesakeof brevity, will be denoted by 1,2, , n. It is convenient tointroducethe set / = (1,2, , n) of all theseplayers. Without yetmaking any predictionsor assumptionsabout thecoursea play of this gameis likelyto take,we observethis:if we group the playersinto two parties,and treateachparty as an absolutecoalition i.e.if we assumefull coopera-tion within eachparty then a zero-sumtwo-persongame results.1 Pre-cisely:Let S be any given subset of 7, S its complementin /. Weconsiderthe zero-sumtwo-persongamewhich results when all players k

belongingto Scooperatewith eachotheron the onehand, and all playersk

belongingto S cooperatewith eachotheron the otherhand.Viewed in this manner F falls under the theory of the zero-sumtwo-

persongameof ChapterIII. Eachplay of this gamehas a well definedvalue (we mean thev' defined in 17.8.1.).Let us denoteby v(S)the valueof a playfor thecoalition of all k belongingto S (which, in our presentinter-pretation, is oneof the players).

1This is exactlywhat we did in the casen = 3 in 23.1.1.Thegeneralpossibility wasalready alluded to at the beginning of 24.1.

238)))

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THECHARACTERISTICFUNCTION 239

Mathematicalexpressionsfor v(S)obtain as follows:125.1.3.Assume the zero-sumn-persongame F in the normalized form of

11.2.3.Thereeachplayerk = 1,2, , n choosesa variable r* (eachoneuninformed about the n 1otherchoices)and getstheamount

3C*(Tl,T2, , Tn).Of course(the game is zero-sum):

(25:1) 3e*(n, , r n) m 0.*-i

The domainsof the variables are:T* = 1, , j8* for k = 1,2, - , n.

Now in the two-persongame which arisesbetweenan absolutecoalition of allplayersk belongingto S (player1')and that one of all playersk belongingto S (player2'),we have the followingsituation:

Thecompositeplayer 1'has the aggregateof variablesT* where k runsover all elementsof S. Itisnecessaryto treatthis aggregateas one variableand we shall therefore designate it by one symbol ra. The compositeplayer2'has the aggregateof variablesrk where k runs over all elementsof S. This aggregatetoo is one variable, which we designateby thesymbolr~a. The player1'getsthe amount))

(25:2) 5c(rV-s) = OC*(n, - , r n) = - 3C(n, - , rn);'kinS km -S

the player2'getsthe negative of this amount.

A mixedstrategy of the playerI/ is a vector of <S0,8 the components

of which we denoteby r . Thus the of S0aarecharacterizedby

t- ^ o, 2) s*a = LT

A mixedstrategy of the player2'is a vector rj of /V,4 the componentsof which we denoteby rj T-a. Thusthe ij of Sp-aarecharacterizedby

rj r-8 ^0, 5J rj T-a = 1.r~

1This is a repetition of the construction of 23.2.,which applied only to the specialcasen 3.

1TheTS, r~5 of the first expressionform together the aggregate of the n, , rn ofthe two other expressions;soTS, r~3 determine thosen, , rn.

The equality of the two last expressionsis, of course,only a restatement of thezero-sum property.

8 ft3 is the number ofpossibleaggregatesTS, i.e.the product of all 0*where k runs over

all elements of S.40~s is the number of possibleaggregatesr\"5, i.e.the product of all 0*where k runs

over all elements of <S.)))

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240 GENERAL THEORY:ZERO-SUMn-PERSONS> >

Thebilinearform K( , t? ) of (17:2)in 17.4.1.is therefore

K(7,7)- #(rV-*)r'*-',rVs

and finally))

v(S) = Max-Min-K(, ) = Min-Max-^K(, ).

25.2.Discussionof the Concept

25.2.1.Theabove function v(S)is defined for all subsets8of I and hasreal numbers as values. Thus it is, in the senseof 13.1.3.,a numericalset function. We call it the characteristicfunction of the game T. As wehave repeatedly indicated,we expectto basethe entire theory of thezero-sumn-persongameon this function.

Itiswell to visualize what this claim involves. We proposeto determineeverything that can be said about coalitionsbetweenplayers,compensa-tions betweenpartners in every coalition,mergersor fights betweencoali-tions,etc.,in terms of the characteristicfunction v(*S) alone. Primafacie,this program may seemunreasonable,particularly in view of thesetwo facts:

(a) An altogetherfictitious two-persongame,which is relatedto therealn-persongameonly by a theoreticalconstruction,was used to definev(S). Thus v(S) is based on a hypothetical situation, and not strictlyon the n-persongameitself.

(b) v(S) describeswhat a given coalition of players (specifically,the setS) can obtain from their opponents (the set AS) but it fails todescribehow the proceedsof the enterpriseare to be divided among thepartners k belongingto S. This division, the \"imputation,\" is indeeddirectlydeterminedby the individual functions 3C*(Ti, , r n), k belong-ing to S,while v(S)dependson much less.Indeed,v(S) is determinedbytheir partial sum JC(r5, r~5) alone,and even by lessthan that sinceit is

the saddlevalue of the bilinearform K( , y ) basedon 5C(r5, r~\"s) (cf.the

formulae of 25.1.3.).25.2.2.In spite of these considerationswe expectto find that the

characteristicfunction v(S) determineseverything, includingthe \"impu-tation\"(cf.(b) above). Theanalysisof the zero-sumthree-persongameinChapter V indicatesthat the directdistribution (i.e.,\"imputation\by means of the 3G*(ri, , r n) is necessarilyoffset by somesystem of\"compensations\"which the playersmust make to eachotherbeforecoali-tions canbe formed. The \"compensations\"shouldbe determinedessen-tially by the possibilitieswhich existfor eachpartner in the coalition S(i.e.for eachk belongingto *S), to forsakeit and to join someothercoalitionT. (Onemay have to consideralso the influence of possiblesimultaneousand concerteddesertionsby setsof several partners in Setc.)I.e.the\"imputation\"of v(/S) to theplayersk belongingto Sshouldbe determined)))

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THE CHARACTERISTICFUNCTION 241by the otherv(T)1 and not by the 3C*(ri, , rn). We have demon-strated this for the zero-sum three-persongamein ChapterV. Oneof themain objectivesof the theory we aretrying to build up is to establishthesame thing for the generaln-persongame.

26.3.Fundamental Properties25.3.1.Beforewe undertake to elucidatethe importanceof the char-

acteristicfunction v(S)for the generaltheory of games,we shallinvestigatethis function as a mathematical entity in itself. We know that it is anumerical set function, defined for all subsets S of / = (1,2, - , n)and we now proposeto determineits essentialproperties.

It will turn out that they arethe following:

(25:3:a) v(0) = 0,(25:3:b) v(-S)= -v(S),(25:3:c) v(Su T) ^ v(S)+ v(T), if S n T = 0.

We prove first that the characteristicset function v(>S) of every gamefulfills (25:3:a)-(25:3:c).

25.3.2.Thesimplestproof is a conceptualone,which can becarriedoutwith practically no mathematical formulae. However, since we gaveexactmathematical expressionsfor v(S) in 25.1.3.,one might desireastrictly mathematical,formalistic proof in terms of the operationsMaxand Min and the appropriatevectorial variables. We emphasizethereforethat our conceptualproof is strictly equivalent to the desiredformalistic,mathematical one, and that the translation can be carriedout without

any realdifficulty. But sincethe conceptualproof makes the essentialideasclearer,and in a briefer and simplerway, while the formalistic proofwould involve a certainamount of cumbersomenotations, we prefer to

give the former. Thereaderwho is interestedmay find it a goodexerciseto constructthe formalistic proof by translatingour conceptualone.

25.3.3.Proof of (25:3:a):2 Thecoalition has no members,so it alwaysgetsthe amount zero,therefore v(0) = 0.

Proof of (25:3:b):v(*S) and v(-S)originatefrom the same(fictitious)zero-sumtwo-persongame, the one played by the coalitionS against

1All this is very much in the senseof the remarks in 4.3.3.on the role of \" virtual\"

existence.1Observethat we aretreating eventhe empty set asa coalition. Thereadershould

think this over carefully. In spite of its strange appearance,the step is harmless and

quite in the spirit ofgeneralset theory. Indeed,it would be technically quite a nuisanceto excludethe empty set from consideration.

Ofcoursethis empty coalition has no moves, no variables, no influence, no gains,and no losses.But this is immaterial.

Thecomplementary setof , the setof all players7, will alsobetreated asa possiblecoalition. This too is the convenient procedurefrom the set-theoreticalpoint of view.Toa lesserextent this coalition alsomay appearto bestrange, sinceit has no opponents.Although it has an abundance of members and henceof moves and variables it will

(in a zero-sum game) equally have nothing to influence, and no gains or losses.But this

too is immaterial.)))

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242 GENERAL THEORY:ZERO-SUMn-PERSONS

the coalition S. Thevalue of a play of this game for its two compositeplayersisindeedv(S)and v( S)respectively. Therefore v( S) = v(S).

Proof of (25:3:c):The coalition S can obtain from its opponents (byusingan appropriatemixedstrategy) the amount v(S)and no more. Thecoalition T can obtain similarly the amount v(77

) and no more. Hencethecoalition S u T can obtain from its opponents the amount v(S)+ v(T),even if the subcoalitionsS and T fail to cooperatewith eachother.1 Sincethe maximum which the coalition S u T can obtain under any conditionisv(Su T) this impliesv(Su T) ^ v(S)+ v(2T).2

25.4.Immediate Mathematical Consequences25.4.1.Beforewegofurther letus draw someconclusionsfrom theabove

(25:3:a)-(25:3:c).Thesewill be derived in the sensethat they hold forany numerical setfunction v(S)which fulfills (25:3:a)-(25:3:c)irrespectiveof whether or not it is the characteristicfunction of a zero-sumn-persongameF.(25:4) v(7) = 0.

Proof:*By (25:3:a),(25:3:b),v(7) = v(-0)= -v(0)= 0.(25:5) v(5i u u Sp) v(5i) + - + v(S9)

if Si, , Sp arepairwisedisjunctsubsetsof 7.

Proof:Immediatelyby repeatedapplicationof (25:3:c).(25:6) v(Si)+ + v(S,) g

if Si, , Sp area decompositionof 7, i.e.pairwisedisjunctsubsetsof 7 with the sum 7.

Proof:We have Siu u Sp = 7, hencev(Siu - - - u Sp) = by(25:4).Therefore (25:6)followsfrom (25:5).

25.4.2.While (25:4)-(25:6)areconsequencesof (25:3:a)-(25:3:c), theyand even somewhat less can replace(25:3:a)-(25:3:c)equivalently.Precisely:(25:A) Theconditions(25:3:a)-(25:3:c)areequivalent to the asser-

tion of (25:6)for the values p = 1,2, 3 only; but (25:6)mustthen be statedfor p = 1,2 with an = sign,and for p = 3 with

a sign.1Observethat we are now using Sn T 0. If Sand T had common elements, we

couldnot break up the coalition SU T into the subcoalitions Sand T.1This proof is very nearly a repetition of the proofofa + b 4-c ^ in 22.3.2.One

couldeven deduceour (25:3:c)from that relation:Considerthe decomposition of / intothe three disjunct subsetsS,T, (SU T). Treatthe three corresponding (hypothetical)absolutecoalitions as the three playersof the zero-sum three-persongame into which thistransforms T. Then v(), v(T),v(SU T) correspondto the -a,-b,c loc.cit.;hencea + b + c means -v(S)- v(T) 4-v(SU T) 0; i.e.v(SU T) v(/S) + v(T).

For a v(S) originating from a game, both (25:3:a)and (25:4)are conceptuallycontained in the remark of footnote 2 on p. 241.)))

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GIVEN CHARACTERISTICFUNCTION 243

Proof:(25:6)for p = 2 with an = sign statesv(S)+ v(~S) =(wewrite Sfor S\\, henceSiis S);i.e.v( S) = v(S)which is exactly(25:3:b).

(25:6)for p = 1with an = signstatesv(/) = (in this caseSimust be/) which is exactly (25:4).Owing to (25:3:b),this is exactly the sameas (25:3:a).(Cf. the above proof of (25:4).)

(25:6)for p = 3 with an g signstatesv(S)+ v(T) + v(-(Su T)) ^(we write S,T for Si,S2; henceSBis -(Su T)),i.e.

-v(-OSuT)) vOS)+v(T).By (25:3:b)this becomesv(Su T) ^ v(S)+ v(T)which is exactly(25:3:c).

Soour assertionsareequivalent preciselyto the conjunction of (25:3:a)-(25:3:c).

26.Constructionof a Gamewith a Given CharacteristicFunction26.1.TheConstruction

26.1.1.We now prove theconverseof 25.3.1.:That for any numerical setfunction v(S) which fulfills the conditions(25:3:a)-(25:3:c)thereexistsazero-sumn-persongameF of which this v(S)is the characteristicfunction.

In orderto avoid confusion it is betterto denotethe given numericalset function which fulfills (25:3:a)-(25:3:c)by v (S). We shall definewith its help a certainzero-sumn-persongameF, and denotethe char-acteristicfunction of this F by v(/S). Itwill then benecessaryto prove that))

Let therefore a numerical set function v (S) which fulfills (25:3:a)-(25:3:c)begiven. We define the zero-sumn-persongameF as follows:1

Eachplayerk = 1,2, , n will, by a personalmove, choosea subsetSk of / which containsk. Eachone makeshis choiceindependentlyof thechoiceof the otherplayers.2

After this the paymentsto be madearedeterminedas follows:Any setSof players,for which

(26:1) Sk = S for every k belongingto Sis calleda ring*** Any two rings with a common elementareidentical.6

1This game F is essentially a more general analogue of the simple majority game ofthreepersons,defined in 21.1. We shall accompany the text which follows with footnotespointing out the details of this analogy.

2Thew-element set 7 has 2n~l subsetsScontaining k, which we can enumerate by anindex r*(/S) 1,2, , 2*~l. If we now let the player A; choose,instead of *, itsindex r* Tk(Sk) 1,2, , 2n~l, then the game is already in the normalized form of11.2.3.Clearlyall fa -2*'1.

8 The rings are the analogues of the couplesin 21.1.Thecontents of footnote 3 onp.222apply accordingly; in particular the rings are the formal conceptin the setof rulesof the game which inducesthe coalitions which influence the actual courseofeachplay.

4 Verbally: A ring is a set of players, in which every onehas chosenjust this set.Theanalogy with the definition of a couplein 21.1.isclear. Thedifferences aredue

to formal convenience:in 21.1.we made eachplayer designate the other element of thecouplewhich he desires;now weexpecthim to indicate the entire ring. A closeranalysisof this divergencewould beeasyenough, but it doesnot seemnecessary.

6Proof:LetSand T be two rings with a common element k\\ then by (26:1)Sk = Sand Si, - T, and soS - T.)))

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244 GENERALTHEORY:ZERO-SUMn-PERSONS

In otherwords:Thetotality of all rings (which have actually formed in aplay) is a systemof pairwisedisjunctsubsetsof /.

Eachplayerwho is containedin none of the ringsthus defined forms byhimself a (one-element)set which is calleda solo set. Thus the totalityof all ringsand solosets(whichhave actually formed in a play)is a decompo-sition of /;i.e.a system of pairwisedisjunct subsetsof I with the sum I.Denotethesesetsby Ci, , Cp and the respectivenumbers of theirelementsby n\\, , np.

Considernow a player k. Hebelongs to preciselyone of thesesetsCi, , Cp say to Cq. Then playerk getsthe amount))

(26:2) _ v.(C.)- v (Cr).1n q n 4r-lThis completesthe descriptionof the game T. We shall now showthat

this F is a zero-sumn-persongame and that it has thedesiredcharacteristicfunction v ().26.1.2.Proof of the zero-sumcharacter:Considerone of the setsCq.Eachoneof the n q playersbelongingto it getsthe sameamount, stated in

(26:2).Hencethe playersof Cq togetherget the amount

(26:3)r-l

In orderto obtain the total amount which all players 1, , n get, wemust sum the expression(26:3)over all setsCq, i.e.over all q = 1, , p.This sum is clearly))

- v,(Cr),Q-l r-l

i.e.zero.2Proof that the characteristicfunction is v (S):Denotethe characteristic

function of T by v(S). Rememberthat (25:3:a)-(25:3:c)hold for v(S)becauseit is a characteristicfunction, and for v (S)by hypothesis. Conse-quently (25:4)-(25:6)alsohold for both v(S)and v (/S).

We prove first that

(26:4) v(iS) ^ v (S) for all subsets8 of /.If S is empty, then both sidesarezero by (25:3:a).Sowe may assumethatSis not empty. In this casea coalition of all playersk belongingto S can

1Thecourseof the play, that is the choicesSi, , Sn or, in the senseoffootnote 2on p.243,the choicesn, , rn determine the Ci, ;Cp, and thus the expression(26:2).Ofcourse(26:2)is the JC*(n, , rn) of the generaltheory.

p2Obviously 2^ n q n.)))

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INESSENTIALAND ESSENTIALGAMES 245

govern the choicesof its Sh so as with certainty to makeS a ring. It suf-fices for every k in S to choosehis Sk = S. Whatever the other players(in S) do, S will thus be oneof the sets(ringsor solosets)Ci, , Cp,say Cg. Thus eachfc in Cq

= S getsthe amount (26:2);hencethe entirecoalition Sgetsthe amount (26:3).Now we know that the system))

p

is a decompositionof /; henceby (25:6)] v (Cr) ^ 0. That is, ther-l

expression(26:3)is ^ v (Cfl ) = VoOS).1 In other words,the playersbelong-ing to the coalition S can securefor themselvesat leastthe amount v (5)irrespectiveof what the playersin S do. Thismeansthat v(S) ^ v (S);i.e.(26:4).

Now we can establishthe desiredformula))

(26:5) v(S)Apply (26:4)to -S. Owing to (25:3:b)this means -v(S)^ -v (S),i.e.(26:6) v(S) g VoOS).

(26:4),(26:6)give together(26:5).2

26.2.Summary

26.2.To sum up:in paragraphs25.3.-26.1.we have obtained a com-plete mathematical characterization of the characteristicfunctions v(S)ofall possiblezero-sumn-persongamesF. If the surmisewhich we expressedin 25.2.1.provesto be true, i.e.if we shall be able to basethe entiretheoryof the game on the globalpropertiesof the coalitionsas expressedby v(S),then our characterization of v(/S) has revealed the exactmathematicalsubstratum of the theory. Thus the characterizationof v(S)and the func-tional relations(25:3:a)-(25:3:c)areof fundamental importance.

We shall therefore undertakea first mathematical analysisof the mean-ing and of the immediatepropertiesof theserelations. We call the func-tions which satisfy them characteristicfunctions even when they areviewedin themselves,without referenceto any game.

27.StrategicEquivalence. Inessentialand EssentialGames27.1.Strategic Equivalence. TheReducedForm

27.1.1.Considera zero-sum n-persongameT with the characteristicfunction v(S). Let alsoa systemof numbersaj, , 2 be given. We

1Observethat the expression(26:3),i.e.the total amount obtained by the coalition S,is not determined by the choicesof the players in Salone. But we derived for it a lowerbound v (),which is determined.

2Observethat in our discussion of the goodstrategies of the (fictitious) two-persongame between the coalitions S and S (our aboveproof really amounted to that), weconsideredonly pure strategies,and no mixed ones. In other words, all these two-persongameshappened to be strictly determined.

This, however, is irrelevant for the end which we arenow pursuing.)))

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246 GENERAL THEORY:ZERO-SUMn-PERSONS

now form a new gameF'which agreeswith F in all detailsexceptfor this:F'is playedin exactlythe sameway as F, but when all is over, playerk getsin F'the amount which he would have got in F (after the sameplay),plusa.(Observethat the aj, , 2 are absolute constants!) Thus if F isbrought into the normalized form of 11.2.3.with the functions))

then F' is also in this normalized form, with the correspondingfunctions3C*(r i>

* ' '>

rn) 3C*(ri, - - , r n) + ajj. Clearly F' will be a zero-sumn-persongame (alongwith F) if and only if

(27:1) i J = 0,fc-1

which we assume.Denotethe characteristicfunction of F'by v'(S),then clearly))

(27:2) v'(S)kinS

Now it is apparent that the strategicpossibilitiesof the two gamesF and F'areexactly the same. The only difference betweenthesetwo gamescon-sists of the fixed payments J after eachplay. And thesepaymentsareabsolutelyfixed;nothing that any or all of the playerscan do will modifythem. Onecouldalsosay that the positionof eachplayerhas beenshiftedby a fixed amount, but that the strategicpossibilities,the inducementsandpossibilitiesto form coalitionsetc.,areentirely unaffected. In otherwords:If two characteristicfunctions v(S)and v'(S) arerelatedto eachotherby(27:2)2, then every gamewith the characteristicfunction v(S)is fully equiva-lent from all strategicpoints of view to somegame with the characteristicfunction v'(S), and conversely. I.e.v(S) and v'(S)describetwo strategi-cally equivalent families of games. In this sensev(S)and v'(S)may them-selvesbeconsideredequivalent.

Observe that all this is independentof the surmiserestatedin 26.2.,accordingto which all gameswith the samev(S) have the same strategiccharacteristics.

27.1.2.Thetransformation (27:2)(we needpay no attention to (27:1),cf. footnote 2 above)replaces,as we have seen,the setfunctions v(/S) by a

1The truth of this relation becomesapparent if one recallshow v(S), v'(S)weredefined with the help of the coalition S. It is alsoeasyto prove (27:2)formalisticallywith the help of the3C*(n, , rn), 3Ci(n, , rn).

sUnder these conditions (27:1)follows and need not be postulated separately.Indeed,by (25:4)in 25.4.1.,v(7) -v'(7) -0,hence(27:2)gives

n

2) <4 -0; i.e. a -0.klal k-l)))

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INESSENTIALAND ESSENTIALGAMES 247

strategically fully equivalent set-function v'(S). We therefore call thisrelationshipstrategicequivalence.

We now turn to a mathematical property of this conceptof strategicequivalenceof characteristicfunctions.

It is desirableto pick from eachfamily of characteristicfunctions v(S)in strategicequivalencea particularly simple representativev(/S). Theideais that, given v(S),this representativev(S)shouldbeeasyto determine,and that on the otherhand two v(S)and v'(S)would be in strategicequiva-lenceif and only if their representativesv(*S) and v'(*S) are identical.Besides,we may try to choosetheserepresentativesv(S) in sucha fashionthat their analysisis simplerthan that of the original v(S).

27.1.3.When we started from characteristicfunctions v(S) and v'(S),then the conceptof strategicequivalencecouldbe basedupon (27:2)alone;(27:1)ensued (cf. footnote 2, p.246). However,we proposeto start nowfrom one characteristicfunction v(S)alone,and to survey all possiblev'(S)which arein strategicequivalencewith it in orderto choosethe representa-tive v(/S) from among them. Therefore the questionariseswhich systemsaj, , ajwe may use,i.e.for which of thesesystems(using(27:2))thefact that v(S) is a characteristicfunction entails the samefor v'(S). Theanswer is immediate,both by what we have said so far, and by directverification: The condition (27:1)is necessaryand sufficient.1

Thus we have the n indeterminate quantities aj, , a at ourdisposalin the searchfor a representativev(S);but the S>

' ' *>

an arsubjectto one restriction:(27:1).So we have n 1free parametersatour disposal.

27.1.4.We may therefore expectthat we can subjectthe desiredrepre-sentative v(S) to n 1requirements.As such we choosethe equations))

(27:3) v((l)) = v((2))= . = v((\.2I.e.we requirethat every one-man coalition every playerleft to himselfshouldhave the samevalue.

We may substitute (27:2)into (27:3)and statethis togetherwith (27:1),and so formulate all our requirementsconcerningthe aj, , a. Sowe obtain:

(27:1*) al = 0,t-i))

(27:2*)It iseasyto verify that theseequationsaresolvedby preciselyonesystemof

'))

1This detailed discussion may seempedantic. We gave it only to make clearthatwhen westart with two characteristicfunctions v(S)and v'(S)then (27:1)is superfluous,but when we start with onecharacteristicfunction only, then (27:1)isneeded.

1Observethat thesearen 1and not n equations.)))

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248 GENERAL THEORY:ZERO-SUMn-PERSONS))

(27:4) a{

Sowe can say:

(27:A) We call a characteristicfunction v(S)reducedif and only if itsatisfies(27:3).Then every characteristicfunction v(S) is instrategicequivalencewith preciselyone reducedv(S). Thisv(S)is given by theformulae (27:2)and (27:4),and we call it thereducedform of v(S).

The reducedfunctions will be the representativesfor which we havebeenlooking.

27.2.Inequalities. TheQuantity r27.2.Let us considera reducedcharacteristicfunction v(S). We

denotethe joint value of the n terms in (27:3)by 7,i.e.(27:5) -

7))

We can state(27:5)also this way:

(27:5*) v(S) = 7 for every one-elementsetS.Combinationwith (25:3:b)in 25.3.1.transforms (27:5*)into

(27:5**) v(S) = 7 for every (n - l)-elementsetS.We re-emphasizethat any one of (27:5),(27:5*),(27:5**)is besides

defining7 justa restatementof (27:3),i.e.a characterizationof the reducednature of v(S).

Now apply (25:6)in 25.4.1.to the one-elementsetsSi= (1), ,<S = (n). (Sop = n). Then (27:5)gives -ny ^ 0,i.e.:(27:6) 7^0.

Considernextan arbitrary subset Sof /. Let p be the number of itselements:S= (fti, , k p). Now apply (25:5)in 25.4.1.to the one-elementsetsSi= (fci), , Sp = (kp). Then (27:5)gives))

Apply this also to S which has n p elements.Owing to (25:3:b)in

25.3.1.,the above inequality now becomes

-*(fl)2S -(n-p)75 i.e. v(/S) S (n -p)y.1 Proof:Denotethe joint value of the n terms in (27:2*)by 0. Then (27:2*)amounts

to al - -v((fc))+ 0,and so(27:1*)becomesn n

n0 -% v((*)) -0; i.e.-fc-i *-i)))

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INESSENTIALAND ESSENTIALGAMES 249

Combiningthesetwo inequalitiesgives:

(27:7) py ^ v(/S) g (n p)y for every p-elementsetS.(27:5*)and v(0) = (i.e.(25:3:a)in 25.3.1.)can also be formulated

this way:

(27:7*) Forp = 0,1we have = in the first relation of (27:7).

(27:5**)and v(7) = (i.e.(25:4)in 25.4.1.)can also be formulatedthis way:

(27:7**) Forp = n 1,n we have = in the secondrelation of (27:7).

27.3.Inessentiality and Essentiality

27.3.1.In analyzing theseinequalitiesit is best now to distinguishtwoalternatives.

This distinctionis basedon (27:6):Firstcase:7 = 0. Then (27:7)gives v(S) =0 for all S. This is a

perfectly trivial case,in which the game is manifestly devoidof further

possibilities. Thereis no occasionfor any strategyof coalitions,no elementof struggleor competition:each player may play a lone hand, sincethereis no advantage in any coalition. Indeed,every playercan get the amountzero for himself irrespectiveof what the others are doing. And in nocoalition can all its memberstogethergetmore than zero. Thus the valueof a play of this game is zero for every player,in an absolutelyunequivocalway.

If a generalcharacteristicfunction v(S) is in strategicequivalencewith such a v(S) i.e.if its reducedform is v(S) = then we have thesameconditions,only shifted by a for the playerk. A play of a game Fwith this characteristicfunction v(S) has unequivocally the value ajj forthe playerk:he can get this amount even alone,irrespectiveof what theothersaredoing. No coalition coulddo betterin toto.

We call a game F, the characteristicfunction v(S)of which has such areducedform v(S)= 0,inessential.1

27.3.2.Secondcase:y >0. Bya changein unit 2 wecouldmakey = I.8This obviously affects none of the strategicallysignificant aspectsof thegame,and it is occasionallyquite convenient to do. At this moment, how-ever, we do not proposeto do this.

In the present case,at any rate,the playerswill have goodreasonstowant to form coalitions.Any playerwho is left to himself losesthe amounty (i.e.he gets -7,cf. (27:5*)or (27:7*)),while any n - 1playerswho

1 That this coincideswith the meaning given to the word inessential in 23.1.3.(in the

specialcaseofa zero-sum three-persongame) will beseenat the end of27.4.1.2Sincepayments aremade, we mean the monetary unit. In a wider senseit might

be the unit of utility. Cf.2.1.1,3This would not have beenpossiblein the first case,where 7 0.)))

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250 GENERAL THEORY:ZERO-SUMn-PERSONS

cooperatewin togetherthe amount 7 (i.e.their coalition gets7, cf. (27:5**)or (27:?**)).1

Hencean appropriatestrategy of coalitionsis now of greatimportance.We call a gameT essentialwhen its characteristicfunction v(S) has a

reducedform v(S)not = O.2

27.4.Various Criteria. Non-additive Utilities

27.4.1.Given a characteristicfunction v(S),we wish to have an explicitexpressionfor the 7 of its reducedform v(S). (Cf.above.)

Now 7 is the joint value of thev((fc)),i.e.of the v((fc)) + aj,and this))

is by (27:4)iV v((j)). Hencej-i

(27:8) 7 = - i))

y-i

Consequentlywe have:(27:B) Thegame F is inessentialif and only if

t v((j)) = (i.e.7 = 0),3-1

and it is essentialif and only if

S v((/)) < (i.e.7 > 0).*;-lFora zero-sumthree-persongamewe have, with the notationsof 23.1.,

v((l)) = -a,v((2))= -6,v((3))= -c;so 7 = iA. Therefore our con-ceptsof essentialand inessentialspecializeto those of 23.1.3.in the caseof a zero-sumthree-persongame. Consideringthe interpretation of theseconceptsin both cases,this was to be expected.

1This is, of course,not the whole story Theremay be other coalitions of > 1but < n 1 players which are worth aspiring to. (If this is to happen, n 1 mustexceed1 by more than 1, i.e.n ^ 4.) This dependsupon the ^(S)of the setsSwith> 1but < n 1 elements. But only a complete and detailedtheory of games canappraisethe roleof thesecoalitions correctly.

Our abovecomparison of isolatedplayers and n 1 player coalitions (the biggestcoalitions which have anybody to oppose!)suffices only for our present purpose:toestablish the importance of coalitions in this situation.

1Cf.again footnote 1 on p. 249.8 So -7 is the of footnote 1 on p.248.

n4 We have seenalready that oneor the other must bethe case,since2}v((/))^ as;'-i

well as 7 0.)))

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INESSENTIALAND ESSENTIALGAMES 25127.4.2. We can formulate, some other criteriaof inessentiality:

(27:C) Thegame F is inessentialif and only if its characteristicfunctionv(S)can begiven this form:

v(S) 2) ajtin 5

for a suitablesystema?, , aJJ.

Proo/:Indeed,this expressesby (27:2)preciselythat v(S)is in strategicequivalencewith v(S)= 0. As this v(S) is reduced,it is then the reducedform of v(S) and this is the meaning of inessentiality.

(27:D) The gameF is inessentialif and only if its-characteristicfunction v(S)has always = in (25:3:c)of 25.3.1.;i.e.when

V (Su T) = v(S)+ v(T) if S n T = 0.Proof:Necessity:A v(S) of the form given in (27:C)above obviously

possessesthis property.Sufficiency: Repeatedapplicationof this equation gives = in (25:5)of

25.4.1.;i.e.v(Si u - - - u Sp) = v(SO + + v(Sp)

if Si, , Sp arepairwisedisjunct.

Consideran arbitrary S,say S = (fci, , kp). Then Si = (fci), ,SP = (k p) give

v(S) = v((*0)+ + v((*,)).Sowe have

v(S) = 5) ak

km 8

with a? = v((l)), , a = v((n))and so F is inessentialby (27:C).27.4.3.Both criteria(27:C)and (27:D) expressthat the values of all

coalitionsariseadditively from thoseof their constituents.1 It will berememberedwhat rolethe additivity of value, or rather its frequent absence,has played in economicliterature.The casesin which value is not gen-erally additive were among the most important, but they offered sig-nificant difficulties to every theoreticalapproach;and one cannot saythat thesedifficulties have ever beenreally overcome.In this connectiononeshould recallthe discussionsof conceptslike complementarity,totalvalue, imputation, etc. We arenow gettinginto the correspondingphaseof our theory; and it is significant that we find additivity only in the unin-

c

1Thereaderwill understand that we areusing the word \"value\" (of the coalition S)for the quantity v(S).)))

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252)) GENERAL THEORY:ZERO-SUMn-PERSONS))

teresting(inessential)case,while the really significant (essential)gameshave a non-additive characteristicfunction. 1

Thosereaderswho arefamiliar with themathematical theory of measurewill make this further observation:the additive v(S) i.e.the inessentialgames are exactly the measurefunctions of /, which give I the totalmeasurezero. Thus the generalcharacteristicfunctions v(S) area newgeneralization of the conceptof measure.Theseremarksarein a deepersenseconnectedwith the precedingonesconcerningeconomicvalue. How-ever, it would lead too far to pursuethis subjectfurther. 2

27.5.TheInequalities in the Essential Case

27.5.1.Let us return to the inequalitiesof 27.2.,in particular to (27:7),(27:7*),(27:7**).For 7 = (inessentialcase) everything is triviallyclear. Assume therefore that y > (essentialcase).))

Figure 50.Abscissa:p, number of elementsof S. Dot at 0, 7, 7, or heavy line:Range of

possiblevalues v(S) for the Swith the corresponding p.

Now (27:7),(27:7*),(27:7**)set a rangeof possiblevalues for v(S)for every number p of elementsin S. This range is pictured for eachp = 0,1,2, - , n 2, n 1,n in Figure50.

We can add the followingremarks:27.5.2.First:It will be observedthat in an essentialgame i.e.when

7 > necessarily n ^ 3. Otherwise the formulae (27:7), (27:7*),(27:7**) or Figure50,which expressestheir content lead to a conflict:Forn = 1or 2 an (n l)-elementset S has or 1elements,henceits

1We are, of course,concernedat this moment only with a particular aspectof thesubject:we areconsidering values of coalitions only i.e.of concertedactsof behaviorand not of economicgoodsor services. Thereaderwill observe,however, that the spe-cialization is not as far reaching as it may seem:goodsand servicesstand really for theeconomicactof their exchange i.e.for a concertedactofbehavior.

*Thetheory of measure reappearsin another connection. Cf.41.3.3.)))

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INESSENTIALAND ESSENTIALGAMES 253

v(S)must on the one hand be 7, and on the other hand or 7, which isimpossible.1

Second:Forthe smallestpossiblenumber of participantsin an essentialgame,i.e.for n = 3,the formulae (27:7),(27:7*),(27:7**) or Figure50determineeverything:they statethe values of v(S)for 0,1,n 1,n~elementsetsS;and for n = 3 the followingareall possibleelementnumbers:0,1,2,3. (Cf. alsoa remark in footnote 1on p.250.) This is in harmony with

the fact which we found in 23.1.3.,accordingto which thereexistsonlyonetype of essentialzero-sumthree-persongames.

Third:Forgreaternumbersof participants,i.e.for n ^ 4, the problemassumes a new complexion.As formulae (27:7),(27:7*),(27:7**) orFigure50 show, the elementnumber of p of the set S can now haveothervalues than 0,1,n 1,n. I.e.the interval

(27:9) 2 g p ^ n - 2

now becomesavailable.2 It is in this interval that the above formulae nolonger determinea unique value of v(S); they set for it only the interval

(27:7) -P7 ^ v(S)S (n - p)7,

the length of which is ny for every p (cf.again Figure50).27.6.3.In this connection the question may be asked whether really

the entireinterval (27:7)is available, i.e.whether it cannot benarrowedfurther by some new, more elaborateconsiderationsconcerningv(S).The answer is:No. It is actually possibleto define for every n 4 asingle game Fp in which, for eachp of (27:9),v(5) assumesboth values

p7 and (n p)y for suitablep-elementsetsS. Itmay sufficeto mentionthe subjectherewithout further elaboration.

To sum up:Therealramifications of the theory of gamesappear onlywhen n ^ 4 is reached.(Cf. footnote 1 on p.250, where the same ideawas expounded.)

27.6.Vector Operationson CharacteristicFunctions

27.6.1.In concludingthis sectionsomeremarksof a more formal natureseemappropriate.

The conditions(25:3:a)-(25:3:c)in 25.3.1.,which describethe charac-teristic function v(S),have a certainvectorial character:they allow ana-loguesof the vector operations,defined in 16.2.1.,of scalarmultiplication, andof vector addition. Moreprecisely:

Scalarmultiplication:Given a constant t ^ and a characteristicfunc-tion v(S), then tv(S) = u(S) is also a characteristicfunction. Vectoraddition:Given two characteristicfunctions v(S),w(S);3 then

1Of course,in a zero-sum one-persongame nothing happens at all, and for the zero-sum two-person games we have a theory in which no coalitions appear. Hencetheinessentiality of all these casesis to beexpected.s It has n 3 elements;and this number is positive assoonasn ^ 4.

8 Everything here must refer to the same n and to the same setof players

/ - (1,2, - - - , n).)))

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254 GENERALTHEORY:ZERO-SUMn-PERSONS

v(S)+ w(S) z(S)is also a characteristicfunction. Theonly differencefrom the correspondingdefinitions of 16.2.is that we had to requiret S> O.1-2

27.6.2.The two operations defined above allow immediate practicalinterpretation:

Scalarmultiplication:If t = 0, then this producesu(/S)as 0,i.e.theeventlessgame consideredin 27.3.1.So we may assume t >0. In thiscaseour operationamounts to a changeof the unit of utility, namely to itsmultiplication by the factor t.

Vector addition:This correspondsto the superpositionof the gamescorrespondingto v(S) and to w(S). One would imagine that the sameplayers 1,2, , n are playing these two games simultaneously,butindependently. I.e.,no move made in one game is supposedto influencethe othergame,as far as the rulesareconcerned.In this casethe charac-teristicfunction of the combinedgame is clearlythe sum of thoseof the twoconstituentgames.3

27.6.3.We do not proposeto enterupon a systematicinvestigation oftheseoperations,i.e.of their influence upon the strategicsituations in thegameswhich they affect. Itmay beuseful, however, to makesomeremarkson this subject without attempting in any way to be exhaustive.

We observefirst that combinationsof theoperationsof scalarmultiplica-tion and vector addition also can now be interpreteddirectly. Thus thecharacteristicfunction

(27:10) z(S)= tv(S) + w(5)

belongsto the game which arisesby superpositionof the gamesof v(S)andw(S)if theirunits of utility arefirst multipliedby t ands respectively.

If s = 1 t y then (27:10)correspondsto the formation of the centerofgravity in the senseof (16:A:c)in 16.2.1.

It will appear from the discussionin 35.3.4.(cf.in particularfootnote 1on p.304 below)that even this seeminglyelementaryoperationcan havevery involved consequencesas regardsstrategy.

We observenextthat therearesomecaseswhere our operationshave noconsequencesin strategy.

First,the scalarmultiplication by a t > alone,beinga merechangeinunit, has no suchconsequences.

indeed,t < would upset (25:3:c)in 25.3.1.Note that a multiplication of theoriginal 3C*(n, , rn) with a t < would be perfectly feasible. It is simplest toconsidera multiplication by t 1,i.e.a change in sign. But a change of sign of the3C*(Ti, ,r)doesnot at all correspondto a changeof sign of the v(S). This should beclearby common sense,as a reversalof gains and lossesmodifies all strategic considera-tions in a very involved way. (This reversaland someof its consequencesare familiarto chessplayers.) A formal corroboration of our assertionmay be found by inspectingthe definitions of 25.1.3.

*Vector spaceswith this restriction of scalarmultiplication are sometimes calledpositive vector spaces. We do not needto enter upon their systematic theory.

3This should be intuitively obvious. An exactverification with the help of 25.1.3.involves a somewhat cumbersome notation, but no real difficulties.)))

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GROUPS,SYMMETRY AND FAIRNESS 255

Second and this is of greatersignificance the strategicequivalencediscussedin 27.1.is a superposition:we pass from the gameof v(S) to thestrategicallyequivalent gameof v'(S) by superposingon the former aninessentialgame.1 (Cf.(27:1)and (27:2)in 27.1.1.and,concerninginessen-tiality, 27.3.1.and (27:C)in 27.4.2.)We may expressthis in the followingway: we know that an inessentialgame is one in which coalitionsplay norole. Thesuperpositionof such a gameon another onedoesnot disturbstrategicequivalence,i.e.it leaves the strategicstructure of that gameunaffected.

28.Groups,Symmetry and Fairness28.1.Permutations, Their Groups,and Their Effect on a Game

28.1.1.Let us now considerthe roleof symmetry, or more generally,theeffects of interchanging the players!,-,n or their numbers in ann-persongameF. This will naturally bean extensionof the correspondingstudy made in 17.11.for the zero-sumtwo-persongame.

This analysisbeginswith what is in the main a repetitionof the stepstaken in 17.11.for n = 2. But sincethe interchangesof the symbols1, , n offerfor a generaln many more possibilitiesthan for n = 2, it isindicatedthat we shouldgo about it somewhatmore systematically.

Considerthe n symbols1, , n. Form any permutation P of thesesymbols. P is describedby stating for every i = 1, , n, into which ip(also= 1, , n),P carriesit. Sowe write:

(28:1) P:i-*ip,

or by way of completeenumeration:))

Among the permutationssomedeservespecialmention:

(28:A:a) The identity In which leaves every i(= 1, * , n) un-changed:

i- iln = i.(28:A:b) Given two permutationsP, Q, their product PQ, which

consistsin carrying out first P and then Q:))

1With the characteristicfunction w(S) 2} <** then in our abovenotatibnsfcinS

v'OSf) - vGSf) +w(S)

(12\\

2* I/Theidentity (cf.below) is / - (!'' ' ' ' ' *\\

\\i, z, , n/)))

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256 GENERAL THEORY:ZERO-SUMn-PERSONS

Thenumber of all possiblepermutationsis the factorial of n r

n\\ = 1 2 . . . n,

and they form togetherthe symmetric group of permutations 2n. Any

subsystemG of 2n which fulfills thesetwo conditions:

(28:A:a*) /n belongsto G,

(28:A:b*) PQ belongsto G if P and Q do,is a group of permutations.1

A permutation Pcarriesevery subsetS of / = (!, , n) into anothersubsetSp *

28.1.2.After thesegeneraland preparatory remarkswe now proceedto apply their conceptsto an arbitrary n-persongame F.

Perform a permutation P on the symbols1, , n denotingthe playersof F. I.e.denotethe playerk = 1, , n by kp insteadof fc; this trans-forms the game F into another game Fp. Thereplacementof F by Fp mustmake its influence felt in two respects:in the influence which eachplayerexerciseson the courseof the play, i.e.in the indexk of the variable r k

which eachplayerchooses;and in the outcomeof the play for him, i.e.inthe indexk of the function 3C*which expressesthis.8 SoTp is again in thenormalized form, with functions 3C(ri, , r n), k = 1, , n. Inexpressing3C(ri, , r n) by meansof 3C*(ri, , r n), we must remem-ber:the playerk in F had 3C*;now he is kp in Fp, sohe has3C>. If we form3C* with the variablesTI, , r n, then we expressthe outcome of thegame Tp when the player whosedesignationin Fp is k choosesr*. Sotheplayerk in F who is kp in Tp choosesT*/>. Sothe variablesin 3C*must beTIP,

* , r np. We have therefore:(28:3)))

1For the important and extensive theory of groups compare L. C. Mathewson:Elementary Theory of Finite Groups, Boston 1930;W. Burnside: Theory of Groups ofFinite Order,2nd Ed.Cambridge 1911;A. Speiser:Theorieder Gruppen von endlicherOrdnung, 3rd Edit. Berlin 1937.

We shall not needany particular results or conceptsof group theory, and mentionthe above literature only for the use of the readerwho may want to acquire a deeperinsight into that subject.

Although we do not wish to tie up our exposition with the intricaciesof group theory,we nevertheless introduced someof its basictermini for this reason:a real understandingof the nature and structure of symmetry is not possiblewithout some familiarity with(at least)the elements of group theory. We want to preparethe readerwho may want toproceedin this direction, by using the correctterminology.

For a fuller exposition of the relationship between symmetry and group theory, cf.H. Weyl: Symmetry, Journ. Washington Acad. of Sciences,Vol. XXVIII (1938),pp.253ff.

*If S - (* , *,),then Sp - (ibf, , kp).8 Cf.the similar situation for n = 2 in footnote 1on p.109.4Thereaderwill observethat the superscript P for the index k of the functions 3C

themselves appearson the left-hand side,while the superscript P for the indicesk of thevariables r* appearon the right-hand side. This is the correctarrangement; and theargument preceding(28:3)was neededto establish it.

Theimportance ofgetting this point faultless and clearliesin the fact that we could)))

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GROUPS,SYMMETRY AND FAIRNESS 257

Denotethe characteristicfunctions of T and Tp by v(S) and vp(S)respectively. Sincethe players,who form in Tp the setSp, arethe sameoneswho form in T the setS,we have

(28:4) vp(Sp) = v(5) for every S.128.1.3.If (for a particularP) T coincideswith Tp, then we say that T is

invariant or symmetric with respectto P. By virtue of (28:3)this isexpressedby(28:5) 3Mn, , r n) ^ 3C*(nP, , T||-).When this is the case,then (28:4)becomes(28:6) v(8p) = v(S) for every S.

Given any T, we can form the systemGr of all P with respectto which Tis symmetric. It isclearfrom (28:A:a),(28:A:b)above, that the identity 7n

belongsto Gr, and that if P, Qbelongto Gr , then their productPQdoestoo.SoGr is a group by (28:A:a*),(28:A:b*)above. We call Gr the invariancegroup of F.

Observethat (28:6)can now be stated in this form:

(28:7) v(S) = v(r) if thereexistsa P in Gr with Sp = T,i.e.which carriesS into T.

The size of Gr i.e.the number of its elements gives somesort of ameasureof \"how symmetric \" F is. If every permutationP (otherthanidentity 7n) changesT, then Gr consistsof 7n alone, r is totally unsymmetric.If no permutation P changesT, then Gr containsall P,i.e.it is the sym-metric group Sn , F is totally symmetric. Thereare,of course,numerousintermediatecasesbetweenthesetwo extremes,and the precisestructureof T'ssymmetry (orlack of it) is disclosedby the groupGr.

28.1.4.The conditionafter (28:7)impliesthat S and T have the samenumber of elements.The converseimplication,however, need not betrue if Gr is small enough, i.e.if F is unsymmetric enough. It is therefore

not otherwise be sure that successiveapplications of the superscripts P and Q (in thisorder) to r will give the same result as a (single) application of the superscript PQ to r.Thereadermay find the verification of this a good exercisein handling the calculus ofpermutations.

For n 2 and P I ' .J , application of P on either sidehad the sameeffect,soit isnot necessaryto beexhaustive on this point. Cf.footnote 1on p.109.

6 In the zero-sum two-person game, 3C 3Ct B 3C2, and similarly 3CF s 3Cf = 3Cf.Hencein this case(cf. above,n -2and P-

\\\\'^j J (28:3)becomes3Cp (n, n) -3C(ri,n).

This is in accordwith the formulae of 14.6.and 17.11.2.But this simplification is possibleonly in the zero-sum two-person game; in all

other caseswe must rely upon the general formula (28:3)alone.1This conceptualproof is clearerand simpler than a computational one,which could

be basedon the formulae of 25.1.3.The latter, however, would causeno difficulties

either, only more extensive notations.)))

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258 GENERAL THEORY:ZERO-SUMn-PERSONS

of interestto considerthose groups G = Gr which permit this converseimplication,i.e.for which the following is true:

(28:8) If S,T have the samenumber of elements,then thereexistsa P in G with Sp = T, i.e.which carriesS into T.

This condition(28:8)is obviously satisfiedwhen G is the symmetricgroupSn , i.e.for the G = Gr = Sn of a totally symmetricF. It is alsosatisfiedfor certainsmallergroups, i.e.for certainF of lessthan total symmetry.1

28.2.Symmetry and Fairness

28.2.1.At any rate,whenever (28:8)holdsfor G = Gr, we can concludefrom (28:7):(28:9) v(S)dependsonly upon the number of elementsin S.That is:(28:10) v(S) = vp

where p is the number of elementsin /S, (p 0,1, , ri).Consider the conditions (25:3:a)-(25:3:c)in 25.3.1.,which give an

exhaustivedescriptionof all characteristicfunctions v(S). It is easy torewrite them for vp when (28:10)holds. They become:(28:ll:a) v = 0,(28:ll:b) vn_p = -vp,(28:ll:c) vp+q ^ vp + v q for p + q g n.

(27:3)in 27.1.4.is clearly a consequenceof (28:10)(i.e.of (28:9)),so that such a v(S) is automatically reduced, with 7 = VL We havetherefore,in particular, (27:7),(27:7*),(27:7**)in 27.2.,i.e.the conditionsof Figure50.

Condition(28:ll:c)can be rewritten, by a procedurewhich is parallelto that of (25:A) in 25.4.2.

(12\\

2' 1 /cf.severalprecedingreferences);soG = 2n is the only possibility of any symmetry.

Considertherefore n 3, and callG set-transitive if it fulfills (28:8).Thequestion,which G T* Sn are then set-transitive, is of a certain group-theoretical interest, but weneednot concernourselveswith it in this work.

For the readerwho is interested in group theory we neverthelessmention:Thereexistsa subgroup of Sn which contains half of its elements(i.e.in!),known as

the alternating group O. Thisgroup is of great importance in group theory and has beenextensively discussedthere. For n ^ 3 it is easilyseento beset-transitive too.

So the real question is this: for which n < 3 do there exist set-transitive groupsG * S;a?

It is easyto show that for n = 3, 4 none exist. For n **5, 6 such groups do exist.(For n 5 a set-transitive group G with 20elements exists, while S6, as have 120,60elements respectively.For n = 6 a set-transitive group G with 120elements exists,while Se,tte have 720,360elementsrespectively.)For n 7, 8 rather elaborategroup-theoretical arguments show that no such groups exist. For n 9 the question is stillopen. It seemsprobablethat no such groups exist for any n > 9, but this assertionhasnot yet beenestablishedfor all thesen.)))

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GROUPS,SYMMETRYAND FAIRNESS 269

Put r = n p q\\ then (28:ll:b)permits us to state (28:ll:c)asfollows:

(28:ll:c*) vp + v, + vr ^ if p + q + r = n.Now (28:ll:c*)is symmetric with respectto p, g, r;1 hencewe may makep ^ q ^ r by an appropriate permutation. Furthermore,when p =(hencer-= n - g), then (28:11:c*) follows from (28:11:a), (28:11:b)(even with =). Thus we may assume p 7*.o. So we need to require(28:ll:c*)only for 1g p ^ g g r, and therefore the same is true for(28:ll:c).Observefinally that, as r = n p q, the inequality g ^ rmeansp + 2q ^ n. We restatethis:(28:12) It sufficesto require(28:ll:c)only when

1 ^ P ^ q, p + 2g ^ n.2

28.2.2.The property (28:10)of the characteristicfunction is a conse-quenceof symmetry, but this property is also important in its own right.Thisbecomesclearwhen we considerit in the simplestpossiblespecialcase:for n = 2.

Indeed,for n = 2 (28:10)simply meansthat the v' of 17.8.1.vanishes.8This means in the terminology of 17.11.2.,that the game T is fair. Weextendthis concept:The n-persongame F is fair when its characteristicfunction v(S) fulfills (28:9),i.e.when it is a vp of (28:10).Now, as in

17.11.2.,this notion of fairness of the game embodieswhat is really essentialin the conceptof symmetry. It must be remembered,however, that theconceptof fairness and similarly that of total symmetry of the gamemay or may not imply that all individual playerscan expectthe samefatein an individual play (provided that they play well). For n = 2 this

implication did hold, but not for n ^ 3 ! (Cf. 17.11.2.for the former, andfootnotes 1and 2 on p.225 for the latter.)

28.2.3.We observe,finally, that by (27:7),(27:7*),(27:7**)in 27.2.,orby Figure50, all reducedgamesaresymmetricand hencefair, when n = 3,but not when n ^ 4. (Cf.the discussionin 27.5.2.) Now the unrestrictedzero-sum n-person game is brought into its reducedform by the fixedextrapayments i, , an (to the players1, , n, respectively),asdescribedin 27.1.Thus the unfairness of a zero-sum three-persongamei.e.what is really effective in its asymmetry is exhaustively expressedby these i, 2, 8 ; that is, by fixed, definite payments. (Cf. also the\"basicvalues/7 a',6',c'of 22.3.4.)In a zero-sumn-persongame with

1 Both in its assertionand in its hypothesis!1Theseinequalities replacethe original p + q ^ n; they areobviously much stronger.

As they imply 3p ^ p + 2g ^ n and 1 -f 2q ^ p -f 2g ^ n, we have

n n - 1P ^ 3' q *^~\"

8 By definition v' v((l)) v((2)). For n 2 the only essentialassertion of(28:9)(which is equivalent to (28:10))is v((l)) v((2)). Dueto the above,this meanspreciselythat v' - -v',i.e.that v' - 0.)))

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260 GENERAL THEORY:ZERO-SUMn-PERSONS

n ^ 4, this is no longer always possible,sincethe reducedform neednot be fair. That is, theremay exist,in such a game, much more funda-mental differences between the strategicpositionsof the players, whichcannot be expressedby the !,,an, i.e.by fixed, definite payments.This will becomeamply clearin the courseof Chapter VII. In the sameconnectionit is alsouseful to recallfootnote 1on p.250.

29.Reconsiderationof the Zero-sumThree-personGame29.1.Qualitative Discussion

29.1.1.We arenow prepared for the main undertaking:To formulatethe principlesof the theory of the zero-sumn-persongame.l Thecharacter-isticfunction v(S),which we have defined in the precedingsections,providesthe necessarytool for this operation.

Our procedurewill be the sameas before:We must selecta specialcaseto serveas a basis for further investigation. This shall be one which wehave already settledand which we neverthelessdeem sufficiently charac-teristicfor the generalcase. By analyzing the (partial) solution foundin this specialcase,we shall then try to crystallizethe rules which shouldgovern the generalcase. After what we said in 4.3.3.and in 25.2.2.,itought to beplausiblethat the zero-sumthree-persongame will be the specialcasein question.

29.1.2.Letus therefore reconsiderthe argument by which our presentsolution of the zero-sumthree-persongame was obtained. Clearly theessentialcasewill be the one of interest. We know now that we may aswell considerit in its reducedform, and that we may also choosey = I.2The characteristicfunction in this caseis completelydetermined,as dis-cussedin the secondcaseof 27.5.2.:))

(29:1) v(5)))-1

1 when S has)) elements.))3))

We saw that in this game everything is decidedby the (two-person)coalitionswhich form, and our discussions4 producedthe following mainconclusions:

Three coalitions may form, and accordinglythe three players will

finish the play with the followingresults:1Of coursethe general n-person game will still remain, but we shall beableto solve

it with the help of the zero-sum games. Thegreateststepis the presentone:the passageto the zero-sum n-persongames.

Cf.27.1.4.and 27.3.2.1In the notation of 23.1.1.this means a b c 1.The general parts of the

discussionsreferred to were those in 22.2.,22.3.,23. Theabove specialization takes usactually backto the earlier(more special)caseof 22.1.Soour considerations of 27.1.(on strategic equivalenceand reduction) have actually this effectin the zero-sum three-persongames:they cany the generalcaseback into the precedingspecialone,as statedabove.

4In 22.2.2.,22.2.3.;but theseare really just elaborations of those in 22.1.2.,22.1.3.)))

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RECONSIDERATION)) 261))

(1,2)))

(1,3)))

(2,3))) -1))

-1))

Figure 51.This \" solution \" callsfor interpretation,and thefollowingremarkssuggest

themselvesin particular:1

29.1.3.(29:A:a) The threedistributionsspecifiedabove correspondto all

strategicpossibilitiesof the game.(29:A:b) Noneof them can be considereda solution in itself;it is the

systemof all threeand their relationshipto eachother which

really constitutethe solution.(29:A:c) The three distributionspossesstogether,in particular, a

\"stability\"to which we have referred thus far only verysketchily. Indeedno equilibrium can be found outside ofthesethreedistributions;and so oneshouldexpectthat anykind of negotiation between the players must always in finelead to one of thesedistributions.

(29:A:d) Again it is conspicuousthat this \"stability\"is only acharacteristicof all threedistributionsviewed together. Noone of them possessesit alone;eachone,taken by itself, couldbe circumvented if a different coalition pattern shouldspreadto the necessarymajority of players.

29.1.4.We now proceedto searchfor an exactformulation of theheuristic principleswhich lead us to the solutionsof Figure51,alwayskeepingin mind the remarks(29:A:a)-(29:A:d).

A more precisestatement of the intuitively recognizable\"stability\"of the systemof threedistributionsin Figure51 which shouldbe a concisesummary of the -discussionsreferred to in footnote 4 on p.260 leads usback to a positionalready taken in the earlier,qualitative discussions.2

It can be put as follows:

(29:B:a) If any otherschemeof distributionshouldbe offered forconsiderationto the three players, then it will meetwith

1Theseremarks take up again the considerations of 4.3.3.In connection with

(29;A:d)the secondhalf of 4.6.2.may alsobe recalled.*Theseviewpoints permeate all of 4.4.-4.G.,but they appearmore specifically in

4.4.Land 4.6.2.)))

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262 GENERAL THEORY:ZERO-SUMn-PERSONS

rejectionfor the following reason:a sufficient number ofplayers1 prefer, in their own interest,at leastone of the dis-tributions of the solution (i.e.of Figure51),and are con-vinced or ean beconvinced2 of the possibilityof obtaining theadvantagesof that distribution.

(29:B:b) If, however, one of the distributions of the solution isoffered, then no suchgroup of playerscan befound.

We proceedto discussthe merits of this heuristicprinciplein a moreexactway.

29.2.Quantitative Discussion

29.2.1.Suppose that 0i, 2, 0s is a possiblemethod of distributionbetweenthe players1,2,3.I.e.

ft + j8> + 0s = 0.

Then,sinceby definition v((i))(= 1)is the amount that playeri can getfor himself (irrespectiveof what all others do),he will certainly blockanydistributionwith ft < v((z)). We assumeaccordinglythat))

We may permutethe players1,2,3so that

Pi ^ A g; ft.Now assume 2 < i- Then a fortiori #3 < . Consequentlythe players

2,3will both prefer the last distributionof Figure51,8 where they both getthehigher amount i.4 Besides,it isclearthat they can getthe advantage ofthat distribution (irrespectiveof what the third player does),sincetheamountsi,i which it assignsto them do not exceedtogetherv((2,3)) = 1.

If, on the otherhand, /32 ^ i,then a fortiori 0i^ . Since 8 ^ 1,this is possibleonly when /3i = 2 = i,0a = 1,i.e.when we have the firstdistributionof Figure51. (Cf.footnote 3 above.)

1Ofcourse,in this case,two.1What this \"convincing\" means was discussedin 4.4.3. Our discussion which

follows will make it perfectly clear.3Sincewe made an unspecified permutation of the players 1,2,3the last distribution

of Fig. 51really stands for all three.4Observethat eachone of thesetwo playersprofits by such a changeseparatelyand

individually. It would not suffice to have only the totality (of thesetwo) profit. Cf.,e.g.,the first distribution of Fig. 51with the second;the players 1,3asa totality wouldprofit by the changefrom the former to the latter, and neverthelessthe first distributionis just as gooda constituent of the solution as any other.

In this particular change,player 3 would actually profit (getting i instead of 1),and for player 1 the changeis indifferent (getting |in both cases). Neverthelessplayer 1will not act unless further compensations aremade and thesecan bedisregardedin thisconnection. For a more careful discussion of this point, cf.the last part of this section.)))

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EXACT FORM OF THEGENERALDEFINITIONS 263

This establishes(29:B:a)at the end of 29.1.4.(29:B:b)loc.cit.isimmediate:in eachof the threedistributionsof Figure51thereis,to besure,one playerwho is desirousof improving hisstanding,1but sincethereisonly one,he is not able to doso. Neitherof his two possiblepartnersgainsanything by forsaking his presentally and joining the dissatisfiedplayer:alreadyeachgetsi,and they can getno more in any alternative distributionof Figure51.2

29.2.2.Thispoint may be clarifiedfurther by someheuristicelaboration.We seethat the dissatisfiedplayer finds no one who desiresspontaneously

to behis partner, and he can offer no positive inducementto anyone tojoin him; certainly none by offering to concedemore than i from theproceedsof their future coalition. The reasonfor regardingsuchan offer as ineffec-tive can beexpressedin two ways:on purely formal groundsthis offer maybe excludedbecauseit correspondsto a distributionwhich is outside theschemeof Figure51;the real subjectivemotive for which any prospectivepartner would considerit unwise3 to accepta coalition undersuchconditionsis most likely the fear of subsequentdisadvantage, theremay be furthernegotiationsprecedingthe formation of a coalition, in which he would befound in a particularly vulnerable position. (Cf. the analysis in 22.1.2.,22.1.3.)

Sothereis no way for the dissatisfiedplayerto overcome the indifferenceof the two possiblepartners. We stress:there is, on the side of the twopossiblepartnersno positivemotive againsta changeinto another distribu-tion of Figure51,but just the indifferencecharacteristicof certaintypes ofstability.4

30.TheExactForm of the GeneralDefinitions30.1.TheDefinitions

30.1.1.We return to the caseof the zero-sumn-persongame F with

generaln. Let the characteristicfunction of T be v(S).We proceedto give the decisivedefinitions.In accordancewith the suggestionsof the precedingparagraphs we

mean by a distribution or imputation a setof n numbers i, , a with

'thefollowingproperties(30:1) a, ^ v((0) for i = 1, - - , n,

(30:2) a, = 0.i-l1Theone who gets 1.1Thereadermay find it a goodexerciseto repeatthis discussion with a general (not

reduced)v(S), i.e.with generala, 6, c, and the quantities of 22.3.4.Theresult is thesame;it cannot be otherwise, sinceour theory of strategic equivalence and reduction iscorrect. (Cf.footnote 3 on p. 260.)

8 Orunsound, or unethical.4 At every change from one distribution of Fig. 51to another, one player is definitely

against, one definitely for it; and so the remaining player blocks the change by hisindifference.)))

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264 GENERALTHEORY:ZERO-SUMn-PERSONS

It may be convenient to view thesesystems !,-,as vectorsin then-dimensionallinear spaceLn in the senseof 16.12.:))

A setS (i.e.a subsetof / = 1, , n) is calledeffectivefor the imputa-

tion a , if

(30:3) \"* * vO81)-iinS-r -

An imputation a. dominatesanotherimputation ft , in symbols))

if thereexistsa setS with the followingproperties:

(30:4:a) S is not empty,

(30:4:b) S is effective for a ,(30:4:c) <* > ft for all i in S.

A setV of imputations isa solutionif it possessesthe followingproperties:

(30:5:a) No ft in V is dominatedby an a in V,(30:5:b) Every ft not in V is dominatedby some a in V.

(30:5:a)and (30:5:b)can be stated as a singlecondition:

(30:5:c) The elementsof V arepreciselythoseimputations which)). _ 4 _ ^areundominatedby any elementof V.))

(Cf.footnote 1on p.40.)30.1.2.The meaning of thesedefinitions can, of course,be visualized

when we recallthe considerationsof the precedingparagraphsand also ofthe earlierdiscussionsof 4.4.3.

Tobeginwith, our distributionsor imputationscorrespondto the moreintuitive notionsof the samename in the two placesreferredto. What wecall an effective setis nothing but theplayerswho \"areconvincedor canbeconvinced\"of the possibilityof obtaining what they areoffered by a ; cf.again 4.4.3and (29:B:a)in 29.1.4.Thecondition(30:4:c)in the definitionof domination expressesthat all theseplayers have a positivemotive for

> >

preferring a to ft . It is therefore apparent that we have defined domi-nation entirely in the spirit of 4.4.1.,and of the preferencedescribedby(29:B:a)in 29.1.4.

Thedefinition of a solutionagreescompletelywith that given in 4.5.3.,aswell aswith (29:B:a),(29:B:b)in 29.1.4.)))

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EXACT FORM OF THEGENERAL DEFINITIONS 265

30.2.Discussionand Recapitulation

30.2.1.The motivation for all thesedefinitions has beengiven at theplacesto which we referredin detail in the courseof the last paragraph.We shall neverthelessre-emphasizesomeof their nlain features particu-larly the conceptof a solution.

We have already seenin 4.6.that our conceptof a solutionof a gamecorrespondspreciselyto that of a \" standard of behavior\" of everydayparlance.Our conditions (30:5:a),(30:5:b),which correspond to theconditions(4:A:a),(4:A:b)of 4.5.3.,expressjust the kind of \"innersta-bility\" which is expectedof workablestandards of behavior. This waselaboratedfurther in 4.6.on a qualitative basis. We can now reformulatethose ideas in a rigorousway, consideringthe exactcharacterwhich thediscussionhasnow assumed. Theremarkswe wish to makearethese:1

30.2.2.(30:A:a) Consider a solution V. We have not excludedfor an

imputation ft in V the existenceof an outsideimputation a '(not in V) with a ' H ft .2 If suchan a ' exists,the attitudeof the players must be imagined like this:If the solution V(i.e.this systemof imputations) is \" accepted\" by the players1,

, n, then it must impressupon their mindsthe idea that

only the imputations ft in V are\" sound\"ways of distribution.

An a ' not in V with a ' H ft , although preferableto aneffective setof players,will fail to attractthem, becauseit is\"unsound.\" (Cf.our detaileddiscussionof thezero-sumthree-persongame,especiallyas to the reasonfor the aversion of eachplayer to acceptmore than the determinedamount in a coali-tion. Cf. the endof 29.2.and its references.)Theview of the

>\" unsoundness\" of a '

may alsobe supportedby the existence>

of an a in V with a H a '(cf.(30:A:b)below). All thesearguments are, of course,circular in a sense and againdependon the selectionof V as a \"standardof behavior,\" i.e.as a criterionof \"soundness.\"But this sort of circularity isnot unfamiliar in everyday considerations dealing with

\"soundness.\"irThe remarks (30:A:a)-(30:A:d)which follow are a more elaborateand precise

presentation of the ideasof 4.6.2.Remark (30:A:e)bears the same relationship to4.6.3.

1Indeed,we shall seein (31:M)of 31.2.3.that an imputation , for which never>

a ' H , exists only in inessential games.)))

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266 GENERAL THEORY:ZERO-SUMn-PERSONS

(30:A:b) If the players1, , n have acceptedthesolutionV as a\" standard of behavior/1 then the ability to discreditwith thehelp of V (i.e.of its elements)any imputation not in Vi isnecessaryin orderto maintain their faith in V. Indeed,for

every outside a '(not in V) theremust existan a in V with

a H a '. (Thiswasour postulate(30:5:b).)(30:A:c) Finally theremust be no inner contradictionin V> i.e.for

a , ft in V, never a H ft . (This was our otherpostulate(30:5:a).)

(30:A:d) Observethat if domination, i.e.therelation H , weretransi-tive, then the requirements (30:A:b)and (30:A:c)(i.e.ourpostulates (30:5:a)and (30:5:b))would excludethe ratherdelicatesituationin (30:A:a).Specifically:In the situationof

(30:A:a),ft belongsto V, a 'doesnot, and a ' *- ft . By

(30:A:b)thereexistsan a in V so that a H a '.Now if

domination were transitive we couldconcludethat a H ft ,

which contradicts(30:A:c)sincea , ft both belongto V-(30:A:e) Theabove considerationsmakeit even more clearthat only

V in its entiretyisa solutionand possessesany kind of stabilitybut none of its elementsindividually. The circularchar-

acterstressedin (30:A:a)makes it plausiblealso that severalsolutionsV may existfor the samegame. I.e.severalstablestandardsof behavior may existfor thesamefactual situation.Eachof thesewould, of course,bestableand consistentin itself,but in conflict with all others. (Cf.alsothe end of 4.6.3.andthe end of 4.7.)

In many subsequentdiscussionswe shall seethat this multiplicity ofsolutionsis, indeed,a very generalphenomenon.

30.3.TheConceptof Saturation

30.3.1.It seemsappropriate to insert at this point someremarks of amore formalistic nature. So far we have paid attention mainly to themeaning and motivation of the conceptswhich we have introduced,but thenotion of solution,as defined above,possessessomeformal featureswhichdeserveattention.

The formal logical considerationswhich follow will be of no imme-diate use,and we shall not dwell upon them extensively,continuing after-wardsmore in the vein of the precedingtreatment.Neverthelesswe deemthat theseremarksareuseful herefor a more completeunderstandingof thestructureof our theory. Furthermore,the proceduresto be used herewill

have an important technicalapplicationin an entirely different connectionin 51.1.-51.4.)))

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EXACT FORM OF THEGENERAL DEFINITIONS 267

30.3.2.Considera domain (set)D for theelementsx,y of which a certainrelation x&y exists. The validity of (R betweentwo elementsx,y of D isexpressedby the formula x&y.1 (R is defined by a statementspecifying

unambiguously for which pairsx,y of D, xfay is true,and for which it isnot.If xGiy is equivalent to 2/(Rrc, then we say that xfay is symmetric. Foranyrelation (R we can define a new relation (R 5 by specifyingx(R8

y to mean theconjunction of xfay and yfax. Clearly(R 5 is always symmetricand coincideswith (R if and only if (R issymmetric. We call (R a the symmetrized form of (R.2

We now define:

(30:B:a) A subset A of D is (Si-satisfactoryif and only if x(Ri/ holdsfor all x,y of A.

(30:B:b) A subset A of D and an elementy of D are(Si-compatibleif and only if x(Ry holdsfor all x ot A.

From theseone concludesimmediately:(30:C:a) A subsetA of D is (Si-satisfactoryif and only if this is true:

They which are(R-compatiblewith A form a supersetof A.

We define next:(30:C:b) A subset A of D is (Si-saturatedif and only if this is true:

They which are(R-compatiblewith A form preciselythe set A .Thus the requirementwhich must be added to (30:C:a) in order to secure(30:C:b)is this:(30:D) If y is not in A, then it is not (R-compatiblewith A; i.e.

thereexistsan x in A such that not x(Ry.

Consequently(R-saturation may be equivalently defined by (30:B:a)and(30:D).

30.3.3.Beforewe investigate theseconceptsany further, we give someexamples.The verification of the assertionsmade in them is easyand will

be left to the reader.First:Let D be any set and x&y the relation x = y. Then (R-satis-

factorinessof A means that A is eitherempty or a one-elementset,while(R-saturation of A meansthat A is a one-elementset.

Second:Let D be a set of real numbersand x(Siy the relation x ^ t/.3

Then (R-satisfactorinessof A meansthe same thing as above,4 while (R-sat-uration of A meansthat A is a one-elementset,consistingof the greatestelementof D. Thus thereexistsno such A if D has no greatestelement

1It is sometimes more convenient to use a formula of the form (R(or, y), but for our

purposesx&y is preferable.2 Someexamples:Let D consistof ail realnumbers. Therelations x y and x 9* y

are symmetric. None of the four relations x y, x ^ y, x < y, x > y \\R symmetric.The symmetrized form of the two former is x y (conjunction of x ^ y and x y),the symmetrized form of the two latter is an absurdity (conjunction of x < y and x > y).

8 Dcouldbeany other set in which such a relation is defined, cf.the secondexample in

65.4.1.4Cf.footnote 1 on p. 268.)))

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268 GENERAL THEORY:ZERO-SUMn-PERSONS

(e.g.for the setof all realnumbers)and A is unique if D has a greatestelement(e.g.when it is finite).

Third:Let D bethe planeand xfay expressthat the points#, y have thesame height (ordinate). Then (R-satisfactoriness of A means that allpoints of A have the same height, i.e.lieon one parallel to the axis ofabscissae.(R-saturationmeansthat A is preciselya line parallelto the axisof abscissae.

Fourth:Let D bethe setof all imputations,and x<S(.y thenegation of thedomination x H y. Then comparisonof our (30:B:a),(30:D)with (30:5:a),(30:5:b)in 30.1.1.,or equallyof (30:C:b)with (30:5:c)id.shows:(R-satura-tion of A meansthat A is a solution.

30*3.4.One look at the condition (30:B:a)suffices to seethat satis-factorinessfor the relationx<S(y is the sameas for the relation yfax and soalso for their conjunction x(&8y. In otherwords:(R-satisfactorinessis thesame thing as (R a-satisfactoriness.

Thus satisfactorinessis a conceptwhich needbe studied only on sym-metric relations.

Thisisdueto the x, y symmetricform of the definitory condition (30:B:a).Theequivalent condition(30:C:a)doesnot exhibit this symmetry, but ofcoursethis doesnot invalidate the proof.

Now the definitory condition(30:C:b)for (R-saturation is very similarin structure to (30:C:a).It is equally asymmetric. However, while

(30:C:a)possessesan equivalent symmetricform (30:B:a),this is not thecasefor (30:C:b).Thecorrespondingequivalent form for (30:C:b)is, aswe know, the conjunction of (30:B:a)and (30:D) and (30:D)is not at allsymmetric. I.e.(30:D) is essentiallyalteredif x(Rt/ is replacedby y(Rx.Sowe see:(30:E) While (R-satisfactorinessin unaffected by the replacementof

(R by (R5, it doesnot appear that this is thecasefor (R-saturation.

Condition (30:B:a)(amounting to (R-satisfactoriness) is the same for(R and (R 5. Condition(30:D)for (R s is impliedby the same for (R since(R 5

implies(R. Sowe see:(30:F) (R s-saturationis impliedby (R-saturation.

Thedifference betweenthesetwo types of saturation referredto aboveis a realone:it is easy to give an explicitexampleof a setwhich is (R s-sat-urated without being(R-saturated.1

Thus thestudy of saturationcannot berestrictedto symmetricrelations.30.3.5.For symmetric relations (R the nature of saturation is simple

enough. In orderto avoid extraneouscomplicationswe assumefor thissectionthat x(Rx is always true.2

1E.g.:The first two examplesof 30.3.3.are in the relation of (R 5 and (R to eachother(cf.footnote 2 on p. 267);their conceptsof satisfactorinessare identical, but those ofsaturation differ.

*This isclearlythe casefor our decisiveexample of30.3.3.:x&y the negation of * H ysincenever x H x.)))

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EXACT FORM OF THEGENERAL DEFINITIONS 269

Now we prove:

(30:G) Let (R be symmetric. Then the (R-saturation of A is equiv-alent to its beingmaximal (R-satisfactory. I.e.it is equivalentto:A is (R-satisfactory,but no propersupersetof A is.

Proof:(R-saturation means (R-satisfactoriness (i.e.condition (30:B:a))togetherwith condition (30:D).So we needonly prove:If A is (R-satis-factory, then (30:D)is equivalent to the non-(R-satisfactorinessof all propersupersetsof A.

Sufficiency of (30:D):If B D A is (R-satisfactory, then any y in B,butnot in A, violates (30:D).1

Necessityof (30:D):Considera y which violates (30:D).Then

B = A u (y) D A.

Now B is (R-satisfactory,i.e.for #', y' in B,always x'<S(y'. Indeed,whenx',y' areboth in A, this followsfrom the (R-satisfactorinessof A. If x',y'areboth = y, we aremerely assertingy&y. If one of z',y' is in A, and theother= y, then the symmetry of (R allows us to assumexf in A, y' = y.Now our assertioncoincideswith the negation of (30:D).

If (R is not symmetric,we can only assertthis:

(30:H) (R-saturation of A impliesits beingmaximal (R-satisfactory.

Proof:Maximal (R-satisfactorinessis the sameasmaximal (R-satisfactori-ness, cf. (30:E).As (R 5 is symmetric,this amounts to (R saturation by(30:G). And this is a consequenceof (R-saturation by (30:F).

The meaning of the result concerninga symmetric (R is the following:Startingwith any (R-satisfactoryset,this setmust beincreasedas long aspossible, i.e.until any further increasewould involve the loss of (R-satis-factoriness. In this way in fine a maximal (R-satisfactorysetis obtained,

i.e.an (R-saturated one by (30:G).2 This argument securesnot onlythe existenceof (R-saturated sets,but it alsopermitsus to infer that every(R-satisfactorysetcan be extendedto an (R-saturated one.

1 Note that none of the extra restrictions on <H has beenused so far.8 This processof exhaustion is elementary i.e.it is over after a finite number of

steps when D is finite.However, sincethe setof all imputations is usually infinite, the caseof an infinite D

is important. When D is infinite, it is still heuristically plausible that the processofexhaustion referred to can be carried out by making an infinite number of steps. Thisprocess,known as transfinite induction, has beenthe objectof extensive set-theoreticalstudies. It can be performed in a rigorous way which is dependent upon the so-calledaxiom of choice.

Thereaderwho is interested will find literature in F.Hausdorff, footnote 1,on p.61.Cf.alsoE. ZermelOj BeweisdassjedeMenge wohlgeordnet werden kann. Math. Ann.Vol. 59 (1904)p.514ff.and Math. Ann. Vol. 65(1908)p. 107ff.

Thesematters carry far from our subjectand are not strictly necessaryfor ourpurposes. We do not therefore dwell further upon them.)))

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270 GENERAL THEORY:ZERO-SUMn-PERSONS

Itshouldbe noted that every subsetof an (R-saturatedsetis necessarily(R-satisfactory.l The above assertionmeans therefore that the conversestatement is alsotrue.

30.3.6.It would be very convenient if the existenceof solutionsin ourtheory couldbe establishedby such methods. The prima facie evidence,however, is againstthis:the relation which we must use,a;(Ry negation ofthe domination x s-1 y, cf. 30.3.3.is clearlyasymmetrical. Hencewe can-not apply (30:G),but only (30:H):maximal satisfactorinessis only neces-sary, but may not be sufficient for saturation,i.e.for beinga solution.

That this difficulty is really deepseatedcan beseenas follows:If wecouldreplacethe above (R by a symmetric one,this could not only be usedto provethe existenceof solutions,but it would also prove in the sameoperationthe possibilityofextendingany (R-satisfactorysetof imputations to a solution(cf. above). Now it is probablethat every game possessesa solution,butwe shall seethat there existgamesin which certain satisfactory setsaresubsets of no solutions.2 Thus the device of replacing(R by somethingsymmetric cannot work becausethis would be equally instrumental in prov-ing the first assertion,which is presumablytrue,and the secondone,whichis certainly not true.8

Thereadermay feel that this discussionis futile, sincethe relation x&ywhich we must use (\"notx H y\") is defacto asymmetric. From a technicalpoint of view, however, it is conceivablethat another relation x$y may befound with the following properties:x$y is not equivalent to x&y; indeed,

issymmetric,while (R is not, but-saturationis equivalent to (R-saturation.In this case(R-saturated setswould have to existbecausethey are the-saturated ones,and the g-satisfactory but not necessarilythe (R-satis-

factory setswould always be extensibleto g-saturated,i.e.(R-saturatedones.4 This program of attack on the existenceproblemof solutionsis notas arbitrary as it may seem. Indeed,we shallseelatera similar problemwhich is solvedin preciselythis way (cf. 51.4.3.).All this is, however, forthe time beingjust a hopeand a possibility.

30.3.7.In the last sectionwe consideredthe question whether every(R-satisfactory set is a subset of an (R-saturated set. We noted that forthe relation x&y which we must use (\"notx H y\" asymmetric)the answeris in thenegative. A brief comment upon this fact seemsto be worth while.

If the answerhad beenin the affirmative it would have meant that anysetfulfilling (30:B:a)can be extendedto one fulfilling (30:B:a)and (30:D);or,in the notationsof 30.1.1.,that any setof imputationsfulfilling (30:5:a)can be extendedto onefulfilling (30:5:a)and (30:5:b).

1Clearly property (30:B:a)is not lost when passing to a subset.1Cf.footnote 2 on p. 285.8 This is a rather useful principle of the technical sideof mathematics. Theinappro-

priatenessof a method can be inferred from the fact that if it were workable at all itwould prove too much.

4Thepoint is that (R- and S-saturation are assumed to be equivalent to eachother,but <H- and S-satisfactorinessarenot expectedto be equivalent.)))

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EXACT FORM OF THEGENERAL DEFINITIONS 271It is instructive to restatethis in the terminology of 4.6.2.Then the

statementbecomes:Any standard of behavior which is free from innercontradictioncan be extendedto one which is stable, i.e.not only freefrom inner contradictions,but also ableto upset all imputationsoutsideof it.

Theobservationin 30.3.6.,accordingto which the above is not true ingeneral,is of some significance:in orderthat a set of rules of behaviorshould be the nucleus (i.e.a subset) of a stablestandard of behavior, itmay have to possessdeeperstructural propertiesthan merefreedom frominner contradictions.1

30.4.ThreeImmediate Objectives30.4.1.We have formulated the characteristicsof a solutionof an unre-

strictedzero-sumn-persongame and can therefore begin the systematicinvestigation of the propertiesof this concept.In conjunctionwith theearly stagesof this investigation it seemsappropriate to carry out threespecialenquiries. Thesedealwith the followingspecialcases:

First:Throughout the discussionsof 4.,the idearecurredthat the unso-phisticatedconceptof a solution would be that of an imputation, i.e.inour presentterminology, of a one-elementsetV- In 4.4.2.we sawspecifi-cally that this would amount to finding a \"first\" elementwith respecttodomination. We saw in the subsequentparts of 4.,as well as in our exactdiscussionsof 30.2.,that it is mainly the intransitivity of our conceptofdomination which defeats this endeavorand forcesus to introducesetsofimputationsV as solutions.

It is, therefore, of interest now that we arein a positionto do it togive an exactanswerto the following question:Forwhich gamesdo one-elementsolutionsV exist? What elsecan be said about the solutionsofsuch games?

Second:The postulatesof 30.1.1.wereextractedfrom our experienceswith the zero-sumthree-persongame,in its essentialcase. It is, therefore,of interestto reconsiderthis casein the light of the present,exacttheory.Of course,we know indeed this was a guidingprinciplethroughout ourdiscussions that the solution which we obtained by the preliminarymethods of 22.,23.,are solutionsin the senseof our presentpostulatestoo. Neverthelessit is desirableto verify this explicitly. The real point,however, is to ascertainwhether the presentpostulates do not ascribetothosegamesfurther solutionsas well. (We have alreadyseenthat it is notinconceivablethat thereshouldexistseveralsolutionsfor the samegame.)

We shall therefore determineall solutions for the essentialzero-sumthree-persongames with results which arerather surprising,but, as weshallsee,not unreasonable.

1If the relation S referred to at the end of 30.3.6.couldbe found, then this $ andnot (R would disclosewhich standards of behavior are such nuclei (i.e.subsets):the

S-satisfactory ones.Cf. the similar situation in 51.4.,where the corresponding operation is performed

successfully.)))

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272 GENERAL THEORY: ZERO-SUMn-PERSONS

30.4.2.Thesetwo itemsexhaustactually all zero-sumgameswith n ^ 3.We observedin the first remarkof 27.5.2.that for n = 1,2, thesegamesareinessential;so this, togetherwith the inessentialand the essentialcasesofn = 3,takes careof everything in n ^ 3.

When this program is fulfilled we are left with the gamesn ^ 4 andwe know already that difficulties of a new kind begin with them (cf. theallusionsof footnote 1,p.250, and the end of 27.5.3.).

30.4.3.Third:We introduced in 27.1.the conceptof strategicequiva-lence. It appeared plausible that this relationship acts as its nameexpresses:two games which are linked by it offer the same strategicalpossibilitiesand inducementsto form coalitions,etc. Now that we haveput the conceptof a solution on an exactbasis, this heuristicexpectationdemandsa rigorousproof.

Thesethreequestionswill be answeredin (31:P)in 31.2.3.; in 32.2.; andin (31:Q)in 31.3.3.,respectively.

31.First Consequences31.1.Convexity, Flatness, and SomeCriteria for Domination

31.1.1.This sectionis devotedto proving various auxiliary resultscon-cerning solutions,and the other conceptswhich surround them, like ines-sentiality, essentiality, domination, effectivity. Sincewe have now putall thesenotionson an exactbasis, the possibilityas well as the obligationarises to be absolutelyrigorousin establishingtheir properties.Someofthe deductionswhich followmay seempedantic,and it may appearoccasion-ally that a verbal explanationcouldhave replacedthe mathematical proof.Suchan approach,however, would be possiblefor only part of the resultsof this sectionand, taking everything into account,the best plan seemstobe to go about the wholematter systematicallywith full mathematical rigor.

Someprincipleswhich play a considerablepart in finding solutionsare(31:A), (31:B),(31:C),(31:F),(31:G),(31:H),which for certain coalitionsdecidea priori that they must always, or never, be taken into consideration.It seemedappropriate to accompanytheseprincipleswith verbal explana-tions (in the senseindicatedabove) in addition to thqir formal proofs.

Theotherresultspossessvarying interestof their own in different direc-tions. Togetherthey give a first orientation of the circumstanceswhichsurroundour newly won concepts.Theanswersto the first and third ques-tions in 30.4.are given in (31:P) and (31:Q).Another question whicharosepreviously is settledin (31:M).

31.1.2.Considertwo imputations a , ft and assumethat it has become

necessaryto decidewhether a H ft or not. This amounts to decidingwhether or not thereexistsa setS with the properties(30:4:a)-(30:4:c)in30.1.1.Oneof these,(30:4:c)is

> ft for all i in S.)))

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FIRSTCONSEQUENCES 273

We call this the main condition. The two others, (30:4:a),(30:4:b),arethe preliminary conditions.

Now one of the major technicaldifficulties in handling this conceptofdomination i.e.in finding solutionsV in the senseof30.1.1.isthe presenceof thesepreliminary conditions.It is highly desirableto beable,so to say,to short circuit them, i.e.to discovercriteriaunderwhich they arecertainlysatisfied,and othersunder which they arecertainly not satisfied. In look-ing for criteriaof the lattertype, it isby no meansnecessarythat they should

involve non-fulfillment of the preliminary conditionsfor all imputations ait sufficesif they involve it for all thoseimputations a which fulfill the main

condition for someother imputation . (Cf.the proofsof (31:A) or (31:F),where exactlythis is utilized.)

We areinterestedin criteria of this nature in connectionwith the ques-tion of determiningwhether a given setV of imputationsis a solutionor not;i.e.whether it fulfills the conditions (30:5:a),(30:5:b)the condition

(30:5:c)in 30.1.1.This amounts to determiningwhich imputationsaredominatedby elementsof V-

Criteriawhich disposeof the preliminary conditionssummarily, in thesituation describedabove, are most desirableif they contain no reference

at all to ^V'2i.e.if they refer to 8alone. (Cf.(31:F),(31:G),(31:H).)But

even criteriawhich involve a may be desirable.(Cf. (31:A).) We shall

considereven a criterion which deals with S and a by referring to the*

behavior of another a '. (Ofcourse,both in V. Cf. (31:B).)In order to cover all these possibilities,we introduce the following

terminology:We considerproofs which aim at the determinationof all imputations

/3 , which aredominatedby elementsof a given set of imputationsV- We

arethus concernedwith the relations a H ft (a in V), and the questionwhether a certainsetS meetsour preliminary requirementsfor sucha rela-tion. We call Scertainlynecessaryif we know (owingto the fulfillment by S

of someappropriatecriterion) that S and a always meetthe preliminaryconditions.We call a set S certainly unnecessary,if we know (again owingto the fulfillment by Sof someappropriatecriterion,but which may now

involve otherthings too,cf. above) that the possibilitythat Sand a meet

1Thepoint being that in our original definition of a ** ft the preliminary conditions

referto 5and to (but not to ft ). Specifically:(30:4:b)does.1Thehypothetical element of V, which should dominate ft .)))

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274 GENERAL THEORY:ZERO-SUMn-PERSONS

the preliminary conditionscan bedisregarded(becausethis never happens,or for any otherreason.Cf. alsothe qualifications madeabove).

Theseconsiderationsmay seemcomplicated,but they expressa quitenatural technicalstandpoint.1

We shall now give certaincriteriaof the certainly necessaryand of thecertainly unnecessarycharacters.After each criterionwe shall give averbal explanationof its content,which, it ishoped,will makeour techniqueclearerto the reader.

31.1.3.First,threeelementarycriteria:

(31:A) Sis certainly unnecessaryfor a given a (in V) if thereexistsan t in Swith a, = v((i)).

Explanation:A coalition neednever be consideredif it doesnot promiseto every participant (individually) definitely more than he can get forhimself.

Proof:If a fulfills the main conditionfor someimputation, then on > ft.

Since ft is an imputation, so ft ^ v((i)). Hencea< > v((i)). This con-tradictscti = v((i)).

(31:B) S is certainlyunnecessaryfor a given a (in V) if it iscertainly

necessary(andbeingconsidered)for anothera'(in V),suchthat

(31:1) ; ^ a, for all i in S.Explanation:A coalition neednot beconsideredif another one,which

has the sameparticipants and promisesevery one (individually) at leastas much, is certainto receiveconsideration.

Proof:Let a and ft fulfill the main condition:a, > ft for aliiin S. Then

a'and ft fulfill it also,by (31:1),a( > ft for all i in S. SinceSand a'are1 For the readerwho is familiar with formal logicweobservethe following:The attributes \"certainly necessary\"and \"

certainly unnecessary\" are of a logicalnature. They arecharacterizedby our ability to show (by any means whatever) that acertain logicalomission will invalidate no proof (of a certain kind). Specifically:Leta

> >

proofbeconcernedwith the domination ofa ft by an element a ofV. Assume that this> >

domination a H occurring with the help of the setS( a in V) beunder consideration.

Then this proof remains correctif we treat S and a (when they possessthe attribute inquestion) as if they always (or never) fulfilled the preliminary conditions, without ouractually investigating these conditions. In the mathematical proofs which we shallcarry out, this procedurewill beapplied frequently.Itcan evenhappen that the sameSwill turn out (by the useof two different criteria)

to beboth certainly necessaryand certainly unnecessary (for the same a , e.g.for all ofthem). This means merely that neither of the two omissions mentioned abovespoils

*any proof. This can happen, for instance, when a fulfills the main condition for noimputation. (An exampleisobtainedby combining (31:F)and (31:G)in the casedescribedin (31::b).Another is pointed out in footnote 1on p.310,and in footnote 1on p.431.))))

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FIRSTCONSEQUENCES 275

beingconsidered,they thus establishthat ft is dominatedby an elementof

Vi and it is unnecessaryto considerSand a .(31:C) S is certainly unnecessaryif another set TzSis certainly

necessary(and is beingconsidered).

Explanation:A coalition neednot beconsideredif a part of it is alreadycertainto receiveconsideration.

Proof:Let a (in V) and ft fulfill themain conditionfor S\\ then they will

obviously fulfill it a fortiori for T S. SinceT and a arebeingconsidered,

they thus establishthat ft isdominatedby an elementofV and it isunneces-

sary to considerSand a .31.1.4.We now introduce some further criteria,and on a somewhat

broader basis than immediately needed.Forthis purposewe begin with

the followingconsideration:Foran arbitrary set S = (k\\, , k p) apply (25:5)in 25.4.1.,with

Si= (ti), , Sp =(fc p). Then

v(S) ^ v((*0)+ ' ' ' + v((fcp))

obtains,i.e.(31:2) v(S) g; % v((fc)).

kinS

Theexcessof the left-hand sideof (31:2)over the right hand sideexpressesthe total advantage (for all participantstogether)inherentin the formationof the coalition S. We call this the convexity of S. If this advantagevanishes,i.e.if(31:3) vGS) = % v((fc)),

kin 3then we call S flat.

Thefollowingobservationsareimmediate:

(31:D) Thefollowingsetsarealwaysflat:(31:D:a) Theempty set,(31:D:b) Every one-elementset,(31:D:c) Every subsetof a flat set.(31:E) Any oneof the followingassertionsis equivalent to the in-

essentialityof the game:(31:E:a) / = (!, , n) is flat,(31:E:b) Thereexistsan S such that both Sand S areflat,(31:E:c) Every 8 is flat.

Proof:Ad (31:D:a),(31:D:b):Forthesesets(31:3)is obvious.)))

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276 GENERALTHEORY: ZERO-SUMn-PERSONS

Ad (31:D:c):AssumeSsT,Tflat. Put R = T -S. Then by (31:2)

(31:4) v(5)))k in S

(31:5) v(ft) X v((fc)).tinR

SinceT is flat, so by (30:3)

(31:6) v(T) - I v((*)).*inT

As Sn ft = 0,S u ft = T;therefore

v(S)+ v() ^ v(T),))

* in S * in R kinT

Hence(31:6)implies

(31:7) v(S)+v()3 v((*))+*inS fcinfl

Now comparisonof (31:4),(31:5)and (31:7)shows that we must haveequality in all of them. But equality in (31:4)expressesjust the flatnessof S.

Ad (31:E:a):The assertioncoincideswith (27:B)in 27.4.1.Ad (31:E:c):The assertioncoincideswith (27:C):a27.4.2.Ad (31:E:b):Foran inessentialgame this is true for any S owing to

(31:E:c).Conversely,if this is true for (at leastone)S,then

v(5) = v((*)), v(-S)= S v((*))fkin S k not in S

henceby addition (use(25:3:b)in 25.3.1.),= t v((*))f

/fe-i

i.e.the game is inessentialby (31:E:a)or by (27:B)in 27.4.1.31.1.6.We arenow in a positionto prove:

(31:F) Sis certainly unnecessaryif it is flat.

Explanation:A coalition neednot beconsideredif the game allows nototal advantage (for all its participantstogether)over what they would getfor themselvesas independentplayers.1

1Observethat this is relatedto (31:A), but not at all identical with it ! Indeed:(31:A)dealswith the at, i.e.with the promises made to eachparticipant individually. (31:F)dealswith v(*S)(which determines flatness), i.e.with the possibilities of the game for allparticipants together. But both criteria correlatethesewith the v((i)), i.e.with whateachplayer individually can get for himself.)))

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FIRSTCONSEQUENCES 277

> >

Proof:If a ^ ft with the help of this S'thenwe have:NecessarilyS 5* . cti>fa for all i in S and fa ^ v((i)), henceon > v((i)). So2) < > v((t)). As S is flat, this means J) <* > v(/S). But Smust

t in 3 t in 5 i in Sbe effective, <* * v(fi>), which is a contradiction.

UnS

(31:G) S is certainly necessaryif Sis flat and 8 ^ 0.Explanation:A coalition must be consideredif it (is not empty and)

opposesone of the kind describedin (31:F).>

Proof:Thepreliminary conditionsarefulfilled for all imputations a .Ad (30:4:a):S ^ Qwas postulated.Ad (30:4:b):Always ^ v((t)), so ^ v((i)). Since

t not in 3 t not in 3n

5) ; = 0,the left-hand side is equal to ^ en. Since S is flat, the))t-lright-hand side is equal to v( S), i.e.(use(25:3:b)in 25.3.1.)to v(S).So- % a.^ -vGS), a< g v(S),i.e.S is effective.

t in S t in SFrom (31:F),(31:G)we obtain by specialization:

(31:H) A p-elementset is certainly necessary if p = n 1,andcertainly unnecessaryif p = 0,1,n.

Explanation:A coalition must be consideredif it has only one opponent.A coalition neednot be considered,if it is empty or consistsof oneplayeronly (!),or if it has no opponents.

Proof:p = n 1: S has only one element,henceit is flat by (31:D)above. Theassertionnow follows from (31:G).

p = 0,1:Immediateby (31:D)and (31:F).p = n: In this casenecessarily>S = / = (l,*--,n)rendering the

main condition unfulfillable. Indeed,that now requiresa< > ft for alln n _^ _^i = 1, , n, hence a > ft. But as a , areimputations,both

t-i t-isidesvanish, and this is a contradiction.

Thus thosep for which the necessityof S is in doubt, arerestrictedtop ?*0,1,n 1,n, i.e.to the interval

(31:8) 2 g p g n - 2.This interval plays a role only when n ^ 4. The situation discussedissimilar to that at the end of 27.5.2.and in 27.5.3.,and the casen = 3appearsoncemore as oneof particularsimplicity.

31.2.TheSystem of All Imputations. One-elementSolutions

31.2.1.We now discussthe structureof the systemof all imputations.(31:1) Foran inessentialgamethereexistspreciselyoneimputation:)))

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278 GENERALTHEORY:ZERO-SUMn-PERSONS

(31:9) a = {i, -,), <* = v((i)) for t = 1, , n.Foran essentialgame thereexistinfinitely many imputations

an (n l)-dimensionalcontinuum but (31:9)is not oneofthem.

Proof:Consideran imputation

7 = {01,- ,/M,and put

ft = v((0) + * for i = 1, , n.

Then the characteristicconditions(30:1),(30:2)of 30.1.1.become(31:10) i ^ for i = 1, , n.

(31:11) - - v ((*))-i-i *-in

If T is inessential,then (27:B)in 27.4.1.gives - v((i)) = 0;sot'-i

(31:10),(31:11)amount to ci = = *n = 0,i.e.(31:9)is the uniqueimputation.

n

If T is essential,the (27:B)in 27.4.1.gives - v((t)) >0,so (31:10),t-i(31:11)possessinfinitely many solutions,which form an (n l)-dimensionalcontinuum; 1sothe sameis true for the imputations ft . But the a. of (31:9)is not oneof them, becauseei = = en = now violate (31:11).

An immediateconsequence:(31:J) A solution V is never empty.

Proof:I.e.the empty set is not a solution. Indeed:Considerany

imputation ft , thereexistsat leastoneby (31:1). ft is not in and no

a in has a H ft . So violates (30:5:b)in 30.1.1.231.2.2.We have pointedout before that the simultaneousvalidity of

(31:12) ~2 *- 7, 7 ** \"^

is by no meansimpossible.3 However:

(31:K) Never a H a.1Thereis only one equation: (31:11).1This argument may seempedantic ; but if the conditions for the imputations con-

flicted (i.e.without (31:1)),then V Qwould bea solution.3ThesetsSof thesetwo dominations would have to be disjunct. By (31:H)these

Smust have ^ 2elements each. Hence(31:12)can occuronly when n ^ 4.By a more detailed consideration even n 4 can beexcluded;but for every n 5

(31:12)is really possible.)))

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FIRSTCONSEQUENCES 279

Proof:(30:4:a),(30:4:c)in 30.1.1.conflict for to = ~p.

(31:L) Given an essentialgame and an imputation a, thereexistsan imputation ft such that ft H a but not a H ft. 1

Proof:Puta = {,, , }.

Considerthe equation(31:13) <* = V((0).Sincethe game is essential,(31:1)excludesthe propositionthat (31:13)be

valid for all i = 1, , n. Let (31:13)fail, say forf = i . Sincea isanimputation, soa S v((i )), hencethe failure of (31:13)me&nsa o > v((i )),i.e.(31:14) <. = v((i )) + e, a >0.Now define a vector))

byft o

= aio- * = v((i )),

ft = a H r for i ^ t' .n 1

n n

Theseequationsmakeit clearthat ft ^ v((i))2 and that ft = ^ a = O.8i-l -lSo is an imputation along with a .

We now prove the two assertionsconcerninga , ft .ft H a :We have ft > a for all i ^ io, i.e.for all i in the setS = (I'D).

>

This sethas n 1elementsand it fulfills the main condition (for ft , a ),hence(31:H)gives H a .

Not a H ft :Assumethat a H . Then asetS fulfilling themain con-dition must exist,which is not excludedby (31:H). So S must haveg: 2 elements.Soan i ^ i'o in S must exist. The former impliesft > a

>1Hence a ?* .1For t - to, we have actually 0, v((*' )). For i 5* i , we have 0, > a v((i)).

n n

3 X ft \"* 2 a* kecausethe difference of ft and a.iscfor onevalue of i (i t* ) and

-i t-ifor n 1values of i (all i ** io).))

n -)))

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280 GENERAL THEORY:ZERO-SUMn-PERSONS

(by the constructionof ft ) ; the latterimpliesa > ft (owing to the main

condition) and this is a contradiction.31.2.3.We can draw the conclusionsin which we wereprimarily inter-

ested:-

(31:M) An imputation a , for which never a 'H a , existsif andonly if the gameis inessential.1

Proof:Sufficiency:If the game is inessential,then it possessesby (31:1)preciselyone imputation a , and this has the desiredpropertyby (31:K).

Necessity:If the game isessential,and a. isan imputation, then a ' =

of (31:L)gives a' = ft H a.(31:N) A game which possessesa one-elementsolution2 is necessarily

inessential.

Proof:Denotethe one-elementsolutionin questionby V = ( a ). ThisV must satisfy (30:5:b)in 30.1.1.This meansunder our present circum-

stances:Every ft otherthan a is dominatedby a. I.e.:> > >

ft j a implies a H ft .Now if the game is essential,then (31:L)providesa ft which violates this

condition.

(31:0) An inessentialgame possessespreciselyonesolutionV. This

is the one-elementsetV = ( a ) with the a of (31:1).Proof:By (31:1)thereexistspreciselyone imputation, the a of (31:1).

A solution V cannot be empty by (31:J);hencethe only possibilityis

V = ( a ). Now V = ( a ) isindeeda solution,i.e.it fulfills (30:5:a),(30:5:b)in 30.1.1.:the former by (31:K),the latterbecausea is the only imputationby (31:1).

We cannow answercompletelythe first questionof 30.4.1.:(31:P) A gamepossessesa one-elementsolution(cf.footnote 2above)

if and only if it is inessential;and then it possessesno othersolutions.

Proof:This is just a combination of the resultsof (31:N)and (31:0).1Cf.the considerations of (30:A:a)in 30.2.2.,and particularly footnote 2 on p. 265.1We do not excludethe possibility that this game may possessother solutions aswell,

which may or may not be one-elementsets. Actually this never happens (under ourpresent hypothesis), as the combination of the result of (31:N)with that of (31:0) orthe result of (31:P) shows. But the present consideration is independent of all this.)))

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FIRSTCONSEQUENCES 281

31.3.TheIsomorphism Which Correspondsto StrategicEquivalence

31.3.1.Considertwo gamesF and F' with the characteristicfunctionsv(S) and v'(S) which are strategically equivalent in the senseof 27.1.We proposeto prove that they are really equivalent from the point of viewof the conceptsdefined in 30.1.1.This will be done by establishinganisomorphiccorrespondencebetweenthe entitieswhich form the substratumof the definitions of 30.1.1.,i.e.the imputations. That is, we wish toestablisha one-to-onecorrespondence,between the imputations of F andthose of F',which is isomorphicwith respectto those concepts,i.e.whichcarrieseffective sets,domination, and solutionsfor F into those for F'.

The considerationsare merely an exactelaboration of the heuristicindicationsof 27.1.1.,hencethe readermay find them unnecessary. How-ever, they give quite an instructive instance of an \" isomorphismproof,\"

and, besides,our previous remarks on the relationshipof verbal and ofexactproofs may be appliedagain.31.3.2.Letthe strategicequivalencebegivenby ai, , ajin the sense

of (27:1),(27:2)in 27.1.1.Considerall imputations a = \\a\\ 9 , an )

of F and all imputations a ' = {a'lf , a^jofF'. We lookfor a one-to-one correspondence(31:15) ~^<=~^'with the specifiedproperties.

What (31:15)ought to be is easilyguessedfrom the motivation at thebeginning of 27.1.1.We describedtherethe passagefrom F to F'by addingto the game a fixed payment of a to the playerk. Applying this principleto the imputations means

(31:16) a'k = ak + ak for * =!,-,n.1

Accordingly we define the correspondence(31:15)by the equations(31:16).31.3.3.We now verify the assertedpropertiesof (31:15),(31:16).Theimputationsof F aremappedon the imputationsof F':This means

by (30:1),(30:2)in 30.1.1.,that

(31:17) a ^ v((t)) for i - 1, - - . n.n

(31:18) , = 0,'-!go over into

(31:17*) a( ^ v'((i)) for i = 1, , n,

(31:18*) 2:-0.t-1l lf we introduce the (fixed) vector a - lJ, , i| then (31:16)may be

> >

written vectorially a ' a + a . I.e.it is a translation (by a ) in the vectorspaceof the imputations.)))

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282 GENERAL THEORY:ZERO-SUMn-PERSONS

This is so for (31:17),(31:17*)becausev'((t)) = v((t))+ ? (by (27:2)inn

27.1.1.),and for (31:18),(31:18*)becauseJ) a? = (by (27:1)id.).t-1Effectivity for F goesover into effectivity for F':This meansby (30:3)

in 30.1.1.,that

2) ^ v(S)

goesover into

2 J s vw.))tin))

Thisbecomesevident by comparisonof (31:16)with (27:2).Domination for F goesover into domination for F':Thismeansthe same

thing for (30:4:a)-(30:4:c)in 30.1.1.(30:4:a)is trivial; (30:4:b)is effec-tivity, which we settled:(30:4:c)assertsthat a > ft goesover into at- > 0Jwhich is obvious. Thesolutionsof F aremapped on the solutionsof F':This meansthe same for (30:5:a),(30:5:b)(or (30:5:c))in 30.1.1.Theseconditionsinvolve only domination, which we settled.

We restatetheseresults:

(31:Q) If two zero-sumgamesF and F'arestrategicallyequivalent,then thereexistsan isomorphismbetween their imputationsi.e.a one-to-onemapping of thoseof F on those of F' whichleavesthe conceptsdefined in 30.1.1.invariant.

32.Determinationof all Solutionsof the EssentialZero-sumThree-personGame

$2.1.Formulation of the Mathematical Problem. TheGraphical Method

32.1.1.We now turn to the secondproblem formulated in 30.4.1.:Thedeterminationof all solutionsfor the essentialzero-sumthree-persongames.

We know that we may considerthis game in the reducedform and thatwe can choose7 = I.1 Thecharacteristicfunction in thiscaseis completelydeterminedas we have discussedbefore:2))

(32:1) v(fl) =))-1

1))

when Shas))

An imputation is a vector))

elements.))

a =))

1Cf.the discussion at the beginning of 29.1.,or the referencesthere given: the end of27.1.and the secondremark in 27.3.1Cf.the discussion at the beginning of 20.1.,or the secondcaseof 27.5.)))

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ALL SOLUTIONSOF THETHREE-PERSONGAME 283

whosethreecomponentsmust fulfill (30:1),(30:2)in 30.1.1.*Thesecon-ditionsnow become(considering(30:1))(32:2) ai -1, a, ^ -1, 8 -1,(32:3) on + a* + 3 = 0.We know, from (31:1)in 31.2.1.,that theseai,<*2, a3 form only a two-dimensionalcontinuum i.e.that they shouldberepresentablein the plane.Indeed,(32:3)makesa very simpleplanerepresentationpossible.))

Figure 52.))

a,- -I))

a* -))

7))Figure 53.

32.1.2.Forthis purposewe takethree axesin the plane,making anglesof 60 with eachother. Forany point of the plane we define on, 2, asby directedperpendiculardistancesfrom these three axes. The wholearrangement,and in particular the signs to beascribedto the <*i, as, as)))

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284)) GENERAL THEORY:ZERO-SUMn-PERSONS))

aregiven in Figure52. It iseasy to verify that for any point thealgebraicsum of thesethree perpendiculardistancesvanishesand that conversely

any triplet a = {i, 2, 8 } for which the sum vanishes,correspondsto apoint.

Sothe planerepresentationof Figure52 expressespreciselythe condition(32:3).The remaining condition (32:2)is therefore the equivalent of a

restrictionimposedupon the point a within the planeof Figure52. Thisrestrictionis obviously that it must lie on or within the triangle formedby the threelines i = 1, 2 = 1,a = 1. Figure53 illustratesthis.

Thus the shadedarea,to be calledthe fundamental triangle, representsthe a which fulfill (32:2),(32:3) i.e.all imputations.32.1.3.We next expressthe relationshipof domination in this graphicalrepresentation.As n = 3,we know from (31:H) (cf. also the discussionof))

Figure 54.)) Figure 55.))

(31:8)at the end of 31.1.5.)that among the subsets 8 of / = (1,2, 3)those of two elementsare certainly necessary, and all others certainlyunnecessary. I.e.,the setswhich we must considerin our determinationof all solutionsV arepreciselythese:))

Thus for

domination

means that))

(1,2);(1,3);(2,3).

a = {ai,a2, as}, ft = {ft,ft, ft}

a H B))

(32:4) Either i > ft, a2 > ft; or ai > fa, a* > ft; or 2 >ft,as >ft.

Diagrammatically: a dominatesthe points in the shaded areas,and noothers,1in Figure54.))

1In particular, no points on the boundary lines of theseareas.)))

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ALL SOLUTIONSOF THETHREE-PERSONGAME 285

>Thus the point a dominatesthree of the six sextants indicated in

Figure55 (namely A, C, B). From this one concludeseasily that a isdominatedby the three other sextants (namely B,D, F). So the only

points which do not dominate a and arenot dominatedby it, lieon thethreelines(i.e.sixhalf-lines) which separatethesesextants.I.e.:

(32:5) If neither of a. , ft dominatesthe other,then the direction>

from a. to ft is parallel to one of the sidesof the fundamentaltriangle.

32.1.4.Now the systematicsearchfor all solutionscan begin.Considera solution V, i.e.a set in the fundamental triangle which

fulfills the conditions(30:5:a),(30:5:b)of 30.1.1.In what followswe shalluse theseconditionscurrently, without referring to them explicitlyon eachoccasion.

Sincethe game is essential,V must contain at leasttwo points1 say

a and ft . By (32:5)the directionfrom a to ft is parallel to one of thesidesof the fundamental triangle;and by a permutation of the numbersof theplayers1,2,3we can arrange this to be the side i = 1,i.e.the horizontal.

*So a , ft lie on a horizontal line I. Now two possibilitiesariseand we treatthem separately:

(a) Every point of V lieson I.(b) Somepointsof V do not lie on L

32.2.Determination of All Solutions

32.2.1.We consider(b) first. Any point not on I must fulfill (32:5)with>

respectto both a and ft , i.e.it must be the third vertex of one of the two>

equilateraltriangleswith the base a , ft :one of the two points a ', a \" of>

Figure56. Soeithera 'or a \" belongsto V. Any point of V which differs

from a, ft and a 'ora \" must again fulfill (32:5),but now with respectto all>

threepoints a , ft and a 'or a \". This, however, is impossible,as aninspectionof Figure56 immediately reveals. SoV consistsofpreciselythesethreepoints, i.e.of the threeverticesof a triangle which is in the positionof triangle / or triangle // of Figure57. Comparisonof Figure57 with

Figures54 or 55 showsthat the verticesof triangle / leave the interior ofthis triangle uridominated. Thisrulesout I.2

1 This is alsodirectly observablein Fig. 54.2This provides the example referred to in 30.3.6.:Thethree verticesof triangle / do

not dominate eachother, i.e.they form a satisfactory set in the senseofloc.cit. Theyarenevertheless unsuitable asa subset of a solution.)))

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286 GENERAL THEORY:ZERO-SUMn-PERSONS))

Figure 57.))

Fundamental Triangle -

Figure 58.))

danienUl Triangle -

Figure 59.)))

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ALL SOLUTIONSOFTHETHREE-PERSONGAME 287

The same comparisonshows that the vertices of triangle // leaveundominatedthe dotted areasindicated in Figure58. Hencetriangle //must beplacedin such a manner in the fundamental triangle that thesedotted areasfall entirely outside the fundamental triangle. This meansthat the threeverticesof II must lieon the threesidesof the fundamentaltriangle,as indicated in Figure59. Thus these three vertices are themiddlepointsof the threesidesof the fundamental triangle.

Comparisonof Figure59 with Figure54 or Figure55 showsthat thisset V is indeed a solution. One verifies immediatelythat thesethreemiddlepointsarethe points (vectors)

(32:6) {-!,*,*},{*,-!,41,It,*,-1|,i.e.that this solution V is the setof Figure51.))

Fundament*!

Triangle))

Figure 60.

32.2.2.Let us now consider(a)in 32.1.4.In this caseall of V lieson thehorizontal line L By (32:5)no two pointsof I dominate eachother, so thatevery point of / isundominatedby V. Henceevery point of I (in the funda-mental triangle)must belong to V. I.e.,V is preciselythat part of /

which is in the fundamental triangle. So the elementsa = {<*i, 2, s}of V arecharacterizedby an equation

(32:7) i c.

Diagrammatically:Figure60.Comparisonof Figure60 with Figures54 or 55 showsthat the line I

leavesthe dotted areaindicatedon Figure60 undominated. HencethelineI must be placedin such a manner in the fundamental triangle that thedotted areafalls entirely outside the fundamental triangle. This meansthat I must lie belowthemiddlepointsof thosetwo sidesof the fundamental)))

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288 GENERAL THEORY:ZERO-SUMn-PERSONS

trianglewhich it intersects.1 In the terminology of (32:7):c <|.On theother hand, c ^ 1 is necessary to make I intersectthe fundamentaltriangleat all. Sowe have:

(32:8) -1S c <iComparisonof Figure60 with Figures54 or 55 showsthat under these

conditions2 the setV i.e.I is indeeda solution.But the form (32:7)of this solutionwas brought about by a suitable

permutation of the numbers1,2,3.Hencewe have two further solutions,characterizedby

(32:7*) a,= c,

and characterizedby

(32:7**) 3 = c,

alwayswith (32:8)32.2.3.Summingup:This is a completelist of solutions:

(32:A) For every c which fulfills (32:8):The three sets (32:7)(32:7*),(32:7**).

(32:B) Theset(32:6).33.Conclusions

33.1.TheMultiplicity of Solutions. Discrimination and Its Meaning

33.1.1.The result of 32.calls for careful considerationand comment.We have determinedall solutionsof the essentialzero-sumthree-persongame. In 29.1.,before the rigorousdefinitions of 30.1.were formulated,we had already decidedwhich solutionwe wanted; and this solution reap-pearsnow as (32:B).But we found other solutionsbesides:the (32:A),which areinfinitely many sets,eachoneof them an infinite setof imputa-tions itself. What do thesesupernumerarysolutionsstand for?

Consider,e.g.,the form (32:7)of (32:A). This gives a solution for

every c of (32:8)consistingof all imputationsa = {i, 2, 3} which fulfill

(32:7),i.e. i = c. Besidesthis, they must fulfill only the requirements,

1The limiting position of J, going through the middle points themselves, must beexcluded. Thereason is that in this position the vertex of the dotted areawould lie onthe fundamental triangle, and this is inadmissable sincethat point too is undominatedby V, i.e.by I.

Observethat the corresponding prohibition did not occurin case (b), i.e.for thedotted areasof Figure 58. Their verticestoo wereundominated by V, but they belongto V. In our present position of V, on the other hand, the vertex under considerationdoesnot belong to V, i.e.to I.

This exclusion of the limiting position causesthe < and not the ^ in theinequality which follows.

* (32:8),i.e.I intersectsthe fundamental triangle, but below its middle.)))

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CONCLUSIONS 289

(30:1),(30:2)of 30.1.1.i.e.(32:2),(32:3)of 32.1.1.In other words:Our solution consistsof all

(33:1) \"^ = {c,a, -c-a}, -1g a 1- c.The interpretation of this solution consists manifestly of this:One

of the players(in thiscase1)isbeingdiscriminatedagainstby the two others(in this case2,3).They assignto him the amount which he gets,c. Thisamount is the samefor all imputationsof the solution,i.e.of the acceptedstandard of behavior. Theplacein societyof player 1is prescribedby thetwo otherplayers;he is excludedfrom all negotiationsthat may lead tocoalitions.Suchnegotiationsdo go on, however, between the two otherplayers: the distribution of their share, c, depends entirely upon theirbargainingabilities.Thesolution,i.e.the acceptedstandard of behavior,imposesabsolutely no restrictionsupon the way in which this share isdividedbetweenthem, expressedby a, c a.1 This is not surprising.Sincethe excludedplayer is absolutely\"tabu,\" the threat of the partner'sdesertionis removed from eachparticipant of the coalition.Thereis noway of determiningany definite division of the spoils.2'3

Incidentally:It is quite instructive to seehow our conceptof a solutionas a setof imputationsis able to take careof this situationalso.

33.1.2.Thereis more that shouldbe said about this \"discrimination\"againsta player.

First,it is not done in an entirely arbitrary manner. Thec, in whichdiscriminationfinds its quantitative expression,is restrictedby (32:8)in32.2.2.Now part of (32:8),c ^ 1,is clearenough in its meaning, but thesignificance of the other part c < 4 is considerablymore recondite(cf.however, below). It all comesback to this:Even an arbitrary system ofdiscriminationscan be compatiblewith a stable standard of behavior i.e.orderof society but it may have to fulfill certainquantitative conditions,in orderthat it may not impair that stability.

Second,the discriminationneed not beclearlydisadvantageousto theplayer who is affected. It cannot be clearlyadvantageous, i.e.his fixedvalue c cannot be equal to or betterthan the bestthe others may expect.This would mean,by (33:1),that c ^ 1 c, i.e.c ^ i, which is exactlywhat (32:8)forbids. But it would be clearly disadvantageousonly forc = 1; and this is a possiblevalue for c (by (32:8)),but not the only one.c = 1means that the player is not only excluded,but also exploitedto

1Exceptthat both must be <z 1 i.e.what the player can get for himself, withoutany outsidehelp.

a, c & ^ 1 is, of course,the 1 ^ a ^ 1 cof (33:1).2 Cf. the discussions at the end of 25.2.Observethat the arguments which we

adducedthere to motivate the primate of v(S)have ceasedto operatein this particularcase and v(S)nevertheless determines the solutions!

8 Observethat due to (32:8)in 32.2.2.,the \"spoils\", i.e.the amount c,can beboth

positive and negative.4 And that is excludedin c < J, but not in c ^ 1.)))

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290 GENERAL THEORY:ZERO-SUMn-PERSONS

100per cent. Theremaining c (of (32:8))with 1< c <i correspondtogradually lessand lessdisadvantageousforms of segregation.

33.1.3.It seemsremarkablethat our conceptof a solutionis able toexpressall thesenuancesof non-discriminatory(32:B),and discriminatory(32:A), standardsof behavior the latterboth in their 100percentinjuriousform, c = 1,and in a continuousfamily of lessand lessinjurious ones

1< c <i. It is particularly significant that we did not look for anysuch thing the heuristicdiscussionsof 29.1werecertainly not in this spirit

but wewereneverthelessforcedto theseconclusionsby the rigoroustheoryitself. And thesesituationsaroseeven in the extremelysimpleframeworkof the zero-sumthree-persongame!

Forn ^ 4 we must expecta much greaterwealth of possibilitiesforall sorts of schemesof discrimination,prejudices,privileges,etc. Besidesthese,we must always lookout for theanaloguesof the solution (32:B),i.e.the nondiscriminating \" objective\" solutions. But we shallseethat the con-ditionsarefar from simple.And we shall alsoseethat it is preciselytheinvestigation of the discriminatory\"inobjective\"solutionswhich leadsto aproper understandingof the generalnon-zero-sumgames and thencetoapplicationto economics.

33.2.Staticsand Dynamics

33.2.At this point it may beadvantageousto recallthe discussionsof4.8.2.concerningstaticsand dynamics. What we said then appliesnow;indeedit was really meant for the phasewhich our theory has now reached.

In 29.2.and in the placesreferred to there,we consideredthe nego-tiations, expectationsand fears which precedethe formation of a coalitionand which determineits conditions.Thesewereall of the quasi-dynamictype describedin 4.8.2.The same applies to our discussionin 4.6.andagain in 30.2.,of how various imputationsmay or may not dominateeachotherdependingon their relationshipto a solution;i.e.,how the conductsapprovedby an establishedstandard of behavior do not conflict with eachother,but can be used to discreditthe non-approvedvarieties.

Theexcuse,and the necessity,for using such considerationsin a statictheory weresetforth on that occasion.Thus it is not necessaryto repeatthem now.)))

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34.Preliminary Survey34.1.GeneralViewpoints

34.1.We are now in possessionof a generaltheory of the zero-sumn-persongame,but the stateof our information is still far from satisfactory.Save for the formal statement of the definitions we have penetratedbutlittle belowthe surface. The applicationswhich we have made i.e.thespecialcasesin which we have succeededin determiningour solutions canbe rated only as providing a preliminary orientation. As pointed out in30.4.2.,theseapplicationscover all casesn ^ 3,but we know from our pastdiscussionshow little this is in comparisonwith the generalproblem. Thuswe must turn to gameswith n ^ 4 and it isonly herethat the full complexityof the interplay of coalitionscan be expectedto emerge.A deeperinsightinto the nature of our problemswill beachievedonly when wehave masteredthe mechanismswhich govern thesephenomena.

The present chapteris devoted to zero-sum four-persongames.Ourinformation about these still presents many lacunae. This compelsaninexhaustive and chieflycasuistictreatment,with its obvious shortcomings.1

But even this imperfect expositionwill disclosevarious essentialqualitativepropertiesof the generaltheory which couldnot be encounteredpreviously,(for n g 3). Indeed,it will be noted that the interpretationof the mathe-matical results of this phase leads quite naturally to specific\"social\"conceptsand formulations.

34.2.Formalism of the Essential Zero -sum Four-personGame34.2.1.In orderto acquirean idea of the nature of the zero-sum four-

persongameswe beginwith a purely descriptiveclassification.Lettherefore an arbitrary zero-sum four-persongameF be given, which

we may as well considerin its reducedform: and alsoletus choosey = I.2Theseassertionscorrespond,as we know from (27:7*)and (27:7**)in 27.2.,to the following statementsconcerningthe characteristicfunctions:))

(34:1) v(5) =))-1

1 when S has)) elements.))

1 E.g.,a considerableemphasis on heuristic devices.2 Cf.27.1.4.and 27.3.2.The readerwill note the analogy between this discussion

and that of 29.1.2.concerning the zero-sum three-persongame. About this more willbesaid later.

291)))

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292 ZERO-SUMFOUR-PERSONGAMES

Thus only the v(/S) of the two-elementsetsS remain undeterminedbythesenormalizations. We therefore directour attention to thesesets.

ThesetI = (1,2,3,4)of all playerspossessessix two-elementsubsetsS:(1,2),(1,3),(1,4),(2,3),(2,4),(3,4).

Now the v(S) of thesesetscannot be treatedas independentvariables,becauseeachone of theseS has another oneof the same sequenceas itscomplement.Specifically:the first and the last, the secondand the fifth,the third and the fourth, are complementsof each other respectively.Hencetheir v(S)arethe negativesof eachother. It is also to be remem-beredthat by the inequality (27:7)in 27.2.(with n = 4, p = 2) all thesev(S)are^2, ^ 2. Henceif we put))

)) = 2xi,(34:2) v((2,4))= 2x2,

1 v((3,4))= 2x3,then we have

f v((2,3))= -2*!,(34:3) v((l,3)) = -2x2,

lv((l,2))= -2*3,and in addition

(34:4) -1 xi,x2, x, ^ 1.Conversely:If any threenumbersXi, x2, x3 fulfilling (34:4)aregiven,

then we can define a function v(S) (for all subsetsS of I = (1,2,3,4))by(34:l)-(34:3),but we must show that this v(S)is the characteristicfunctionof a game. By 26.1.this meansthat our presentv(S)fulfills the conditions(25:3:a)-(25:3:c)of 25.3.1.Of these,(25:3:a)and (25:3:b)areobviouslyfulfilled, so only (25:3:c)remains. By 25.4.2.this meansshowing that

v(Si)+ v(S2) + v(S3) ^ if Si,S2, Ssarea decompositionof /.(Cf. also (25:6)in 25.4.1.)If any of the setsSi,S2, S8 is empty, the twoothersarecomplementsand so we even have equalityby (25:3:a),(25:3:b)in 25.3.1.So we may assume that none of the setsSi,S2, S8 is empty.Sincefour elementsareavailable altogether,oneof thesesets,say Si = S,must have two elements,while the two othersareone-elementsets.Thusour inequality becomes

v(S)-2^0, i.e. v(S) ^ 2.If we expressthis for all two-elementsetsS,then (34:2),(34:3)transformthe inequality into

2*ig 2, 2x2 ^ 2, 2x3 ^ 2,-2*!g 2, -2*,g 2, -2x,g 2,

which isequivalent totheassumed(34:4). Thuswe have demonstrated:)))

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PRELIMINARY SURVEY)) 293))

(34:A) Theessentialzero-sumfour-persongames (in their reducedform with the choice7 = 1) correspondexactly to the tripletsof numbers xi, a?2, z8 fulfilling the inequalities (34:4). Thecorrespondencebetween such a game,i.e.its characteristicfunction, and its x\\, z2, x* isgiven by the equations(34:l)-(34:3).1

34.2.2.Theabove representationof the essentialzero-sumfour-persongamesby triplets of numbersx\\, z2, x8 canbe illustratedby a simplegeo-metrical picture. We can view the numbers x\\, x$, x3 as the Cartesiancoordinatesof a point.2 In this casethe inequalities(34:4)describea part))

Figure 61.of spacewhich exactly fills a cubeQ. This cube is centeredat the originof the coordinates,and its edgesareof length 2 becauseits sixfacesarethesixplanes

Xi = 1, Z 2 = 1, X3 = 1,as shown in Figure61.

Thus each essentialzero-sum four-persongame F is representedbypreciselyone point in the interior or on the surface of this cube,and viceversa. It is quite useful to view thesegamesin this manner and to try tocorrelatetheir peculiaritieswith the geometricalconditions in Q. Itwill be particularlyinstructive to identify thosegameswhich correspondtodefinite significant pointsof Q.

1 Thereadermay now compareour result with that of29.1.2.concerning the zero-sumthree-persongames. It will benoted how the variety of possibilities has increased.

2 We may alsoconsider these numbers as the components of a vector in L$ in thesenseof 16.1.2.et seq. This aspectwill sometimes be the more convenient, as in foot-note 1 on p. 304.)))

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294 ZERO-SUMFOUR-PERSONGAMES

But even before carryitig out this program, we propose to considercertain questions of symmetry. We want to uncover the connectionsbetweenthe permutationsof the players1,2,3,4,and the geometricaltrans-formations (motions)of Q. Indeed:by 28.1.the former correspondto thesymmetriesof the gameT, while the latterobviously expressthe symmetriesof the geometricalobject.

34.3.Permutations of the Players34.3.1.In evolving the geometricalrepresentationof the essential

zero-sumfour-persongame we had to perform an arbitrary operation,i.e.one which destroyed part of the symmetry of the original situation.Indeed,in describingthe v(S)of the two-elementsetsS,we had to singleout three from among thesesets (which aresix in number), in order tointroducethe coordinatesx\\ y x2, x3. We actually did this in (34:2),(34:3)by assigningthe player4 a particularrole and then settingup a correspond-encebetween the players 1,2,3and the quantities Xi, x2, x3 respectively(cf.(34:2)). Thus a permutationof the players1,2,3will inducethe samepermutation of the coordinatesXi, x2, x3 and so far the arrangement issymmetric. But theseareonly six permutationsfrom among the total of24 permutationsof the players1,2,3,4.*Soa permutation which replacesthe player4 by another one is not accountedfor in this way.

34.3.2.Let us considersuch a permutation. For reasons which will

appear immediately,considerthe permutation A, which interchangestheplayers1and 4 with eachother and alsothe players2 and3.2 A lookat theequations (34:2),(34:3)suffices to show that this permutation leaves Xi

invariant, while it replacesx2, x3 by x2, x3. Similarlyone verifies:Thepermutation 5,which interchanges2 and 4, and also 1 and 3, leavesx2

invariant and replacesXi, x3 by Xi, x3. The permutation C, which

interchanges3 and 4 and also1and 2, leavesxs invariant and replacesXi, x2

by -Xi,-x2.Thus eachone of the threepermutationsA, B,C affects the variables

Xi, x2, x8 only as far as their signsareconcerned,eachchanging two signsand leaving the third invariant.

Sincethey alsocarry4 into 1,2,3,respectively,they produceall permuta-tions of the players 1,2,3,4,if combinedwith the six permutationsof theplayers 1,2,3.Now we have seen that the lattercorrespondto the sixpermutations of Xi, x2, x3 (without changesin sign). Consequentlythe24 permutationsof 1,2,3,4correspondto the six permutationsof Xi, x2, x3,eachone in conjunctionwith no changeof sign or with two changesofsign.8

1 Cf.28.1.1.,following the definitions (28:A:a),(28:A:b).2 With the notations of 29.1.:

1,2,3,4\\ B _ /1,2,3,4\\ _ /1,2,3,4B ~ c))

3Thesesign changesare 1 + 3 4 possibilities in eachcase,sowe have 6 X 4 24operationson Xi, x^x\\ to representthe 24permutations of 1,2,3,4, as it should be.)))

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SPECIAL POINTSIN THECUBEQ)) 295))

34.3.3.We may alsostatethis as follows:If we considerall movementsin spacewhich carry the cube Q into itself, it is easilyverified that theyconsistof the permutationsof the coordinateaxesxi,x2, x* in combinationwith any reflections on the coordinateplanes(i.e.the planesx*, x9 ; x\\ t Xf ;#1,xj). Mathematicallythesearethe permutationsof x\\ y X*, x\\ in combina-tion with any changesof signamong thexi,X2, x8. Theseare48 possibili-ties.1 Only half of these,the 24 for which the number of sign changesiseven (i.e.or 2),correspondto the permutationsof the players.))

IV))

Figure 62.))

It is easilyverified that thesearepreciselythe 24 which not only carrythe cubeQ into itself, but alsothe tetrahedronI,V, VI, VII,as indicatedin

Figure62. Onemay alsocharacterizesucha movement by observingthatit always carriesa vertex of Qinto a vertex ; and equallya vertex ointo avertex o,but never a into a o.2

We shall now obtain a much more immediate interpretation of thesestatementsby describingdirectly the gameswhich correspondto specificpointsof the cubeQ:to thevertices or o,to thecenter(the origin in Figure61),and to the main diagonalsof Q.

35.Discussionof SomeSpecialPointsin the CubeQ35.1.TheCorner/ (and 7, VI, VII)

35.1.1.We begin by determiningthe gameswhich correspondto thefour corners :7, V, VI,VII. We have seenthat they arisefrom eachotherby suitable permutations of the players 1,2,3,4.Therefore it suffices toconsideroneof them,say /.

1For eachvariable *i, xtt x\\ there are two possibilities:changeor no change. Thisgivesaltogether 21- 8possibilities. Combination with the sixpermutations of Xi, t, x*yields 8 X 6 48operations.*This group of motions is well known in group theory and particularly in crystallog-raphy, but we shall not elaboratethe point further.)))

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296 ZERO-SUMFOUR-PERSONGAMES

Thepoint 7 correspondsto the values 1,1,1of the coordinatesx\\, xz,Thus the characteristicfunction v(>S) of this game is:))

(35:1)v(S)))

-12))

-21

when S has))

12 (and 4 belongsto S)

elements2 (and 4 doesnot belongto S)34))

(Verification is immediatewith the helpof (34:1),(34:2),(34:3)in 34.2.1.)Insteadof applyingthe mathematical theory of ChapterVI to this game,letus first seewhether it doesnot allow an immediate intuitive interpretation.

Observefirst that a player who is left to himself losesthe amount 1.This is manifestly the worst thing that can ever happen to him sincehecan protect himself against further losseswithout anybody else'shelp.1Thus we may considera player who gets this amount 1 as completelydefeated.A coalition of two pla3rers may be consideredas defeated if it

getsthe amount 2, sincethen eachplayerin it must necessarilyget I.2- 3

In this game the coalition of any two players is defeatedin this senseif itdoesnot compriseplayer4.

Let us now pass to the complementarysets.If a coalition is defeatedin the above sense,it is reasonableto consider the complementarysetasa winning coalition. Therefore the two-elementsets which contain theplayer 4 must be rated as winning coalitions.Also sinceany player whoremainsisolatedmust be rated as defeated,three-personcoalitionsalwayswin. This is immaterial for those three-elementcoalitionswhich containthe player 4, sincein thesecoalitionstwo membersare winning alreadyif the player 4 is among them. But it is essentialthat 1,2,3be a winning

coalition,sinceall its propersubsetsaredefeated.4

1This view of the matter is corroboratedby our results concerning the three-persongame in 23.and 32.2.,and more fundamentally by our definition of the imputation in

30.1.1.,particularly condition (30:1).2Sinceneither he nor his partner need acceptlessthan 1,and they have together

2, this is the only way in which they can split.8 In the terminology of 31.1.4.:this coalition is flat. Thereis of courseno gain, and

therefore no possiblemotive for two players to form such a coalition. But if it happensthat the two other players have combined and show no desireto acquirea third ally, wemay treat the remaining two as a coalition even in this case.

4 We warn the readerthat, although we have used the words \" defeated\"and \" win-

ning\" almost as termini technici, this is not our intention. Theseconceptsare, indeed,very well suited for an exact treatment. The \"defeated\" and \"winning\" coalitionsactually coincidewith the setsSconsideredin (31:F)and in (31:G)in 31.1.5.;those forwhich S is flat or S is flat, respectively. But we shall considerthis question in sucha way only in Chap.X.

For the moment our considerations are absolutely heuristic and ought to be takenin the samespirit as the heuristic discussionsof the zero-sum three-persongame in 21.,22.Theonly difference is that we shall beconsiderably briefer now, sinceour experienceandroutine have grown substantially in the discussion.

As we now possessan exact theory of solutions for games already, we are under)))

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SPECIALPOINTSIN THECUBE Q 297

35.1.2.So it is plausibleto view this as a strugglefor participation inany one of the various possiblecoalitions:(35:2) (1,4),(2,4),(3,4),(1,2,3),where the amounts obtainablefor thesecoalitionsare:

(35:3) v((l,4))= v((2,4))= v((3,4))= 2, v((l,2,3))= 1.Observe that this is very similar to the situation which we found in

the essentialzero-sum three-persongame, where the winning coalitionswere:

(35:2*) (1,2),(1,3),(2,3),and the amounts obtainablefor thesecoalitions:

(35:3*) v((l,2))= v((l,3))= v((2,3))= 1.In the three-persongame we determinedthe distributionof the pro-

ceeds(35:3*)among the winners by assuming:A player in a winning coali-tion shouldget the sameamount no matter which is the winning coalition.Denoting theseamounts for the players1,2,3by a,$, y respectively,(35:3*)gives

(35:4*) a + = a + 7 =:0+ 7 = lfrom which follows

(35:5*) a = ft = y = iThesewere indeedthe values which thoseconsiderationsyielded.

Let us assume the same principlein our present four-person game.Denoteby a,0,7, 5, respectively,the amount that eachplayer1,2,3,4getsif he succeedsin participatingin a winning coalition. Then (35:3)gives

(35:4) a + 6 = + 6 = 7 + $ = 2, a + + 7 = 1,from which follows

(35:5) = = 7 =i * =iAll the heuristicarguments usedin 21.,22.,for the three-persongame couldbe repeated.1

35.1.3.Summing up:

(35:A) This is a game in which the player4 is in a speciallyfavoredpositionto win:any one ally sufficesfor him to form a winning

coalition. Without his cooperation,on the other hand, threeplayersmust combine. This advantage also expressesitself in

obligation to follow up this preliminary heuristic analysis by an exactanalysis which

is basedrigorously on the mathematical theory. We shall cometo this. (Cf.loc.cit.above, and also the beginning of 36.2.3.)

1 Ofcourse,without making this thereby a rigorous discussion on the basisof 30.1.)))

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298)) ZERO-SUMFOUR-PERSONGAMES))

the amounts which eachplayer 1,2,3,4should get when he isamong the winners if our above heuristic deduction can betrusted. These amounts are i, |,i,|respectively. It isto benoted that the advantageof player 4 refersto the caseofvictory only;when defeated,all playersarein the sameposition(i.e.get -1).

The last mentionedcircumstanceis, of course,due to our normaliza-tion by reduction. Independentlyof any normalization, however, this gameexhibits the following trait: One player's quantitative advantage overanother may, when both win, differ from what it is when both lose.

This cannot happen in a three-persongame,as is apparent from theformulation which concludes22.3.4.Thus we get a first indicationof animportant new factor that emergeswhen the number of participantsreachesfour.

35.1.4.One last remark seemsappropriate.In this gameplayer 4'sstrategicadvantage consistedin the fact that he neededonly one allyfor victory, whereaswithout him a total of threepartners was necessary.Onemight try to passto an even more extremeform by constructinga gamein which every coalition that doesnot containplayer 4 is defeated.It isessentialto visualize that this is not so,or rather that such an advantageis no longerof a strategicnature. Indeedin such a game))

123))

if 8 has))

hence))

if Shas))

elementsand 4 doesnot belongto S,))

12 elementsand 4 belongsto S.o4))

This is not reduced,as

v((I)) = v((2)) v((3))= -1,)) v((4))= 3.))

If we apply the reductionprocessof 27.1.4.to this v(S) we find that itsreducedform is

0.))

i.e.thegame is inessential. (Thiscouldhave beenshown directlyby (27:B)in 27.4.)Thus this gamehas a uniquely determinedvalue for eachplayer1,2,3,4:1, 1, 1,3,respectively.

In otherwords:Player 4'sadvantage in this gameis one of a fixedpayment (i.e.of cash),and not one of strategicpossibilities.Theformeris,of course,more definite and tangiblethan thelatter,but of lesstheoreticalinterestsinceit can beremovedby our processof reduction.)))

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SPECIAL POINTSIN THECUBEQ 299

35.1.5.We observedat the beginning of this sectionthat the cornersV, VI,VIIdiffer from J only by permutationsof the players. It is easilyverified that the specialroleof player4 in / is enjoyedby the players1,2,3,in V, VI,VII,respectively.

35.2.TheCorner VIII(and //,///,77). TheThree-personGameand a \"Dummy91

35.2.1.We nextconsiderthe gameswhich correspondto thefour cornerso://,///,IV,VIII. As they arisefrom eachotherbysuitablepermutationsof the players1,2,3,4,it sufficesto considerone of them, say VIIL

Thepoint VIIIcorrespondsto the values 1, 1, 1of thecoordinatesxi,x2, x*. Thus the characteristicfunction v(S)of this game is:))

(35:6) v(S) =))

-1-2))

21

when S has))

12 (and 4 belongsto S)elements

2 (and4 doesnot belongto 5)34))

(Verification is immediatewith the help of (34:1),(34:2),(34:3)in 34.2.1.)Again, insteadof applying to this game the mathematical theory of ChapterVI, let us first see whether it does not allow an immediate intuitiveinterpretation.

The important feature of this game is that the inequality (25:3:c)in25.3.becomesan equality,i.e.:(35:7) v(Su T) = v(S)+ v(T) if S n T = 0,when T = (4). That is:If S representsa coalition which doesnot containthe player 4, then the addition of 4 to this coalitionis of no advantage;i.e.it does not affect the strategicsituation of this coalition nor of itsopponents in any way. This is clearly the meaning of the additivityexpressedby (35:7).1

35.2.2.This circumstancesuggests the following conclusion, whichis of coursepurely heuristic.2 Sincethe accessionof player 4 to any

1Note that the indifference in acquiring the cooperationof 4 is expressedby (35:7),and not by

v(SU T) - vGS).

That is, a player is \"indifferent\" asa partner, not if his accessiondoesnot alter the valueof a coalition but if he brings into the coalition exactly the amount which and no morethan he is worth outside.

This remark may seemtrivial; but there existsa certain danger ofmisunderstanding,particularly in non-reduced games where v((4))> 0, i.e.where the accessionof 4(although strategically indifferent!) actually increasesthe value ofa coalition.

Observealsothat the indifference of Sand T (4) to eachother is a strictly recip-rocalrelationship.1We shall later undertake exactdiscussion on the basisof 30.1.At that time it willbe found also that all thesegames are specialcasesof more general classesof someimportance. (Cf.Chap.IX, particularly 41.2.))))

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300 ZERO-SUMFOUR-PERSONGAMES

coalition appears to he a matter of completeindifference to both sides, itseemsplausibleto assumethat player 4 has no part in the transactionsthatconstitute the strategy of the game. lieis isolatedfrom the others andthe amount which he can getfor himself v(/S) = 1 is the actual valueof the game for him. The other players 1,2,3,on the other hand, playthe gamestrictly among themselves;hencethey are playing a three-person game. The values of the original characteristicfunction v(S)which describesthe original three-persongame are :))

=v((3)) = -1,))(35:6*) v((l,2))= v((l,3)) = v((2,3))= 2,

v((l,2,3))= 1,))

/' = (1,2,3)is now the setof all players.))

(Verify this from (35:6).)At first sight this three-persongame representsthe oddity that v(/')

(/' is now the setof all players!)is not zero. This, however, is perfectlyreasonable:by eliminating player4 we transform the game into one whichis not of zero sum;sincewe assessedplayer4 a value 1,the othersretaintogethera value 1. We do not yet proposeto deal with this situationsystematically. (Cf. footnote 2 on p.299.) It is obvious, however, thatthis condition can be remediedby a slight generalization of the transforma-tion used in 27.1.We modify the game of 1,2,3by assuming that eachone got the amount i in cash in advance,and then compensatingfor thisby deductingequivalent amounts from the v(S) values in (35:6*).Justas in 27.1.,this cannot affect the strategy of the game,i.e.it producesastrategicallyequivalent game.l

After considerationof the compensationsmentionedabove2 we obtainthe new characteristicfunction :

v'(0)= 0,v'((l)) = v'((2))= v'((3))= - *,

(35:6**) v'((l,2)) = v'((l,3)) = v'((2,3))= *,v'((l,2,3))= 0.

This is the reducedform of the essentialzero-sum three-persongamedis-cussedin 32. exceptfor a difference in unit :We have now y = $ instead

1In the terminology of 27.1.1.:a? = al = al |.The condition there which

we have infringed is (27:1): aj = 0. This is necessarysincewe started with a non-t

zero-sum game.Even 2^ai \" could be safeguarded if we included player 4 in our considerations,

tputting aj = 1.This would leave him just as isolated as before, but the necessarycompensation would make v((4)) = 0, with results which areobvious.

Onecan sum this up by saying that in the present situation it is not the reducedform of the game which provides the best basisof discussion among all strategicallyequivalent forms.

2I.e.deduction of asmany times J from v(S) as8 has elements.)))

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SPECIALPOINTSIN THECUBEQ 301of the 7 = 1of (32:1)in 32.1.1.Thus we can apply the heuristicresultsof 23.1.3.,or the exactresults of 32.l Let us restrictourselves,at anyrate,to the solution which appearsin both casesand which is the simplestone:(32:B)of 32.2.3.Thisis the setof imputations(32:6)in 32.2.1.,whichwe must multiply by the presentvalue of y = ; i.e.:

(-*,*,!},ft, -it!,lit,-*}(Theplayersare,of course,1,2,3.)Inotherwords:Theaim of the strategyof the players1,2,3is to form any coalition of two; a playerwho succeedsin this, i.e.who is victorious, getsf , and a playerwho is defeatedgets $.Now eachof the players1,2,3of our original game getsthe extraamounti beyond this, hencethe above amounts f , 1must be replacedby1,-1.

36.2.3.Summingup:(35:B) This is a game in which the player 4 is excludedfrom all

coalitions. The strategicaim of the other players 1,2,3is toform any coalition of two. Player4 gets 1at any rate. Anyotherplayer 1,2,3getsthe amount 1when he is among the win-ners,and the amount 1when he is defeated.All this is basedon heuristicconsiderations.

One might say more conciselythat this four-persongameis only an\"inflated\" three-persongame:the essentialthree-persongameof the players1,2,3,inflated by the additionof a \"dummy\" player4. We shallseelaterthat this conceptis of a more generalsignificance.(Cf. footnote 2 onp.299.)

36.2.4.One might comparethe dummy roleof player 4 in this gamewith the exclusiona playerundergoesin the discriminatorysolution(32:A)in 32.2.3.,as discussedin 33.1.2.Thereis, however, an important differ-encebetween thesetwo phenomena.In our presentset-up,player 4 hasreally no contribution to make to any coalition at all;he stands apart byvirtue of the characteristicfunction v(S). Our heuristic considerationsindicatethat he should be excludedfrom all coalitionsin all acceptablesolutions. We shall see in 46.9.that the exacttheory establishesjustthis. The excludedplayer of a discriminatorysolution in the senseof33.1.2.is excludedonly in the particularsituationunderconsideration.Asfar as the characteristicfunction of that gameis concerned,his roleis thesame as that of all otherplayers. In otherwords:The \"dummy\" in ourpresentgameis excludedby virtue of the objectivefacts of the situation(the characteristicfunction v(S)).2 The excludedplayer in a discrimi-natory solution is excludedsolely by the arbitrary (though stable)\"prejudices\"that the particularstandard of behavior (solution)expresses.

1 Of coursethe present discussion is heuristic in any event. As to the exacttreat-ment, cf. footnote 2 on p. 299.

2This is the \"physical background,'

1 in the senseof 4.6.3.)))

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302 ZERO-SUMFOUR-PERSONGAMES

We observedat the beginning of this sectionthat the corners//, 777,IV differ from VIIIonly by permutationsof the players. It is easilyverified that the specialrole of player4 in VIIIis enjoyedby the players1,2,3in 77,777,77, respectively.

85.3.SomeRemarks Concerning the Interior of Q35.3.1.Let us now considerthe game which correspondsto the centerof

Q, i.e.to the values 0,0,0of the coordinatesx\\ } 2, g. Thisgame isclearlyunaffected by any permutation of the players1,2,3,4,i.e.it is symmetric.Observethat it is the only such game in Q, sincetotal symmetry meansinvariance underall permutationsof x\\, 2, x* and sign changesof any twoof them (cf. 34.3.);henceXi = xa = xs = 0.

Thecharacteristicfunction v(S)of this game is:))

(35:8) v(S) =))

-1when S has

1

12 elements.34))

(Verification is immediatewith the helpof (34:1),(34:2),(34:3)in 34.2.1.)Theexactsolutionsof this game are numerous;indeed,one must say thatthey areof a rather bewilderingvariety. It has not beenpossibleyet toorder them and to systematizethem by a consistentapplicationof the exacttheory, to such an extentas onewould desire. Neverthelessthe known

specimensgive someinstructive insight into the ramifications of the theory.We shall considerthem in somewhat more detail in 37.and 38.

At present we make only this (heuristic)remark:The idea of this(totally) symmetricgame isclearly that any majority of the players(i.e.anycoalition of three)wins, whereasin caseof a tie (i.e.when two coalitionsform, eachconsistingof two players)no paymentsaremade.

35.3.2.Thecenterof Q representedthe only (totally) symmetric gamein our set-up:with respectto all permutations of the players1,2,3,4.Thegeometricalpicture suggestsconsiderationof another symmetry as well:with respectto all permutationsof the coordinatesx\\ 9 x^ x8. In this waywe selectthe pointsof Q with

(?5:9) xi = xi = si,which form a main diagonal of Q, the line

(35:10) 7-center-y777.

We saw at the beginning of 34.3.1.that this symmetry meanspreciselythat the gameis invariant with respectto all permutationsof the players1,2,3.In otherwords:

1This representation shows oncemore that the game is symmetric, and uniquelycharacterizedby this property. Cf.the analysis of 28.2.1.)))

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SPECIAL POINTSIN THECUBEQ 303

The main diagonal (35:9),(35:10)representsall thosegameswhich aresymmetric with respectto the players1,2,3,i.e.where only player 4 mayhave a specialrole.

Q hasthreeothermain diagonals(//-center-F,///-center-F/,/F-centerF//),and they obviously correspondto thosegameswhere another player(players1,2,3,respectively)alone may have a specialrole.

Let us return to the main diagonal (35:9),(35:10).Thethreegameswhich we have previously considered(/,F///,Center)lie on it;indeedinall thesegamesonly player4 had a specialrole.1 Observethat the entirecategoryof gamesis a one-parametervariety. Owing to (35:9),such agame is characterizedby the value x\\ in

(35:11) -1g xi ^ 1.The threegamesmentioned above correspondto the extremevalues x\\ = 1,Xi = -1 and to the middlevalue Xi = 0. In order to getmore insight intothe working of the exacttheory, it would be desirableto determineexactsolutionsfor all thesevalues of Xi, and then to seehow thesesolutionsshiftas x\\ varies continuously along (35:10).It would be particularly instruc-tive to find out how the qualitatively different kindsof solutionsrecognizedfor the specialvalues x\\ = 1,0,1 go over into eachother. In 36.3.2.we shallgive indicationsabout the information that is now available in thisregard.

35.3.3.Another question of interestis this:Considera game, i.e.apoint in Q, where we can form someintuitive pictureof what solutionstoexpect,e.g.the cornerF///. Then considera game in the immediateneighborhoodof F///,i.e.one with only slightly changedvalues of Xi, x,x3. Now it would be desirableto find exactsolutionsfor theseneighboringgames,and to seein what details they differ from the solutionsof theoriginal game, i.e.how a small distortion of x\\, xj,xs distorts the solu-tions.2 Specialcasesof this problemwill be consideredin 36.1.2.,and atthe end of 37.1.1.,as well as in 38.2.7.

35.3.4.Sofar we have consideredgamesthat arerepresentedby pointsof Q in more or lessspecialpositions.8 A more general,and possiblymoretypical problem ariseswhen the representative point X is somewherein the interior of Q, in \"general\"position, i.e.in a position with noparticular distinguishingproperties.

Now onemight think that a good heuristiclead for the treatment ofthe problemin suchpointsis providedby the following consideration.Wehave someheuristicinsight into the conditionsat the corners/-F/// (cf.35.1.and 35.2.).Any point X of Q is somehow\"surrounded\"by thesecorners;more precisely,it is their centerof gravity, if appropriateweights

1In the centernot even he.1This procedureis familiar in mathematical physics, where it is used in attacking

problems which cannot besolvedin their general form for the time being:it is the analysisof perturbations.

8 Corners,the center,and entire main diagonals.)))

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304 ZERO-SUMFOUR-PERSONGAMES

areused. Henceonemight suspectthat the strategy of the games,repre-sentedby X, is in someway a combination of the strategiesof the (morefamiliar) strategiesof the gamesrepresentedby 1-VIII. One might evenhopethat this \" combination \" will in somesensebesimilar to the formationof the centerof gravity which relatedX to I-VIII.1

We shall seein 36.3.2.and in 38.2.5.7.that this is true in limited partsof Q, but certainly not over all of Q. In fact, in certaininterior areasof Qphenomenaoccurwhich arequalitatively different from anything exhibitedby I'VIII. All this goesto show what extremecaremust beexercisedindealingwith notions involving strategy, or in making guessesabout them.Themathematical approachis in such an early stageat present that muchmore experiencewill be neededbefore one can feel any self-assuranceinthis respect.

36.Discussionof the Main Diagonals36.1.ThePart Adjacent to the Corner VIII.:Heuristic Discussion

36.1.1.The systematic theory of the four-persongame has not yetadvancedso far as to furnish a completelist of solutionsfor all the gamesrepresentedby all pointsof Q. We arenot ableto specifyeven onesolutionfor every such game. Investigations thus far have succeededonly indeterminingsolutions(sometimesone,sometimesmore)in certainparts ofQ. It is only for the eight corners1-VIII that a demonstrablycompletelist of solutionshas beenestablished.At the presentthe parts of Q inwhich solutionsareknown at all form a ratherhaphazardarray of linear,planeand spatial areas. They aredistributed all over Q but do not fill itout completely.

Theexhaustive list of solutionswhich areknown for the cornersI-VIIIcan easilybe establishedwith help of the results of Chapters IX and X,where thesegameswill be fitted into certainlargerdivisionsof the generaltheory. At presentwe shall restrictourselvesto the casuisticapproach

1Consider two points X =|a?i, z2, xt \\ and Y (t/i, y^ 2/s) in Q. We may view

these as vectorsin L8 and it is indeed in this sensethat the formation of a centerofgravity

tX + (1- t)Y - (toi + (1- Ol/i, tx* + (1- Oyt, to, + (1- <)y,}is understood. (Cf.(16:A:c)in 16.2.1.)

Now if X [xi, z2, z|and Y = \\yi, y^ 2/3) define the characteristic functionsv(S) and w(S) in the senseof (34:l)-(34:3)in 34.2.1.,then tX + (1- t)Y will give, bythe same algorithm, a characteristicfunction

u(S)s tv(S) + (1- /)wOSf).

(It is easyto verify this relationship by inspection of the formulae which we quoted.)And this same u(S)was introduced as centerof gravity of v(S)and w() by (27:10)in27.6.3.

Thus the considerations of the text are in harmony with those of 27.6. That wearedealing with centersof gravity ofmore than two points (eight: I-VIII)instead of onlytwo, is not essential:the former operation can beobtained by iteration of the latter.Itfollows from theseremarks that the difficulties which arepointed out in the textbelow have a direct bearing on 27.6.3.,as was indicated there.)))

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DISCUSSIONOF THEMAIN DIAGONALS)) 305))

which consistsin describingparticular solutionsin caseswhere such areknown. It would scarcelyserve the purposeof this expositionto give apreciseaccountof the momentary stateof these investigations1 and itwould take up an excessiveamount of space. We shall only give someinstanceswhich, it is hoped,arereasonablyillustrative.

36.1.2.We considerfirst conditionson the main diagonal /-Center-VIIIin Q nearits end at VIII,x\\ = x2 = x3 = 1(cf.35.3.3.),and we shall try))

VIII))

The diagonal / Center VIII redrawn

Center))

Figure 63.

to extendover the x\\ = #2 = 3 > 1as far as possible.(Cf.Figure63.)On this diagonal))

(36:1)))

_ ^2*!))

1

when S has))

12 (and 4 belongsto S)

elements2 (and4 doesnot belongto S)34))

(Observethat this gives(35:1)in 35.1.1.for xi = 1and (35:6)in 35.2.1.forx\\ = 1.) We assume that x\\ > 1but not by too much, just howmuch excessisto bepermittedwill emergelater. Letus first considerthissituation heuristically.

SinceXi is supposedtobenot very far from 1,the discussionof 35.2.may still give someguidance. A coalition of two players from among))

1This will be done by one of us in subsequent mathematical publications.)))

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306 ZERO-SUMFOUR-PERSONGAMES

1,2,3may even now be the most important strategicaim, but it is no longerthe only one:the formula (35:7)of 35.2.1.is not true,but instead

(36:2) v(Su T) > v(S)+ v(T) if SnT=

when T = (4).1 Indeed,it is easily verified from (36:1)that this excessis always2 2(1+ 1). Forx\\ = 1this vanishes,but we have x\\ slightly> 1,so the expressionis slightly > 0. Observethat for the precedingcoalition of two players other than player 4, the excessin (36:2)8 is by(36:1)always 2(1 Xi). Forx\\ = 1 this is 4, and as we have x\\ slightly> 1,it will be only slightly <4.

Thus the first coalition (between two players, other than player 4),is of a much strongertexturethan any other(whereplayer4 entersinto thepicture), but the lattercannot be disregardednevertheless.Sincethefirst coalition is the strongerone,it may be suspectedthat it will form firstand that onceit is formed it will actas one player in its dealingswith thetwo others. Then somekind of a three-persongame may be expectedtotake placefor the final crystallization.

36.1.3.Taking,e.g.(1,2)for this \"first\" coalition, the surmisedthree-persongame is betweenthe players(1,2),3,4.4 In this game the a, 6, c of23.1.are a = v((3,4))= 2xl9 b = v((l,2,4))= 1, c = v((l,2,3))= I.6Hence,if we may apply the resultsobtainedthere (all of this is extremely

heuristic!)the player (1,2)gets the amount a = ^ = 1 x\\,&

if successful(in joining the last coalition),and a = 2xi if defeated.The

player3 getsthe amount = ^ = x\\ if successful,and b = 1z

if defeated.The player4 getsthe amount 7 = ~ = x\\ if success-ft

ful, and c = 1 if defeated.Since\"first\" coalitions(1,3),(2,3)may form, just as well as (1,2),

therearethe sameheuristicreasonsas in the first discussionof the three-persongame (in 21.-22.)to expectthat the partners of thesecoalitionswill

split even. Thus, when such a coalition is successful(cf. above),itsmembersmay be expectedto get -^ -1 each,and when it is defeated

&

Xi each.36.1.4.Summing up:If thesesurmisesprove correct,the situation is as

follows:

1 UnlessS * or T, in which casethere is always in (36:2).I.e.in the pres-ent situation Smust have one or two elements.

2By footnote 1above,Shas oneor two elementsand it doesnot contain 4.8 I.e.,now S, T are two one-elementsets,not containing player 4.4Onemight say that (1,2)is a juridical person,while 3,4are, in our picture, natural

persons.*In all the formulae which follow, remember that Xi is near to 1, i.e.presumably

negative; hence Xi is a gain, and x\\ is a loss.)))

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DISCUSSIONOF THE MAIN DIAGONALS 307

If the \"first\" coalition is (1,2),and if it is successfulin finding an ally,and if the player who joins it in the final coalition is player 3,then the

1 _ /* 1 . Tplayers1,2,3,4getthe amounts ~ S ~ S Xi, 1respectively. If the

2i i

player who joins the final coalition is player 4, then theseamounts are

replacedby \"IXl ,

1~ XS -1,XL If the \"first\" coalition (1,2)is not& &

successful,i.e.if theplayers3,4combineagainst it, then the playersgettheamounts Xi, xi,Xi, x\\ respectively.

If the \"first\" coalition is (1,3)or (2,3),then the correspondingpermuta-tion of the players1,2,3must beappliedto the above.

36.2.ThePart Adjacent to the Corner VIII.: Exact Discussion

36.2.1.It is now necessaryto submit all this to an exactcheck. Theheuristicsuggestionmanifestly correspondsto the followingsurmise:

Let V be the setof theseimputations:~~>, (l - si 1-))

>

(36:3) a\" =))

- si 1- xi J2 ' 2 I and the imputations which

\\originate from theseby per-' ~~*1 Xl | muting the players, (i.e.the))

x i x \

components)1,2,3.))

(Cf.footnote 5, p.306.) We expectthat this V is a solutionin the rigoroussenseof 30.1.if Xi is near to 1and we must determinewhether this is so>and preciselyin what interval of the x\\.

This determination,if carriedout, yieldsthe followingresult:

(36:A) The setV of (36:3)is a solution if and only if

-1 xi g -*.This then is the answer to the question,how far (from the starting pointxi = 1,i.e.the cornerVIII)the above heuristicconsiderationguidestoa correctresult.1

36.2.2.The proof of (36:A) can be carriedout rigorouslywithout anysignificant technicaldifficulty. It consistsof a rathermechanical disposal

1We wish to emphasize that (36:A) doesnot assertthat V is (in the specifiedrange ofXi) the only solution of the game in question. However, attempts with numeroussimilarly built setsfailed to disclosefurther solutions for x\\ ^ J (i.e.in the range of(36:A)). For x\\ slightly > I (i.e.slightly outside the range of (36:A)),where theV of (36:A) is no longer a solution, the same is true for the solution which replacesit.Cf. (36:B)in 36.3.1.

We do not question, of course,that other solutions of the \"discriminatory\" type,as repeatedlydiscussedbefore,always exist. But they are fundamentally different fromthe finite solutions V which arenow under consideration.

Thesearethe arguments which seemto justify our view that somequalitative changein the nature of the solutions occursat

x\\ G (on the diagonal /-center-VIII).)))

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308 ZERO-SUMFOUR-PERSONGAMES

of a seriesof specialcases,and doesnot contributeanything to theclarifica-tion of any questionof principle.1 Thereadermay therefore omit readingit if he feelssodisposed,without losingthe connectionwith the main courseof the exposition.Heshouldrememberonly the statement of the resultsin (36:A).

Neverthelesswe give the proof in full for the followingreason:ThesetV of (36:3)was found by heuristic considerations,i.e.without using theexacttheory of 30.1.at all. The rigorousproof to be given is based on30.1.alone,and thereby bringsus back to the only ultimately satisfactorystandpoint, that of the exacttheory. The heuristic considerationswereonly a deviceto guessthe solution,for want of any bettertechnique;andit isa fortunate feature of the exacttheory that its solutionscan occasionallybe guessedin this way. But such a guess must afterwards be justifiedby the exactmethod,or ratherthat methodmust be used to determineinwhat domain (of the parametersinvolved) the guesswas admissible.

We give the exactproof in order to enable the readerto contrast andtocompareexplicitlythesetwo procedures,the heuristicand the rigorous.

36.2.3.Theproof is as follows:If x\\ = 1,then we arein the cornerVIII,and the V of (36:3)coincides

with the set which we introducedheuristically(as a solution) in 35.2.3.,and which can easilybe justified rigorously(cf.alsofootnote 2 on p.299).Thereforewe disregardthis casenow, and assumethat

(36:4) xi > -1.We must first establish which sets Ss/=(1,2,3,4)are certainly

necessaryor certainly unnecessaryin the senseof 31.1.2.sincewe arecarryingout a proof which is preciselyof the type consideredthere.

Thefollowingobservationsareimmediate:

(36:5) By virtue of (31:H) in 31.1.5.,three-elementsetsS arecer-tainly necessary,two-elementsetsare dubious, and all othersetsarecertainly unnecessary.2

(36:6) Whenever a two-element set turns out to be certainlyneces-sary, we may disregardall those three-elementsetsof which it isa subset,owing to (31:C)in 31.1.3.

Consequentlywe shall now examine the two-elementsets.This of coursemust bedone for all the a in the setV of (36:3).

1Thereadermay contrast this proof with somegiven in connection with the theory ofthe zero-sum two-person game, e.g.the combination of 16.4.with 17.6.Such a proof ismore transparent, it usually coversmore ground, and gives somequalitative elucidationof the subjectand its relation to other parts of mathematics. In somelater parts of thistheory such proofshave beenfound, e.g.in 46. But much of it is still in the primitiveand technically unsatisfactory stateof which the considerationswhich follow aretypical.*This is due to n - 4.)))

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DISCUSSIONOF THEMAIN DIAGONALS 309

Consider first those two-elementsetsS which occur in conjunction>

with a V As aj = 1 we may excludeby (31:A) in 31.1.3.the possibilitythat S contains 4. S = (1,2)would be effective if a{+ a'2 ^ v((l,2)),i.e.1-xi ^ -2xi,xi ^ -1which is not the caseby (36:4).S = (1,3)is effective if a( + J ^ v((l,3)), i.e. -^ -2*!,Xl - f Thus

46the condition

(36:7) Xl ^ - iwhich we assumeto be satisfiedmakesits first appearance.S = (2,3)we

do not need,since1and 2 play the samerolein a ' (cf.footnote 1above).We now passto a \". As a'8'= 1wenow excludethe Swhich contains3

(cf.above). S = (1,2)is disposedof as before,sincea 'and a \" agreeinthese components.S = (1,4)would be effective if a\" + a\" g v((l,4)),i.e. ~ ^ 2xi,xi ^|which, by (36:7),is not the case. S = (2,4)isdiscardedin the sameway.

Finally we take ~Z r \". S = (1,2)is effective: a'/'+ a'2\" = v((l,2))i.e. 2xj.= 2xi. S = (1,3)neednot be consideredfor the followingreason:We arealready consideringS = (1,2)for a '\",if we interchange2and 3 (cf.footnote 1above) this goesover into (1,3),with the components-Xi,-XL Our original S = (1,3)for a'\"with the components-Xi,Xiis thus renderedunnecessaryby (31:B) in 31.1.3.,as x t ^ xi owing to(36:7).S = (2,3) is discarded in the same way. S = (1,4) would beeffective if a'/'+ a'4\" g v((l,4)) i.e. g 2xb xi ^ 0,which, by (36:7),isnot the case. S = (2,4)is discardedin the sameway. S = (3,4)is effec-tive:ai\" + a'/'= v((3,4)),i.e.2xi = 2Xl .

Summingup:(36:8) Among the two-element setsS the three given below are

certainly necessary,and all othersarecertainlyunnecessary:

(1,3)for ^',(1,2)and (3,4)for T'\".Concerningthree-elementsetsS:By (31:A)in 31.1.3.we may exclude

those containing 4 for a ' and 3 for a \". Consequentlyonly (1,2,3)is left

for ' and (1,2,4)for a\". Of these the former is excludedby (36:6),as it contains the set (1,3)of (36:8).For^'\"every three-elementset

1 Here,and in the entire discussion which follows, we shall make useof the freedom toapply permutations of 1,2,3asstatedin (36:3),in order to abbreviate the argumentation.Hencethe readermust afterwards apply thesepermutations of 1,2,3to our results.)))

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310 ZERO-SUMFOUR-PERSONGAMES

containsthe set(1,2)or the set(3,4)of (36:8);hencewe may excludeit by(36:6).

Summingup:(36:9) Among the three-elementsets <S, the one given below is

certainlynecessary,and all othersarecertainly unnecessary:1

(1,2,4)for 7\".36.2.4.We now verify (30:5:a)in 30.1.1.,i.e.that no a of V dominates

any of V.^ = a ':By (36:8),(36:9)we must useS = (1,3).Can a ' dominate

with this S any 1,2,3permutation of a ' or a \" or a '\"? Thisrequiresfirst

the existenceof a component< x\\ (this is the 3 component of a. ') among

the 1,2,3componentsof the imputation in question. Thus a' and a'\"areexcluded.2 Even in a \" the 1,2componentsareexcluded(cf.footnote 2)but the 3 componentwill do. But now another one of the 1,2,3componentsof this imputation a \" must be < ~ -1 (this is the 1componentof a ')

Z

and this is not the case;the 1,2componentsof a. \" areboth = -1-&

a. = a \":By (36:8),(36:9)we must useS = (1,2,4).Can a \" dominate

with this S any 1,2,3permutation of a ' or a \" or a '\"? Thisrequiresfirstthat the 4 componentof the imputation in questionbe < x\\ (this is the

4 componentof a\-") Thus a \"and a \"' areexcluded.For a 'wemust

requirefurther that two of its 1,2,3componentsbe < -^(this is the 1

as well as the 2 componentof a \,") and this isnot the case;only one of these

componentsis 5* ~ *\"

\"a* = 1?'\":By (36:8),(36:9)we must useS = (1,2)and then S = (3,4).5= (1,2):Can a'\"dominatewith this asdescribedabove? Thisrequiresthe existenceof two components< zi (this is the 1as well as the 2 com-

~\"*

ponent of a'\")among the 1,2,3componentsof the imputation in question.This is not the casefor a'\",as only oneof thesecomponentsis j& x\\

1 As every three-elementset is certainly necessaryby (36:5)above,this is anotherinstance of the phenomenon mentioned at the end of footnote 1 on p. 274.

1Indeed 5^ a?i, i.e.x\\ rand -x\\ x\\, i.e.Xi both by (36:7).V)))

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DISCUSSIONOFTHEMAIN DIAGONALS 311* *

there. Nor is it the casefor a ' or a \", as only one of thosecomponentsis1 x\\ *

?* s there.1 S = (3,4):Can a '\" dominatewith this S as describedt

above? This requires first that the 4 componentof the imputation in>

questionbe < Xi (thisis the 4 componentof a \"').Thus a \" and a '\" areexcluded.For a ' we must requirefurther the existenceof a component< x\\ (this is the 3 componentof a '\")among its 1,2,3components,andthis is not the case;all thesecomponentsare^ Xi (cf.footnote 2 on p.310).

This completesthe verification of (30:5:a).36.2.5.We verify next (30:5:b)in 30.1.1.,i.e.that an imputation ft which

is undominated by the elementsof V must belongto V.Considera ft undominated by the elementsof V. Assume first that

04 < Xi. If any one of 0i, 2, 03were < Xi, we couldmake (by permuting

1,2,3)3 < XL This gives 7'\"H 7 with S = (3,4)of (36:8).Hence

ft i, 02,03 ^ XL

If any two of 0i, 2, 03were < ~ 1

, we couldmake (by permuting 1,2.3)))

0i,02 < ~V^' This gives a \" H with S = (1,2,4)of (36:9).Hence,2

at most one of 0i, 2, 3 i

1,2,3,we can thus make))

at most one of 0i, 2, 3 is < ~- i.e.two are^ Xl - By permuting))

Clearly 4 ^ 1. Thuseachcomponent of is ^ the correspondingcom-

ponentof a ', and sinceboth areimputations 2 it followsthat they coincide:

= a ',and so it is in V.Assume next that 4 ^ x\\. If any two of 0i, 2, 0s were < Xi, we

* >

couldmake (by permuting 1,2,3)0i, 2 < XL Thisgives a\"'H with

S = (1,2)of (36:8).Hence,at most one of 0i, 2, 0sis < Xi, i.e.two are^ XL By permuting 1,2,3we can make

01,02 S -Xi.If 0s ^ *i,then all this impliesthat eachcomponentof is ^ the cor-

1And i-^-1 -xi,i.e.xi -1.2 Consequently for both the sum of all components is the same:zero.)))

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312 ZERO-SUMFOUR-PERSONGAMES

respondingcomponentof a '\",and sincebothareimputations(cf.footnote 2,on p.311)it followsthat they coincide:ft = a '\",and so it is in V.

Assume therefore that fa < XL If any one of 0i,fa were < ~ \"'

I x\\ * *we could make (by permuting 1,2)fa < ^ This gives a'H ft

&

with 5= (1,3)of (36:8).Hence

A, A ^ ^J-))Clearly 8 ^ 1. Thuseachcomponentof ft is g: the correspondingcom-

ponent of a\",and sinceboth areimputations (cf. footnote 2, p.311),it

followsthat they coincide:ft = a \", and so it is in V.This completesthe verification of (SOiSib).1

Sowe have establishedthe criterion(36:A).2

36.3.Other Partsof the Main Diagonals

36.3.1.When Xi passesoutside the domain (36:A) of 36.2.1.,i.e.whenit crossesits borderat x\\ = i,then the V of (36:3)id.ceasesto be asolution. Itis actually possibleto find a solutionwhich is valid for a certaindomain in x\\ > i (adjoiningx\\ = i),which obtains by adding to theV of (36:3)the further imputations

(36:10)a lv = I j^-1, xi, s -1, xi\\ and permutations as in))

(36:3).3

Theexactstatementis actually this:(36:B) ThesetV of (36:3)and (36:10)is a solutionif and only if-i < x, < O.4

1Thereaderwill observethat in the courseof this analysis all setsof (36:8),(36:9)have beenusedfor dominations, and had to beequatedsuccessivelyto all three a ',a \",

\"7\"' of (36:3).f Concerning zi = 1,cf. the remarks made a-t the beginning of this proof.3 An inspection of the aboveproof shows that when x\\ becomes> J, this goes

wrong: ThesetS (1,3)(and with it (2,3))is no longer effective for a '. Ofcoursethisrehabilitates the three-elementset = (1,2,3)which was excludedsolelybecause(1,3)(and (2,3))is contained in it.

.Thusdomination by this element of V, a ',now becomesmore difficult, and it istherefore not surprising that an increaseof the set V must be consideredin the searchfor a solution.

4Observethe discontinuity at x\\ J which belongsto (36:A) and not to (36:B)lTheexacttheory is quite unambiguous, even in such matters.)))

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THECENTER AND ITSENVIRONS 313The proof of (36:B)is of the sametype as that of (36:A) given above,

and we do not proposeto discussit here.Thedomains(36:A) and (36:B)exhaust the part xi ^ of the entire

available interval 1g x\\ g 1 i.e.the half F/77-Centerof the diagonalV/77-Center-/.

36.3.2.Solutionsof a nature similar to V describedin (36:A) of 36.2.1.and in (36:B)of 36.3.1.,have been found on the otherside x\\ > i.e.the half Center-/of the diagonal as well. It turns out that on this half,qualitative changesoccurof the samesort as in the first half coveredby(36:A) and (36:B).Actually three such intervals exist,namely:

(36:C) ^ xi <i,(36:D) i < xi g *,(36:E) |g Xl g 1.(Cf.Figure64, which is to be comparedwith Figure63.)

VIII Omtw J

x *))

*. - -i -* o * 4 i

Figure 64.

We shall not discussthe solutionspertaining to (36:C),(36:D),(36:E).1The readermay however observethis:x\\ = appears as belonging

to both (neighboring)domains (36:B)and (36:C),and similarly xi = i toboth domains(36:D)and (36:E). This is so because,as a closeinspectionof the correspondingsolutionsV showsthat while qualitative changesin thenature of V occurat x\\ = and i,thesechangesarenot discontinuous.

The point Xi = i,on the other hand, belongsto neither neighboringdomain (36:C)or (36:D). It turns out that the typesof solutionsV whicharevalid in thesetwo domainsareboth unusableat x\\ = i. Indeed,theconditionsat this point have not beensufficiently clarified thus far.

37.TheCenterand ItsEnvirons$7.1.First Orientation Concerning the Conditions around the Center

37.1.1.Theconsiderationsof the last sectionwererestrictedto a one-dimensionalsubset of the cubeQ:ThediagonalFJ/7-center-/.By usingthe permutationsof the players 1,2,3,4,as describedin 34.3.,this can bemade to disposeof all four main diagonalsof Q. By techniquesthat aresimilarto thoseof the last section,solutionscan alsobefound along someotherone-dimensionallines in Q. Thus thereis quite an extensivenet oflines in Q on which solutionsare known. We do not proposeto enumeratethem, particularlybecausethe information that isavailable now correspondsprobablyto only a transient stateof affairs.

1Another family of solutions, which alsocoverpart of the same territory, will bediscussedin 38.2.Cf.in particular 38.2.7.,and footnote 2 on p. 328.)))

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314 ZERO-SUMFOUR-PERSONGAMES

This, however, shouldbe said:sucha searchfor solutionsalong isolatedone-dimensionallines,when the whole three-dimensionalbodyof the cubeQwaits for elucidation,cannot be more than a first approachto the problem.If we can find a three-dimensionalpart of the cube even if it is a smallone for all points of which the same qualitative type of solutionscan beused, we shall have someideaof the conditionswhich areto be expected.Now sucha three-dimensionalpart existsaround the centerof Q. Forthisreasonwe shalldiscussthe conditionsat the center.

37.1.2.The centercorrespondsto the values 0,0,0of the coordinatesx\\, #2, #3 and represents,as pointedout in 35.3.1.,the only (totally) sym-metric game in our set-up. Thecharacteristicfunction of this game is :))

(37:1) v(S) =))

-1))when S has

1

12 elements.34))

(Cf.(35:8)id.) As in thecorrespondingcasesin 35.1.,35.2.,36.1.,we beginagain with a heuristicanalysis.

This game is obviously one in which the purposeof all strategiceffortsis to form a three-personcoalition. A player who is left alone is clearlyaloser,any coalition of 3 in the samesensea winner, and if the game shouldterminate with two coalitionsof two playerseachfacing eachother,thenthis must obviously be interpretedas a tie.

The qualitative question which ariseshere is this:The aim in thisgame is to form a coalition of three. It is probablethat in the negotiationswhich precedethe play a coalition of two will be formed first. Thiscoalitionwill then negotiatewith the two remaining players, trying to securethecooperationof one of them against the other. In securingthe adherenceof this third player, it seemsquestionablewhether he will be admitted intothe final coalition on the same conditionsas the two original members.If the answeris affirmative, then the total proceedsof the final coalition, 1,will be divided equally among the three participants:i,|, . If it isnegative, then the two original members(belongingto the first coalitionof two) will probablyboth getthe sameamount, but more than . Thus 1will bedividedsomewhatlike this:i + c,i + c,i 2e with an e > 0.

37.1.3.The first alternative would be similar to the one which weencounteredin the analysis of the point / in 35.1.Herethe coalition(1,2,3),if it forms at all, contains its three participants on equal terms.Thesecondalternative correspondsto the situation in the interval analyzedin 36.1.-2. Hereany two players(neitherof them beingplayer4)combinedfirst, and this coalition then admittedeitherone of the two remaining play-erson lessfavorable terms.

37.1.4.Thepresentsituation is not a perfect analogue of eitherof these.)))

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THECENTER AND ITSENVIRONS 315In the first casethe coalition (1,2)could not make stiff termsto player 3

becausethey absolutelyneededhim:if 3 combinedwith 4, then 1and 2would be completelydefeated;and (1,2)could not, as a coalition, combinewith 4 against 3,since4 neededonly one of them to be victorious (cf.thedescriptionin 35.1.3.).In our presentgame this is not so:the coalition(1,2)can use 3 as well as 4, and even if 3 and 4 combine against it, only atie results.

In the secondcasethe discrimination against the memberwho joinsthe coalition of threeparticipantslast is plausible,sincethe original coali-tion of two is of a much strongertexture than the final coalition of three.Indeed,asx\\ tendsto 1,the latter coalition tendsto becomeworthless;cf.the remarksat the end of 36.1.2.In our presentgame no suchqualitativedifference can be recognized:the first coalition (of two) accountsfor thedifferencebetweendefeat and tie, while formation of the final coalition (ofthree)decidesbetweentie and victory.

We have no satisfactory basisfor a decisionexceptto try both alterna-tives. Beforewe do this, however,an important limitation of our consider-ationsdeservesattention.

37.2.TheTwo Alternatives and the Role of Symmetry

37.2.1.It will be noted that we assumethat the sameone of the twoalternatives above holdsfor all four coalitionsof threeplayers. Indeed,weare now looking for symmetric solutions only, i.e.solutionswhich contain,

along with an imputation a = {i, 2, a, QU}, all its permutations.Now a symmetry of the game by no means implies in general the

correspondingsymmetry in eachone of its solutions. The discriminatorysolutions discussedin 33.1.1.make this clearalready for the three-persongame. We shall find in 37.6.further instancesof this for the symmetricfour-person game now under consideration.

Itmust beexpected,however,that asymmetric solutionsfor a symmetricgame are of too reconditea characterto be discoveredby a first heuristicsurvey like the presentone. (Cf. the analogous occurrencein the three-persongame, referred to above.) This then is our excusefor looking atpresentonly for symmetric solutions.

37.2.2.One more thing ought to be said:it is not inconceivable that,while asymmetric solutions exist, general organizational principles,likethflse correspondingto our above two alternatives, are valid either for the

totality of all participantsor not at all. Thissurmisegains somestrengthfrom the considerationthat the number of participantsis still very low, and

may actually be too low to permit the formation of several groupsof par-ticipants with different principlesof organization. Indeed,we have onlyfour participants,and ample evidencethat three is the minimum numberfor any kind of organization. Thesesomewhat vague considerationswill

find exactcorroboration in at leastone specialinstancein (43:L)et seq.of43.4.2.Forthe presentcase,however, we are not ableto support them byany rigorousproof.)))

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87.3.TheFirst Alternative at the Center37.3.1.Let us now considerthe two alternatives of 37.1.2.We take

them up in reverseorder.Assume first that the two original participants admit the third one

under much less favorable conditions. Then the first coalition (of two)must be consideredas the coreon which the final coalition (of three)crystal-lizes. In this last phase the first coalition must therefore be expectedtoactas one player in its dealingswith the two others,thus bringing aboutsomethinglike a three-persongame. If this view is sound, then we mayrepeatthe correspondingconsiderationsof 36.1.3.

Taking, e.g.(1,2),as the \"first\" coalition, the surmisedthree-persongameis between the players (1,2), 3,4. The considerationsreferred toabove therefore apply literally, only with changednumerical values:a = 0,6 = c = 1 and so a = 1,ft = y = O.1

Sincethe \"first\" coalition may consist of any two players, thereareheuristicreasonssimilarto thosein the discussionof the three-persongame(in 21.-22.)to expectthat the partners in it will split even:when an ally isfound, as well as when a tieresults,the amount to be dividedbeing1orrespectively.2

37.3.2.Summing up:if the above surmisesprove correct,the situation isas follows:

If the \"first\" coalition is (1,2)and if it is successfulin finding anally, and if the playerwho joinsit in the final coalition is3,then the players1,2,3,4get the amounts , ,0, 1 respectively. If the \"first\" coalitionis not successful,i.e.if a tie results, then theseamounts arereplacedby0,0,0,0.

If the distribution of the players is different, then the correspondingpermutation of the players1,2,3,4must be appliedto the above.

It is now necessaryto submit all this to an exactcheck. Theheuristicsuggestionmanifestly correspondsto the followingsurmise:

Let V be the setof thesefollowingimputations

C\\7-9\\a '= ^'i> 0> ~M and the imputations which originate from

^\"=(00001 theseby permuting the players (i.e.thecomponents)1,2,3,4.

We expectthat this V is a solution.1Theessentialdifference between this discussion and that referredto, is that player 4

is no longer excludedfrom the \" first\" coalition.1Theargument in this caseis considerably weaker than in the casereferredto (or in

the corresponding application in 36,1.3.)sinceevery \"first\" coalition may now wind upin two different ways (tie or victory). Theonly satisfactory decisionas to the value ofthe argumentation obtains when the exacttheory is applied. Thedesiredjustification isactually contained in the proof of 38.2.1.-3.;indeed, it is the specialcase

y\\ - y* - y* - y* - lof (38:D)in 38.2.3.)))

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THECENTER AND ITSENVIRONS 317A rigorousconsideration,of the same type as that which constitutes

36.2.,showsthat this V is indeeda solution in the senseof 30.1.We donot give it here,particularly becauseit is contained in a more generalproofwhich will be given later. (Cf. the reference of footnote 2 on p.316.)

37.4.TheSecondAlternative at the Center37.4.1.Assume next that the final coalition of three contains all its

participantson equalterms. Then if this coalition is,say (1,2,3),the players1,2,3,4get the amounts i,i;i, 1respectively.

It would be rash to concludefrom this that we expectthe setof impu-tations V to which this leads,to be a solution;i.e.the setof theseimputa-

tions a = {i, 2, <*3,a*}>

(37:3) a '\" = (i,i,i,-1} and permutations as in (37:2).We have madeno attempt as yet to understandhow this formation of thefinal coalition in \"onepiece\" comesabout, without assumingthe previousexistenceof a favored two-personcore.

37.4.2.In the previous solution of (37:2)suchan explanationis discern-able. Thestratified form of the final coalition is expressedby the imputa-

tion a 'and the motive for just this schemeof distributionliesin the threat~-* \"-*

of a tie,expressedby the imputation a \". To put it exactly:the a 'form a

solution only in conjunction with the a \", and not by themselves.In (37:3)this secondelement is lacking. A directcheckin the sense

of30.1.disclosesthat the a \"'fulfill condition (30:5:a)there,but not (30:5:b).

I.e.they do not dominate eachother, but they leave certainotherimputa-tions undominated. Hencefurther elementsmust be added to V.1

This addition can certainly not be the a\"= {0,0,0,0)of (37:2)since

that imputation happens to be dominatedby a '\".2 In other words the

extension(i.e.stabilization,in the senseof 4.3.3.)of a '\" to a solutionmustbe achievedby entirely different imputations (i.e.threats)in the caseof

the ~^'\"of (37:3)as in the caseof the ~'of (37:2).It seemsvery difficult to find a heuristicmotivation for the stepswhich

are now necessary.Luckily, however, a rigorousprocedureis possiblefrom hereon, thus renderingfurther heuristicconsiderationsunnecessary,

1Toavoid misunderstandings :It isby no means generally true that any setof imputa-tions which do not dominate eachother can be extended to a solution. Indeed,the

problem of recognizing a given set of imputations as being a subset of some (unknown)solution is still unsolved. Cf.30.3.7.

In the presentcasewe arejust expressing the hope that such an extension will provepossiblefor the V of (37:3),and this hopewill befurther justified below.

With S - (1,2,3).)))

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318 ZERO-SUMFOUR-PERSONGAMES

Indeed,one can prove rigorously that thereexistsone and only one sym-metricextensionof the V of (37:3)to a solution. This is the addition of

theseimputations a = {ai,a2, as,<**}

(37:4) ~2IV = (i,i,-i,-*} and permutationsas in (37:2).37.4.3.If a common-senseinterpretation of this solution, i.e.of its

constituent a lv of (37:4),is wanted,it must be said that it doesnot seemto

be a tieat all (likethe correspondinga \" of (37:2)) rather,it seemsto besome kind of compromisebetween a part (two members)of a possiblevictorious coalition and the othertwo players. However,as stated above,we do not attempt to find a full heuristicinterpretationfor the V of (37:3)and (37:4); indeedit may wellbe that this part of theexacttheory isalreadybeyond such possibilities.1 Besides,some subsequent exampleswill

illustrate the peculiaritiesof this solutionon a much widerbasis. Againwe refrain from giving the exactproof referredto above.

37.5.Comparison of the Two Central Solutions

37.6.1.The two solutions (37:2)and (37:3),(37:4),which we foundfor the game representingthe center,presenta new instanceof a possiblemultiplicity of solutions. Of coursewe had observed this phenomenonbefore,namely in the caseof the essentialthree-persongame in 33.1.1. Butthereall solutionsbut one werein someway abnormal (wedescribedthis byterming them \" discriminatory\.") Only one solution in that casewas afinite setof imputations;that solution alone possessedthe samesymmetryas the gameitself (i.e.was symmetricwith respectto all players). Thistime conditionsarequitedifferent. We have found two solutionswhich areboth finite setsof imputations,2 and which possessthe full symmetry of thegame. Thediscussionof 37.1.2.showsthat it is difficult to considereithersolutionas \" abnormal\"or \"discriminatory\"in any sense;they aredistin-guishedessentiallyby the way in which the accessionof the last participantto the coalition of threeis treated,and therefore seemto correspondto twoperfectlynormal principlesof socialorganization.

37.5.2.If anything, the solution(37:3),(37:4)may seemthe lessnormalone. Both in (37:2)and in (37:3),(37:4)the characterof the solutionwas

determinedby thoseimputationswhich describeda completedecision,a '> t ^

and a \"' respectively. To thesethe extra\" stabilizing\"imputations, a \"

> t >

and a /v , had to beadded. Now in the first solutionthis extraa \" had an

1This is, of course,a well known occurrencein mathematical-physical theories,evenif they originate in heuristic considerations.

*An easycount of the imputations given and of their different permutations showsthat the solution (37:2)consistsof 13elements,and the solution (37:3),(37:4)of 10.)))

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THECENTER AND ITSENVIRONS 319obvious heuristicinterpretationas a tie,while in the secondsolution the

nature of the extraa IV appearedto be more complex.A more thorough analysis discloses,however, that the first solution

is surroundedby somepeculiarphenomena which can neitherbe explainednor foreseenby the heuristicprocedurewhich providedeasyaccessto thissolution.

Thesephenomena arequite instructive from a generalpoint of view too,becausethey illustrate in a rather striking way some possibilitiesandinterpretations of our theory. We shall therefore analyze them in somedetail in what follows. We add that a similar expansionof the secondsolution has not beenfound up to now.

37.6.Unsymmetrical Central Solutions

37.6.1.To beginwith, thereexistsomefinite but asymmetricalsolutionswhich arecloselyrelatedto (37:2)in 37.3.2.becausethey contain someof theimputations{i,i,0, 1J.1 One of thesesolutionsis the one which obtainswhen we approachthe centeralong the diagonal /-Center-V7// from eitherside,and use there the solutionsreferred to in 36.3.I.e.:it obtains bycontinuous fit to the domains (36:B) and (36:C) there mentioned. (Itwill be rememberedthat the point Xi = 0,i.e.the center,belongsto boththesedomains,cf. 36.3.2.)Sincethis solution can betaken also to expressa sui generisprincipleof socialorganization, we shalldescribeit briefly.

This solutionpossessesthe samesymmetry as thosewhich belongto thegameson thediagonal/-Center-VIII,as it isactually one of them :symmetricwith respectto players 1,2,3,while player 4 occupiesa specialposition.2

We shall therefore describeit in the same way we did the solutionsonthe diagonal, e.g.in (36:3)in 36.2.1.Hereonly permutations of theplayers1,2,3aresuppressed,while in the descriptionsof (37:3)and (37:4)wesuppressedall permutationsof the players1,2,3,4.

37.6.2.Forthe sakeof a bettercomparison,we restatewith thisnotation(i.e.allowing for permutationsof 1,2,3only) the definition of our first fully-symmetricsolution(37:2)in 37.3.2.It consistsof theseimputations:8

7'= (i,i,o,-i}. ft\" = {i,i, 1,0} and the imputationswhich originate from

\"?/// (in 1 \\\\ theseby permuting the players1,2,3.P = tiiU, l,i>

7/v = {0,0,0,0}1I.e.somebut not all of the 12permutations of this imputation.*That the position of player 4 in the solution is really different from that of the others,

is what distinguishes this solution from the two symmetric onesmentioned before.

Our7',7\",7\"'exhaust the a'of (37:2)in 37.3.2.,while ^ is a \", id._ _* .a ' had to be representedby the three imputations ', \", ft

'\" becausethis

system of representation makes it necessaryto state in which one of the three possiblepositions of that imputation (i.e.the values },0, 1)the player 4 is found.)))

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320 ZERO-SUMFOUR-PERSONGAMES

Now the (asymmetric)solution to which we refer consistsof theseimputations:

?' \"a*\" \"?\" QQintt7'9*^ and the imputationswhich origi-(37:5) V P ' P asm^/.z; natefrom theseby permuting the

F = (4,0,-*,Op players 1,2,3.Oncemore we omit giving the proof that (37:5)is a solution. Instead

we shall suggestan interpretation of the difference betweenthis solutionand that of (37:2) i.e.of the first (symmetric)solutionin 37.3.2.

37.6.3.This difference consistsin replacing

7'\"= (4,0,-i,i)by

7F = (4,0,-4,0}.That is:the imputation ft

'\" in which the player 4 would belongto the\"first\" coalition (cf. 37.3.1.),i.e.to the group which wins the maximum

amount is removed,and replacedby another imputation fty . Player4

now getssomewhatlessand the losingplayeramong 1,2,3(in this arrange-ment player 3) getssomewhatmore than in ft '\".This difference is pre-cisely4, so that player4 is reducedto the tie position0,and player3 movesfrom thecompletelydefeatedposition 1to an intermediateposition 4

Thus players1,2,3form a \" privileged\" groupand no onefrom the out-side will be admitted to the \"first\" coalition. But even among the threemembersof the privilegedgroup the wrangle for coalition goeson, sincethe \"first\" coalition has room for two participantsonly. It is worth notingthat a memberof the privilegedgroupmay even be completelydefeated,as

in ft \", but only by a majority ofhis \" class\" who form the \" first \" coalitionand who may admit the \"unprivileged\"player 4 to the third membershipof the \"final\" coalition, to which he is eligible.

37.6.4.Thereaderwill notethat this describesa perfectly possibleformof socialorganization.This form is discriminatoryto be sure,althoughnot in the simpleway of the \"discriminatory\"solutionsof thethree-persongame. It describesa more complexand a more delicatetype of socialinter-relation,due to the solution rather than to the gameitself.2 Onemay think it somewhatarbitrary, but sincewe areconsideringa \"society\"of very smallsize,all possiblestandardsof behavior must beadjustedratherpreciselyand delicatelytothe narrownessof its possibilities.

We scarcely need to elaboratethe fact that similar discriminationagainstany otherplayer (1,2,3instead of 4) couldbe expressedby suitable

>1This imputation ft

v is reminiscent in its arrangement of a /F in (37:4)of 37.4.2.,but it has not beenpossibleto make anything of that analogy.1As to this feature, cf.the discussionof.35.2.4.)))

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NEIGHBORHOODOF THECENTER 321solutions,which could then be associatedwith the three other diagonalsof the cubeQ.

38.A Family of Solutionsfor a Neighborhoodof the Center38.1.Transformation of the Solution Belonging to the First Alternative at the Center

38.1.1.We continue the analysisof the ramifications of solution(37:2)in 37.3.2.It will appear that it can be subjectedto a peculiartransforma-tion without losingits characteras a solution.

This transformation consistsin multiplying the imputations (37:2)of37.3.2.by a common (positive)numerical factor z. In this way the follow-ing setof imputationsobtains:

f^K-n ^ '= |2'2* ^ ~Z| anc ^e im Putations which originate from__ theseby permuting the players1,2,3,4.7\" = {0,0,0,0}

In orderthat thesebe imputations, all their componentsmust be ^ 1(i.e.the common value of the v((i)). As z > this means only that

z ^ 1,i.e.we must have

(38:2) < z ^ 1.Forz = 1our (38:1)coincideswith (37:2)of 37.3.2.Itwould not seem

likely a priori that (38:1)shouldbe a solution for the samegamefor anyotherz of (38:2).And yet a simplediscussionshowsthat it is a solutionif and only if z >$ i.e.when (38*:2)is replacedby

(38:3) f < z g 1.The importanceof this family of solutionsis further increasedby the factthat it can be extendedto a certainthree-dimensionalpiecesurroundingthe centerof the cubeQ. We shall give the necessarydiscussionin full,becauseit offersan opportunity to demonstratea techniquethat may be ofwiderapplicabilityin theseinvestigations.

Theinterpretationof theseresultswill beattempted afterwards.38.1.2.We beginby observingthat considerationof the setV defined

by the above (38:1)for the game describedby (37:1)in 37.1.2.(i.e.thecenterof Q), could be replacedby considerationof the original set V of(37:2)in 37.3.2.in another game. Indeed,our (38:1)was obtained from

(37:2)by multiplying by z. Insteadof this we could keep(37:2)andmultiply the characteristicfunction (37:1)by l/z;this would destroy thenormalization 7 = 1which was necessaryfor the geometricalrepresentationby Q (cf.34.2.2.)but we proposeto acceptthat.

What we arenow undertakingcan be formulated therefore as follows:Sofar we have startedwith a given game,and have lookedfor solutions.

Now we proposeto reversethis process,starting with a solution and looking)))

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322 ZERO-SUMFOUR-PERSONGAMES

for the game. Precisely:we start with a given setof imputationsVi andask for which characteristicfunction v(S)(i.e.games)this V is a solution.1

Multiplicationof the v(S)of (37:1)in 37.1.2.by a commonfactor meansthat we stilldemand

(38:4) v(/S) = when S is a two-elementset,but beyondthis only the reducedcharacterof the game (cf.27.1.4.),i.e.(38:5) v((l)) = v((2))- v((3))- v((4)).

Indeed,this joint value of (38:5)is l/zand therefore (38:4),(38:5)and(25:3:a)(25:3:b)in 25.3.1.yield that this v(S) is just (37:1)multipliedbyl/z. Our assertion(38:3)above meansthat the V of (37:2)in 37.3.2.is asolutionfor (38:4),(38:5)if and only if the joint value of (38:5)(i.e.-l/z)is ^ 1and > f.

38.1.3.Now we shall go one step further and drop the requirementofreduction,i.e.(38:5).So we demand of v(S) only (38:4),restrictingitsvalues for two-elementsetsS. We restatethe final form of our question:(38:A) Considerall zero-sumfour-persongameswhere

(38:6) v(S) =0 for all two-elementsetsS.Forwhich among theseis thesetV of (37:2)in 37.3.2.a solution?

It will benoted that sincewe have droppedthe requirementsof normal-ization and reductionof v(S)all connectionswith the geometricalrepresen-tation in Q aresevered.A specialmanipulation will be necessarythereforeat the end,in orderto put the results which we shall obtain back into theframework of Q.

38.2.Exact Discussion

38.2.1.Theunknowns of the problem(38:A) areclearlythe values

(38:7) v((l = -2/1, v((2))= -2/2, v((3))= -2/3, v((4))= -y4.We proposeto determinewhat restrictionsthe condition in (38:A) actuallyplaceson thesenumbers2/1, 2/*> 2/a, 2/4.

1This reversedprocedureis quite characteristicof the elasticity of the mathematicalmethod for the kind and degreeof freedom which existsthere. Although initially itdeflectsthe inquiry into a direction which must be consideredunnatural from any butthe strictest mathematical point of view, it is neverthelesseffective;by an appropriatetechnical manipulation it finally disclosessolutions which have not beenfound in anyother way.

After our previous exampleswhere the guidance camefrom heuristic considerations,it is quite instructive to study this casewhere no heuristic help is reliedon and solutionsarefound by a purely mathematical trick, the reversalreferredto above.

For the readerwho might bedissatisfied with the useof such devices(i.e.exclusivelytechnicaland non-conceptual ones),we submit that they arefreely and legitimately usedin mathematical analysis.

We have repeatedlyfound the heuristic procedureeasierto handle than the rigorousone. Thepresent caseoffers an example of the oooosite.)))

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NEIGHBORHOODOF THECENTER 323

This gameis no longersymmetric.1 Hencethe permutationsof theplayers1,2,3,4arenow legitimateonly if accompaniedby thecorrespondingpermutationsof yi, y^ y&, y^

Tobegin with, the smallestcomponentwith which a given player k isever associatedin thevectorsof (37:2)in 37.3.2.,is 1. Hencethevectorswill be imputationsif and only if 1*z v((fc))i.e.(38:8) y k *l for k = 1,2,3,4.

Thecharacterof V as a setof imputations is thus established;let usnow seewhether it is a solution. This investigation is similarto the proofgiven in 36.2.3-5.

38.2.2.Theobservations(36:5),(36:6),of 36.2.3.,apply again. A two-

elementsetS = (i,j) is effective for a = {i,#2, s, ouj when a + ay ^

(cf.(38:A)). Hencewe have for the a ', a\"of (37:2):In a \" every two-

elementsetS iseffective. In a ':No two-elementsetS which doesnot con-tain the player 4 is effective, while those which do contain him, S = (1,4),(2,4),(3,4)clearlyare. However,if we considerS = (1,4),we may discardthe two others;S = (2,4)arises from it by interchanging1and 2, which

does not affect a ';3 S = (3,4)is actually inferior to it after 1and 3 areinterchanged,sincei ^ O.4

Summing up:(38:B) Among the two-element setsS,thosegiven belowarecer-

tainly necessary,and all othersarecertainlyunnecessary:

(1,4)for 7',6 all for 7\".Concerningthree-elementsets:Owing to the above we may excludeby

(36:6)all three-elementsetsfor a \", and for a ' thosewhich contain (1,4)or

(2,4).6 This leavesonly S = (1,2,3)for a '.Summingup:

(38:C) Among the three-elementsetsS, the one given below iscertainly necessary,and all othersarecertainlyunnecessary:

(1,2,3)for 7'.1 Unless y\\ y* y\\ y<.2 But there is nothing objectionablein such usesof the permutations of 1,2,3,4as we

made in the formulation of (37:2)in 37.3.2.8 This permutation and similar oneslater are clearly legitimate devicesin spite of

footnote 1above. Observefootnote 1on p.309and footnote 2 above.4 As i 1 we coulddiscardall thesesets,including S (1,4),when v((4)) 1;i.e.when 2/4 1,which is a possibility. But we areunder no obligation to do this. We

prefernot to do it, in orderto beableto treat 3/4 1and j/ 4 > 1together.6 And all permutations of 1,2,3,4;thesemodify a ' too.6Thelatter obtains from the former by interchanging 1and 2,which doesnot affect a '.)))

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324 ZERO-SUMFOUR-PERSONGAMES

We leave to the readerthe verification of (30:5:a)in 30.1.1.,i.e.that no>

a 'of V dominatesany ft of V. (Cf.the correspondingpart of the proof in36.2.4.Actually the proof of (30:5:b),which follows, also contains thenecessarysteps.)

38.2.3.We next verify (30:5:b)in 30.1.1.,i.e.that an imputation ft

which is undominatedby the elementsof V must belongto V.>

Considera ft undominatedby the elementsof V. If any two of fa, 182,

0s, ft 4 were < 0,we could make these (by permuting 1,2,3,4)fa, 2 <0.>

This gives a \" H ft with S = (1,2)of (38:B).Henceat mostone of fa, 2,

03, 4 is < 0. If none is < 0,then all are ^ 0. Soeachcomponentof ft

is < the correspondingcomponentof a\",and sinceboth areimputations

(cf.footnote 2 on p.311),it follows that they coincide,ft = a\";and soit is in V.

Hencepreciselyone of 0i, 2, 0s, ft 4 is <0. By permuting 1,2,3,4wecan make it 4.

If any two of fa, ft*, 3 were <i,we couldmake these(by permuting1,2,3)ft i, 02 <i. Besides,4 <0. So the interchangeof 3 and 4 givesa 'H ft with S = (1,2,3)of (38:C).Henceat mostone of fa, fa, 3 is <i.Ifnoneis<,then0i,02, 3 ^i. Hence4 ^ f. But04 ^ v((4))= -y4,so this necessitates 1/4 ^ $, i.e.$/4 ^ f . Thus we need i/ 4 < f toexcludethis possibility,and as we arepermuting freely 1,2,3,4,we even need

(38:9) y k <} for k = 1,2,3,4.If this conditionis satisfied,then we can concludethat preciselyoneof

fa, 02,0a is <i. By permuting 1,2,3,we can make it 3.So0i,02 ^ i,03 ^ 0. If 4 ^ -1,1 then eachcomponentof is ^ the

>

correspondingcomponentof a ', and sinceboth areimputations (cf. foot-

note2 on p.311)it follows that they coincide: = a'and so it is in V.Hence 4 < 1. Also 3 <i. So interchangeof 1 and 3 gives

~^' H 7 with S = (1,4)of (38:B).This, at last, is a contradiction,and thereby completesthe verification

of (30:5:b)in 30.1.1.Thecondition(38:9),which we neededfor this proof, is really necessary:

it is easy to verify that

7'= i*,*,*,-i}1 If v((4)) 1,i.e.if y 4 1,then this is certainly the case;but we do not wish

to assumeit. (Cf.footnote 4 on p. 323.))))

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NEIGHBORHOODOF THECENTER 325

is undominated by our V, and the only way to prevent it from being animputation is to have |< v((4))= y 4, i.e.2/4 <f.1 Permuting 1,2,3,4then gives (38:9).

Thus we needprecisely(38:8)and (38:9).Summing up:(38:D) TheV of (37:2)in 37.3.2.is a solutionfor a gameof (38:A)

(with (38:6),(38:7)there)if and only if

(38:10) 1 ^ y k <| for h = 1,2,3,4.38.2.4.Let us now reintroducethe normalization and reductionwhich

we abandonedtemporarily,but which arenecessaryin orderto refer theseresults to Q, as pointedout immediately after (38:A).

The reductionformulae of 27.1.4.show that the shareof the player fc

must bealteredby the amount ajj where))

= yk- i(2/i+ 2/2 + 2/3 + 2/4)

and))

7 = -= i(2/i+ 2/2 + 2/3 + 2/0-

Fora two-elementsetS = (i,j),v(S)is increasedfrom its original value to

<*? + a =2A + 2/7

- i(2/i+ 2/2 + 2/3 + 2/4)= i(2/ + 2/y

-yk- yi)

(fc, I arethe two playersotherthan i,j).Theabove y is clearly^ 1> (by (38:10)),hencethe game isessential.

Thenormalization isnow carriedout by dividing the characteristicfunction,as well as every player's share,by y. Thus for S = (t, j),v(S) is nowmodified further to))

,y 2/i + y* + 2/3 + 2/4

This then is the normalized and reducedform of the characteristicfunction, as used in 34.2.1.for the representationby Q. (34:2)id.gives,togetherwith the above expression,the formulae

1Observethat the failure of V to dominate this 'couldnot becorrectedby adding

ft' to V (when j/ 4 ^ !). Indeed, ft

' dominates a \" - (0,0,0,0)with S (1,2,3),so

it would be necessaryto remove a \" from V, thereby creating new undominatedimputations, etc.

If t/j t/j 2/t y 4 ss|t then a changeof unit by f brings our game back to the

form (37:1)of 37.1.2.,and it carriesthe above ft' into the o IV -

{ J, i,i,-1|of (37:3)in 37.4.1.Thus further attempts to make our V over into a solution would probablytransform it gradually into (37:3),(37:4)in 37.4.1-2.This is noteworthy, sincewestarted with (37:2)in 37.3.2.

Theseconnectionsbetween the two solutions (37:2)and (37:3),(37:4)should beinvestigated further.)))

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326)) ZERO-SUMFOUR-PERSONGAMES))

(38:11)))

x _ yi~ 2/2 - 2/8 +

2/4^12/1 + 2/2 + 2/3 + 2/4'

= -2/i+ 2/2 2/8 + 2/4;*2/1 + #2 + 2/8 + 2/4

_ -2/i--2/2 + ys + 2/4;8yi + 2/2 + 2/8 + 2/4

'))

for the coordinatesx\\, x*, X* in Q.38.2.5.Thus (38:10)and (38:11)togetherdefine the part of Q in which

thesesolutions i.e.the solution(37:2)in 37.3.2.,transformed as indicatedabove can beused. This definition is exhaustive,but implicit.Let usmake it explicit.I.e.,given a point of Q with the coordinateszi, x^ 3,letus decidewhether (38:10)and (38:11)can then be satisfiedtogether(byappropriate2/1, 2/2, 2/s, 2/4).

We put for the hypotheticalyi, 2/2, 2/8, i/ 4))

2/i + 2/2 + 2/s + 2/4= -))(38:12)

with z indefinite. Then the equations(38:11)become))

(38:12*)))

, xi2/i y* ~ 2/3 + 2/4 =

>))

,-2/i+ 2/2 - 2/3)) 2/4 = >

z))

-2/i- 2/2 \"h 2/8 + 2/4 = 8>))

(38:12)and (38:12*)can be solvedwith respectto 2/1, 2/2, 2/s,

1+ Xi xz x9i/ \\

~~~

(38:13)))I))

z-*+))

2/2 =))

2/4 =))1+ Xi + X 2))

Now (38:11)is satisfied,and we must use our freedom in choosingz tosatisfy (38:10).

Letw bethe greatestand v the smallestof the four numbers))

(38:14)))+))

Theseareknown quantities,sinceZi, x2, 0:3areassumedto begiven.Now (38:10)clearlymeansthat 1^ v/z and that w/z <$, i.e.it means

that

(38:15) $w <zgv.Obviously this conditioncan befulfilled (for z) if and only if

^38:16) fu; < v.)))

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NEIGHBORHOODOF THECENTER 327

And if (38:16)is satisfied,then condition (38:15)allows infinitely manyvalues an entireinterval for z.

38.2.6.Beforewe draw any conclusionsfrom (38:15),(38:16),we givethe explicitformulae which expresswhat has becomeof the solution (37:2)of 37.3.2.owing to our transformations. We must takethe imputations

* *a ', a \", loc.cit.,add the amount a* to the componentk (i.e.to the playerfc'sshare),and divide this by y.

Thesemanipulations transform the possiblevalues of the componentkwhich arei,0, 1in (37:2) as follows. We considerfirst k = 1,and

use the above expressionsfor ak and y as well as (38:13).Then:))

1g0e8 into *5!= 2 + 4y. - (y. + y. + y, +

2 y i + a + 3 + 4))

Ogoesinto= **- (if. + If. + If. + *) = Xl - Xt - Xt ,y yi + yt + yt + yt

_ !goesinto Jll+Jf = -4+ 4.-(yt+ *+ *+ .)))

2/1 + 2/2 + ^8 + 2/4= Z + Xi ~ Zj ~ X|.

Fortheotherk = 2,3,4theseexpressionsare changedonly in sofar that theirXi X2 zs is replacedby Xi + x2 z8, Zi 2 + Zs, Zi + 2 + z$,respectively.1

Summing up (and recalling (38:14)):(38:E) Thecomponentfc is transformed as follows:

i goesinto z/2+ w* 1,goesinto uk 1,1goesinto z + uu 1,

with the Wi, w 2, MS, u4 of (38:14).We leave it to the readerto restate(37:2)with the modification (38:E),

paying due attention to carrying out correctly the permutations 1,2,3,4which arerequiredthere.

It will be noted that for the center i.e.x\\ = x8 = X* = (38:E)reproducesthe formulae (38:1)of 38.1.1.,as it should.

38.2.7.We now return to the discussionof (38:15),(38:16).Condition(38:16)expressesthat the four numbers1*1,W2,u*, M4 of (38:14)

arenot too far apart that their minimum is more than f of their maximum

i.e.that on a relative scaletheir sizesvary by lessthan 2:3.This is certainly true at the center,where x\\ = x* = X* = 0; there

ui, t*2, MS, ^4 are all =1. Hencein this casev = w = 1,and (38:15)1This is immediate, owing to the form of the equations (38:13),and equally by con-

sidering the influence of the permutations of the players 1,2,3,4on the coordinatesXi, xj, xs as describedin 34.3.2.)))

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328 ZERO-SUMFOUR-PERSONGAMES

becomesf < z g 1proving the assertionsmade earlierin this discussion(cf.(38:3)in 38.1.1.).

Denotethe part of Q in which (38:16)is true by Z. Then even asufficiently small neighborhoodof the centerbelongsto Z.1 So Z is athree-dimensionalpiecein the interior of Q, containing the centerin its owninterior.

We can alsoexpressthe relationshipof Z to the diagonalsof Q, say to/-Center-VIIL Z contains the following parts of that diagonal. (UseFigure64):on one side preciselyC, on the othera little less than half ofJ3.* We add that thesesolutionsare different from the family of solu-tionsvalid in (36:B)and (36:C)which were referredto in 36.3.

38.3.Interpretation of TheSolutions

38.3.1.Thefamily of solutionswhich we have thus determinedpossessesseveralremarkablefeatures.

We note first that for every gamefor which this family is a solutionat all (i.e.in every point of Z) it gives infinitely many solutions.1 And allwe said in 37.5.1.appliesagain:thesesolutionsarefinite setsof imputations*and possessthe full symmetry of the game.6 Thus thereis no \" discrimina-tion \" in any one of thesesolutions. Nor can the differences in \"organi-zational principles,\"which wediscussedloc.cit.,beascribedto them. Thereis neverthelessa simple\" organizational principle'1 that can bestated in aqualitative verbal form, to distinguish these solutions. We proceedtoformulate it.

38.3.2.Consider (38:E),which expressesthe changesto which (37:2)in 37.3.2.is to be subjected.It is clearthat the worst possibleoutcomefor the playerk in this solution is the last expression(sincethis correspondsto 1), i.e. z + Uk 1. This expressionis > or = 1,accordingtowhether z is < or = u k. Now Wi, w 2, MS, w4 arethe four numbersof (38:14),thesmallestof which isv. By (38:15)z v, i.e.always z + w* 1^ 1,and = occursonly for the greatestpossiblevalue of z, z = v, and thenonly for thosek for which Uk attains its minimum, v.

1If Xi, xs,Xi differ from by <A then eachof the four numbers MI, MJ, MI, u 4of (38:14)is < 1 -f A I and > 1 A 1; henceon a relative sizethey vary by < J :t f.Sowe are still in Z. In other words:Z contains a cubewith the same centeras Q,butwith A of Q's(linear) size.

Actually Z is somewhat bigger than this, its .volume is about yAo of the volume of Q.*On that diagonal x\\ x xt, so the MI, MI, MI, M 4 are:(three times) 1 x\\ and

1 -f 3xi. So for zi 0, v - 1 - xb w - 1 + 3xi, hence (38:16)becomesx\\ < J.And for si g 0, v - 1 + 3xi, w - 1 - Xi, hence(38:16)becomes> - A. So theintersection is this:

Xi < t (this is preciselyC)xi > - A (B is xi > - i).

1Thesolution which we found contained four parameters:y\\, yi, y t, y* while the gamesfor which they arevalid had only three parameters:&i, x>, xv.

4Eachonehas 13elements,like (37:2)in 37.3.2.1In the centerxi - xt -xt - we have y\\ - y* - y* - y 4 (cf.(38:13)),i.e.sym-

metry in 1,2,3,4.On the diagonal Xi - xt - x9 we have j/i - y f - y t (cf.(38:13)),i.e.symmetry in 1,2,3.)))

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NEIGHBORHOODOF THECENTER 329We restatethis:

(38:F) In this family of solutions,even as the worstpossibleoutcome,a player k faces, in general,somethingthat is definitely betterthan what he could get for himself alone, i.e.v((fc)) = 1.This advantage disappearsonly when z has its greatestpossiblevalue, z = v, and then only for thosek for which thecorrespond-ing number iii, w 2, w 8, w 4 in (38:14)attains the minimum in(38:14).

In otherwords:In thesesolutionsa defeatedplayer is in generalnot11 exploited\" completely,not reducedto the lowestpossiblelevel the levelwhich he could maintain even alone,i.e.v((fc)) = 1. We observedbeforesuch restraint on the part of a victorious coalition,in the \"milder\"kind of \"discriminatory\"solutionsof the three-persongamediscussedin33.1.(i.e.when c > 1,cf. the end of 33.1.2.).But thereonly oneplayercouldbe theobjectof this restraintin any one solution,and this phenomenonwent with his exclusionfrom the competitionfor coalitions.Now thereisno discriminationor segregation;insteadthis restraint appliesto all playersin general,and in the centerof Q ((38:1)in 38.1.1.,with z < 1)the solutionis even symmetric!1

38.3.3.Even when z assumesits maximum value v, in generalonly oneplayerwill losethis advantage, sincein generalthe four numbersui, wj, MS,

Ui of (38:14)aredifferent from eachotherand only oneis equal to theirminimum v. All four playerswill loseit simultaneouslyonly if u\\, w, Wt, u\\

are all equal to their minimum v i.e.to each other and one look at(38:14)suffices to show that this happens only when x\\ = x\\ = x\\ = 0,i.e.at the center.

This phenomenon of not \"exploiting\"a defeatedplayercompletelyis avery important possible(but by no meansnecessary)feature of our solu-tions, i.e.of socialorganizations. It is likelyto play a greaterrolein thegeneraltheory also.

We concludeby stating that someof the solutionswhich we mentioned,but failed to describein 36.3.2.,also possessthis feature. Thesearethesolutionsin C of Figure64. But neverthelessthey differ from the solutionswhich we have consideredhere.

1 Thereis a quantitative difference of somesignificance aswell. Both in our presentset-up(four-person game, centerof Q)and in the one referredto (three-persongame inthe senseof 33.1.),the besta player can do (in the solutions which we found) is J, andthe worst is 1.

Theupper limit of what he may get in caseofdefeat,in thoseof our solutions wherehe is not completely

\"exploited,\" is now - J (i.e.-zwith \\ <z 1)and it was \\ then

(i.e.cwith 1 & c < J). Sothis zonenow coversthe fraction .__ ( 1) **4 \"*

9*

i.e.22|% of the significant interval, while it then covered100%.)))

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CHAPTER VIIISOME REMARKSCONCERNINGn 5 PARTICIPANTS

39.TheNumberof Parametersin Various Classesof Games39.1.TheSituation for n = 3,4

39.1.We know that the essentialgames constituteour real problemand that they may always be assumedin the reducedform and with 7 = 1.In this representation there exists preciselyone zero-sum three-persongame,while the zero-sum four-persongames form a three-dimensionalmanifold.1 We have seenfurther that the (unique)zero-sumthree-persongame is automatically symmetric,while the three-dimensionalmanifoldof all zero-sum four-person gamescontainspreciselyone symmetricgame.

Let us expressthis by stating, for eachoneof the above varieties ofgames,how many dimensionsit possesses,i.e.how many indefinite param-etersmust be assignedspecific(numerical) values in order to characterizeagameof that class. Thisis best donein the form of a table,given in Figure65 in a form extendedto all n ^ 3.2 Our statementsabove reappearinthe entriesn = 3,4 of that table.

39.2.TheSituation for All n 3

39.2.1.We now complete the table by determining the number ofparametersof the zero-sum n-persongame,both for the classof all thesegames,and for that of the symmetricones.

The characteristicfunction is an aggregateof as many numbers v(S)astherearesubsetsS in / = (1, , n), i.e.of 2n. Thesenumbersaresubjectto the restrictions(25:3:a)-(25:3:c)of 25.3.1.,and to thosedue to thereducedcharacterand the normalization 7 = 1,expressedby (27:5)in27.2.Of these (25:3:b)fixes v(-S)whenever v(S) is given, hence ithalves the number of parameters:3 so we have 2n~l instead of 2n. Next(25:3:a)fixesoneof the remaining v (S):v(); (27:5)fixesn of the remainingv(S):v((l)), , v((n)); hencethey reducethe number of parametersby n + I.4 Sowe have 2n~l n 1parameters.Finally (25:3:c)neednot beconsidered,sinceit containsonly inequalities.

39.2.2.If thegameis symmetric,then v(S)dependsonly on the numberof elementsp of S:v(S) = vp, cf. 28.2.1.Thus it is an aggregateof asmany numbers vp as there are p = 0, 1, , n, i.e.n + 1. These

1Concerning the general remarks, cf. 27.1.4.and 27.3.2.;concerning the zero-sumthree-persongamecf.29.1.2.;concerning the zero-sum four-person gamecf.34.2.1.

*Thereare no essentialzero-sum gamesfor n 1,2!8 Sand Sarenever the sameset!4S Qt (1), , (n) differ from eachother and from eachother'scomplements.

330)))

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NUMBER OF PARAMETERS)) 331))

numbersaresubjectto the restrictions(28:ll:a)-(28:ll:c)of 28.2.1.;thereducedcharacteris automatic, and we demand also Vi = y = 1.(28:11:b) fixes vw_p when vp is given; henceit halves thenumbersof thoseparametersfor which n-p^p.When n -p = p l i.e.n = 2p,which hap-pensonly when n is even, and then p = n/2 (28:11:b)showsthat this v

p))

must vanish. Sowe have ^ parametersif n is odd and 5 if n is even,instead of the original n + 1. Next (28:ll:a)fixes oneof the remainingvp: Vo) and Vi = 7 = 1 fixes another one of the remaining vp: Vi;

hencethey reducethe number of parametersby 2:2 so we have)) -2))

n))

or ^~ 2 parameters.Finally (28:ll:c)neednot be consideredsinceit

containsonly inequalities.39.2.3.We collectall this information in the tableof Figure65. We also

stateexplicitly the values arising by specializationto n = 3,4,5,6,7,8,the first two of which werereferred to previously.))

Numberof players)

All games) Symmetric) games)

3) 0*) 0*)

4) 3) 0*)

5) 10) 1)

6) 25) 1)

7) 56) 2)

8) 119) 2)

. . .) . . .)

n) Of) or n odd

n even)

2

|-2for))* Denotesthe game is unique.

Figure 65. Essential games. (Reducedand y 1.)

The rapid increaseof the entriesin the left-hand column of Figure65may serveas anotherindication,if onebeneeded,how the complexityof agameincreaseswith the number of its participants. It seemsnoteworthy

1Contrast this with footnote 3 on p.330!1p - 0, 1differ from eachother and from eachother'sn - p. (Thelatter only

becauseof n 3.))))

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332 REMARKS CONCERNING n 5 PARTICIPANTS

that thereis an increasein the right-hand column too,i.e.for the symmetricgames,but a much slowerone.

40.TheSymmetricFive-personGame40.1.Formalism of the Symmetric Five-personGame

40.1.1.We shallnot attempt a directattackon the zero-sum five-persongame. The systematic theory is not far enough advanced to allow it;and for a descriptiveand casuisticapproach (as used for the zero-sum,four-persongame)the numberof its parameters,10,is ratherforbidding.

It is possiblehowever to examinethe symmetriczero-sumfive-persongamesin the lattersense. The number of parameters,1,is smallbut notzero,and this is a qualitatively new phenomenon deservingconsideration.For n = 3,4 there existedonly one symmetricgame,so it is for n = 5that it happensfor the first time that the structureof the symmetricgameshowsany variety.

40.1.2.Thesymmetriczero-sumfive-persongame ischaracterizedby thevp, p = 0,1,2,3,4,5of 28.2.1.,subjectto the restrictions(28:ll:a)-(28:ll:c)formulated there. (28:ll:a),(28:ll:b)state(with 7 = 1)

(40:1) v = 0, vi=-l, v4 = 1, v 6 =

and v2 = v3, i.e.(40:2) v2 = -ij, v 3 =

7j

Now (28:ll:c)statesvp+g ^ vp + v q for p + q ^ 5 and we can subjectPJ q to the further restrictionsof (28:12)id. Therefore p = 1,q = 1,2,*and so thesetwo inequalitiesobtain (using(40:1),(40:2)):

p = 1,q = 1: -2^ -17; p = 1,q = 2: 1 i? ^ iy;

i.e.(40:3) - i g rj g 2.

Summing up:(40:A) The symmetriczero-sumfive-person game is characterized

by one parametert\\ with the helpof (40:1),(40:2).Thedomainof variability of y is (40:3).

40.2.TheTwoExtreme Cases40.2.1.Itmay beuseful to give a directpictureof the symmetricgames

describedabove. Let us first considerthe two extremesof the interval(40:3):

1= 2,-*.1This is easily verified by inspection of (28:12),or by using the inequalities of foot-

note 2 on p. 250. Thesegive 1 p f, 1 ^ 4 j 2; henceas p, q are integers, p 1,q -

1,2.)))

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THESYMMETRIC FIVE-PERSONGAME 333

Consider first 17= 2:In this casev(S) = 2 for every two-element

setS;i.e.every coalition of two players is defeated.1 Thus a coalition ofthree(beingthe setcomplementaryto the former) is a winning coalition.This tells the whole story: In the gradual crystallizationof coalitions,thepoint at which the passagefrom defeat to victory occursis when the sizeincreasesfrom two to three,and at this point the transition is 100%.2

Summing up:(40:B) 77

= 2 describesa gamein which the only objectiveof allplayersis to form coalitionsof threeplayers.

40.2.2.Considernext 17= i. In this casewe argueas follows:))

1))

when S has))

4))

elements.))

A coalition of four always wins.3Now the above formula showsthat a coalition of two is doingjust as

well, pro ratdj as a coalition of four; henceit is reasonableto considertheformer just as much winning coalitionsas the latter. If we take thisbroaderview of what constitutes winning, we may again affirm that thewhole story of the game has beentold:In the formation of coalitions,thepoint at which the passagefrom defeat to victory occursis when the sizeincreasesfrom one to two; at this point the transition is 100%.4

Summing up:(40:C) rj

= i describesa gamein which the only objectiveof allplayersis to form coalitionsof two players.

40.2.3.On the basisof (40:B)and (40:C)it would be quiteeasyto guessheuristicallysolutions for their respectivegames.This, as well as theexactproof that thosesetsof imputationsarereally solutions,is easy; butwe shall not considerthis matter further.

Beforewe passto the considerationof theotherrj of (40:3)letus remarkthat (40:B)and (40:C)areobviously the simplest instancesof a general

1 Cf.the discussion in 35.1.1.,particularly footnote 4 on p. 296.2Oneplayer is just as much defeatedas two, four areno more victorious than three.

Ofcoursea coalition of three has no motive to take in a fourth partner ; it seems(heuristi-cally) plausible that if they do they will accepthim only on the worst possibleterms. Butneverthelesssuch a coalition of four wins if viewed asa unit, sincethe remaining isolatedplayer is defeated.

8 In any zero-sum n-person game any coalition of n 1wins, sincean isolatedplayeris always defeated. Cf.loc.cit. above.

4Oneplayer is defeated,two or four players are victorious. A coalition of threeplayers is a compositecasedeserving someattention :v(S)is J for a three-element setSt i.e.it obtains from the i of a two-element setby addition of 1.Thus a Coalition ofthree is no better than a winning coalition of two (which it contains) plus the remainingisolatedand defeatedplayer separately. This coalition is just a combination of a.win-ning and a defeatedgroup whose situation is entirely unaltered by this operation.)))

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334 REMARKS CONCERNINGn 5 PARTICIPANTS

method of defining games.Thisprocedure(which is more general than thatof Chapter X, referred to in footnote 4 on p.296) will be consideredexhaus-tively elsewhere(for asymmetricgamesalso). It issubjectto somerestric-tionsof an arithmetical nature ; thus it isclearthat therecan beno (essentialsymmetriczero-sum)n-persongame in which every coalition of p is winningif p is a divisor of n, sincethen n/p suchcoalitionscouldbeformed and every-bodywould win with no loserleft. On the other hand the samerequirementwith p = n 1 doesnot restrictthe game at all (cf. footnote 3,p.333).40.3.Connection between the Symmetric Five-personGameand the 1,2,3-symmetric

Four-personGame40.3.1.Considernow the rj in the interior of (40:3).The situation is

somewhat similar to that discussedat the end of 35.3.We have someheuristicinsight into the conditionsat the two ends of (40:3)(cf. above).Any point ry of (40:3)is somehow \" surrounded\" by theseend-points. Moreprecisely,it is their centerof gravity if appropriate weights are used.1Theremarksmadeloc.cit.apply again:while this constructionrepresentsall gamesof (40:3)as combinationsof the extremecases(40:B),(40:C),itis neverthelessnot justified to expectthat the strategiesof the former canbe obtainedby any directprocessfrom thoseof the latter. Our experiencesin the caseof the zero-sum four-persongame speakfor themselves.

Thereis, however, another analogy with the four-person game whichgivessomeheuristicguidance. Thenumber of parametersin our caseis thesame as for those zero-sumfour-persongames which aresymmetric with

respectto the players1,2,3;we have now the parameter 77 which runs over

(40:3) -i ^ T? g 2,

while the gamesreferred to had the parameterx\\ which varies over

(40:4) -1 xi I.2This analogy betweenthe (totally) symmetricfive-persongame and the

1,2,3-symmetricfour-persongame is so far entirely formal. There is,however, a deepersignificancebehindit. To seethis we proceedas follows:

40.3.2.Considera symmetric five-person gameF with its 77 in (40:3).Let us now modify this game by combining the players4 and 5 into oneperson,i.e.one player4'. Denotethe new game by F'. It is important torealize that F' is an entirely new game:we have not assertedthat in F

players4 and 5 will necessarilyacttogether,form a coalition, etc.,or thatthereareany generally valid strategicalconsiderationswhich would moti-vate just this coalition.8 We have forced 4 and 5 to combine;we did thisby modifying the rulesof the game and thereby replacingF by F'.

1Thereadercaneasilycarry out this composition in the senseof footnote 1on p.304,relying on our equations (40:1),(40:2)in 40.1.2.

*Cf.35.3.2.In the representation in Qused there, x\\ = xt xt.9 This ought to becontrasted with the discussion in 36.1.2.,where a similar combina-

tion of two players was formed under such conditions that this merger seemedstrategi-cally justified.)))

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THESYMMETRIC FIVE-PERSONGAME 335Now T is a symmetricfive-person game,while r' is a 1,2,3-symmetric

four-persongame.1 Given the > of T we shall want to determine thexiof I\" in orderto seewhat correspondenceof (40:3)and (40:4)thisdefines.Afterwards we shall investigate whether therearenot, in spite of what wassaid above,someconnectionsbetweenthe strategies i.e.the solutions ofT and T'.

The characteristicfunction v'(S) of r' is immediately expressibleinterms of the characteristicfunction v(S)of T. Indeed:

v'((l)) = v((l)) = -1, v'((2))= v((2))= -1,v'((3))= v((3))= -1, v'((4')) = v((4,5))= -i,;

v'((l,2)) = v((l,2)) = -,, v'((l,3)) = v((l,3)) = -n,v'((2,3))= v((2,3)).-,, v'((l,4'))= v((l,4,5))- ,,v'((2,4'))= v((2,4,5))= , v'((3,4'))= v((3,4,5))= ,;

v'((l,2,3))= v((l,2,3))= n, v'((l,2,4'))= v((l,2,4,5))= 1,v'((l,3,4'))= v((l,3,4,5))= 1, v'((2,3,4'))= v((2,3,4>5)) = 1;

and of coursev'(0)= v'((l,2,3,4'))= 0.

While F was normalized and reduced,r\" is neither;and we must bringT'into that form sincewe want to computeits x\\, 2, 3, i-e.refer it to theQ of 34.2.2.

Let us therefore apply first the normalization formulae of 27.1.4.Theyshow that the share of the player k = 1,2,3,4'must be altered by theamount ajj where

al =, -and

7 = -Hence))

_3 l 5This 7 is clearly ^ ^~ =

^ > (by (40:3));hencethe game is

essential.The normalization is now carriedout by dividing every player'sshare by 7.

Thus for a two-elementsetS = (i,j),v'(S) is replacedby))

7Consequentlya simplecomputation yields))

o -t- i?

v\"((l,4'))= v\"((2,4'))= v\"((3,4'))= 2(3\" ~ 1}-

1 Theparticipants in r areplayers 1,2,3,4,5,who all have the samerolein the originalT. Theparticipants in r'areplayers 1,2,3and the compositeplayer (4,5):4'. Clearly1,2,3have still the same role, but 4' is different.)))

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336 REMARKS CONCERNING n 5 PARTICIPANTS

This then is the normalized and reducedform of the characteristicfunction, as used in 34.2.for the representationby Q. (34:2)in 34.2.1gives,togetherwith the aboveexpression,the formula))

Takingx\\ = x* = x* for granted,this relation can alsobewritten asfollows:

(40:5) (3 - xO(3+ n) = 10.Now it is easy to verify that (40:5)maps the Tj-domain (40:3)on the

Xi-domain (40:4). The mapping is obviously monotone. Itsdetails areshown in Figure66 and in the adjoiningtableof correspondingvalues ofxiand 17. Thecurve in this figure representsthe relation (40:5)in the x\\ 9

ly-plane. This curve is clearly (an arcof) a hyperbola.40.3.3.Our analysis of the 1,2,3-symmetricfour-person game has

culminated in the result statedin 36.3.2.:The game,i.e.the diagonal7-Center-V7/7in Q which representsthem, is divided into five classesA-E, each of whrch is characterizedby a certain qualitative type ofsolutions. The positions of the zones A-E on the diagonal 7-Center-VIII,i.e.the interval 1g x\\ ^ 1,areshown in Figure64.

The presentresults suggesttherefore the considerationof the cor-respondingclassesof symmetricfive-person gamesF in the hopethat someheuristic leadfor the detectionof their solutionsmay emergefrom theircomparisonwith the 1,2,3-symmetricfour-persongamesF, classby class.

Usingthe tablein Figure66 we obtain the zonesA-E in ^ 17 g 2,which are the images of the zones A-E in 1 x\\ ^ 1. The detailsappear in Figure67.

A detailedanalysisof the symmetricfive-person gamescanbecarriedout on this basis. It disclosesthat the zonesA, B do indeedplay the rolewhich we expect,but that the zonesC, D, E must be replacedby others,<?',D'. ThesezonesA-D'in ^ t\\

z* 2 and their inverse imagesA-D'in 1^ Xi ^ 1 (again obtained with the help of the table of Figure66)areshown in Figure68.

It is remarkablethat theZi-diagramof Figure68 showsmoresymmetrythan that of Figure67,although it is the latterwhich is significant for the1,2,3-symmetricfour-persongames.

40.3.4.The analysis of symmetric five-person games has also someheuristic value beyond the immediate information it gives. Indeed,bycomparing the symmetric five-person game F and the 1,2,3-symmetricfour-persongameF'which correspondsto it, and by studyingthedifferencesbetween their solutions,one observesthe strategiceffects of the mergerof players4 and 5 in one(composite)player4'. Totheextentto which thesolutionspresentno essentialdifferences (which is the casein the zonesA,B,as indicatedabove)onemay say that this mergerdid not affect the really)))

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THE SYMMETRICFIVE-PERSONGAME 337))

Figure 66.Correspondingvalues of Xi and 17:*i: -1 -I -i } J i J 1*:-!-i i A ! 1 2))

1) A \\ I)< |C|D \\ B |)

-I) t i 1)

1 A) 1 B \\C\\ D I fi)1))

o t i A I i

Figure 67.))

I B | C-))

-t-))

-l)) -i o

A 1*1*1))Figure 68.)))

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338 REMARKS CONCERNINGn 5 PARTICIPANTS

significant strategicconsiderations.1 On the otherhand, when such differ-encesarise(this happens in the remaining zones)we face the interestingsituation that even when 4 and 5 happen to cooperatein T, their jointpositionis dislocatedby the possibilityof their separation.2

Spaceforbidsa fullerdiscussionbasedon the rigorousconceptofsolutions.1Ofcourseonemust expect,in solutions of F, arrangements where the players4 and 5

are ultimately found in opposing coalitions. It is clearthat this can have no parallelin F'. All we mean by the absenceof essentialdifferences is that those imputations in asolution of Twhich indicatea coalition of4and 5 should correspondto equivalent imputa-tions in the solution of r'.

Theseideasrequire further elaboration, which is possible,but it would leadtoo farto undertake it now.

1Already in 22.2.,our first discussion of the three-persongame disclosedthat thedivision of proceedswithin a coalition is determined by the possibilities of eachpartnerin caseof separation. But this situation which we now visualize is different. In ourpresentr it can happen that even the total share of player 4 plus player 5 is influencedby this \"virtual\" fact.

A qualitative ideaof such a possibility is bestobtained by considering this :When apreliminary coalition of 4 and 5 is bargaining with prospectivefurther allies, their bar-gaining position will be different if their coalition is known to be indissoluble (in r')than when the oppositeis known to be a possibility (in r).)))

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CHAPTER IXCOMPOSITIONAND DECOMPOSITIONOF GAMES

41.Compositionand Decomposition41.1.Searchfor n-personGamesfor Which All Solutions Can BeDetermined

41.1.1.Thelast two chapters will have conveyeda specificideaof therapidity with which the complexityof our problemincreasesas the numbern of participants goesto 4,5, etc. In spite of their incompleteness,those considerationstended to be so voluminous that it must seemcom-pletelyhopelessto pushthis casuistic approachbeyondfive participants.1Besides,the fragmentary characterof the resultsgainedin this manner veryseriouslylimits their usefulnessin informing usabout the generalpossibilitiesof the theory.

On the other hand, it is absolutelyvital to get someinsight into theconditionswhich prevail for the greatervalues of n. Quiteapart from thefact that thesearemost important for the hoped for economicand socio-logicalapplications,there is also this to consider:With every increaseofn, qualitatively new phenomenaappeared.This was clearfor each ofn = 2,3,4(cf. 20.1.1.,20.2.,35.1.3.,and also the remarks of footnote 2on p.221),and if we did not observeit for n = 5 this may be due to ourlack of detailedinformation about this case. It will developlater,(cf. theend of 46.12.),that very important qualitative phenomenamake theirfirst appearancefor n = 6.

41.1.2.Forthesereasonsit is imperative that we find sometechniquefor the attackon gameswith higher n. In the presentstateof things wecannot hope for anything systematic or exhaustive. Consequentlythenatural procedureis to find somespecialclassesof gamesinvolving manyparticipants2 that can be decisivelydealt with. It is a generalexperiencein many parts of the exactand natural sciencesthat a completeunder-standing of suitablespecialcases which aretechnicallymanageable,butwhich embodythe essentialprinciples has a goodchanceto be the pace-maker for the advance of the systematicand exhaustivetheory.

We will formulate and discusstwo such families of specialcases. Theycan be viewed as extensivegeneralizationsof two four-persongames sothat eachoneof thesewill be the prototype of oneof the two families.Thesetwo four-persongamescorrespondto the 8 cornersof the cubeQ,introduced in 34.2.2.:Indeed,we saw that thosecornerspresentedonly

1As was seen in Chapter VIII, it was already necessaryfor five participants torestrict ourselvesto the symmetric case.

1And in such a manner that eachoneplays an essentialrole!339)))

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340 COMPOSITIONAND DECOMPOSITIONOF GAMES

two strategically different types of games the corners/, Vj VI, VII,discussedin 35.1.and the corners//, ///, IV, VIII,discussedin 35.2.Thus the corners/ and VIIIof Q arethe prototypesof thosegeneraliza-tions to which this chapterand the followingone will be devoted.

41.2.TheFirst Type. Composition and Decomposition

41.2.1.We first considerthe cornerVIIIof Q, discussedin 35.2.Aswas brought out in 35.2.2.this gamehas the followingconspicuousfeature:Thefour participantsfall into two separatesets(oneof threeelementsandthe otherof one)which have no dealingswith eachother. I.e.the playersof eachsetmay be consideredas playing a separategame,strictly amongthemselvesand entirelyunrelated to the other set.

The natural generalization of this is to a game F of n = A; + I par-ticipants,with the followingproperty:Theparticipantsfall into two setsofk and I elements,respectively,which have no dealingswith eachother.I.e.the playersof eachsetmay be consideredplaying a separategame,sayA and H respectively,strictly among themselvesand entirely unrelatedto the other set.1

We will describethis relationshipbetween the gamesT, A, H by thefollowing nomenclature:Compositionof A, H producesF, and converselyF can be decomposedinto the constituentsA, H.2

41.2.2.Beforewe deal with the above verbal definitions in an exactway, somequalitative remarksmay be appropriate:

First,it shouldbenotedthat our procedureof compositionand decompo-sition is closelyanalogousto one which has beensuccessfullyapplied in

many parts of modern mathematics.3 As thesematters are of a highlytechnicalmathematical nature, we will not say more about them here.Sufficeit to statethat our presentprocedurewaspartly motivated by thoseanalogies.The exhaustivebut not trivial results, which we shall obtain

1In the original game of 35.2.the secondsetconsistedofoneisolatedplayer, who wasalsotermed a \"

dummy.\" This suggestsan alternative generalization to the aboveone:A game in which the participants fall into two setssuch that those of the first set play agame strictly among themselves etc.,while thoseof the secondsethave no influence uponthe game, neither regarding their own fate, nor that of the others. (Theseare then the\"dummies.\

This is, however, a specialcaseof the generalization in the text. It is subsumed init by taking the game H of the secondsetasan inessential one,i.e.onewhich hasa definitevalue for eachoneof its participants that cannot beinfluenced by anybody. (Cf.27.3.1.and the end of 43.4.2.A player in an inessential game could conceivably deterioratehisposition by playing inappropriately. We ought to excludethis possibility for a\"dummy\" but this point is of little importance.)

Thegeneraldiscussion,which wearegoing to carry out (both gamesA and H essen-tial) will actually disclosea phenomenon which doesnot arisein the specialcaseto whichthe corner VIIIof 35.2.belongs i.e.the caseof \"dummies\" (H inessential). Thenew phenomenon will be discussedin 46.7.,46.8.,and the caseof \"dummies\" wherenothing new happens in 46.9.

2Itwould seemnatural to extend the conceptsof composition and decomposition tomore than 2 constituents. This will becarriedout in 43.2.,43.3.

3Cf.G. Birkhoff & S.MacLane:A Survev of Modern Ahrebra. New York. 1941.Chapt.XIII.)))

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COMPOSITIONAND DECOMPOSITION 341and alsobe able to use for further interpretationsarea rather encouragingsymptomfrom a technicalpoint of view.

41.2.3.Second,the readermay feel that the operationof compositionis of an entirely formal and fictitious nature. Why should two games,A and H,playedby two distinct setsof playersand having absolutelynoinfluence upon eachother,be consideredas one game F?

Our result will disclosethat the completeseparation of the games A

and H, as far as the rules areconcerned,does not necessarilyimply thesame for their solutions. I.e.:Although the two setsof players cannotinfluence eachotherdirectly,neverthelesswhen they areregardedas oneset, one society there may be stable standards of behaviour whichestablishcorrelationsbetweenthem.1 Thesignificanceof this circumstancewill be elaboratedmore fully when we reachit loc.cit.

41.2.4.Besides,it shouldbe noted that this procedureof compositionis quite customary in the natural sciencesas well as in economictheory.Thus it is perfectly legitimateto considertwo separatemechanical systemssituated, to take an extremecase,say one on Jupiterand one on Uranusas one. It is equally feasibleto considerthe internal economiesof twoseparatecountries the connectionsbetweenwhich aredisregarded asone.This is, of course,the preliminary step beforeintroducingthe interactingforcesbetweenthosesystems. Thus we couldchoosein our first exampleas those two systems the two planets Jupiter and Uranus themselves(bothin the gravitational fieldof the Sun),and then introduceas interactionthe gravitational forces which the planets exerton eachother. In OUT

secondexample,the interaction enters with the considerationof inter-national trade,international capitalmovements, migrations,etc.

We could equally use the decomposablegameF as a stepping stoneto othergamesin its neighborhood,which, in their turn, permit no decompo-sition.2

In our present considerations,however, theselattermodifications will

not be considered.Our interestis in the correlationsintroducedby thesolutionsreferred to at the beginning of this paragraph.

41.3.Exact Definitions

41.3.1.Let us now proceedto the strictly mathematical descriptionof the compositionand decompositionof games.

Let k players1', , fc', forming the setJ = (!', , k f

) play thegame A; and I players 1\", , I\", forjning the setK = (1\", , V)play the gameH. We re-emphasizethat A and H are disjoint setsofplayersand8 that the gamesA and H arewithout any influence upon each

1 Thereis sortie analogy between this and the phenomenon noted before (cf.21.3.,37.2.1.)that a symmetry of the game neednot imply the samesymmetry in all solutions.

1Cf.35.3.3.,applied to the neighborhood of corner/.,which according to 35.2.is adecomposablegame. The remark of footnote 2 on p. 303on perturbations is alsopertinent.

8 If the same players 1, t n are playing simultaneously two games, then an

entirely different situation prevails. That is the superposition of games referred to in)))

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342 COMPOSITIONAND DECOMPOSITIONOF GAMES

other. Denotethe characteristicfunctions of thesetwo games by v&(S)and vH(!T)respectively,where S zJ and T K.

In forming the compositegameF, it is convenient to use the samesymbols 1', ,*',1\", ,/\" for its n = k + I players.1 Theyform the set/ = J u K = (!', -,*',1\", , V).

Clearlyevery setR / permitsa unique representation

(41:1) # = SuT, SsJ, TzK;the inverseof this formula being

(41:2) S = #n/, T = R n K*Denotethe characteristicfunction of the game F by vr(fi) with R /.

The intuitive fact that the games A and H combinewithout influencingeachotherto F has this quantitative expression:The value in F of a coalitionR I obtainsby additionof the value in A of its part S (sJ) in J and ofthe value in H of its part T (sK) in K. Expressedby a formula:

(41:3)vr(fl) = vA (S)+ vR (T) where R, S, T are linked by (41:1),i.e.(41:2).8

41.3.2.Theform (41:3)expressedthe compositevr(R) by meansof itsconstituents VA(), vH(T). However,it also contains the answer to theinverseproblem:ToexpressVA(/S), vH(T)by v r(#).

Indeed vA (0)= v H(0)= O.4 Hence putting alternately T =and S = Q in (41:3)gives:

(41:4) vA (5)=vr(S) for SsJ,(41:5) v H(r)= v r(r) for T <= #.6

We arenow in a positionto expressthe fact of the decomposdbilityof thegameF with respectto the two setsJ and K. I.e.:the given game F

(among the elementsof / = J u K) is such that it can be decomposedintotwo suitablegamesA (among the elementsJ) and H (among the elementsof K). As stated,this is an implicitproperty of F involving the existenceof the unknown A, H. But it will be expressedas an explicitpropertyof F.

Indeed:If two such A, H exist,then they cannot be anything but thosedescribedby (41:4),(41:5).Hencethe propertyof F in questionis, that the

27.6.2.and alsoin 35.3.4.Its influences on the strategy are much more complex andscarcelydescribableby generalrules, aswaspointed out at the latter loc.cit.

1Insteadof the usual 1, , n.1Theseformulae (41:1),(41:2)have an immediate verbal meaning. Thereadermay

find it profitable to formulate it.8 Of course,a rigorous deduction on the basisof 25.1.3.is feasiblewithout difficulty.

All of 25.3.2.appliesin this case.4 Note that the empty set is a subset of both / and K\\ sinceJ and K aredisjoint,

it is their only common subset.6This is an instance of the technical usefulness of our treating the empty set asa

coalition. Of.footnote 2 on p. 241.)))

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COMPOSITIONAND DECOMPOSITION 343

A, H of (41:4),(41:5)fulfill (41:3).Substituting,therefore, (41:4),(41:5)into (41:3),using(41:1)to expressR in termsof /S, T gives this:(41:6) vr(Su T) = vr (S)+ v r(T) for S J, T7 S#.Or, if we use (41:2)(expressingS,,rin termsof R) in placeof (41:1)(41:7) vr(fi) = v r (fi n J) + v r (fl n X) for Bel.

41.3.3.In order to seethe role of the equations (41:6),(41:7)in theproper light, a detailedreconsiderationof the basic principlesupon whichthey rest,is necessary. This will be done in sections41.4.-42.5.2.whichfollow. However, two remarks concerningthe interpretation of theseequationscan be madeimmediately.

First:(41:6)expressesthat a coalition betweena setS sJ and a setT SK has no attraction that while there may be motives for playerswithin J to combine with eachother,and similarly for playerswithin K,thereareno forces acting acrossthe boundariesof J and K.

Second:To thosereaderswho are familiar with the mathematical theoryof measure,we make this further observation in continuation of thatmade at the end of 27.4.3.:(41:7)is exactly Carath6odory'sdefinition ofmeasurability. Thisconceptisquite fundamental for the theory of additivemeasureand Carath6odory'sapproachto it appears to be the technicallysuperiorone to date.l Itsemergencein the presentcontextisa remarkablefact which seemsto deservefurther study.

41.4.Analysis of Decomposability

41.4.1.We obtained the criteria (41:6),(41:7)of Y's decomposabilityby substituting the v A (/S),vH(Tr

) obtained from (41:4),(41:5)into thefundamental condition (41:3).However,this deductioncontainsa lacuna:We did not verify whether it is possibleto find two gamesA, H which pro-ducethe vA (S),vH (5P) formally defined by (41:4),(41:5).

Thereis no difficulty in formalizing theseextrarequirements.As weknow from 25.3.1.they mean that v*(S)and vH(T) fulfill the conditions(25:3:a)-(25:3:c)eod. It must be understoodthat we assume the givenVr(B) to originate from a game F, i.e.that vp(fi) fulfills theseconditions.Hencethe following questionpresentsitself:(41:A) v r(B) fulfills (25:3:a)-(25:3:c)in 25.3.1.togetherwith the

above (41:6),i.e.(41:7).Will then the vA (S) and vH(T) of(41:4),(41:5)also fulfill (25:3:a)-(25:3:c)in 25.3.1.?Or, if thisis not the case,which further postulate must be imposeduponvr(B)?

Todecidethis, we check(25:3:a)-(25:3:c)of 25.3.1.separatelyfor v A (S)and vH(77). It is convenient to takethem up in a different order.

41.4.2.Ad (25:3:a):By virtue of (41:4),(41:5),this is the samestate-ment for VA(*S) and vH(T)as for v r(#).

1Cf. C.Carathtodwy: Vorlesungen liber ReelleFunktionen, Berlin, 1918,Chapt.V.)))

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344 COMPOSITIONAND DECOMPOSITIONOF GAMES

Ad (25:3:c):By virtue of (41:4),(41:5),this carriesover to vA (S)andVn(T) from Vr(fi) it amounts only to a restriction from the R cI toS J and T sK.

Beforediscussingthe remaining (25:3:b),we insert a remarkconcerning(25:4)of 25.4.1.Sincethis is a consequenceof (25:3:a)-(25:3:c),it islegitimatefor us to draw conclusionsfrom it and it will be seenthat thisanticipationsimplifiesthe analysisof (25:3:b).

From hereon we will have to use promiscuouslycomplementarysetsin

7,J,K. It is, therefore, necessaryto avoid the notation S,and to writeinsteadI S,J S,K S,respectively.

Ad (25:4):ForvA (S)and vH(T) the role of the setI is taken over by thesetsJ and K, respectively. Hencethis condition becomes:

vA (J) = 0,0.))

Owing to (41:4),(41:5),this means

(41:8) VrO/) =0,(41:9) vp(K) = 0.SinceK = I /, therefore (25:3:b)(applied to v rOS)for which it wasassumedto hold)gives))

(41:10) vr (J) + Vr(K) = 0.Thus (41:8)and (41:9)imply eachotherby virtue of the identity (41:10).

In (41:8)or (41:9)we have actually a new condition,which does notfollow from (41:6)or (41:7).

Ad (25:3:b):We will derive its validity for vA (S) and vH(7) from theassumedonefor VF(#). By symmetry it sufficesto considerv&(S).

Therelation to beproven is

(41:11) vA (S)+ v&(J -S) = 0.By (41:4)this means

(41:12) vrOS)+ v T(J -S) 0.Owing to (41:8),which we must requireanyhow, this may be written

(41:13) vrOS)+ vr(J - S) = v r(J)(Of course,S J.)

Toprove (41:13),apply (25:3:b)for v r (fl) to R = J - S and R = J.Forthesesets/ - R = SuK and / - R = K, respectively. So (41:13)becomes

VrOS) - VrOSutf) = -vr(X),i.e.VrOS u K) = vrOS)+ v r(X),

and this is the specialcaseof (41:6)with T = K.)))

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MODIFICATIONOF THETHEORY 345

Thus we have filled in the lacuna mentioned at the beginning of thisparagraphand answeredthe questionsof (41:A).

(41:B) The further postulate which must be imposedupon vr(fi)is this:(41:8),i.e.(41:9).

All theseput togetheranswerthe questionof 41.3.2.concerningdecom-posability:

(41:C) The game F is decomposablewith respectto the setsJ and K(cf. 41.3.2.)if and only if it fulfills theseconditions:(41:6),i.e.(41:7)and (41:8),i.e.(41:9).

41.5.Desirability of a Modification

41.5.1.The two conditionswhich we proved equivalent to decompos-ability in (41:C)areof very different character.(41:6)(i.e.(41:7))is thereally essentialone,while (41:8)(i.e.(41:9))expressesonly a rather inciden-tal circumstance.We will justify this rigorously below,but first a quali-tative remarkwill beuseful. The prototypeof our conceptofdecompositionwas the game referred to at the beginning of 41.2.1.:the game representedby the corner VIIIof 35.2.Now this game fulfilled (41:6),but not(41:8).(Theformer followsfrom (35:7)in 35.2.1.,the latterfrom v(J) =v((l,2,3))= 17* 0.) We neverthelessconsideredthat game as decom-posable (with J = (1,2,3),K = (4)) how is it then possible, that itviolates the condition (41:8)which we found to be necessaryfor the decom-posability?

41.5.2.The answeris simple:Forthe above game the constituentsA

(in J = (1,2,3))and H (in K = (4))do not completelysatisfy (25:3:a)-(25:3:c)in 25.3.1.To be precise,they do not fulfill the consequence(25:4)in 25.4.1.:VA (/) = VH(#) = is not true (and it was from thisconditionthat we derived (41:8)).In other words:the constituentsof Fare not zero-sumgames.This point, of course,was perfectly clearin35.2.2.,where it receiveddue consideration.

Consequentlywe must endeavor to get rid of the condition (41:8),recognizing that this may force us to considerother than zero-sum games.

42.Modification of the Theory42.1.No CompleteAbandoning of the Zero-sum Condition

42.1.Completeabandonment of the zero-sum condition for our games1would mean that the functions 3C*(ri, , r n) which characterizeditin the senseof 11.2.3.areentirely unrestricted. I.e.that the requirement

(42:1) 2) JC*(n, , rn) BE

Jb-l

1We again denote the playersby 1, , n.)))

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346 COMPOSITIONAND DECOMPOSITIONOF GAMES

of 11.4.and 25.1.3.is dropped,with nothing elseto take its place. Thiswould necessitatea revision of considerablesignificance,sincethe construc-tion of the characteristicfunction in 25.dependedupon (25:1),i.e.(42:1),andwould therefore have to betaken up de novo.

Ultimately this revision will becomenecessary(cf.ChapterXI) but notyet at the presentstage.

In orderto geta preciseidea of just what is necessarynow, letus makethe auxiliary considerationscontainedin 42.2.1.,42.2.2.below.

42.2.StrategicEquivalence. Constant-sum Games

42.2.1.Consider a zero-sum game F which may or may not fulfill

conditions(41:6)and (41:8).Passfrom F to a strategicallyequivalentgamer' in the senseof 27.1.1.,27.1.2.,with the a?, , aj describedthere. Itis evident,that (41:6)for F isequivalent to the samefor F'.1

The situation is altogetherdifferent for (41:8).Passagefrom F toF' changesthe left hand side of (41:8)by ajj, hencethe validity of

kinJ

(41:8)in onecaseis by no means impliedby that in the other. Indeedthis is true:

(42:A) Forevery F it is possibleto choosea strategicallyequivalentgame F'so that the latterfulfills (41:8).

n

Proof:Theassertionis1that we can chooseaj, , a with ] aj =*i(this is (27:1)in 27.1.1.)so that

v(J) + aj =kinJ

Now this is obviously possibleif J 7* or /, sincethen ] a can be*m J

given any assignedvalue. ForJ = Q or J, thereis nothing to prove, asthenv(J) = by (25:3:a)in 25.3.1.and (25:4)in 25.4.1.

This result can be interpretedas follows:If we refrain from consideringother than zero-sumgames,2 then condition (41:6)expressesthat whilethe gameF may not be decomposableitself, it is strategicallyequivalentto somedecomposablegameF'.3

42.2.2.The above rigorousresult makes it clearwhere the weaknessof our presentarrangementlies.Decomposabilityis an important strategicpropertyand it is therefore inconvenient that of two strategicallyequivalentgamesonemay be termeddecomposablewithout theother. It is, therefore,

1By (27:2)in 27.1.1.Observethat the v r GS), v r 'OS)of (42:A)are the v(S),v'(S)of(27:2)loc.cit.

*I.e.we require this not only for r,but alsofor its constituents A, H.*The treatment of the constituents in 35.2.2.amounts to exactly this, as an inspec-

tion of footnote 1 on p.300shows explicitly.)))

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MODIFICATIONOF THETHEORY 347

desirable to widen theseconceptsso that decomposabilitybecomesaninvariant understrategicequivalence.

In otherwords:We want to modify our conceptso that the transforma-tion (27:2)of 27.1.1.,which definesstrategicequivalence,doesnot interferewith the relationshipbetweena decomposablegame F and its constituentsA and H. This relationshipis expressedby (41:3):(42:2) vr(Su T) = vA (S)+ vK (T) for S cJ, T cK.Now if weuse (27:2)with the samecfk for all threegamesF, A, H then (42:2)is manifestly undisturbed. The only trouble is with the preliminarycondition (27:1).Thisstatesfor F, A, H that))

2 = 0, afc

- 0,* in / k in J k in K

respectively and while wenow assumethe first relation true,thetwo othersmay fail.

Hencethe natural way out is to discard (27:1)of 27.1.1.altogether.I.e.to widen the domain of games,which we consider,by including allthose gameswhich arestrategicallyequivalent to zero-sumonesby virtueof the transformation formula (27:2)alone without demanding(27:1).

As was seenin 27.1.1.this amounts to replacingthe functions))

* \" \"

>Tn)

of the latterby new functions

3Ci(n, ' ' , r n) -JC*(n, - , rn) + al(The a?, , aj are no longer subject to (27:1)).The systems offunctions 3C(ri, , r n) which areobtainedin this way from the systemof functions 3Cfc(ri, , r n) which fulfill (42:1)in 42.1.areeasy to char-acterize.Thecharacteristicis (in placeof (42:1)loc.cit.)the property

(42:3) 3C;(n,- , r n) EE .i*iSumming up:

(42:B) We arewidening the domain of gameswhich we consider,bypassing from the zero-sum games to the constant-sum games.*At the sametime, we widen the conceptof strategicequivalence,

>18 is an arbitrary constant = 0. In the transformation (27:2)which producesthis

<game from a zero-sum one, there is obviously))

1This gives a precisemeaning to the statement at the beginning of42.1.accordingtowhich we arenot yet preparedto considerall gamesunrestrictedly.)))

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348 COMPOSITIONAND DECOMPOSITIONOF GAMES

introduced in 27.1.1.,by defining it again by transformation(27:2)loc.cit.,but droppingthe condition(27:1)eod.

42.2.3.It is essentialto recognizethat our above generalizationsdo notalterour main ideason strategicequivalence.This is best done by con-sideringthe followingtwo points.

First,we stated in 25.2.2.that we proposedto understand all quanti-tative propertiesof a gameby means of its characteristicfunction alone.Onemust realize that the reasonsfor this arejust as good in our presentdomain of constant-sumgames as in the original (and narrower) one ofzero-sumgames.Thereasonis this:(42:C) Every constant-sumgameis strategicallyequivalent to a

zero-sumgame.Proof:The transformation (27:2)obviously replacesthe a of (42:3)

n

above by s + aj. Now it is possibleto choosethe aj, , aj so*-i

n

as to make this s + J) = 0,i.e.to carry the given constant-sumgameJb-i

into a (strategicallyequivalent) zero-sumgame.Second,our new conceptof strategicequivalencewas only necessary

for thesakecf thenew (non-zero-sum)gamesthat we introduced. Fortheold (zero-sum)gamesit is equivalent to the old concept.In otherwords:If two zero-sumgamesobtain from eachotherby meansof the transforma-tion (27:2)in 27.1.1.,then (27:1)is automatically fulfilled. Indeed,thiswasalreadyobservedin footnote 2 on p.246.

42.3.TheCharacteristicFunction in the New Theory

42.3.1.Given a constant-sum game F\" (with the 3C(ri, , rn)fulfilling (42:3)),we could introduce its characteristicfunction v'(S) byrepeatingthe definitions of 25.1.3.1 On the otherhand, we may followthe proceduresuggestedby the argumentation of 42.2.2.,42.2.3.:We canobtain F' with the functions 3Cl(ri, , rn) from a zero-sumgameFwith the functions JCjb(ri, , r w) as in 42.2.2.,i.e.by))

(42:4) jei(n, , rn) s3C*(r,,- , r n) + <tf

with appropriateotj, , aj (cf.footnote 1on p.246),and then define thecharacteristicfunction v'(S)of Tf by meansof (27:2)in 27.1.1.,i.e.by

(42:5) v'(S) ^ V (5) + aj.kinS

1Thewhole arrangement of 25.1.3.can berepeatedliterally, although T is no longerzero-sum,with two exceptions. In (25:1)and (25:2)of 25.1.3.we must add A to theextreme right hand term. (This is so,becausewe now have (42:3)in placeof (42.1).)This difference is entirely immaterial.)))

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MODIFICATIONOF THETHEORY 349

Now the two proceduresareequivalent, i.e.the v'(/S)of (42:4),(42:5)coincideswith the oneobtainedby the reapplicationof 26.1.3.Indeed,aninspectionof the formulae of 25.1.3.showsimmediately,that the substitu-tion of (42:4)thereproducesthe result (42:5).l *f

42.3.2.v(S)is a characteristicfunction of a zero-sumgame,if and onlyif it fulfills the conditions(25:3:a)-(25:3:c)of 25.3.1.,as was pointed outthereand in 26.2.(Theproof was given in 25.3.3.and 26.1.)What dotheseconditionsbecomein the caseof a constant-sumgame?

In orderto answerthis question,letus remember,that (25:3:a)-(25:3:c)loc.cit.imply (25:4)in 25.4.1.Hence,we can add (25:4)to them, andmodify (25:3:b)by adding v(7) to its right hand side (this is no changeowing to (25:4)).Thus the characterizationof the v(S) of all zero-sumgamesbecomesthis:(42:6:a) v() = 0,(42:6:b) v(S)+ v(-S)= v(7),(42:6:c) v(5) + v(T) g v(Su T) if S n T = 0,and(42:6:d) v(7) = 0.Now thev'(S)of all constant-sumgamesobtain from thesev(/S)by subject-ing them to the transformation (42:5)of 42.3.1.Howdoesthis transforma-tion affect (42:6:a)-(42:6:d)?

Oneverifies immediately,that (42:6:a)-(42:6:c)areentirely unaffected,while (42:6:d)is completelyobliterated.3 Sowe see:(42:D) v(S) is the characteristicfunction of a constant-sumgame

if and only if it satisfiesthe conditions(42:6:a)-(42:6:c).(We write from now on v(S)for v'GS)).

As mentionedabove, (42:6:d)is no longer valid. However,we have

(42:6:d*) v(7) = s.Indeed,this is clearfrom (42:3),consideringthe procedureof 25.1.3.Itcan also be deducedby comparingfootnote 1on p.347 and footnote 3above (our v(S) is the v'(S) there). Besides(42:6:d*)is intuitively clear:A coalitionof all playersobtainsthe fixed sum s of the game.

1Theverbal equivalent of this consideration is easily found.1Had we decidedto define v'(*S>) by means of (42:2),(42:5)only, a question of ambi-

guity would have arisen. Indeed:A given constant-sum game r' can obviously beobtained from many different zero-sum gamesr by (42:4),will then (42:5)always yieldthe same v;(5)?

It would beeasyto prove directly that this is the case. This is unnecessary, how-ever,becausewe have shown that the v'(S)of (42:5)is always equal to that oneof25.1.3.and that v'(S)is defined unambiguously, with the help of F' alone.

n1According to (42:5),the right hand sideof (42:6:d)goesoverinto ?> i-e ^ ?

t in / t - 1and this sum is completely arbitrary.)))

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350 COMPOSITIONAND DECOMPOSITIONOF GAMES

42.4.Imputations, Domination, Solutions in the New Theory

42.4*1.From now on, we areconsideringcharacteristicfunctions of anyconstant-sumgame,i.e.functions v(S)subjectto (42:6:a)-(42:6:c)only.

Our first task in this widerdomain, is naturally that of extendingto itthe conceptsof imputations,dominations,and solutionsas defined in 30.1.1.

Let us begin with the distributions or imputations. We can take overfrom 30.1.1.their interpretationas vectors

>

a = {ai, -,).Of the conditions(30:1),(30:2)eod.we may conserve(30:1):(42:7) a, v((t))unchanged the reasonsreferred to there1 arejust as valid now as then.(30:2)eod.,however, must be modified. The constant-sumof the gamebeings (cf.(42:3)and (42:6:d*)above),eachimputation shoulddistributethis amount i.e.it is natural to postulate

(42:8)

By (42:6:d*)this is equivalent to

(42:8*) a, = v(/).-iThedefinitions of effectivity, domination,solutionwe take over unchanged

from 30.1.1.3 the supportingargumentsbrought forward in the discussionswhich led up to thosedefinitions, appear to loseno strength by our presentgeneralization.

42.4.2.Theseconsiderationsreceivetheir final corroborationsby observ-ing this:(42:E) Forour new conceptof strategicequivalenceof constant-sum

gamesF, F',4 thereexistsan isomorphismof their imputations,i.e.a one-to-onemappingof those of F on those of F',whichleavesthe conceptsof 30.1.1.5 invariant.

This is an analogue of (31:Q)in 31.3.3.and it can be demonstrated inthe sameway. As there,we define the correspondence

(42:9) 7^7'1a < v((t)) would beunacceptable,cf.e.g.the beginning of 29.2.1.2 For the specialcaseof a zero-sum game v(7) = so(42:8),(42:8*)coincide as

they must with (30:2)loc.cit.'I.e.(30:3);(30:4:a)-(30:4:c);(30:5:a),(30:5:b),or (30:5:c)loc.cit.,respectively.4 As defined at the end of42.2.2.,i.e.by (27:2)in 27.1.1.,without (27:1)eod.

As redefined in 42.4.1.)))

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MODIFICATIONOF THETHEORY 351

betweenthe imputations a = \\a h , an j of F and the imputations\"?'= K - , a'n ] of I\" by

(42:10) a'k = ak + J

where the a?, - , a arethoseof (27:2)in 27.1.1.Now the proof of (31:Q)in 31.3.3.carriesover almost literally. Theone

difference is that (30:2)of 30.1.1.is replacedby our (42:8)but since(27:2)))n))

in 27.1.1.gives v'(/) = v(7) + <*?,this too takes careof itself.1 Thet-i

readerwho goesover 31.3.again, will seethat everything elsesaid thereappliesequallyto the presentcase.

42.5.Essentiality, Inessentiality, and Decomposability in the New Theory

42.6.1.We know from (42:C)in 42.2.3.that every constant-sumgame isstrategicallyequivalent to a zero-sum game. Hence(42:E) allowsus tocarryover the general resultsof31.from the zero-sumgamesto the constant-sum onesalways passingfrom the latter classto the former one by strategicequivalence.

This forces us to define inessentiality for a constant-sum game bystrategicequivalenceto an inessential zero-sumgame. We may statetherefore:

(42:F) A zero-sum game is inessentialif and only if it is strategicallyequivalent to the game with v(S) = 0. (Cf.23.1.3.or (27:C)in 27.4.2.)By the above, the same is true for a constant-sumgame. (But we must use our new definitions of inessentialityand of strategicequivalence.)

Essentialityis, of course,defined as negation of inessentiality.Application of the transformation formula (42:5)of 42.3.1.to the

criteriaof 27.4.shows,that thereareonly minor changes.(27:8)in 27.4.1.must be replacedby))

(42:11)))

sincethe right hand side of this formula is invariant under (42:5)and it

goesover into (27:8)loc.cit.for v(I) = (i.e.the zero-sumcase).Thesubstitution of (42:11)for (27:8)necessitatesreplacementof the

on the right hand sideof both formulae in the criterion (27:B)of 27.4.1.by

n

1And this was the only point in the proof referredto, at which 2, S - (i.e.(27:1)t-iin 27.1.1.,which we no longer require) is used.)))

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352 COMPOSITIONAND DECOMPOSITIONOF GAMES

v(7). The criteria(27:C),(27:D)of 27.4.2.are invariant under (42:5),and henceunaffected.

42.5.2.We can now return to the discussionof compositionand decom-positionin 41.3.-41.4.,in the wider domain of all constant-sumgames.

All of 41.3.can berepeatedliterally.When we cometo 41.4.,,the question (41:A) formulated there again

presents itself. In orderto determinewhether any postulates beyond(41:6),i.e.(41:7)of 41.3.2.arenow needed,we must investigate(42:6:a)-(42:6:c)in 42.3.2.,instead of (25:3:a)-(25:3:c)in 25.3.1.(for all threeofvr (/Z), vA (S),v,(I*)).

(42:6:a),(42:6:c)are immediately disposed of, exactly as (25:3:a),(25:3:c)in 41.4.As to (42:6:b),the proof of (25:3:b)in 41.4.is essentiallyapplicable,but this time no extraconditionarises(like(41:8)or (41:9)loc.cit.).Tosimplify matters,we give this proof in full.

Ad (42:6:b):We will derive its validity for vA (S) and vH(T) from theassumedone for vrCR). By symmetry it sufficesto consider

Therelation to be proven is))

(42:12) vA (S)+ vA (J - S) =

By (41:4)this means

(42:12*) VrOS) + v r(J - S) = v r(J).To prove (42:12*)apply (42:6:b)for v r (fl) to R = / - S and R = J.

For these I - R = SuK and I - R = K, respectively. So (42:12*)becomes

Vr(S)+ v r (J) - Vr(Su K) = v r (7)- v r(X),i.e.

v r(Su K) = Vr(S)+ v r(X),

and this is the specialcaseof (41:6)with T = K.Thus we have improved upon the result (41:C)of 41.4.as follows:

(42:G) In the domain of all constant-sum games the game F isdecomposablewith respectto the setsJ and K (cf. 41.3.2.)if

and only if it fulfills the condition(41:6),i.e.(41:7).42.5.3.Comparisonof (41:C)in 41.4.and of (42:G)in 42.5.2.showsthat

the passagefrom zero-sumto constant-sumgamesrids us of the unwantedcondition(41:8),i.e.(41:9)for decomposability.

Decomposabilityis now defined by (41:6),i.e.(41:7)alone,and it isinvariant under strategicequivalence as it shouldbe.

We alsoknow that when a gameF is decomposedinto two (constituent)gamesA and H (all of them constant-sumonly!),we can make all thesegameszero-sumby strategicequivalence.(Cf. (42:C) in 42.2.3.for F,and then (42:A) in 42.2.1.et sequ.for A, H.))))

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THEDECOMPOSITIONPARTITION 363Thus we can always use oneof the two domainsof games zero-sum

or constant-sum whichever is more convenient for the problemjust underconsideration.

In the remainder of this chapter we will continue to con-sider constant-sum games, unless the opposite is explicitly stated.

43.TheDecompositionPartition43.1.Splitting Sets. Constituents

43.1.We defined the decomposabilityof a gameF not per se,but with

respectto a decompositionof the set/ of all playersinto two complementarysets,J,K.

Thereforeit is feasible to take this attitude:Consider the game Fas given, and the setsJ',K as variable. SinceJ determinesK (indeedK = 7 J), it suffices to treat J as the variable. Then we have thisquestion:

Given a game F (with the setof players/) for which setsJ s/ (and thecorrespondingK = / J) is F decomposable?

We call thoseJ(QI) for which this is the casethe splitting setsof T.The constituent game A which obtains in this decomposition(cf.41.2.1.and (41:4)of 41.3.2.)is the J-constituentof F.1

A splittingsetJ is thus defined by (41:6),i.e.(41:7)in 41.3.2.,whereK = 7 J must be substituted.

Thereaderwill note that this concepthas a very simpleintuitive mean-ing:A splittingsetisa self containedgroupof players,who neither influence,nor areinfluencedby, the othersas far as the rulesof thegameareconcerned.

43.2.Propertiesof the System of All Splitting Sets43.2.1.Thetotality of all splittingsetsof a given gameis characterized

by an aggregateof simple properties.Most of these have an intuitiveimmediatemeaning, which may makemathematical proof seemunnecessary.We will neverthelessproceedsystematicallyand give proofs,stating theintuitive interpretations in footnotes. Throughout what follows we writev(S)for v r (S)(thecharacteristicfunction of F).

(43:A) J is a splittingset if and only if its complementK = / Jis one.2

Proof:Thedecomposabilityof F involves J and K symmetrically.

(43:B) and / aresplittingsets.8

Proof:(41:6)or (41:7)with J = , K = / are obviously true, asv(0) = 0.

1By the same definition the game H (cf.41.2.1.and (41:5)in 41.3.2.)is then theK-constituent (K - / - J) of T.

1That a setof players is self-containedin the senseof 43.1.,is clearlythe samestate-ment, as that the complement is self-contained.

1That theseareself-containedis tautological.)))

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354 COMPOSITIONAND DECOMPOSITIONOF GAMES

43.2.2.(43:C) /'n J\"and J' u J\"aresplittingsets if J',J\"are.1

Proof:Ad J' u J\":As J',J\"are splittingsets,we have (41:6)for J,Kequal to J', 7 -J' and J\",/ -J\". We wish to prove it for Jf u J\",I - (J'uJ\.") Consider therefore two S J1 u J\", T 7 - (J9 u /\Let S'bethe part of S in J', then S\"= S S'lies in the complementofJ', and as S J' u J\",S\" also lies in J\". So S = S'+ S\",S'cJ',S\"S/\". Now S' j',S\"c/ - J' and (41:6)for J',I - J' give

(43:1) v(5) = v(S')+ v(S\.Next S\" 7 - J' and T s I - (J'u J\")s 7 - J' so S\"uTsI-J'.Also S'cj'.ClearlyS'u (S\"u T) = S u T. Hence(41:6)for J',7 - J'alsogives

(43:2) v(Su T) = v(S')+ v(S\"u 71).Finally S\"<=J\"and Ts7 - (J'u J\") 7 - J\". Hence (41:6)forJ\",7 - J\"gives

(43:3) v(S\"u T7

) = v(S\")+ v(T).Now substitute (43:3)into (43:2)and then contractthe right hand side

by (43:1).This gives

v(Su T) = v(5) + v(T),which is (41:6),as desired.

Ad J' n J\":Use (43:A) and the above result. As J',J\"aresplittingsets,thesameobtainssuccessivelyfor 7 J',I J\",(I J')u (7 /\which is clearly7 (/'n 7\")

2, and J' n /\" the last one beingthe desiredexpression.43.3.Characterization of the System of All Splitting Sets. TheDecomposition Partition

43.3.1.Itmay be that thereexistno othersplittingsetsthan the trivialones , 7 (cf.(43:B)above). In that case,we call the game F indecompos-able* Without studying this question any further, 4 we continue toinvestigatethe splittingsets of F.

1The intersection J'n J\":It may strike the readeras odd, that two self-containedsetsJ',J\" should have a non-empty intersection at all. This is possible,however, as theexample /' J\" shows. Thedeeperreasonis that a self-containedsetmay well be thesum of smaller self-containedsets (propersubsets). (Cf.(43:H)in 43.3.) Ourpresentassertionis that if two self-containedsetsJ',/\" have a non-empty intersection /' n /\",then this intersection is such a self-containedsubset. In this form it will probably appearplausible.

Thesum /'U J\":That the sum of two self-containedsetswill again beself-containedstands to reason. This may be somewhat obscuredwhen a non-empty intersection/'n J\" exists, but this caseis really harmless as discussedabove. The proof whichfollows is actually primarily an exactaccountof the ramifications of just this case.

1Thecomplement of the intersection is the sum of the complements.1Actually most gamesare indecomposable;otherwise the criterion (42:G)in 42.5.2.

requires the restrictive equations (41:6),(41:7)in 41.3.2.4Yetl Cf.footnote 3 and its references.)))

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THEDECOMPOSITIONPARTITION 355

(43:D) Considera splittingsetJ of F and the /-constituentA of F.Then a J' J is a splittingsetof A if and only if it is one of F.1

Proof:Considering(41:4),J' is a splittingsetof A by virtue of (41:6)when

(43:4) v(Su T) = v(S)+ v(7) for 8 J', T fi J - J'.(We write v(S)for v r(/S)). Again by (41:6)J' is a splittingsetof F when

(43:5) v(Su T) = v(S)+ v(T) for S J', J1 c/ - J'.We must prove the equivalence of (43:4)and (43:5).As J J, so

(43:4)is clearly a specialcaseof (43:5) hencewe needonly prove that(43:4)implies(43:5).

Assume,therefore, (43:4). We may use (41:6)for F with J,K = / J.Considertwo S J', T s7 - J'. Let Tf be the part of T in J, then

T'' = T - T' lies in 7 - J. So T = T'u T\", Tf cJ, T\" c/ - J and(41:6)for F with J, I - J give

(43:6) v(r) = v(2\") + v(T\.Next SzJ'zJand T7' cJ so SuT'sJ.Also T\" / - J. Clearly(Su T;

) u T77 ' = S u T. Hence(41:6)for F with J, 7 -J alsogives

(43:7) v(Su T) = v(5 u T') + v(T\.Finally Scj'and T'&I-J'and T'cJ, so T'sJ-J'.Hence(43:4)gives

(43:8) v(Su T') = v(S)+ v(2\.Now substitute (43:8)into (43:7)and then contractthe right hand side

by (43:6).This gives preciselythe desired(43:5).43.3.2.(43:D)makes it worth while to considerthose splittingsetsJ,

for which J ^ 0,but no propersubsetJ1 Qof J is a splittingset. Wecall sucha setJ, for obvious reasons,a minimal splitting set.

Consider our definitions of indecomposabilityand of minimality.(43:D)impliesimmediately:

(43:E) The J-componentA (of F) is indecomposableif and only if

J is a minimal splittingset.The minimal splitting sets form an arrangement with very simple

properties,and they determine the totality of all splitting sets.Thestatement follows:

(43:F) Any two different minimal splittingsetsaredisjunct.(43:G) The sum of all minimal splittingsetsis 7.

1 Tobe self-containedwithin a self-containedset, is the same thing as to be such inthe original (total) set. Thestatement may seemobvious; that it is not so,will appearfrom the proof.)))

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356 COMPOSITIONAND DECOMPOSITIONOF GAMES

(43:H) By forming all sums of all possibleaggregatesof minimal

splitting sets,we obtain preciselythe totality of all splittingsets.1

Proof:Ad (43:F):Let /', J\"be two minimal splitting setswhich arenot disjunct. Then J' n J\" 7* Q is splitting by (43:C),as it is J' andSJ\". So the minimality of J' and J\" implies that Jr n J\" is equal toboth J1 and J11. Hence/' = J\".

Ad (43:G):It sufficesto show that every k in / belongsto someminimal

splittingset.Thereexistsplittingsetswhich contain the playerk (i.e./) ; letJ be the

intersectionof all of them. J issplittingby (43:C).If J werenot minimal,then there would exista splitting set J' ?* , J, which is J. NowJ\"= J -J' = J n (/ -J') is also a splitting setby (43:A), (43:C),andclearly also J\" j* , J. EitherJ' or J\"= / J1 must contain k saythat J1 does.Then J' is among the setsof which J is the intersection.HenceJ1aJ. But as J' J and J' 7* J, this is impossible.

Ad (43:H):Every sum of minimal splitting setsis splitting by (43:C),so we needonly prove the converse.

LetKbea splittingset. If J is minimal splitting,then J n K is splittingby (43:C),also J n K J henceeitherJnK = QoTjnK= J. Inthe first case/, K aredisjunct, in the secondJ K. Sowe see:(43:1) Every minimal splitting set J is eitherdisjunct with K or))

LetK'bethe sum of the formerJ,and K\" thesum of the latter. K'u K11

isthe sum of all minimal splittingsets,henceby (43:G)

(43:9) K'uK\"= 7.

By theirorigin K'is disjunct with K, and K\" is K. I.e.(43:10) #'/-#, X\"KNow (43:9),(43:10)togethernecessitateK\" = K\\ henceIf is a sum of asuitableaggregateof minimal sets,as desired.

43.3.3.(43:F),(43:G) make it clearthat the minimal splitting setsform a partition in the senseof 8.3.1.,with the sum 7. We call this thedecompositionpartition of T, and denoteit by Hr. Now (43:H) can beexpressedas follows:

(43:H*) A splitting set Ksl is characterizedby the followingproperty: The points of eachelementof IIr go togetherasfar as K is concernedi.e.eachelementof IIr liescompletelyinsideor completelyoutsideof K.

1Theintuitive meaning of theseassertionsshould be quite clear. They characterizethe structure of the maximum possibilitiesofdecomposition of F in a plausible way.)))

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Thus n r expresseshow far the decompositionof T in / canbepushed,without destroyingthosetieswhich the rulesof T establishbetweenplayers.1

By virtue of (43:E) the elementsof IIr arealso characterizedby the factthat they decomposeF into indecomposableconstituents.

43.4.Propertiesof the DecompositionPartition

43.4.1.Thenature of the decompositionpartition Ilrbeingestablished,it is natural to study the effectof the finenessof this partition. We wish toanalyze only the two extremepossibilities:When IIr is as fine as possible,i.e.when it dissects/ down to the one-elementsets and when Erisascoarseas possible,i.e.when it doesnot dissectI at all. In otherwords:In the firstcaseHr is the systemof all one-elementsets(in 7) in the secondcaseIlrconsistsof / alone.

Themeaning of thesetwo extremecasesis easilyestablished:

(43:J) n r is the systemof all one-elementsets(in 7) if and only ifthe gameis inessential.

Proof:It is clearfrom (43:H) or (43:H*)that the statedproperty ofHr is equivalent of sayingthat all setsJ(s.7)aresplitting. I.e.(by 43.1.)that for any two complementarysets J and K(= I J) the gameF isdecomposable.This means that (41:6)holds in all those cases. Thisimplies,however, that the condition imposedby (41:6)on S,T (i.e.S Si J,T K) meansmerelythat S,T aredisjunct. Thus our statementbecomes

v(Su T) = v(S)+ v(!T) for S n T =Now this is preciselythe condition of inessentialityby (27:D)in 27.4.2.

(43:K) n r consistsof 7 if and only if the gameF is indecomposable.Proof:It is clearfrom (43:H)(or (43:H*)),that the statedproperty of

Ilr is equivalent to saying that 0,7 arethe only splittingsets.But thisis exactly the definition of indecomposabilityat the beginningof 43.3.

Theseresult^ show that indecomposabilityand inessentialityare two

oppositeextremesfor a game. In particular,inessentialitymeansthat thedecompositionof F, describedat the end of 43.3.,canbe pushed thrdughto the individual players,without ever severingany tiethat the rulesof thegameF establish.2 The readershould comparethis statementwith ouroriginal definition of inessentialityin 27.3.1.

43.4.2.The connectionbetween inessentiality, decomposability,andthe numbern of playersis asfollows:

n = 1:This caseis scarcelyof practicalimportance.Sucha gameisclearlyindecomposable,8 and it is at the sametime inessentialby the firstremark in 27.5.2.

1 I.e.without impairing the self-containednessof the resulting sets.2I.e.that every player is self-containedin this game.3 As / is a one-elementset , / are its only subsets.)))

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358 COMPOSITIONAND DECOMPOSITIONOF GAMES

It should then be noted that indecomposabilityand inessentialityareby (43:J),(43:K)incompatiblewhen n ^ 2,but not when n = 1.

n = 2:Sucha game,too,is necessarilyinessentialby the first remarkoi27.5.2.Henceit is decomposable.

n ^ 3:Forthesegamesdecomposabilityis an exceptionaloccurrence,Indeed,decomposabilityimplies(41:6)with someJ ^ Q,I]henceK =/ - J T*0,/. Sowe can choosej in J,k in K. Then (41:6)with S = (j),T = (k) gives

(43:11) v((j, A;)) = v((j)) + v((fc)).Now the only equationswhich the values of v(S)must satisfy,are(25:3:a),(25:3:b)of 25.3.1.(if zero-sumgamesareconsidered)or (42:6:a),(42:6:bjof 42.3.2.(43:11)is neitherof these,sinceonly the sets(j), (fc), (j,k]occurin (43:11)and thesearenone of the setsoccurringin thoseequationsi.e. or 7 or complementsas n ^ 3.1 Thus (43:11)is an extraconditionwhich is not fulfilled in general.

By the above an indecomposablegamecannot have n = 2 henceit hasn = 1 or n ^ 3. Combiningthis with (43:E),we obtain the following

peculiarresult:(43:L) Every elementof the decompositionpartition Tlr is eithera

one-elementset,or elseit has n ^ 3 elements.Note that the one-elementsetsin IIr arethe one-elementsplittingsets'

i.e.they correspondto those players who areself-contained,separateefrom the remainderof the game(from the point of view of the strategy oi

coalitions).They arethe \" dummies\" in the senseof 35.2.3.and footnote 1

on p.340.Consequently,our result (43:L)expressesthis fact :Thoseplayenwho arenot \"duYnmies,\" aregroupedin indecomposableconstituentgame*of n ^ 3 playerseach.

This appears to be a generalprincipleof socialorganization.

44. DecomposableGames. FurtherExtensionof the Theory44.1.Solutions of a (Decomposable)Gameand Solutions of Its Constituents

44.1.We have completedthe descriptivepart of our study of composition anddecomposition.Let usnow passto the centralpart of theproblemTheinvestigation of the solutionsin a decomposablegame.

Considera gameT which is decomposablefor J and / J = K, wit!the J- and /^-constituentsA and H. We use strategicequivalence,as

explainedat the beginningof 42.5.3.,to make all threegameszero-sum.Assume that the solutionsfor A as well as thosefor H areknown;doei

this then determinethe solution for T? In other words:How do th<

solutionsfor a decomposablegameobtain from thosefor its constituents?Now thereexistsa surmisein this respectwhich appearsto be the prime

facie plausibleone,and we proceedto formulate it.1For n = 2 it is otherwise;(j,k) /,(j) and (A;) arecomplements.1Sucha splitting setis,ofcourse,automatically minimal.)))

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DECOMPOSABLEGAMES 359

44.2.Composition and Decomposition of Imputations and of Setsof Imputations

44.2.1.Let us use the notations of 41.3.1.But as we write v(S) forVr(S)this alsoreplacesby (41:4),(41:5),v A (S),vH(5).

On the other hand, we must distinguish between imputations forF, A, H.1 In expressingthis distinction,it is betterto indicatethe setofplayersto whom an imputation refers,insteadof the gamein which they areengaged. I.e.we will affix to them the symbols/, /,K ratherthan F, A, H.In this sensewe denotethe imputationsfor I (i.e.T) by

(44:1) a j = {i>, , *>, ai, * , r},and those for J,K (i.e.A, H) by

(44:2) 7/= {0i', ,0*},(44:3) 7*= {7i\", ' ,7r).If threesuch imputations arelinkedby the relationship

(44.4^ *' = &' for *' = r>

' ' '>

k '>(** A) a,.= 7,\" for j\" = 1\",' , I\",

> > +then we say that a / obtainsby compositionfor /, 7 *, that /, 7 * obtain

by decompositionfrom a / (for J, If), and that /, 7 K arethe (/-,K-)constituentsof a /.

Sincewe arenow dealing with zero-sumgames,all theseimputationsmust fulfill the conditions (30:1),(30:2)of 30.1.1.Now one verifies

immediatelyfor a/, ft j, y K linkedby (44:4).Ad (30:1)of 30.1.1.:Thevalidity of this for /, y K is clearlyequivalent

to its validity for a /.Ad (30:2)of 30.1.1.:For\"/?/, ~y K this states(using(44:4))

*'(44:5) X ** =

>t'-ri\"

(44:6) a, = 0.y-i\"

For a / it amounts to*' i\"

(44:7) J) <*' + ,-= 0.'!' ;_!/,1 It is now convenient to re-introducethe notations of41.3.1.for the players.)))

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360 COMPOSITIONAND DECOMPOSITIONOF GAMES

Thus its validity for /3 j, y K impliesthe same for a /, while its validity

for a i doesnot imply the samefor /, y K indeed(44:7)doesimply theequivalenceof (44:5)and (44:6),but it fails to imply the validity of eitherone.

Sowe have:

(44:A) Any two imputations ft /, y K can be composedto an a/,while an imputation a / can bedecomposedof two /, y K if andonly if it fulfills (44:5),i.e.(44:6).

We call suchan a / decomposable(for J,K).44.2.2.This situation is similar to that, which prevails for the games

themselves:Compositionis always possible,while decompositionis not.Decomposabilityis again an exceptionaloccurrence.1

Itought to be noted, finally, that theconceptof compositionof imputa-tionshasa simpleintuitive meaning. Itcorrespondsto the sameoperationof \" viewing as one\"two separateoccurrences,which playedthe correspond-ing rolefor games in 41.2.1.,41.2.3.,41.2.4.Decompositionof an a /

(into ft /, y K) ispossibleif and only if the two self-containedsetsof players

J,K aregiven by the setsof imputations a / preciselytheir \" just dues\"which arezero. Thisis the meaning of the condition(44:A) (i.e.of (44:5),(44:6)).

44.2.3.Considera setV/ of imputations / anda setW/c of imputations> >

y K. Let U/ be the setof thoseimputationsa / which obtain by composition>

of all ft j in V/ with all 7 K in Wx. We then say that U/ obtainsby compo-sition from V/, Wxi that V/, W* obtain by decompositionfrom U/ (forJT, K\\ and that V/, W* arethe (J-,K-)constituentsof U/.

Clearlythe operationof compositioncan always be carriedout, what-ever V/t WK whereasa given U/ neednot allow decomposition(for J,K).If U/ canbe decomposed,we call it decomposable(for J,K).

Note that this decomposabilityof U/ restrictsit very strongly;it implies,

among otherthings that all elementsa / of U/ must be decomposable(cf.the interpretationat the end of 44.2.2.).

In orderto interpret theseconceptsfor the setsof imputationsU/f V/,Wx more thoroughly, it is convenient to restrictourselvesto solutionsof the gamesr, A, H.

1Therearegreat technical differences between the conceptsof decomposability etc.,for games and for imputations. Observe,however, the analogy between (41:4),(41:5)in 41.3.2.;(41:8),(41:9),(41:1(V) in 41.4.2.;and our (44:4),(44:5),(44:6),(44:7).)))

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44.3.Composition and Decomposition of Solutions.

TheMain Possibilitiesand Surmises

44.3.1.Let V/f W* be two solutionsfor the gamesA, H respectively.Their compositionyields an imputation set U/ which one might expectto bea solution for the game F. Indeed,U/ is the expressionof a standardof behavior which can beformulated asfollows. We givethe verbal formula-tion in the text under (44:B:a)-(44:B:c),stating the mathematical equiva-lents in footnotes, which, as the readerwill verify, add up preciselyto ourdefinition of composition.

(44:B:a) The players of J always obtain togetherexactly their\"justdues\"(zero),and the sameis true for the playersof K.1

(44:B:b) Thereis no connectionwhatever betweenthe fate of play-ersin the set/ and in the setK.2

(44:B:c) Thefate of the playersin J is governed by the standard ofbehavior Vj> 3 the fate of the playersin K is governed by thestandard of behavior W*. 4

If the two constituentgamesareimagined to occurabsolutelyseparatefrom eachother,then this is the plausibleway of viewing their separatesolutionsVy, W/c as one solution U/ of the compositegameF.

However,sincea solution is an exactconcept,this assertionneeds aproof. I.e.we must demonstratethis:(44:C) If V/, W* aresolutionsof A, H, then their compositionU/ is

a solution of F.

44.3.2.This, by the way, is another instanceof the characteristicrelationshipbetweencommon senseand mathematical rigour. Althoughan assertion(in the presentcasethat U/ is a solution whenever V/, W* are)is requiredby common sense,it has no validity within the theory (in thiscasebasedon the definitions of 30.1.1.)unlessproved mathematically. Tothis extentit might seemthat rigour is more important than commonsense.This, however, is limited by the further considerationthat if the mathe-matical proof fails to establishthe common senseresult, then there is astrong casefor rejectingthe theory altogether. Thus the primate of themathematical procedureextendsonly to establishcheckson the theoriesin a way which would not be opento common sensealone.

1 Every element a / of U/ is decomposable.*Any / which is used in forming U/ and any 7 *, which is used in forming U/, give

>

by composition an element a / of U/.*Theabove mentioned / arepreciselythe elementsof V/.4Theabove mentioned y K arepreciselythe elements of W/c.)))

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362 COMPOSITIONAND DECOMPOSITIONOF GAMES

It will be seenthat (44:C)is true,although not trivial.One might be tempted to expectthat the converseof (44:C) is also

true,i.e.to demanda proof of this:(44:D) If U/ is a solution of F, then it can be decomposedinto solu-

tionsV,,W* of A, H.This is prima facie quite plausible:SinceF is the compositionof what

arefor all intents and purposestwo entirely separate-games,how couldanysolutionof F fail to exhibittjiis compositestructure?

The surprising fact is, however, that (44:D) is not true in general.Thereadermight think that this shouldinduceus toabandon or at leasttomodify materially our theory (i.e.30.1.1.)if we take the above method-ologicalstatement seriously. Yet we will show, that the \" common sense\"basis for (44:D) is quite questionable.Indeed,our result, contradicting(44:D) will provide a very plausible interpretation which connectsit

successfullywith well known phenomena in socialorganizations.44.3.3.The proper understandingof the failure of (44:D) and of the

validity of the theory which replacesit, necessitatesrather detailed con-siderations.Beforewe enterupon these,it might be useful to make, in

anticipation,someindicationsas to how the failure of (44:D)occurs.It is natural, to split (44:D)into two assertions:

(44:D:a) If U/ is a solution of F, then it is decomposable(for J,K).(44:D:b) If a solutionU/ of F is decomposable(for /, K), then its

constituentsV/, W/c aresolutionsfor A, H.Now it will appear that (44:D:b)is true, and (44:D:a)is false. I.e.

it can happen that a decomposablegame F possessesan indecomposablesolution.1

However,the decomposabilityof a solution (or of any setof imputations)is expressedby (44:B:a)-(44:B:c)in 44.3.1.Soone or more of thesecondi-tions must fail for the indecomposablesolution referred to above. Nowit will beseen(cf.46.11.)that the conditionwhich is not satisfiedis (44:B:a).This may seemto be very grave, because(44:B:a)is the primary conditionin the sensethat when it fails, the conditions(44:B:b),(44:B:c)cannoteven be formulated.

The conceptof decompositionpossessesa certain elasticity. Thisappeared_jn 42.2.1.,42.2.2.and 42.5.2.,where we succeededin riddingourselvesof an inconvenient auxiliary conditionconnectedwith the decom-posabilityof a gameby modifying that concept.It will be seenthat ourdifficulties will again be met by this procedure so that (44:D) will bereplacedby a correctand satisfactory theorem.Hencewe must aim atmodifying our arrangements,sothat the condition(44:B:a)can bediscarded.

We will succeedin doing this, and then it will appear that conditions(44:B:b),(44:B:c)make no difficulties and that a completeresult can beobtained.

1This is similar to the phenomenon that a symmetric game may possessan asym-metric solution. Cf.37.2.1.)))

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DECOMPOSABLEGAMES 383

44.4.Extension of the Theory. OutsideSources44.4.1.It is now time to discardthe normalization which we introduced

(temporarily)in 44.1.:That the gamesunder considerationarezero-sum.We return to the standpoint of 42.2.2.accordingto which the gamesareconstant-sum.

Thesebeing understood, considera game F which is decomposable(for J,K) with J-,-K-constituentsA, H.

The theory of composabilityand decomposabilityof imputations, asgiven in 44.2.1.,44.2.2.couldnow be repeatedwith insignificant changes.(44:l)-(44:4)may be taken over literally, while (44:5)-(44:7)are onlymodified in their right hand sides. Since(30:2)of 30.1.1.hasbeenreplacedby (42:8*)of 42.4.1.those formulae (44:5)-(44:7)now become:

k'

(44:5*) a,- = v(J),i'-r(44:6*) rv(K)f/'!\"and

(44:7*) |,+ or = v(7) = v(J) + v(K).i'-i' /'-i\"(Thelast equationon the right hand sideby (42:6:b)in 42.3.2.,or equallyby (41:6)in 41.3.2.with S = /, T = K.) Thesituation is exactlyas in

44.2.1.,indeed,it really arisesfrom that oneby the isomorphismof 42.4.2.Thus a i fulfills (44:7*),but for its decomposability(44:5*),(44:6*)areneeded and (44:7*)does imply the equivalenceof (44:5*)and (44:6*),but it fails to imply the validity of either.

Sothe criterion of decomposability(44:A) in 44.2.1.is again true,onlywith our (44:5*),(44:6*)in placeof its (44:5),(44:6). And the final con-

clusionof 44.2.2.may be repeated:Decompositionof an a / (into ft /, y K )is possibleif and only if the two self containedsetsof playersJ, K are

given by this imputation a / preciselytheir just dues which arenow v(J),))

Sincewe know that this limitation of the decomposabilityof imputa-tions the reasonfor (44:B:a)in 44.3.1.is a sourceof difficulties,we haveto remove it. This means removal of the conditions(44:5*),(44:6*),i.e.of the condition(42:8*)in 42.4.1.from which they originate.

44.4.2.According to the above,we will attempt to work the theory of aconstant-sumgame F with a new conceptof imputations,which is basedon(42:7)of 42.4.1.(i.e.on (30:1)of 30.1.1.)alone,without (42:8*)in 42.4.1.In otherwords2

1Insteadof zero,as loc.cit.*We again denote the playersby 1, , n.)))

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364 COMPOSITIONAND DECOMPOSITIONOF GAMES

An extendedimputation is a system of numbersai, , an with thisproperty:(44:8) <* ^ v((t)) for i = I, , n.

n

We imposeno conditionsupon -. We view theseextendedimputa-

tions, too,as vectors))

44.4.3.Itwill now be necessaryto reconsiderall our definitions which arerootedin the conceptsof imputation i.e.thoseof 30.1.1.and 44.2.1.But,before we do this, it is well to interpret this notion of extendedimputations.

Theessenceof this conceptis that it representsa distributionof certainamounts between the players, without demandingthat they should totalup to the constant sum of the gameF.

Suchan arrangementwould beextraneousto the picturethat the playersareonly dealingwith eachother. However,we have always conceivedofimputations as a distributiveschemeproposedto the totality of all players.(This ideapervades,e.g.all of 4.4.,4.5.;it is quite explicitin 4.4.1.)Sucha proposalmay comefrom one of the players,1but this is immaterial. Wecan equally imagine, that outside sourcessubmit varying imputations tothe considerationof the playersof F. All this harmonizes with our pastconsiderations,but in all this, those \"outsidesources\"manifested them-selvesonly by making suggestions without contributing to,or withdrawingfrom, the proceedsof the game.

44.5.TheExcess44.5.1.Now our presentconceptof extendedimputationsmay be taken

to expressthat the \" outsidesources\" can makesuggestionswhich actuallyinvolve contributions or withdrawals, i.e.transfers. For the extended

imputation a = {i, ,}the amount of this transfer isn

(44:9) e = % a, - v(/)-iand will be calledthe excessof a . Thus

e > for a contribution,(44:10) e = if no transfer takesplace,

e < for a withdrawal.1Who tries to form a coalition. Sincewe considerthe entire imputation as his

proposal,this necessitatesour assuming that he is even making propositions to thoseplayers,who will not beincluded in the coalition. Tothesehe may offer their respectiveminima v((t)) (possibly more, cf. 38.3.2.and 38.3.3.).Theremay alsobe players inintermediate positions \"between included and excluded\" (cf.the secondalternative in37.1.3.).Of course,those lessfavored players may make their dissatisfaction effective,this leadsto the conceptof domination, etc.)))

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DECOMPOSABLEGAMES 365

It will be necessaryto subjectthis to certain suitable limitations, in

orderto obtain realisticproblems;and we will take due account of this.It is important to realize how thesetransfers interact with the game.

The transfers arepart of the suggestionsmade from outside, which areacceptedor rejectedby the players,weighedagainsteachother,accordingto the principlesof domination, etc.1 In the courseof this process,anydissatisfiedset of players may fall back upon the game F, which is thesolecriterion of the effectivity of their preferenceof their situation in one(extended)imputation against another.2 Thus the game, the physicalbackground of the social processunder consideration,determines thestabilityof all detailsof the organization but the initiative comesthroughthe outside suggestions,circumscribedby the limitations of the excessreferredto above.

44.5.2.Thesimplestform that this \"limitation\" of the excesscan take,consistsin prescribingits value e explicitly. In interpretingthis prescrip-tion, (44:10)shouldbe remembered.

Thesituation which existswhen e . may at first seemparadoxical.Thisis particularly true when e <0,i.e.when a withdrawal from outside

is attempted.Why shouldthe players,who couldfall back on a game ofconstant sum v(/) acceptan inferior total? I.e.how can a \"standardofbehavior/'a \"socialorder,\"basedon sucha principlebestable? Thereis,neverthelessan answer:The game is only worth v(l) if all playersform acoalition,act in concert.If they are split into hostile groups, then eachgroup may have to estimate its chancesmore pessimisticallyand such adivision may stabilizetotals that areinferior to v(/).3

The alternative e > 0, i.e.when the outside interferenceconsistsof afree gift, may seemlessdifficult to accept. But in this casetoo, it will be

1 This is, of course,a narrow and possibly even somewhat arbitrary description of thesocialprocess. It should be remembered, however, that we use it only for a definite andlimited purpose:Todetermine stable equilibria, i.e.solutions. Theconcluding remarksof 4.6.3.should make this amply clear.

2 We are,of course,alluding to the definitions of effectivity and domination, cf.4.4.1.and the beginning of 4.4.3. given in exactform in 30.1.1.We will extend the exactdefinitions to our present conceptsin 44.7.1.

3 For a first quantitative orientation, in the heuristic manner: If the players aregrouped into disjunct sets(coalitions) S\\, , Sp , then the total of their own valua-tions isv(Si) + + v(SP). This is v(J) by (42:6:c)in 42.3.2.

Oddly enough, this sum is actually v(7) when p - 2 by (42:6:b)in 42.3.2.i.e.in this model the disagreements between three or more groups arethe effective sourcesof damage.

Clearly by (42:6:c)in 42.3.2.the above sums v(8i) + + v(Sp) are alln

g> V v ((*')). On the other hand, this latter expressionis one of them (put p n,t-i

SK (t)). Sothe damage isgreatestwhen eachplayer is isolatedfrom all others.n

Thewhole phenomenon disappears,therefore, when ^ v((i)) v(7) ,i.e.when thet-igame is inessential. (Cf.(42:11)in 42.5.1.))))

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366 COMPOSITIONAND DECOMPOSITIONOF GAMES

necessaryto study the gamein order to seehow the distributionof thisgift among the playerscan be governed by stablearrangements.Ithas tobe expected,that the optimisticappraisal of their own chances,derivedfrom the possibilitiesof the various coalitionsin which they might par-ticipatewill determine the players in making their claims.The theorymust then provide their adjustment to the available total.

44.6.Limitations of the Excess.TheNon-isolatedCharacterof a Gamein the New Setup

44.6.1.Theseconsiderationsindicate that the excesse must be neithertoo small (when e < 0), nor too large(when e > 0). In the former caseasituation would arise where eachplayer would prefer to fall back on thegame,even if the worst shouldhappen, i.e.if he has to play it isolated.1

In the lattercaseit will happenthat the \"free gift\" is \"toolarge,\"i.e.thatno playerin any imaginedcoalition can makesuchclaimsas to exhausttheavailable total. Then the very magnitude of the gift will actas a dissolventon the existingmechanismsof organizations.

We will see in 45. that thesequalitative considerationsarecorrectand we will get from rigorousdeductionsthe detailsof their operationandthe precisevalue of the excessat which they becomeeffective.

44.6.2.In all theseconsiderationsthe game F can no longerbe con-sideredas an isolatedoccurrence,sincethe excessis a contribution or awithdrawal by an outside source.This makes it intelligible that thiswhole train of ideasshouldcome up in connectionwith the decompositiontheory of the game F. TheconstituentgamesA, H areindeedno longerentirely isolated,but coexistentwith eachother.2 Thus, there is a goodreasonto lookat A, H in this way whether the compositegame F shouldbetreatedin the old manner (i.e.as isolated),or in the new one,may bedebatable.We shall see,however, that this ambiguity for F does notinfluence the result essentially,whereasthe broaderattitude concerningA,H provesto be absolutelynecessary(cf.46.8.3.and also46.10.).

When a gameF is consideredin the above sense,as a non-isolatedoccurrence,with contributionsor withdrawals by an outside source,onemight be tempted to do this:Treatthis outside sourcealso as a player,includinghim togetherwith the otherplayersinto a largergame F'. Therulesof F' (which includesF) must then be devisedin such a manner as toprovidea mechanismfor the desiredtransfers. We shall be able to meetthis demand with the help of our final results, but the problemhas someintricaciesthat arebetterconsideredonly at that stage.

n

1 This happens when the proposedtotal v(/) + eis < v((i)). As the last expres--ision isequal to v(7) ny (by (42:11)in 42.5.1.)this means e < ny.

We will seein 45.1.that this is preciselythe criterion for ebeing \"too small/''This in spite of the absenceof \"interactions,\" as far as the rules of the game areconcerned;cf.41.2.3.,41.2.4.)))

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DECOMPOSABLEGAMES 367))

44.7.Discussionof the New Setup (i ),44.7.1.The reconsiderationof our old definitions mentioned at the

beginningof 44.4.3.is a very simplematter.For the extended imputations we have the new definitions of 44.4.2.

The definitions of effectivity and domination we takeover unchangedfrom30.1.1.1 the supportingarguments broughtforward in thediscussionwhichledup to thosedefinitions appearto loseno strengthby our presentgeneral-izations. The sameapplies to our definition of solutions eod.2 with onecaution:The definition of a solution referred to makes the conceptof asolution dependent upon the set of all imputationsin which it is formed.Now in our presentsetup of extendedimputationswe shallhave to considerlimitations concerningthem notably concerningtheir excessesas indi-catedin 44.5.1.Theserestrictionswill determinethe set of all extendedimputationsto beconsideredand thereby the conceptof a solution.

44.7.2.Specificallywe shall considertwo types of limitations.First,we shall considerthe casewhere the value of the excessis pre-

scribed.Then we have an equation(44:11) e = eowith a given e . Themeaning of this restrictionis that the transfer fromoutsideis prescribed,in the senseof the discussionof 44.5.2.

Second,we shallconsiderthe casewhere only an upper limit of the excessis prescribed.Thenwe have an inequality

(44:12) e g e

with a given e . Themeaning of this restrictionis that the transfer fromoutsideis assigneda maximum (from the point of view of the playerswhoreceiveit).

Thecasein which we arereally interestedis the first one,i.e.that oneof 44.5.2.The secondcasewill prove technicallyuseful for the clarificationof the first one although its introduction may at first seemartificial.We refrain from consideringfurther alternatives becausewe will be able tocompletethe indicateddiscussionwith thesetwo casesalone.

Denotethe setof all extendedimputations fulfilling (44:11)(first case)by E(eo). Considering(44:9)in 44.5.1.,we can write (44:11)as

n

(44:11*) = v(7)+ e .-iDenotethe setof all extendedimputations fulfilling (44:12)(secondcase)by F(eQ). Considering(44:9)in 44.5.1.,we can write (44:12)as

n

(44:12*) g v(7)+ e,.i-l1 I.e.(30:3);(30:4:a)-(30:4:c)loc.cit.,respectively.8 I.e.(30:5:a),(30:5:b)or (30:5:c)eod.)))

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368 COMPOSITIONAND DECOMPOSITIONOF GAMES

Forthesakeof completeness,we repeatthe characterizationof an extendedimputation which must beadded to (44:11*),as well as to (44:12*):(44:13) a, v((0), for i = 1, - , n.

Notethat the definitionsof (44:9)aswellas (44:11*),(44:12*)and (44:13)areinvariant under the isomorphismof 42.4.2.

44.7.3.Now the definition of a solution can be taken over from 30.1.1.Becauseof thecentral roleof this conceptwerestatethat definition,adjustedto the presentconditions.Throughout the definition which follows, E(e^)can be replacedby F(e<>),as indicatedby [ ].

A setV E(eo)[F(ed)]isa solution forE(eQ) [F(eo)]if it possessesthe follow-ing properties:

(44:E:a) No in V is dominatedby an a in V.

(44:E:b) Every /} of E(e*)[F(e9)] not in V is dominatedby some ainV.

(44:E:a)and (44:E:b)can bestated as a singlecondition:(44:E:c) Theelementsof V arethoseelementsof E(e<>)[F(eo)]which

areundominatedby any elementof V.It will benoted that E(0) takes us back to the original 30.1.1.(zero-sumgame)and 42.4.1.(constant-sumgame).

44.7.4.The conceptsof composition, decomposition and constituentsof extendedimputations can again be defined by (44:l)-(44:4)of 44.2.1.As pointedout in 44.4.2.the technical purposeof our extendingthe conceptof imputation is now fulfilled. Decompositionas well as compositioncannow alwaysbecarriedout.

Theconnectionof theseconceptswith the setsE(e) and F(eo)is not sosimple;we will deal with it as the necessityarises.

Forthe composition,decompositionand constituentsof setsof extendedimputationsthe definitions of 44.2.3.can now be repeatedliterally.

45.Limitations of the Excess.Structureof the ExtendedTheory45.1.TheLowerLimit of the Excess

45.1.In the setups of 30.1.1.and of 42.4.1.imputationsalways existed.It is now different: Either set E(e), F(eQ) may be empty for certaine .Obviously this happens when (44:11*)or (44:12*)of 44.7.2.conflict with

(44:13)eodemand this is clearlythe casefor

v(7) + eo < v((i))-iin both alternatives. As the right hand side is equal to v(7) ny by(42:11)in 42.5.1.,this means

(45:1) e < -ny)))

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STRUCTUREOF THEEXTENDEDTHEORY 369If J(e) [F(e<>)]is empty, then the empty setis clearly a solution for it

and sinceit is its only subset, it is alsoits only solution.1 If, on the otherhand, E(e) [F(e )] is not empty, then none of its solutionscan be empty.This followsby literal repetitionof the proof of (31:J)in 31.2.1.

Theright hand sideof the inequality (45:1)is determinedby the gameT; we introducethis notation for it (with the oppositesign,andusing(42:11)in 42.5.1.):(45:2) |rK= ny = v(/) - v((t)).-i

Now we can sum up our observationsas follows:

(45:A) Ifo <- |r|,,

then E(eQ), F(e) are empty and the empty set is their onlysolution. Otherwiseneither E(e<j) nor F(e) nor any solution ofeithercan beempty.

This result gives the first indication, that \"toosmall\" values of e(i.e.e) in the senseof 44.6.1.exist. Actually, it corroboratesthe quanti-tative estimateof footnote 1 on p.366.45.2.The Upper Limit of the Excess.Detachedand Fully DetachedImputations

45.2.1.Let us now turn to those values of e (i.e.e), which are \"toolarge\" in the senseof 44.6.1.When does the disorganizinginfluence ofthe magnitude of e, which we thereforesaw, manifest itself?

As indicated in 44.6.1.,the criticalphenomenonis this:The excessmay be too largeto beexhaustedby the claimswhich any player in anyimaginedcoalition can possiblymake. We proceedto formulate this ideain a quantitative way.

It is best to considerthe extendedimputations a themselves,instead

of their excessese. Suchan a is past any claimswhich may be made in

any coalition,if it assigns to the players of each (non-empty)set S Imore than thoseplayerscouldgetby forming a coalition in F, i.e.if(45:3) 2v cti > v(S) for every non-emptysetS & I.

in S

Comparingthis with (30:3)in 30.1.1.showsthat our criterion amounts to

demandingthat every non-emptysetS beineffective for a .In our actual deductions it will prove advantageousto widen (45:3)

somewhatby includingthe limiting caseof equality. Thecondition thenbecomes

1In spite of its triviality, this circumstance should not be overlooked.The textactually repeatsfootnote 2 on p. 278.)))

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370 COMPOSITIONAND DECOMPOSITIONOF GAMES

(45:4) <xi v(S) for every SsL1

iinS*It is convenient to give these a a name. We call the a of (45:3)fully

detached, and those of (45:4)detached. As indicated, the latterconceptwill be really neededin our proofs both termini aremeant to expressthatthe extendedimputation is detachedfrom the game,i.e.that it cannot beeffectively supportedwithin the game by any coalition.

45.2.2.Onemore remarkis useful:The only restrictionimposedupon extendedimputations is (44:13)of

44.7.2.:(45:5) at ^ v((i)) for i = 1, , n.

Now if the requirement (45:4)of detachednessis fulfilled and hencea fortiori if the requirement(45:3)of full detachednessis fulfilled then it isunnecessaryto postulate the condition (45:5)as well. Indeed,(45:5)is the specialcaseof (45:4)for S = (i).

This remark will be made use of implicitly in the proofs which follow.45.2.3.Now we can revert to the excesses,i.e.characterizethosewhich

belong to detached(or fully detached)imputations. This is the formalcharacterization :(45:B) The game F determinesa number |r|2 with the following

properties:(45:B:a) A fully detachedextendedimputation with the excesse

existsif and only if

e > |r|2.(45:B:b) A detachedextendedimputation with the excesse exists

if and only if

e ^ |r|,.Proof:Existenceof a detacheda 3:Let a be the maximum of all v(S),

SSi(so a ^ v(0) = 0). Put ^ = {aj, , aj} = (a, , a }.Then for every non-empty S SI we have a? ^ a ;> v(5). This is

inS

(45:4),so a is detached.1Itis no longer necessaryto excludeS - , since(45:4)unlike (45:3)is true when8 . Indeed,then both sidesvanish.J The intuitive meaning of thesestatements is quite simple:It is plausible that in

order to producea detachedor a fully detachedimputation, a certain (positive) minimumexcessis required. |r|t is this minimum, or rather lower limit. Sincethe notions\"detached\" and \"fully detached\"differ only in a limiting case(the - sign in (45:4)),it stands to reasonthat their lower limits be the same. Thesethings find an exactexpressionin (45:B).

8 Note that it is necessaryto prove this! Theevaluation which we give hereiscrude,for more preciseonescf. (45:F)below.)))

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STRUCTUREOF THEEXTENDEDTHEORY 371

Propertiesof the detached a :According to the above, detached

a = {!, , an ]

n

exist,and with them their excessese = a< v(7). By (45:4)(witht-i8 = 1)all thesee are 2> 0. Henceit followsby continuity, that theseehave

a minimum e*. Choosea detached a * = {af, , a*}with this excesse*.1

We now put

(45:6) |r|,= e*.

Proof ot (45:B:a),(45:B:b):If V = {ai, , an } isdetached,then byn .

definition e = on v(7) ^ e*. If a = {i, , an ) is fully detached,i-1then (45:3)remains true if we subtract a sufficiently small 6 > from

eachoti. So a' = {on 5, , an 8} is detached.Henceby defini-n

tion e - nb = (a< - ) - v(7) ^ e*,e >e*.i-iConsidernow the detached a * = {af, , a*) with

a*- v(7) = e*.-iThen (45:4)holdsfor a*;hence(45:3)holds if we increaseeacha* by a

$ >0. So a\" = (a*+ 5, , a*+ 6j is fully detached.Itsexcessisn

= (a*+ 5) - v(7) = e* + n5. So every e = e* + n, 6 > 0, i.e.i-levery e > e*,is the excessof a fully detachedimputation hencea fortioriof a detachedone;and e* is, of course,the excessof a detachedimputation

a*.Thus all parts of (45:B:a),(45:B:b)hold for (45:6).46.2.4.The fully detachedand the detachedextendedimputationsare

also closely connectedwith the conceptof domination. The propertiesinvolved aregiven in (45:C)and (45:D)below. They form a peculiaranti-thesisto eachother. Thisis remarkable,sinceour two conceptsarestronglyanalogousto eachother indeed,the secondonearisesfrom the first onebythe inclusion of its limiting cases.

1This continuity argument is valid becausethe = sign is included in (45:4).)))

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372 COMPOSITIONAND DECOMPOSITIONOF GAMES

(45:C) A fully detachedextendedimputation a dominatesno otherextendedimputation ft .

Proof:If a H 0,then a must possessa non-emptyeffectiveset.

(45:D) An extendedimputation a is detachedif and only if it is

dominatedby no otherextendedimputation ft .

Proof:Sufficiencyof beingdetached:Let a = {ai, ,an } bedetached.Assume a contrario ft H a , with the effectivesetS. ThenSis not empty;

< fti for i in S. So 2} < 2} ft ^ v(5) contradicting(45:4).t in 8 f in 8

Necessityof beingdetached:Assumethat a = {i, , an } is notdetached.Let S be a (necessarilynon-empty)set for which (45:4)fails,i.e. oti < v(S). Thenfor a sufficiently small 6 > 0,even

(<* + *) v(S).tin S

Put = {0i, , ft n \\= {ai+ 6, , an + $},then always a< < ft

and 5is effective for ft : ft g v(5). Thus H a .

45.3.Discussionof the Two Limits |r|i,|rt|.Their Ratio

45.3.1.The two numbers |r|iand |F|2, as defined in (45:2)of 45.1.and in (45:B)of 45.2.3.areboth in a way quantitative measuresof theessentialityof F. Moreprecisely:

(45:E) If F is inessential,then JF^ = 0,|F|2 = 0.If F is essential,then |F|i> 0,|F|2 >0.

Proof:Thestatementsconcerning|F|i,which is = ny by (45:2)of 45.1.,coincidewith the definitions of inessentialityand essentialityof 27.3.,asreassertedin 42.5.1.

The statementsconcerning|F|2 follow from thoseconcerning|F|i,bymeansof the inequalitiesof (45:F),which we canuse here.

45.3.2.Thequantitative relationshipof |F|iand |F|2 is charactenzedasfollows:

Always))

(45:F))) -^rirai)))

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STRUCTUREOF THEEXTENDEDTHEORY 373

Proof:As we know, |F|iand |r|8 areinvariant understrategicequivalence,hencewe may assumethe game T to bezero-sum,and even reducedin thesenseof 27.1.4.We can now use the notationsand relationsof 27.2.

Since|T|i= ny, we want to prove that

(45:7) jJLjy g |r|,* Sfef^ y.

Proof of the first inequality of (45:7):Let a = {ai, , aw ) bedetached.Then (45:4)gives for the (n - l)-elementset 8 = I - (fc),

n

5) ai ctk = 5) on ^ v(S) = 7, i.e.-1))

r))

(45:8) ,- * rn r.

Summing (45:8) over k = 1, . n, gives n ] a a*i-l *-ln n n

i.e.(n 1)^ a ^ 717, ^ a : -y 7. Now v(7) = 0,so e = ^ a*.

Thus e ^ r 7 for all detachedimputations;hence|F|t ; r 7.n 1 w 1Proof of the secondinequality of (45:7):Put a00 = ^-y^7, and ^ = {a?, - , aj }

= {a00, , a00}.This a is detached,i.e.it fulfills (45:4)for all Sfi I. Indeed:Let p bethe number of elementsof S. Now we have:

p = Q:5 = , (45:4)is trivial.p = 1:S = (t), (45:4)becomesa00 ^ v((i)),n 2 . i - i ii.e. s y ^ ~~~7 which is obvious.

p ^ 2:(45:4)becomespa00 ^ v(S),but by (27:7)in 27.2.vGS) ^ (n -p)7,

n 2so it suffices to prove pa00 2> (n p)7 i.e.p ^ 7 ^ (w p)7. This

amounts to p 7 ^ ^7> which followsfrom p ^ 2.

Thus a is indeeddetached.As v(/) = 0,the excessis

eoo na oo = n(n^\"

2)7.

Hence|r|i 7-)))

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374 COMPOSITIONAND DECOMPOSITIONOF GAMES

46.3.3.It is worth while to consider the inequalitiesof (45:F) forn = 1,2,3,4, successively:

n = 1,2:In thesecasesthe coefficient j of the lower bound of then

inequality is greaterthan the coefficient 5 of the upperbound.,1 Thiszmay seemabsurd. But sinceF is necessarilyinessentialfor n = 1,2(cf.the first remark in 27.5.2.),we have in thesecases|r|i= 0,|r|i= 0,andso the contradictionsdisappear.

1 ^ on = 3:In this casethe two coefficients T and s coincide:

n I 2Both areequal to i. Sothe inequalitiesmergeto an equation:

(45:9) |r|,= ilrK.

n ^ 4:In thesecasesthe coefficient ^ of the lowerboundisdefinitely^ o

smallerthan the coefficient = of the upperbound.2 Sonow the inequal-JL

ities leave a non-vanishing interval openfor |F2|.The lower bound |F|2 = ^r |F|iis precise,i.e.thereexistsfor each

ft L.

n ^ 4 an essentialgame for which it is assumed. Therealsoexistfor eachn ^ 4 essentialgames with |F|2 > r |F|i,but it is probablynot pos-

n 2sible to reachthe upper bound of our inequality, |F|2 = ~ lr li- The

i

precisevalue of the upperboundhas not yet beendetermined.We do notneedto discussthesethings hereany further. 8

45.3.4.In a more qualitative way, we may therefore say that |F|i,|F|jareboth quantitative measuresof the essentialityof the game F. Theymeasureit in two different, and to a certainextent,independentways.Indeed,the ratio |F|2/|F|i,which never occursfor n = 1,2(no essentialgames!),and is a constant for n = 3 (its value is ^), is variable with F foreachn ^ 4.

We saw in 45.1.,45.2.,that thesetwo quantities actually measurethelimits, within which a dictated excesswill not \" disorganize\" the players,in the senseof 44.6.1.Judgingfrom our results, an excesse < |F|iis\"toosmall\"antl an excesse > |F|2 is \"toogreat\"in that sense. Thisview will becorroboratedin a much more precisesensein 46.8.

1They are ,-}for n - 1; 1, for n 2. Note alsothe paradoxicalvalues <* and-i!1 ti 2s __ < .- means 2 < (n l)(n 2) which is clearly the casefor all n 4.

1For n 4 our inequality is J |r|i |r|i |r|i.As mentioned above,we know anitial game with |r|8 - J |r|iand alsoone with |r| - i |r|i.)))

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STRUCTUREOF THEEXTENDEDTHEORY 375

46.4.DetachedImputations and Various Solutions.

TheTheoremConnecting (e ), F(e)45.4.1.(44:E:c)in the definition of a solution in 44.7.3.and our result

(45:D)in 45.2.4.give immediately:

(45:G) A solution V for E(e^ [F(e)] must contain every detachedextendedimputation of E(e<>)[F(e)].

Theimportanceof this resultis due to its role in the followingconsidera-tion.

After what wassaid at the beginning of 44.7.2.about the rolesof E(e^)and F(e), the importanceof establishingthe completeinter-relationshipbetween these two caseswill be obvious. I.e.we must determine theconnectionbetweenthe solutionsfor E(e) and F(e).

Now the whole difference betweenE(eQ) and F(eQ) and their solutionsis not easy to appraise in an intuitive way. It is difficult to seea prioriwhy thereshouldbe any differenceat all:In the first casethe \"gift,\" madeto the playersfrom the outside,has the prescribedvalue e , in the secondcaseit has the prescribedmaximum value e . It is difficult to seehow the\"outsidesource/'which is willing to contributeup to e can ever be allowedto contributelessthan e in a \" stable\" standard of behavior (i.e.solution).However,our past experiencewill caution us againstrashconclusionsin thisrespect.Thus we saw in 33.1.and 38.3.that alreadythreeand four-persongames possesssolutionsin which an isolated and defeated player is not\" exploited\" up to the limit of the physicalpossibilities and the presentcasebearssomeanalogy to that.

45.4.2.(45:G)permitsus to makea more specificstatement:A detachedextendedimputation a belongsby (45:G)to every solutionfor

F(e), if it belongsto F(e). On the otherhand, a clearly cannot belongtoany solutionfor E(eQ) if it doesnot belongto E(eQ). We now define:

(45:10) D*(e<>) is the setof all detachedextendedimputations a in

F(e), but not in E(e).Sowe see:Any solution of F(e) containsall elementsof D*(eo);any solu-tion of E(e<>)containsno elementof D*(e). ConsequentlyF(e) and E(eQ)have certainly no solution in common if D*(e) is not empty.

Now the detacheda of D*(e) arecharacterizedby having an excesse 6 , but not e = e i.e.by

(45:11) e < e .Fromthis we conclude:(45:H) D*(e<>)is empty if and only if

e, |r|t .)))

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376 COMPOSITIONAND DECOMPOSITIONOF GAMES

Proof:Owing to (45:B) and to (45:11)above, the non-emptinessofZ>*(e ) is equivalent to the existenceof an e with |r|jg e < e i.e.to

o > |r|j. Hencethe emptinessof D*(eQ) amounts to eQ ^ |r|a.Thus the solutionsfor F(e) and for E(e$)aresureto differ,when eo>|F|i.

This is further evidencethat e<> is \"toolarge\" for normal behavior when it is> |r|.

45.4.3.Now we can prove that the differenceindicatedabove is the onlyonebetweenthe solutionsfor E(e*)and for F(e). Moreprecisely:(45:1) Therelationship

(45:12) V^W = VuD*(e)

establishesa one-to-onerelationship between all solutions Vfor E(eQ) and all solutionsW for F(e).

This will be demonstratedin the next section.46.5.Proof of the Theorem

45.5.1.We beginby proving someauxiliary lemmas.The first one consistsof a perfectly obvious observation,but of wide

applicability:(45:J) Let the two extendedimputations y = {71, , yn \\

and

5 == {81, , dn ] bear the relationship

(45:13) 7, ^ 5, for all i = 1, , n\\

then for every a , a H y impliesa H 5 .The meaning of this result is, of course,that (45:13)expressessome

>

kind of inferiority of 6 to y in spite of the intransitivity of domination.This inferiority is, however, not as completeas onemight expect. Thus

onecannot makethe plausibleinferenceof y ** ft from 5 H ft , becausethe

effectivity of a setS for & may not imply the same for y . (Thereadershouldrecallthe basicdefinitions of 30.1.1.)

It should also be observed,that (45:J)emergesonly becausewe haveextendedtheconceptof imputations. Forour olderdefinitions (cf. 42.4.1.)

n n

we would have had 2) 7 = ^ i<; hencey< ^ fl for all i = 1, , nt-i i-i

necessitates% = 5, for all i == 1, , n, i.e.y = & .45.5.2.Now four lemmasleadingdirectlyto the desiredproof of (45:1).

y >, y ^

(45:K) If a H ft with a detachedand in F(e) and ft in/?(e), then

thereexistsan a 'H ft with a ' detachedand in E(eQ ).)))

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STRUCTUREOF THEEXTENDEDTHEORY 377

Proof:Let S bethesetof (30:4:a)-(30:4:c)in 30.1.1.for the domination

a H ft . S = / would imply a{ > ft for all i = 1, , n so

,- v(J) > ft - v(/).))

n n))

But as o is in F(e) and ft in U(e), so % o - v(7) g e = ft v(7),

contradictingthe above.So S T* 7. Choose,therefore, an t' = 1, , n, not in S. Define

a' = {a'j, , a'n } with

A ~*choosinge ^ so that 2) <

~~ V CO= go. Thus all o{^ ; hence a 'is detachedand it is clearly in (e ). Again, as a( = on for i 5^ t'o hencefor all i in S,so our a H /3 impliesa 'H /3 .(45:L) Every solution W for F(e) has the form (45:12)of (45:1)

for a unique V sE(eo).1

Proof:Obviously the V in question if it existsat all is the intersectionW n E(ev), so it is unique. In order that (45:12)shouldhold for))

we needonly that the remainderof W beequal to Z)*(eo),i.e.(45:14) W - E(e<>)= D*(e).Let us therefore prove (45:14).

Everyelementof D*(e<>)isdetachedand in F(e) so it is in W by (45:G).Again, it is not in E(e*\\ so it is in W - E(e<>). Thus))

(45:15) W -E(e)If also

(45:16) W-(eo)D*(eo),then (45:15),(45:16)togethergive (45:14),as desired.Assume therefore,that (45:16)is not true.

Accordingly,consideran a = {i, , an|in W E(eJ)and notn

in D*(e). Then a is in F(e), but not in E(e<>),so a< -v(J) < e . As-i1We do not yet assertthat this V is a solution for f( t) that will comein (45:M).)))

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378 COMPOSITIONAND DECOMPOSITIONOF GAMES

a is not in >*(e), this excludesits beingdetached.Hencethereexistsa

non-empty set8 with a < v(S).tin S

Now form a '= {&(, - , a'n } with

a'i = cti + c for i in S,a< = a< for i not in S,

n

choosing > so that still <*(- v(7) g e and ; ^ v(S). So a '

\\ I in Sis in F(e). If it is not in W, then (asW is a solution for F(eQ)) thereexists>>> > >

a ft in W with ft H a'. As all ^ , this implies ft H a by (45:J).Thisis impossible,sinceboth ft , a belongto (thesolution)W. Hencea 'must be in W. Now ; > at for all i in S, and <*< ^ v (5)- So

tin S

a'H a . But as both a ', a belongto (thesolution)W, this is a contra-diction.(45:M) TheV of (45:L)is a solution for E(e).

Proof:V cE(e) is clear,and V fulfills (44:E:a)of 44.7.3.along with W

(which is a solution for F(eQ)), sinceV W. So we need only verify(44:E:b)of 44.7.3.

Considera in #(<)),but not in V. Then /3 isalsoin F(eQ) but not in

W| hencethereexistsan a in W with a. H ft (W is a solution forF(e)!).>

If this a belongsto E(e), then it belongsto W n E(eQ) = V, i.e.we have

an a in E(eQ) with a H ft .If a doesnot belongto E(e$),then it belongsto W #(e) = D*(e),

and so it isdetached.Thus a H ft , a detachedand in f(eo). Hencethere

existsby (45:K)an a 'H , a 'detachedand in E(e<>). By (45:G)this a 'belongsto W, (E(eo)sF(e), W is a solution for F(eQ)\\); henceit belongsto W n E(eQ) = V. Sowe have an a ' in E(eQ) with a '*- .]

Thus (44:E:b)of 44.7.3.holdsat any rate.(45:N) If V is a solution for JZ(<* ), then the W of (45:12)in (45:1)

is a solutionfor F(e).Proof:WsF^o) is clear,so we must prove (44:E:a),(44:E:b)of 44.7.3.Ad (44:E:a):Assume a H ft for two a , ft in W. a 8- ^3*and (45:D)

excludethat ft be detached.So is not in D*(e), henceit is in

W - D*(6) = V.)))

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STRUCTUREOF THEEXTENDEDTHEORY 379

Hencea H ft excludesthat a too be in (the solution)V. So a is in

W - V = D*(o).Consequentlya is detached.

Now (45:K)producesan a ' H ft which isdetachedand in E(eQ). Being> - ^

detached,a' belongsby (45:G)to (thesolution for E(eQ)) V. As a ', ft

both belongto (thesolution) V and a ' H ft , this is a contradiction.Ad (44:E:b):Considera T = {0i, , ftn] in F(c), but not in W.

Now form ft (c) = {^(e), , n(e)}= {fr + t, , n + } for every6^0.Let increasefrom until one of thesetwo things occursfor thefirst time:))

(45:17) (0is in

(45:18) \"/?()is detached.2

We distinguishthesetwo possibilities:(45:17)happensfirst, say for c = ei ^ 0:ft (ci)is in E(eQ), but it is not

detached.If 1 = 0, then ft = ft (0) is in E(eQ). As ft is not in V W, there

>

existsan a H in (the solution for E(e<>))V. A fortiori a in W.

Assume next*i > 0, and (ci) in V. As ft (ei) is not detached,thereexists a (non-empty) S c/ with ^ ft(ei) < v(S). Besides,always

tin S

fti(ci) > fti. So ft (ci) H . And ft (1)is in V, hencea fortiori in W.

Assume,finally, *i > and ft (*0 not in V. As ft (i) is in E(e<>)t there

existsan a ^ ( l) in (thesolution for E(e*))V. Sincealways ft i(i)> ft,

a H (ei)impliesa ^ ft by (45:J). And a is in V, hencea fortiori in W.

(45:18)happensfirst, or simultaneously with (45:17),say for = ej ^ 0:ft (j) is still in F(e), and it is detached.

If 7(s) is in J(e), then it is by (45:G)in (thesolution for ())V.If 6 (e2) is not in #(e), then it is in D*(e). So ft (*2) is at any ratein W.))

1I.e.the excessof ft () is - c . For (0) - is in F(c), i.e.its excessb e ,*

and the excessof (e)increaseswith ,I.e.5) 0,(e) vOS)forallS7.Each ft (t) increaseswith c.))

% in S in S)))

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380 COMPOSITIONAND DECOMPOSITIONOF GAMES

This excludescs == 0,sinceft = ft (0)is not in W. So61 >0.For < < i, ft (e) is not detached,so thereexistsa non-empty8 /

with 2 0*W < V OS)- Hencethere existsby continuity a non-empty))

S / even with &() v(S). Besides,always ftfo) > ft, hence))

(i)H - And ()belongsto W.

Summing up:In every casethereexistsan a H ft in W. (This a was* *

a , ft (ci),a , (ej)above,respectively.) So(44:E:b)is fulfilled.We can now give the promisedproof:Proof of (45:1):Immediate,by combining (45:L),(45:M),(45:N).

45.6.Summary and Conclusions

46.6.1.Our main results,obtainedso far, can be summarizedas follows:

(45:0) If(45:O:a) e < -|r|i,

then U(e), F(eQ) are empty and the empty set is their onlysolution.

If

(45:0:b) -|r|i e ^ |r|,f))

then J5(e), ^(^o)arenot empty, both have the samesolutions,which areall not empty.

If

(45:0:c) eQ > |r|,,

thenE(eo),F(e) arenot empty,they have no solutionin common,all theirsolutionsarenot empty.

Proof:Immediateby combining (45:A), (45:1)and (45:H).This result makes the critical characterof the points e<>

= |F|i,|r|*quite clearand it further strengthens the views expressedat the end of45.1.and following (45:H)in 45.4.2.concerningthesepoints:That it is herewhere60becomes\"toosmall\"or \"toolarge\" in the senseof 44.6.1.

46.6.2.We arealsoablenow to prove somerelationswhich will beusefullater(in 46.5.).(45:P) LetW be a non-empty solution for F(e), i.e.assume that

eo -|r|i.Then

(45:P:a) Max-. e(O = e*\" iii iff)))

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DETERMINATION OF ALL SOLUTIONS 381

(45:P:b) Min-^ w e(7)- Min (.,|r|,).Also

(45:P:c)Max-*. e(~a)- Min-. e(a)= Max (0,e - |r|,).** in TT ot in iff

Proof:(45:Prc)followsfrom (45:P:a),(45:P:b)sincee -Min (e , |T|2) = Max (e -e , e - |r|2) = Max (0,e - |r|2).

We now prove (45:P:a),(45:P:b).Write W = VuD*(e), V a solution for J0(e), following (45:1).As

eQ ^ |r|i,soVisnot empty (by (45:A) or (45:0)).As weknow e(a) = e

throughout V and e(a) < e throughout D*(e).Now for e ^ |T|2, D*(6) is empty (by (45:H)),so

(45:19) Max-*. e(a)= Max-*. w e(a)= ,in W a m V

(45:20) Min-. e(T)= Min-. w e(\"^)= .in w a m V

And for eQ > |T|2, D*(e) is not empty (again by (45:H)),it is the setof all>

detacheda with e(a ) < e . Henceby (45:B:b)in 45.2.3.thesee(a ) havea minimum, |T|2. Sowe have in this case:

(45:19*) Max-.w/ e(2) = Max-.w e(2)= ,in W in V))

(45:20*) Min 7 ^ w e()= Min-.fa ^^ e()= |r|,.

(45:19),(45:19*)togethergive our (45:P:a),and (45:20),(45:20*)givetogetherour (45:P:b).

46.Determinationof All Solutionsin a DecomposableGame46.1.Elementary Propertiesof Decompositions

46.1.1.Let us now return to the decompositionof a game T.Let F bedecomposablefor J, K( I J) with A, H as its J-,IC-con-

stituents.

Given any extendedimputation a = (ai, , <x} for 7, we form its

J-,^-constituentsft , 7 (ft = a, for i in J, 7^ = a for i in K),and theirexcesses

1Ourassertionincludes the claim that these Max %w,and Min . exist.in W in W

1Verbally: Themaximum excessin the solution W is the maximum excessallowed in

FMi o. Theminimum excessin the solution W is again c , unless e > |r|t,in whichcaseit isonly |r|t. I.e.the minimum is asnearly e aspossible,considering that it mustnever exceed|r|t.

The \"width\" of the interval of excessesin W ia the excessof e over IrU, if any.)))

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382 COMPOSITIONAND DECOMPOSITIONOF GAMES))

(46:1)))

Excessof a. in /:e = e(a ) = ] at-i))

Excessof ft in J:/ = /(a ) = J/ a ~~v (^)>tin/

Excessof 7 in JRC: g = g(a) = ]? a v(X).1tin X

Since(46:2) v(J) + v(K) = v(7)

(by (42:6:b)in 42.3.2.,or equallyby (41:6)in 41.3.2.with S= J, T = K)therefore(46:3) e=f + g

(46:A) We have(46:A:a) |r|i= |A|i + |H|i,(46:A:b) |r|2 = |A|,+ JHJ2.(46:A :c) F is inessentialif and only if A, H areboth inessential.

Proof:Ad (46:A:a):Apply the definition (45:2)in 45.1.to T, A, H ir

turn.(46:4) |r|i= v(7) - v((t)),

tin /(46:5) |A|i = v(J) - v((i)),

t in J(46:6) |H|,= v(K)- % v((i)).

Comparing(46:4)with the sum of (46:5)and (46:6)gives (46:A:a),owing t<

(46:2).Ad (46:A:b):Let ~2,~P , y be as above (before(46:1)).Then a i:

detached(in /) if

2) a, S v(R) for all R S 7.tin R

Recalling(4t:6)in 41.3.2.we may write for this/ A G. f7\\ \\^ \\ \\^ \"^ /C\\ I ,~/fTi\\ **.-* 11 C r~ T H^ r V(4o:7; 2/ a i 2/ a* v w) + v \\^ ) *or a\" o /, 1 IK.

t in S t in r

Again , 7 aredetached(in J,X) if

(46:8) % a, ^ v(S) for all SsJ,iin S

(46:9) J) a, ^ v(T) for all TsX.tinT

1Up to this point it was not necessaryto give explicit expressionto the dependencof a 'sexcesseupon a . We do this now.for easwell asfor/, g.)))

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DETERMINATION OF ALL SOLUTIONS 383

Now (46:7)isequivalent to (46:8), (46:9). Indeed:(46:7)obtainsby adding(46:8)and (46:9);and (46:7)specializesfor T = to (46:8)and for S = 9to (46:9).

Thus a isdetached,if and only if its (/-,K-)constituentsft , 7 arebothdetached.As their excessese and /, g arecorrelatedby (46:3),this givesfor their minima (cf.(45:B:b))

|r|,= |A| 2 + |H|2,i.e.our formula (46:A:b).

Ad (46:A:c):Immediateby combining (46:A:a)or (46:A:b)with (45:E)as appliedto F, A, H.

Thequantities|r|i,|F|2 areboth quantitative measuresof the essentialityof the game F, in the senseof 45.3.1.Our above resultstatesthat both areadditive for the compositionof games.

46.1.2.Another lemma which will be useful in our further discussions:> >

(46:B) If a H ft (for F),then the setSof30.1.1.for this dominationcan be chosenwith S J or S K without any lossof generality.l

Proof:Considerthe set S of 30.1.1.for the domination a H ft. IfaccidentallySSiJ or S K, then there is nothing to prove, so we mayassume that neither S J nor S K. ConsequentlyS = Siu Ti, whereSi /, Ti S K, and neither Sinor T\\ is empty.

We have a > ft for all i in S,i.e.for all i in Si,as well as for all i in T\\.

Finally

2 at ^ v(S).tin 8

Theleft hand sideis clearlyequalto ^ a + ^ a,while the right hand*in Si t in TI

side is equal to v(Si)+ v(Ti)by (41:6)in 41.3.2.Thus))

i in Si

henceat leastone of))

in Sl t in 2*,

must be true.>

Thus of the three conditionsof domination in 30.1.1.(for a ^ ft )(30:4:a),(30:4:c)holds for both of Si,TI and (30:4:b)for at leastone ofthem. Hence,we may replaceour original S by eitherSi(sJ) or Ti(fi -K\.

This completesthe proof.1I.e.this extra restriction on S doesnot (in this case!)modify the conceptof

domination.)))

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384 COMPOSITIONAND DECOMPOSITIONOF GAMES

46.2.Decompositionand Its Relation to the Solutions:First Results Concerning F(e )

46.2.1.We now directour coursetowards the main objectiveof thispart of the theory:Thedeterminationof all solutionsU/ of the decomposablegameT. Thiswill be achievedin 46.6.,concludinga chain of sevenlemmas.

We beginwith somepurely descriptiveobservations.Considera solution U/ for F(e) of T. If U/ is empty, thereis nothing

more to say. Let us assume, therefore, that U/ is not empty owing to(45:A) (orequallyto (45:0))this is equivalent to))

Usingthe notationsof (46:1)in 46.1.1.we form:

Max-../()= *>,a m U/

Min-*. , , /(a ) = p,in U/(46:10) 'Max-. flf() = ftm U/

inU, )=*-'1That all thesequantities can be formed, i.e.that the maxima and the minima in

question exist and areassumed,can beascertainedby a simple continuity consideration.

Indeed/( a ) 2/ a ~~ v (^) and 0r( a ) = 2^ a v(K) are both continuous'in J i in X

functions of a , i.e.of its components 01, , . Theexistenceof their maxima andminima is therefore a well known consequenceof the continuity propertiesof the domain

of a the set U/.For the readerwho is acquainted with the necessarymathematical background

topology we give the precisestatement and its proof. (Theunderlying mathematicalfacts are discussede.g.by C.Carathtodory, loc.cit.,footnote 1 on p. 343. Cf. therepp.136-140,particularly theorem 5).

U/ is a set in the n-dimensional linear spaceLn (Cf.30.1.1.).In order to be surethat every continuous function has a maximum and a minimum in U/, we must know thatU/ is bounded and closed.

Now we prove:(*) Any solution U for F(e ) [E(eQ)] of an n-persongame r is a bounded and closed

set in Ln.Proof:Boundedness:If a {on, , <x) belongsto U, then every a, 2> v((i))

n

and 2/ ~ v(7) =s o, hencea, ^ v(7) + eQ 2} / ^ VCO+ *o ^ v((t)).

Soeach is restricted to the fixed interval

and sothese a form a bounded set.Closedness:This is equivalent to the opennessof the complement of U. That set is,

> >

by (30:5:c)in 30.1.1.,the set of all which aredominated by any a of U* (Observe)))

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DETERMINATION OFALL SOLUTIONS 385

Given two a = {!,- , an j, ft = {ft, , n } there existsa> >

unique 7 = {71, , -yn } which has the sameJ-componentas a , and

the sameJC-componentas ft :(4Q.H) 7. = for tin/,

7 = ft for i in K.46.2.2.We now prove:

(46:C) If ^T, V belongto U/, then the V of (46:11)belongsto U/if and only if

(46:C:a) /(*)+0(7)S o.

Incidentally

(46:C:b) 6(7)=/U)+0(7).Proo/:Formula (46:C:b):By (46:3)in 46.1.1.e(V) = /(V)+ 0(V),

and clearly/(7)=/(),j(7 ) = fif(7)-*

Necessityof (46:C:a):SinceU/fif^o),therefore e(y)^ e is neces-sary and by (46:C:b)this coincideswith (46:C:a).

Sufficiency of (46:C:a):y is clearly an extendedimputation, along

with a , ft , and (46:C:a),(46:C:b)guaranteethat y belongsto F(e).1

Now assumethat y is not in U/. Thenthereexistsa 8 H 7 in U/.The set /S of 30.1.1.for this domination may be chosenby (46:B) with

>Now clearly 5 H 7 implies,when S J that 6 H a

,))

that weareintroducing the solution characterof U at this point!)

For any a denotethe set of all H a by D-+. Then the complement of U is> <*

the sum of all -, a of U.a

Sincethe sum of any number (even of infinitely many) open setsis again open, it

suffices to prove the opennessof eachD-+,i.e.this: If ft ^ a , then for every ft'

a

which is sufficiently near to ft , we have also ft' H a . Now in the definition of dom-

* > ->

ination, ft ^ a by (30:4:a)-(30:4:c)in30.1.1.,appearsin the condition (30:4:c)only.And the validity of (30:4:c)is clearlynot impaired by a sufficiently small change of 0<,since(30:4:c)is a < relation.

(Note that the same is not true for a , becausea appearsin (30:4:b)also,and

(30:4:b)might be destroyed by arbitrary small changes,since(30:4:b)is a ^ relation.

But we neededthis property for ft , and not for a !)1This is the only useof (46:C:a).)))

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386 COMPOSITIONAND DECOMPOSITIONOF GAMES

and when S K that d ^ ft . As 5 , a , ft belongto U/,both alterna-tives areimpossible.

Hence7 must belongto U/> as asserted.We restate(46:C)in an obviously equivalent form:

(46:D) Let V/ be the setof all /-constituentsand Wx the setof allIf-constituentsof U/.

Then U/ obtainsfrom theseV/ and Wx as follows:

U/ is the setof all those y , which have a /-constituenta 'in V/ and a /^-constituent ft

'in W* such that

(46:12) e(O + e(7')^ eQ.1

46.3.Continuation

46.3.Recalling the definition of U/'sdecomposability(for/,/) in (44:B)in 44.3.1.,one seeswith little difficulty, that it is equivalent to this:

U/ obtains from the V/, Wx of (46:D)as outlined there,but withoutthe condition (46:12).

Thus (46:12)may be interpreted as expressingjust to what extentU/is not decomposable.This is of someinterestin the light of what was saidin 44.3.3.about (44:D:a)there.

One may even go a step further: Thenecessityof (46:12)in (46:D) iseasy to establish. (Itcorrespondsto (46:C:a),i.e.to the very simplefirsttwo steps in the proof of (46:C)). Hence(46:D)expressesthat U/ is nofurther from decomposability,than unavoidable.

All this, in conjunction with (44:D:b)in 44.3.3.,suggestsstrongly thatV/, WK ought to be solutionsof A, H. With our presentextensionsof allconceptsit is necessary,however, to decidewhich F(/o),F(go) to take;/o beingthe excesswe proposeto use in /, and go the one in K.2 It will

appearthat the p, # of 46.2.1.arethese/o, g<>.

Indeed,we can prove:

(46:E)(46:E:a) V/ is a solution of A for(46:E:b) W/c is a solutionof H for

It is convenient, however, to derive first another result:

1Note that these a ', ' arenot the a , ft of (46:C) they are their /-,K-constit-uents as well as those of y . e(a ;), e( ft ') are the excessesof a ', ft

' formed in /,K. But they areequal to /( a ), g( ft ) as well as to /(7 ), g(y ). (All of this is relatedto (46:C)).

*Thereaderwill note that this is something like a question of distributing the givenexcesse in / between / and K.)))

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DETERMINATIONOF ALL SOLUTIONS 387

(46:F)(46:F:a) p + = e ,(46:F:b) $ + }= e .

Note that in (46:E),as well as in (46:F),the parts (a), (b) obtain fromeachother by interchanging J,A, p, f with Kt H,#, $. Henceit sufficestoprove in eachcaseonly one of (a), (b) we chose(a).

>

Proof of (46:F:a):Choosean a in U/ for which /(a ) assumesits maxi-* . *

mum p . Sincenecessarilye(a ) ^ e>

and sinceby definition g(a.) ^ ,therefore (46:3)in 46.1.1.gives

(46:13) $+ $ e .Assume now that (46:F:a)is not true. Then (46:13)would imply

further

(46:14) 9+ < 6 .

Usethe above a in U/ with /( a ) = p, and chosealsoa in U/ for which

0( ) assumes its minimum . Then /(a ) + 0( ) = 9+ ^^00 (by

(46:13)or (46:14)).Thus the V of (46:C) belongsto U/, too, Again(46:C)togetherwith (46:14)gives

e(y)-/(\")+ |/(7)= 9+ * < o,

n

i.e. 7i < v(7) + e . Now definet-i

5 = {i, ,}= {71+ >'>7 + c),n ^

choosing > so that 5<= v(/) + e . Thus 6 belongsto F(e).-i

If 6 did not belong to U/, then an TJ H. 5 would existin U/. By> ^ >

(45:J) t? H 7 , which is impossible,since y , 7 areboth in U/. Hence7 belongsto U/. Now 8 - v(J) > ^ 7 ~ v(J) = ^ a,- v(J),

t in J iinJ i in /i.e./( d ) >/(a ) = ?,contradictingthe definition of p.

Consequently(46:F:a)must be true and the proof is completed.Proof of (46:E:a):If a ' belongsto V/j> then it is the J-constituentof an

7 of U/. Hence(cf. footnote 1on p.386)e(a ') =*/(a ) g ?,so that a 'belongsto F($). Thus V/)))

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388 COMPOSITIONAND DECOMPOSITIONOF GAMES

Soour task is to prove (44:E:a),(44:E:b)of 44.7.3.>

Ad (44:E:a):Assume,that a ' H ft' happenedfor two a ', ' in V/.

> > > > >

Then a ', ' arethe J-constituentsof two 7 , 5 in U/. But a ' H '>

clearlyimplies 7 *< 6 , which is impossible.Ad (44:E:b):Consideran a ' in F(<p) but not in V/. Then by definition>

e(a ') g . Use the in U/ mentionedin the above proof of (46:F:a),for which g( ) = . Let ' be the jfiC-constituent of this ft , so that

~ft' is in W* and e(T')= gCft) = f Thus6(T')+ e(7')^ ^ + i = e

(use (46:F:a)).Formthe y (for /), which has the J-, ^-constituents

\"^',T' Then e(7 ) = e(a ') + e(]*') eQ i.e.~y belongsto F(eQ).y does not belongto U/ becauseits /-constituenta ' doesnot belong

to V/. Hence,thereexistsa d H y in (thesolution for F(e)) U/.Let Sbethe setof 30.1.1.for the domination 5 H y . By (46:B)we

may assumethat SsJ or ScK.Assumefirst that S K. As y has the same Jf-constituentsft

' as

ft , we can concludefrom d H 7 that 5 H . Sinceboth 5 ,belongto U/,this is impossible.

ConsequentlySsJ. Denotethe J-constituentof d by 6';as 6

belongsto U/, therefore 6 ' belongsto V/. 7 has the J-constituenta '.Hencewe canconcludefrom 6 H 7 that 5 'H a '.

Thus we have the desired 6 'from V/ with 5 'H a '.46.4.Continuation

46.4.1.(46:D),(46:E)expressedthe generalsolution U/ of T in termsof appropriate solutionsof V/, Wx of A, H. It is natural, therefore, totry to reversethis procedure:To start with the V/, W* and to obtain U/.

It must beremembered,however, that the V/, W* of (46:D) arenotentirely arbitrary. If we reconsiderthe definitions (46:10)of 46.2.1.inthis light of (46:D),then we seethat they canalsobestatedin this form:))

(46:15))))

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DETERMINATIONOF ALL SOLUTIONS 389And (46:F)expressesa relationshipof thesep, tf>, #, which aredeterminedby V/,W/c with eachother and with e .

46.4.2.We will show that this is the only restraint that must be imposedupon the V/, W*. To do this, we start with two arbitrary non-emptysolutionsV/i W/c of A, H (which neednot have beenobtainedfrom anysolutionU/ of T), and assertas follows:(46:G) Let V/ be a non-empty solution of A for F($)and Wx a non-

empty solution of H for F(#). Assume that p, ^ fulfill (46:15)above, and also that with the ^>, ^ of (46:15)

(46:16) + = ?+ = eo.>

Forany a ' of V/ and any ft' of W/c with

(46:17) (O+e(7') 60,

form the 7 (for /) which has the 7-,^-componentsa ', '.>

Denotethe set of all these 7 by U/.The U/ which are obtained in this way are preciselyall

solutionsof F for F(e).Proof:All U/ of the stated characterareobtained in this way: Apply

(46:D)to U/ forming its V/, W/c. Then all our assertionsarecontainedin(46:D),(46:E),(46:F)togetherwith (46:15).

All U/ obtained in this way have the stated character:Consideran U/constructedwith the help of V/, W/c as describedabove. We have toprove that this U/ is a solution T for F(eQ).

Forevery 7 of U/ our (46:17)gives e(y ) = e(a ') + e(ft ') ^ e , so))

that belongsto F(e<>). Thus U;<= F(Soour task is to prove (44:E:a),(44:E:b)of 44.7.3.Ad (44:E:a):Assume that rj H y happened for two i? , y in U/.

Let a ', ' be the /-,K-constituentsof 7 and 6 ', 6 ' the J-, X-con-

stituents of 77 from which they obtain as describedabove. Let S be the

setof 30.1.1.for the domination y H y . By (46:B)we may assumethat> >

S cJ or SSi K. Now S cJ would cause T; H 7 to imply 5 ' H a ',which is impossible,since 6 ', a ' both belong to V/; and S X would

cause if H 7 to imply' H /3

' which is impossible,since e ', 'bothbelongto W*.

>

Ad (44:E:b):Assume per absurdum, the existenceof a 7 in F(e) but>

not in U/,such that thereis no 17 of U/ with 17 H 7 . Let a ', 'bethe

/-,It-constituentsof 7)))

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390 COMPOSITIONAND DECOMPOSITIONOF GAMES))

Assume first e(a ') ^ <f>. Then a' belongs to F(p). Consequentlyeither a 'belongsto V/ or thereexistsa 6 ' in V/ with 5 ' H a '. In the

lattercasechoosean 6 ' in Wx for which e( ') assumesits minimum value .Form the 17 with the /-, /^-constituents 5 ', '. As 6 ', e ' belong to

V/, W*, respectively,and as e( d ') + e(*') g + = e , therefore i?

belongsto U/. BesidesT; H y owing to 6 ' H a ' (thesebeing their J-constituents).Thus 17 contradictsour original assumptionconcerningy .Hencewe have demonstrated, for the caseunder consideration,that a 'must belongto V/.

In otherwords:

(46:18) Either ~'belongsto V/, or e(7')> ?.Observe that in the first casenecessarilye(a ') ^ <f>,

and of coursein the secondcasee(a ') > ^ <p. Consequently:

(46:19) At any ratee(a. ') ^ ^>.

InterchangingJ and K carries(46:18),(46:19)into these:

(46:20) Either ~f$

r belongsto Wx or e(V') > #

(46:21) At any ratee(~^f ) ^ ^.Now if we had the secondalternative of (46:18),then this gives in con-

junction with (46:21)

e(7)= e(O+ e(7')> ^ + ^ = c ,

which is impossible,as y belongsF(ed). The secondalternative of (46:20)is equallyimpossible.

Thus we have the first alternatives in both (46:18)and (46:20),i.e.))

a ', ft'belongto V/, W*. As y belongsto F(e), therefore

e(~O+ e(7;) = e(7)^ e .

Consequently7 must belongto U/ contradictingour originalassumption.

46.5.TheCompleteResult in F(eQ)

46.5.1.The result (46:G)is, in spite of its completeness,unsatisfactoryin one respect:Theconditions(46:16)and (46:17)on which it dependsarealtogetherimplicit.We will, therefore, replacethem by equivalent, butmuch moretransparent conditions.)))

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DETERMINATIONOF ALL SOLUTIONS 391To do this, we beginwith the numbers<p, $which we assumeto be given

first. Which solutionsV/, W* of A, H for F(p),F(#) can we then use inthe senseof (46:G)?

First of all, V/, W* must be non-empty;applicationof (45:A) or (45:0)to A, H (insteadof F) showsthat this means

(46:22) * -|A|i, # -|H|i.Considernext (46:15).Apply (45:P)of 45.6.1.to A, H (insteadof T).

Then (45:P:a)securesthe two Max-equationsof (46:15),while (45:P:b)transforms the two Min-equationsof (46:15)into

(46:23) $ = Min (?, |A| 2), * = Min (},|H|2).Let us, therefore, define v>, \\l/ by (46:23).

Now we express(46:16),i.e.(46:16) + ^ =

_<,+ = e .

The first equation of (46:16)may alsobe written as-<P

= # - ^,i.e.by (46:23)(46:24) Max (0,?- |A| 2) = Max (0,# - |H|2).1

46.5.2.Now two casesarepossible:Case(a):Both sidesof (46:24)are zero. Then in eachMax of (46:24)

the 0-termis ^ than the other term, i.e.<p |A| 2 ^ 0,# |H|2 ^ 0,i.e.,(46:25) ? ^ |A| 2, ^ |H|,.Conversely:If (46:25)holds, then (46:24)becomes = 0,i.e.it is auto-matically satisfied. Now the definition (46:23)becomes(46:26) ?= v, ^ = ^,

and so the full condition (46:16)becomes2

(46:27) ? + * = eQ.(46:25)and (46:27)give also

(46:28) e g |A| 2 + |H|2 = |r|2.Case(b): Both sides of (46:24)arenot zero. Then in eachMax of

(46:24)the 0-termis < than the other term i.e.?- |A|2 >0,#- |H|,>0,i.e.(46:29) ?> |A| 2, # > |H|2.'

1Cf.(45:P:c)and its proof.1Of which we used only the first part to obtain (46:24),on which this discussion is

based.1Note that the important point is that (46:25),(46:29)exhaust all possibilities i.e.

that wecannot have * |A|a, $ > |H|2, or * > |A|i, ? |H|j.This is,ofcourse,due tothe equation (46:24),which forcesthat both sidesvanish or neither.

Themeaning of this will appearin the lemmas which follow.)))

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392 COMPOSITIONAND DECOMPOSITIONOF GAMES

Conversely:If (46:29)holds,then (46:24)becomes$- |A|2 = # -|H|,i.e.it is not automatically satisfied. We can express(46:24)by writing

(46:30) ?* |A|i+ , # = |H|2 + o>,

and then (46:29)becomessimply

(46:31) co >0.Now the definition (46:23)becomes

(46:32) ? = |A|2, * = |H|2,and so the full condition(46:16)*becomes

|A|2 + |H|t+ co = 6 ,i.e.(46:33) e = |r|2 + w.

(46:31)and (46:33)give also

(46:34) 6 > |F|2.46.6.3.Summingup :

(46:H) Theconditions(46:16),(46:17)of (46:G)amount to this:Oneof the two followingcasesmust hold:Case(a):(1) -|r|i e |r|,

togetherwith

(2) -|A|I*S|A|,,[(3) -|H|i**S|H|,,

and(4) 9+ * = .-

Case(b): (1) e,> |r|,,togetherwith

(2) P > |A|i,(3) * > |H|J(

and(4) 6 - |r|,= 9- |A|t = * ~ |H|,.

Proo/:Case(a):We knew all along,that 6 ^ |r|iand<f> ^ |A|i,

^ ^ -|H|i.The otherconditionscoincidewith (46:28),(46:25),(46:27)which contain the completedescriptionof this case.

Case(b): These conditions coincidewith (46:34),(46:29),(46:30),(46:33)which contain thecompletedescriptionof thecase(after eliminationof w which subsumes(46:31)under (l)-(3)).

1Cf.footnote 2 on p.391.1Thereaderwill note that while (l)-(3)for (a) and for (b) show a strong analogy,

the final condition (4) is entirely different for (a) and for (b). Nevertheless,all this wasobtained by the rigorous discussion ofoneconsistenttheory!

Morewill be said about this later.)))

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DETERMINATIONOF ALL SOLUTIONS 393

46.6.TheComplete Result in E(e )46.6.(46:G)and (46:H)characterizethe solutionsof T for F(e) in a

completeand explicitway. It is now apparent, too,that the cases(a), (b)of (46:H) coincidewith (45:0:b),(45:0:c)in 45.6.1.:Indeed(a), (b) of(46:H) aredistinguishedby their conditions(1),and theseareprecisely(45:O:b),(45:O:c).

We now combinethe results of (46:G), (46:H) with thoseof (45:1),(45:0).Thiswill give us a comprehensivepictureof the situation,utilizingall our information.

(46:1) If(46:I:a) (1) eo < -|r|i,

then the empty set is the only solution of r, for E(e) as well asfor F(e9).

If(46:1:b) (1) -|r|i e |r|,,

then T has the samesolutionsO/ for E(eQ) and for F(e). TheseO/ arepreciselythosesets,which obtain in the followingmanner:

Chooseany two p, # so that

(2) -|A|i p |A| 2,(3) -|H|i *g |H|2,

and(4) ?+ # = eo.

Chooseany two solutionsV/, W* of A, H for JE(p), E($).Then U/ is the compositionof V/ and WK in the senseof

44.7.4.If

(46:I:c) (1) e > |r|2,))

then F doesnot have the samesolutionsO/ for E(e^)and U/ forF(e). TheseO/ and U/ are preciselythose setswhich obtainin the followingmanner:Form the two numbers , # with

(2) 9> |A|i,(3) f > |H|2,

which aredefined by(4) eo- |r|2 = 9- |A|,= *- |H|2.

Chooseany two solutionsV/, WK of A, H for E(v),E$).Then O/ is the sum of the followingsets:Thecompositionof

V/ and of the setof alldetached7'(in K)with e(ft ') = |H|a ; the

compositionof thesetof all detacheda ' (in J) with e(a ') =|A|*)))

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394 COMPOSITIONAND DECOMPOSITIONOF GAMES

and of WA:; the compositionof the setof all detacheda '(in J)

with e(a ') =<p and of the set of all detached '

(in K) with

e(ft ') = ^, taking all pairs ^>, ^ with

(5) |A| 2 < *, < 9, |H|i< * < *,and

(6) ?+ ^ = e .U/ obtainsby the sameprocess,only replacingthe condition(6)by

(7) ?+ * g 6 .Proo/:Ad (46:I:a):This coincideswith (45:O:a).Ad (46:I:b):This is a restatementof case(a) in (46:H)exceptfor the

following modifications:First:The identification of the E and F solutionsfor F, A, H. This is

justified by applying(45:O:b)to F, A, H which is legitimate by (1),(2),(3)of (46:I:b).

Second:Theway in which weformed O/ = U/ from V/ = V/, W/c = Wxwhich differed from the one describedin (46:H) insofar as we omittedthe condition (46:17).This is justified by observing that (46:17)isautomatically fulfilled: V/ = Vj #(?), W* = W* E($), hence for

a ' in V/ and ft' in W* always e(a. ') = 9,e(ft ') = # and so by (4)

e(O+ (7')= *o.Ad (46:I:c):This is a restatementof case(b) in (46:H),exceptfor this

modification:We considerboth E and F solutionsfor F (not only F solutionsas in

(46:H)),and use only E solutionsfor A, H (not F solutionsas in (46:H)).Theway in which the former O/, U/ of F areformed from the latter(Vj ofA, Wx of H) is accordinglydifferent from the onedescribedin (46:H).

In orderto remove thesedifferences, one has to proceedas follows:Apply (45:1)and (45:O:c)to F, A, H which is legitimateby (1),(2), (3)of (46:I:c).Then substitute the definingfor the definedin (46:H). If thesemanipulations arecarriedout on (46:H) (in the presentcase(46:I:c)),thenpreciselyour above formulation results.l

46.7.Graphical Representation of a Part of the Result

46.7.Theresultsof (46:1)may seemcomplicated,but they areactuallyonly the preciseexpressionof several simple qualitative principles.Thereasonfor going through the intricaciesof the precedingmathematicalderivation was, of course,that theseprinciplesareriot at all obvious,andthat this is the way to discoverand to prove them. On the otherhand ourresult can be illustratedby a simplegraphicalrepresentation.

1If the readercarriesthis out, he will seethat this transformation, although somewhatcumbersome, presents absolutely no difficulty.)))

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DETERMINATION OF ALL SOLUTIONS)) 395))

We beginwith a more formalistic remark.A lookat the threecases(46:I:a)-(46:I:c)disclosesthis:While nothing

more can be said about (46:I:a),the two other cases(46:I:b),(46:I:c)have somecommon features. Indeed,in both instancesthe desiredsolu-tions O/, U/ of F areobtainedwith the helpof two numbers , # and certaincorrespondingsolutionsV/, W* of A, H. Thequantitative elementsof therepresentationof O/,U/ arethe numbers , #. As waspointedout in foot-note2 on p.386,they representsomethinglikea distributionof the givenexcesse in / betweenJ and K.))

Figure 69.

, # arecharacterizedin the cases(46:I:b)and (46:I:c)by their respec-tive conditions(2)-(4). Let us comparetheseconditionsfor (46:I:b)andfor (46:I:c).

They have this common feature :They force the excesses, # to belongto the samecaseof A, H as the one to which the excesse belongsfor r.

They differ, however, very essentiallyin this respect:In (46:I:b)theyimposeonly one equationupon , # while in (46:1:c)they imposetwo equa-tions.1 Of course,the inequalitiestoo,may degenerateoccasionallytoequations(cf. (46:J)in 46.8.3.),but the generalsituationis as indicated.

The connectionsbetweene and p, # are representedgraphicallybyFig.69.))

1(2),(3) are inequalities in both cases,for two equations in (46:I:c).))

(4) stands for oneequation in (46:I:b)and)))

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396 COMPOSITIONAND DECOMPOSITIONOF GAMES

This figure showsthe p, #-planeand under it the 6o-line. On the latterthe points -|r|i,|r|imark the division into the threezonescorrespondingtocases(46:I:a)-(46:I:c).The

<?>, ^-domainwhich belongsto case(46:I:b)coversthe shadedrectanglemarked(b) in the p, #-plane;the , ^-domain,which belongsto case(46:I:c)coversthe line marked (c)in the p, ^-plane.

Given any $,#-point,followingthe line leadsto its e valuethus 6, V yielda, a',respectively. Given any e -value the reverseprocessdisclosesall its 9,^-points, thus a producesan entireinterval at 6, whilea'yieldsthe unique point b'.1

46.8.Interpretation :TheNormal Zone. Heredity of Various Properties

46.8.1.Figure69 callsfor further comments,which areconduciveto afuller understandingof (46:1).

First:Therehave beenrepeatedindications(for the last time in the com-ment following(45:0)),that the cases(46:I:a)and (46:I:c),i.e.e < -|rkand 60> |F|j,respectively arethe \"toosmall\"or \"toolarge\"values of ein the senseof 44.6.1.;i.e.,that case(46:I:b),|F|i e g |r|2, is in someway the normal zone. Now our pictureshows that when the excesseof F lies in the normal zone, then the correspondingexcesses, # of A,H liealsowithin their respectivenormal zones.2 In otherwords:

Thenormal behavior (positionof the excessin (46:I:b))ishereditaryfromT to A, H.

Second:In the case(46:I:b)the normal zone<f>, $arenot completely

determinedby eo, as we repeatedlysaw before. In case(46:I:c),on theotherhand, they are. This is picturedby the fact that the former domainis therectangle(b) in the <p, #-plane,while the latterdomain isonly a line (c).

It is worth noting, however, that at the two endsof the case(46:I:b)for eQ

= |T|i,|r|f the interval available for p, # is constrictedto a point.8Thus the transition from the variable 9,# of (46:I:b)to the fixed onesof(46:I:c)iscontinuous.

Third:Our first remarkstated that normal behavior (i.e.,that the positionof the excesscorrespondsto (46:I:b))is hereditary from T to A, H. It isremarkablethat, in general,no such heredityholdsfor the vanishing of theexcess,i.e.that eo = 4 doesnot in generalimply p = 0,# = 0. It ispre-cisely the vanishing of the excesswhich specializesour presenttheory (of44.7.)to the olderform (of 42.4.1.which, as we know, is equivalent to theoriginal one of 30.1.1.).We will examinethe variability of p, # when e =morecloselyin thelast (sixth)remark. Beforewedo that, however, wegiveour attention to the connectionbetweenour presenttheory and the olderform.

1We leaveto the readerthe simple verification that the geometricalarrangements ofFig. 69.express,indeed,the condition of (46:I:b),(46:I:c).

f I.e.-lrUS e |r|,implies -|A|i * l*kH#li** S l#k cf. (46:I:b).1This is onecaseof degeneration,alluded to ftt the end of 46.7.4Ofcourse,eQ - liesin the normal case(46:I:b):-|r|i |r|s.)))

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DETERMINATION OF ALL SOLUTIONS 397

Fourth:It is now evident that the present, wider form of the theorymust of necessityreceiveconsideration,even if our primary interest iswith the original form alone. Indeed:in order to find the solutionsof adecomposablegame T in the original sense(for eQ

= 0), we need solutionsfor the constituentgamesA, H in the wider,new sense(for p, # which maynot be zero).

This gives the remarks of 44.6.2.a more precisemeaning:It is nowspecificallyapparent how the passagefrom the old theory to the new onebecomesnecessarywhen the game (A or H) is lookedupon as non-isolated.Theexactformulation of this idea will comein 46.10.

46.8.2.Fifth: We can now justify the final statements about (44:D)in 44.3.2.and (44:D:a),(44:D:b)in 44.3.3.(46:I:b)shows that (44:D)is true in the case(44:I:b),if we relinquishthe old theory for the new one;(44:I:c)shows that (44:D) is not true in the case(44:I:c)even at thatprice. Thus the desireto securethe validity of the plausibleschemeof(44:D)motivates the passageto the new theory as well as the restrictionto case(44:I:b)the normal case.

If we insist interpreting (44:D),(44:D:a),(44:D:b)by the old theory,then (44:D),(44:D:a)fail,1 while the conditional statementof (44:D:b)remainstrue.2

46.8.3.Sixth:We saw that e = does not in generalimply p = 0,# = 0. What doesthis \"in general\"mean?

, # aresubjectto the conditions(2)-(4)of (46:I:b).As eo = 0,so(4) meansthat # = and permitsus to expressthe remaining (2),(3)interms of <p alone. They becomethis:))

Now apply (45:E) to A, H. Then we see:If A, H areboth essential,then the lower limits of (46:35)are< and

the upperlimits are> 0,sop can really be 7* 0. If eitherA or H is inessen-tial, then (46:35)impliesp = and hence# = 0.

We statethis explicitly:(46:J) e = implies9= 0,# = 0,i.e.(44:D)of 44.3.2.holdseven

in the senseof oldtheory if and only if eitherA or H is inessential.46.9.Dummies

46.9.1.We can now disposeof the narrower type of decomposition,describedin footnote 1on p.340 the additionof \"dummies\" to the game.

Considerthe game A of the players1', ,*'.*\" Inflate \" it by addingtoit a seriesof \"dummies\"K;i.e.composeA with an inessentialgame Hof the players1\", , I\". Then the compositegame is F.

1As we may have e -0, ^ 0, # & 0. Then the decomposability requirement(44:B:a)of 44.3.1.is violated, asstatedin 44.3.3.

*Representing the specialcasee \"- 0, 0, # -0.1It is now convenient to reintroduce the notations of41.3.1.for the players.)))

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398 COMPOSITIONAND DECOMPOSITIONOF GAMES

We will use the old theory for all thesegames.By (31:1)in 31.2.1.there existspreciselyone imputation for the inessentialgameH say

Vi = M\", , 7?\"}.1 By (31:0)or (31:P) in 31.2.3.H possessesa

unique solution:Theoneelementset ( y ).Now by (46:J)and (46:I:b)the generalsolution of F obtains by com-

posingthe generalsolutionof A with the generalsolution of H and thelatterone is unique!

In otherwords:\" Inflate \" every imputation / = {j8r, , fly) of J (i.e.A) to an

imputation a / of I (i.e.T) by composingit with y J,i.e.by addingto it the

components7?\",* ' , 7?-: / = [Pi',' ' ' , A*, 7?\", , 7?\"}.Then

this processof \"inflation\" i.e.of composition producesthe generalsolu-tion of F from the generalsolutionof A.

This resultcan be summedup by sayingthat the \"inflation\" of a gameby the additionof \"dummies\"doesnot affect its solution essentially it isonly necessaryto add to every imputation componentsrepresentingthe\"dummies,\"and the values of thesecomponentsare the plausibleones:What each \"dummy\" would obtain in the inessentialgame H, whichdescribestheir relationshipto eachother.

46.9.2.We concludeby adding that (46:J)statesthat if and only ifthe compositionis not of the specialtype discussedabove, the old theoryceasesto have the simplepropertiesof the new one,and its hereditarycharacterfails, as indicatedin the third remarkof 46.8.1.

46.10.Imbedding of a Game46.10.1.In the fourth remarkof 46.8.1.we reaffirmed the indicationsof

44.6.2.,accordingto which the passagefrom the old theory to the new onebecomesnecessarywhen the game is lookedupon as non-isolated.We will

now give this ideaits final and exactexpression.It is more convenient this time to denotethe game under consideration

by A and the setof its playersby J. Itought to be understoodthat this A

is perfectly general no decomposabilityof A is assumed.We beginby introducingthe conceptswhich areneededto treata given

gameA as a non-isolatedoccurrence:This amounts to imbeddingit withoutmodifying it, into a wider setup, which it is convenient to describeasanothergame F. We define accordingly:A is imbeddedinto F, or F is animbedding of A, if F is the compositionof A with another game H.2 Inotherwords,A is imbeddedin all thosegamesof which it is a constituent.1

1Recallthe notations of 44.2.*Thegame H and the Bet of its playersK areperfectly arbitrary, exceptthat K and J

must be disjunct.1Sincea constituent of a constituent is itself a constituent (recallthe appropriatedefinitions, in particular (43:D)in 43.3.1.),an imbedding of an imbedding is again animbedding. In other words; Imbedding is a transitive relation. This relievesus fromconsidering any indirect relationships basedupon it.)))

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DETERMINATIONOF ALL SOLUTIONS 399

46.10.2.Let us now investigate the solutionsof A viewing A as a non-isolatedoccurrence.In the light of the above,this amounts to enumeratingall solutionsof all imbeddingsF of A, and interpretingthem, as far as A isconcerned.The last operation must be the taking of the J-constituentin the senseof 44.7.4.We know from the fifth remark in 46.8.2.that thisisonly feasible,if weconsiderno solutionsfrom outsidethe normal zone (b).

One might hesitatewhether the solutionsof F shouldbe taken in thesenseof the old or the new theory. The former may appear to bemorejustified on the standpointof 44.6.2.:Theoutsideinfluencesupon the gamehaving beenaccountedfor by the passagefrom A to H, thereis no longerany excusefor going outside the old theory.1 It happens,however, thatwe need not settlethis point at all, becausethe result for A will be thesame,irrespectivelyof which theory we use for F. But if we use the newtheory for F, we must restrict ourselvesto the case(46:I:b),as discussedabove.

Thus the questionpresentsitself ultimately in this form:

(46:K) Considerall imbeddingsF of A, and all solutionsof theseF:(a) in the senseof the old theory, i.e.for (0),(b) in the senseof the new theory in the normal zone, i.e.for))

Which arethe J-constituentsof the solutions?

46.10.3.The answeris very simple:

(46:L) The J-constituents(of the F solutions)referredto in (46:K)arepreciselythe following sets:All solutionsfor A in the normalzone, i.e.for any JE(p) of (46:I:b).This is true for both (a)and(b) of (46:K).

Proof:Q= belongsto case(46:I:b)(cf.footnote 4 on p.396),hence

(a) is narrower than (b). Therefore, we needonly show that all the setsobtained from (b) areamong the onesdescribedabove,and that all thesesetscan even be obtainedwith the help of (a).

Thefirst assertionis only that of the hereditarycharacterof the normalzone (b).

Thesecondassertionfollows from (46:I:b),if we can do this:Given a $with -|A|i 9 |A|,find a game H and with -|H|i # g |H|t , suchthat $+ # = and that H possessessolutionsfor E(ifr). Now such anH exists,and it can even be chosenas a three-persongame.

Indeed:Let H be the essentialthree-persongamewith general7 >0.Then by (455)in 45.1.|HK= 87and by (45:9)in 45.3.3.|H|,= J|H|i= %y.We have required# =

<p and what we know now amounts to

-87^ * ;Sfy.1Besides,the transitivity pointed out in footnote 3 on p.398,shows that any further

imbedding of r can be regardeddirectly as an imbedding of A.)))

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400 COMPOSITIONAND DECOMPOSITIONOF GAMES

Thiscanclearlybemet by choosingy sufficiently great. Then wealsoneedasolutionof H for E($).Theexistenceofsucha solution(for -87^ # g fy)will beshown in 47.

46.10.4.To this result two more remarksshouldbe added:First:If we wanted to handlethe processof imbeddingin sucha manner

that the old theory remainshereditary,we would have to seeto this:Thecompositionof T from A and H has to be such that eQ

= implies<f>=

(and hence# = 0). By (46:T) this meansthat eitherA or H areinessen-tial. The lattermeans (cf. eod),that only

\" dummies\"areadded to A.

Summing up:(46:M) The old theory remainshereditary if and only if eitherthe

original gameA is inessential,or the imbeddingis restrictedtothe additionof \" dummies\" to A.

Second:It was suggestedalready in 44.6.2.to treatthe outsidesource,which createsthe excessesand paves the way for the transition of our oldtheory to the new one,as another player.

Our above result (44:L)justifiesa slightly modified view:Theoutsidesourceof 44.6.2.is the game H which is added to A or rather the setKofits players.

Now wehave seenthat the game H must beessential,in orderto achievethedesiredresult. Furthermorewe know that an essentialgamemust haven ^ 3 participants,and the proof of (44:L)showedthat a suitableH with

n = 3 participantsdoesindeedexist.Sowe see:

(46:N) Theoutsidesourceof 44.6.2.can be regardedas a group ofnew players but not as one player. Indeed,the minimumeffective number of membersof this groupis 3.

46.10.5.The foregoing considerationshave justified our passagefromthe old theory to the new one (within the normal zone (b)) and clarifiedthe nature of this transition. We seenow that the \" common sense\"surmiseof 44.3.fails to hold in the old theory, but that it is true in preciselythat new domain to which we changed. This rounds out the theory in asatisfactorymanner.

The leading principleof the discussionsof 44.4.3.-46.10.4.was this:Thegameunder considerationwas originally viewed as an isolatedoccur-rence,but then removed from this isolationand imbedded,without modifica-tion, in all possibleways into a greatergame. This orderof ideas is notalien to the natural sciences,particularly to mechanics.The first stand-point correspondsto the analysisof the so-calledclosedsystems,thesecondto their imbedding,without interaction,into all possiblegreaterclosedsystems.)))

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DETERMINATIONOF ALL SOLUTIONS 401Themethodical importance of this procedurehasbeenvariously empha-

sized in the modern literature on theoretical physics,particularly in the

analysisof the structureof Quantum Mechanics. It is remarkablethat it

couldbe madeuse of so essentiallyin our present investigation.

46.11.Significance of the Normal Zone

46.11.1.The result (46:I:b)defines for every solution of the compositegame T in the normal zone i.e.a fortiori for every solution in the senseof the old theory numbers p, #. This and the immediate propertiesof

p, # in connection with the solution, appear to be so fundamental, as todeservea fuller non-mathematical exposition.

We areconsideringtwo gamesA, H playedby two disjunctsetsof players/ and K. The rulesof thesegamesstipulateabsolutelyno physicalcon-nection betweenthem. We view them neverthelessas onegame T but

this game,of course,is composite,with the two isolatedconstituentsA, H.Let us now find all solutionsof the entire arrangement, i.e.of the com-

positegame T. Sinceit is not desiredto consideranything outsideof T, we

adhereto the original theory of 30.1.1.and 42.4.1.1 Then we have shownthat any suchsolution U/ determinesa number p 2 with the following prop-

erty: Forevery imputation a of U/ the players of A (i.e.in J) obtain

togetherthe amount , and the playersof H (i.e.in K) obtain togetherthe

amount -p. Thus the principleof organization embodiedin U/ must

stipulate (among other things) that the players of H transfer under all

conditionsthe amount to the playersof A.

The remainderof the characterization of Uj i.e.of the principleof

organization or standard of behavior embodiedin it is this:First:Theplayersof A, in their relationship with eachother,must be

regulated by a standard of behavior which is stable, providedthat the

transfer of 9from the other group is placedbeyonddispute.3

Second:The playersof H, in their relationshipwith eachother, must

be regulatedby a standard of behavior which is stable, providedthat the

transfer of to the other group is placedbeyonddispute.4

Third:The octroyed transfer z> must lie between the limits (46:35)of 46.8.3.

(46:35)

46.11.2.The meaning of theserules is clearly that any solution, i.e.any stable socialorderof T is basedupon payment of a definite tribute byone of the two groups to the other. The amount of this tribute is an

integral part of the solution. The possibleamounts, i.e.those which can

1I.e. -0.*Since$ + # - e -0, we do not introduce - -$.1I.e.that the /-constituent V/ of U/ is a solution of A for E($).I.e.that the ^-constituent W*of U/ is a solution of H for E(-$).)))

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402 COMPOSITIONAND DECOMPOSITIONOF GAMES

occurin solutions,arestrictly determinedby (46:35)above. This condi-tion showsin particular:

First:The tribute zero, i.e.the absenceof a tribute is always among thepossibilities.

Second:The tribute zero is the only possibleone if and only if one ofthe two gamesA, H is inessential(cf.the sixth remarkin 46.8.3.).

Third:In all other casesboth positive and negative tributesarepossiblei.e.both the playersof A and the playersof H may be the tribute paying

group.Thelimits of (46:35)aresetby both gamesA, H,i.e.,by the objective

physical possibilitiesof both groups.1 These limits expressthat eachgroup hasa minimum belowwhich no form of socialorganization can depressit: |A|i, |H|i;and, eachgroup has a maximum, above which it cannotraiseitself under any form of socialorganization:|A|i, |H|2.

Thus, for a particularphysicalbackground,i.e.a game, say A, the twonumbers|A|i, |A| 2 can be interpretedthis way: |A|i is the worst that will

be enduredunder any conditionsand |A|* is the maximum claim whichmay find outsideacceptanceunder any conditions.2

The results (45:E) and (45:F) of 45.3.1.-2.now acquirea new signifi-cance:Accordingto thesethe two numberscan only vanish together(whenA is inessential)and their ratio always liesbetweendefinite limits.

46.12.First Occurrenceof the Phenomenon of Transfer:n - 6

46.12.We have seen repeatedly (thus in (46:J)in 46.8.3.and in thesecondand third remarksin 46.11.2.)that the characteristicnew elementof the theory of a compositegame T manifests itself only when both con-stituents A, H areessential. This is the occurrenceof eo= 0,but

?= -** 0,i.e.a non-zero tribute in the senseof 46.11.

Now we know that in order to be essentiala game must have 3players. If this is to be true for both A, H, then the compositegame Tmust have ^ 6 players.

Sixplayersareactually enough as the following considerationshows:Let A, H both be the essential three-persongames with 7 = 1. Then|A|i= |H|i= 3,|A|,= |H|,= f. (Cf.in 46.10.3.).Hencefor -|S9 f,both p and # = $lie between 3 and|.Thisimplies,as will be shownin 47.,theexistenceof solutionsV/,W* of A, H for U($),E($).Theircom-))

1But where the actual amount ? liesbetween those limits, is not determined by thoseobjectivedata, but by the solution, i.e.the standard of behavior which happens to begenerally accepted.1Itmust berecalledthat all this takes the value of the coalition of all playersof A,v(/),as zero;i.e.,we are discussing the losseswhich are purely attributable to lack ofco-operationamong the group, and unfavorable general socialorganisations and gains,which are purely attributable to lack of co-operationin outside groups and favorablegeneralsocialorganizations.)))

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THREE-PERSONGAME 403

position U/ is then a solution of the compositegame T with the given 9.Sincep was only restrictedby f g p ^ f , we can chooseit non-zero.

Thus we have demonstrated:

(46:0) n = 6 is the smallestnumber of playersfor which the char-acteristicnew element of our theory of compositegames (thepossibility of eQ

= with $= -# ^ 0, cf. above) can beobservedin a suitablegame.

We have repeatedlyexpressedthe belief that an increasein the numberof playersneed not only causea more involved operationof the conceptswhich occurredfor smallernumbers,but that it also may originate quali-tatively new phenomena. Specifically such occurrenceswere observedas the number of playerssuccessivelyincreasedto 2,3,4.It is, therefore,of interestthat the samehappensnow as the number of playersreachessix.1

47. TheEssentialThree-personGamein the NewTheory47.1.Needfor This Discussion

47.1.It remainsfor us to discussthe solutionsof the essentialthree-persongame,accordingto the new theory.

This is necessary,sincewe have already made use of the existenceofthesesolutions in 46.10.and 46.12.,but the discussionpossessesalso aninterestof its own. In view of the interpretationwhich we were inducedto put on thesesolutionsin 46.12.and also of their central role in thetheory of decomposition,2 it seemsdesirableto acquirea detailedknowledgeof their structure. Furthermore, a familiarity with thesedetails will leadto otherinterpretationsof somesignificance. (Cf.47.8.and 47.9.) Finally,we shall find that the principlesused in determining the solutionsin ques-tion areof widerapplicability. (Cf.60.3.2.,60.3.3.)

47.2.Preparatory Considerations

47.2.1.We considerthe essentialthree-persongame,to be denotedbyT, in the normalization 7 = 1. Thus |r|i= 3, |r|t = f (cf. 46.12.).Wewish to determinethe solutionsof this T for (e ).8 In the applications,referredto above we neededonly the normal zone 3 ^ e ^ f but we pre-fer to discussnow all e .

This discussionwill be carried out with the graphicalmethod, whichwe used in treating the old theory in 32. We will, therefore, follow theschemeof 32.in severalrespects.

1For someother qualitatively new phenomena which emergeonly when there aresixplayers, cf.53.2.

*This is the only problem ofabsolutely general character,ofwhich we have a completesolution at present!1We arewriting r, e although the applications employed the notations A, ? and H,*(--.

Of course,the present r has nothing to do with the decomposabler consideredbefore.)))

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404 COMPOSITIONAND DECOMPOSITIONOF GAMES

Thecharacteristicfunction is the sameas in 32.1.1.:))

(47:1) v(S) =))-1

1 when S has)) elements.))

An (extended)imputation is a vector))

(X l^l> ^2> ^8} >

whosethreecomponentsmust fulfill (44:13)in 44.7.2.,which becomesnow:

(47:2) i ^ -1, 2 ^ -1, s ^ -1.Besides,in E(e) the excessmust be e , accordingto (44:11*)in 44.7.2.

and this is now

/j/7.O\\ ^J_xvJ_xv />!yxt &) d.\\ ~T~ #2 I ^3 c/0*

47.2.2.We wish to representthese a by the graphicaldeviceof 32.1.2.But that procedurepicturesonly number triplets of sum zero. Thereforewe define

/ A*7 A\\ 1 ^0 9 ^0 ft(47:4) a1 = ai -^, a2 = a2 -^ = as - ~-

Then (47:2),(47:3)become

(47:2*) a1 -

(47:3*) a1 + a2 + a3 = O.2

Now the representationof 32.1.2.becomesapplicable,we needonlyreplace i, 2, 8 by a1, a2, a8. With this reservation,Figure52 can beused.

Forthesereasons,we form for every vector a = (i, 2, 3} of I?(e)not only its componentsin the ordinary sensebut alsoits quasi-componentsin the senseof (47:4):a1, a2, a3; and with the helpof the quasi-components,we utilize the graphicalrepresentationof Figure52.

So this plane representationexpressespreciselythe condition (47:3*).The remaining condition (47:2*)is therefore equivalent to a restriction

imposedupon the point a , within the planeof Figure52. This restriction*The reader should compare (47:l)-(47:3)with (32:l)-(32:3)of 32.1.1.the sole

difference liesin (47:3).*Comparing these (47:2*),(47:3*)with (32:2),(32:3)of 32.1.1.,it appearsthat

(47:3*)and (32:3)coincide,and that (47:2*)and (32:2)differ only by the factor of

proportionality 1 -f)))

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THREE-PERSONGAME 406

obtains in the sameway as the similaronein 32.1.2.:a must lie within

the triangle formed by the threelinesa1 = - (l+ |M,a2 = -M + 5lY

a8 = ( 1+ sr ) This is preciselythe situationof Fig.53.,exceptfor the

proportionalityfactor 1+ |V and it is representedon Figure70. Theo

shadedareato be calledthe fundamental triangle, representsthe a whichfulfill (47:2*),(47:3*),i.e.thoseof E(e9 ).))

a'-O))

Figure 70.

47.2.3.We expressthe relationship of domination in this graphicalrepresentation.As we areusingthe new theory, theconsiderationsof 31.1.concerningthe set8 of 30.1.1.for a domination a H ft i.e.concerningits certainlynecessaryor certainly unnecessarycharacterno longerapply.Sowe discussS de novo.

Itis stilltrue,that 8cannot bea one-or a three-elementset. In thefirstcaseS = (i), so by 30.1.1.on ^ v((t)) = -1,on > ft, henceft < -1,contradictingft ^ -1by (47:2). In the secondcaseS = (1,2,3),so by30.1.1.ai > 181, a > ft, as > ft, hence i + a,+ 8 > ft + ft + ft,contradictingai + 2 + as = Pi + ft + ft = e by (47:3).

1Cf.footnote 2on p.404. HereWe assume,of course,that 1 -f =r ^ 0,i.e.o))

If 1+ ^ < 0,i.e.6 < -3- -|r|i,then the conditions of (47:2*),(47:3)conflict, and5

indeed we know from (45:A) that E(eQ) is empty in this case.)))

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406 COMPOSITIONAND DECOMPOSITIONOF GAMES

Thus S must be a two-elementset, S = (i, j).1 Then dominationmeansthat a, + a, ^ v((z, j)) = 1and > ft, a, > ft, i.e.that

_i_ i < 1 2e* + a1 S 1 >

and > /#, ' > 0>. By (47:3*)the first conditionmay bewritten))

We restatethis:Domination

>

a H |3meansthat

eithera1 > 1, a2 > 2 and a3 ^ - (l - ^-))

(47:5))) or a1 > 1, a8 > 8 and a2 ^ - U -^or a2 > )32, a8 > 8 and a1 ^ - ( 1 - ^M-2

\\ * /47.3.TheSix Casesof the Discussion. Cases(I)-(III)

47.3.1.After thesepreparationswe can proceedto discussthe solutionsV of T for E(e), for all values of eQ.

It will be found convenient to distinguishsix cases. Of theseCase(I)correspondsto (45:O:a),Cases(II)-(IV)and one point of (V) to (45:0:b)(thenormal zone),and Cases(V) and (VI)(without that point) to (45:0:c)(all in 45.6.1.).

47.3.2.Case(I):e < -3. In this case1+|< 0,so (47:2*),(47:3*)oconflictandE(eQ) isempty (cf.footnote 1on p.405)soV mustbeemptytoo.

Case(II):e = -3. In this case1+ ^ = 0, so (47:2*),(47:3*)o

imply a1 = a2 = a8 = 0,i.e.ai = a2 = 8 = ^ = 1,o))

SoE(eo)is a one-elementset,and V must be = E(e<>)by thesameargumentas in the proof of (31:0)in 31.2.3.Thus the conditionsarevery similarto thoseencounteredin an inessentialgame,cf. loc.cit.

1i, j, k a permutation of 1,2,3.This differs from the corresponding (32:4)in 32.1.3.only by the extra condition

at the end of eachline.)))

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THREE-PERSONGAME 407

Case(III):-3< e ^ 0. In this case1+ ^ > 0, so we can use

Figure70. Also 1+ ~ ^ 1- -~;so the extraconditionsof (47:5)in47.2.3.are automatically fulfilled throughout the fundamental triangle.So(47:5)coincideswith (32:4)in 32.1.3.(cf. footnote 2 on p.406). Conse-quently the entirediscussionof 32.1.3.-32.2.3.applies again, if the pro-

portionality factor 1+ e~ is inserted.oThus we obtain the solutionsof (e ) in this casesimplyby taking those

describedin 32.2.3.,multiplying eachcomponent by 1+ > and addingo- (to pass from a* to at)-o

47.4.Case(IV):First Part

47.4.1.Case (IV): < e < ? In this case < 1- ? < 1+ ^-& O Of

Consequentlythe lines

i (\\ 2en\\ 2 /i 2e \\ 3 (^ 2e<>\\a = ~

V1- 3)'a - -

v1- 3)'a = - (l -17

(which bound the extraconditionsof (47:5))aresituated with respecttothe fundamental triangle of Figure70 as indicated on Figure71. Theysubdividethe fundamental triangle into seven areas,eachof which can becharacterizedby stating which two-element setsS areeffective in it in thesenseof (47:5). The list is given belowFigure71. Now we can drawthe analogueof Figure54, indicating for eachpoint of the fundamentaltriangle the shadedareas1 which it dominates. This is done in Figure72accordingto (47:5). It is necessaryto treateachoneof the sevenareasofFigure71separately,and every shadedareaof Figure72 must be continuedacrossthe entirefundamental triangle.

It is clearfrom Figure72, that no point of thearea can bedominatedby a point outside that area.2 Hencethe condition (44:E:c)of 44.7.3.,which characterizesthe solution V for (e ), i.e.for theentirefundamentaltriangle,must alsohold for the part of V in when taken for (in placeof the entirefundamental triangle,i.e.<E(e)). But is a triangle likethefundamental triangle of Figure53, exceptfor the proportionalityfactor

1 ^- 8 Comparisonof Figure54 with in Figure72 showsthat the3

conditionsof domination arethe same.1 Excluding their boundaries.1Including its boundary.

Note that 1 - > 0.)))

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408 COMPOSITIONAND DECOMPOSITIONOF GAMES

47*4.2.Consequently the entire discussionof 32.1.3.-32.2.3.applies2eo

to the part of V in , if the proportionalityfactor 1 5- is inserted.))

'))

\\ \\))

Area:))

Figure 71.))

Effective two-elementsets-S:))

(1,2),(1,2),(1,2),))

(1,2)))

(1,3),(1,3)(1,3),

(1,3)))

(2,3)(2,3)(2,3)(2,3)))

/T^nty / ,-.-. \\ ^..-....\\

^6) / \\ SA.))

Figure 72.))

or the set))Hencethe part of V in must beeitherthe setindicatedin Figure73. (Theline can be in any position)))

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THREE-PERSONGAME)) 409))

belowthe points .) Howeverpermutationsof 1,2,3,i.e.to rotations ofthe triangle by 0,60,120, to produceall solutions. (Cf. 32.2.3., is (32:B),

is (32:A) there.)Having found the part of V in , we

proceedto determine the remainder of V.SinceV is a solution, this remaindermustliein the areawhich is undominated by thepart of V in . Comparisonof Figure73with Figure 72 showsthat this undominatedareais the following one:

Forthe set it consistsof the three i))

must be subjectedto all))

/ / \\\\))Figure 73.

^ trianglesof Fig.74, for the

trianglesof Figure75.l))set it consistsof the three &,It is clearfrom Figure72, that no point in any oneof thesetriangles

can bedominatedby a point in another one.2 Hencethe condition (44:E:c)of 44.7.3.,which characterizesthe solution V for E(e), i.e.for the entirefundamental triangle and which holds for the part of V in taken for))

\\ \\))

Figure 74. Figure 75.

CD (in placeof the entire fundamental triangle, i.e.E(e9)) too states pre-taken))cisely this:(44:E:c)holds for the part of V in each triangle

for that triangle.47.5.Case(IV):SecondPart

47.5.1.Let us therefore take one of those triangles,denotingit by T.Itspositionin the fundamental triangle,8 and the shadedareasdominated

1Theposition of all these triangles are clearly indicated by the drawings, exceptfor

the lower triangle in Figure 75. This triangle lies certainly outside the inner triangle

(area ) this is equivalent to the restriction (32:8)in 32.2.2.,cf.alsoFigure 60there.Itsposition with respectto the outer (fundamental) triangle islessdefinite :It may shrink

to a point or even disappearaltogether.It is not difficult to seethat the latter phenomenon is excluded,unless the (linear)

sizeof the inner triangle is ^ i of the outer one this means 1 ~ * 4 ^* \"*\" 3/i.e.e 1. We do not proposeto discussthis subjectfurther.

1All this refersto Figure 74,or all to Figure 75 but, of course,never to both in the

same argument!*Up to a rotation by 0,60,or 120.For the lower triangle of Figure 75 the apexdoesnot lie on the inner triangle, but below it, (cf. footnote 1 above) but this doesnot

alter out discussion.)))

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410 COMPOSITIONAND DECOMPOSITIONOFGAMES

by a given point in it (taken over from Fig.72)areshown on Figure76.We may now restrictourselvesto this triangle T, and to the conceptofdomination which is valid in it and determinethe solution of (44:E:c)with respectto this. We redraw T and the setup in it separately, alsointroducinga systemof coordinatesx,y in it. (Figure77.)

Note that the apexo is undominatedby points of T henceit must

belongto V.1'2))

-LiM*))

Figure 77.))

Point o))

Line/))

Figure 78.47.5.2.Now considertwo points of V in T at different heightsy. In

orderthat theupperone shouldnot dominatethe lowerone,the lattermustnot liein the two shaded sextants,belongingto the former, i.e.the lowerpoint must be in the middlesextantbelowthe upper one,and viceversa.Thus, if a point of V in T is given, then all points of V in T at differentheightsy must lie in one of the two sextantsmi indicatedin Figure78.

1For other triangles J^ (i.e.T) than the lower one of Figure 75, this follows fromanother consideration, too:As Figures 74, 75show, the apexof such a triangle lieson theborderof the inner triangle (area ) and belongsto what we know to be the part olV in .

1When the lower triangle ^ (i.e.T) of Figure 75 degeneratesto onepoint (cf.footnote 1on p.409)which is, of course,o then this determines the part of V in T.)))

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THREE-PERSONGAME)) 411))

47.6.3.Now assumethat a y\\ is the height of more than onepoint of V.Let then p and q be two different pointsof V with this height yi (Figure79).Now choosea point r in the interior of the triangle ^ Comparisonof Figure79 with Figure77 showsthat this r dominatesboth p and q. Asp, q belongto V t r cannot belongto it. Hencetheremust exista point *in V which dominatesr. Now a secondcomparisonof Figure79 with

Figure77 showsthat a point which dominatesr must alsodominateeitherp or q. Sinces, p, q all belongto V this is a contradiction.))

Point*))

V -))

-Line/))

Figure 79.))

Point o))

-Line*))

Figure 80.

47.5.4.Next assumethat a y\\ (in triangle T, i.e.betweenthe baseI andthe apexo) is the height of no point of V. ThereexistcertainlypointsofV with heightsy ^ t/i, e.g.the apexo is such a point. Choosea point pof V with a height y ^ y\\ as low as possible,i.e.with its y minimum. 1

(Figure80.) Denotethis minimum value with y = i/ 2. Clearlyyi < y*By the definition of y^ no point of V has a height y with yi ^ y < y* andby the above p is the only point of V with a height y = y*.

Now projectp perpendicularlyon y = yi, obtainingq. q cannot bein Vhenceit is dominatedby an 8 in V. Hencethis s cannot liebelowq, i.e.

1Thisis possiblesinceV is a closedset. Cf.(*) of footnote 1on p.384.)))

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412 COMPOSITIONAND DECOMPOSITIONOF GAMES))

its height y : y\\. Consequentlyy J j/i. Comparisonof Figure80with

Figure77 showsthat p doesnot dominateq. Hence* 7* p, necessitatingy T* yi. Thus y > yi, i.e.a lies (definitely) above p. Now a secondcomparisonof Figure80with Figure77 showsthat if a point a above pdominatesq, then it must also dominatep. Sinces,p both belongto V,this is a contradiction.

47.5.5.Summing up:Every y (betweenI and o) is theheight of preciselyonepoint of V. If y varies,then this point changeswithin the restrictions))

Point o))

CurveV))

-Line/))

Figure 81.))

V:Thepoints ewithe curvet /))

V: The lineand the curvet

,/))

Figure 82.)) Figure 83.))

of Fig.78.,i.e.without leaving the sextantsHH indicatedthere,words:))

In other))

(47:6) V (in T) is a curve from o to i, the directionof which neverdeviatesfrom the vertical by more than 3001 (cf.Figure81).

Conversely,if any curve accordingto (47:6)is given then comparisonof Figure81and Figure77 makes it clearthat the areasdominatedbythe points of V sweepout preciselythe complementof V in T. So(47:6)is the exactdeterminationof the part of V in T.2

We cannow obtain the generalsolutionV for E(eQ) (i.e.for the funda-mental triangle)by insertingcurvesaccordingto Figure81into eachtriangle

1Henceit is continuous.1It is equally true when Tdegeneratesto a point, cf.footnote 1on page409.)))

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THREE-PERSONGAME 413A of Figures74 and 75. The results areshown on Figures82 and 83,respectively.1

It will be observedthat thesefigures showstill marked similarity with

thosepertainingto the solutionsof the essentialthree-persongamein theold theory (cf. 32.2.3.,shown in the inner triangle of Figure73). Thenewelementconsistsof the curves in the smalltriangles,all of which aresituatedin the fringe between the two major trianglesof Figures82and 83. Thewidth of this fringe, as shown in Figure71,et sequ.,is measuredby e .2Sowhen e tends to zero, our new solutionstend to the old ones.

It is also worth pointing out that the variety of the solutionsis muchgreaternow than ever before:Entire curves can bechosenfreely (withinthe limitations of (47:6)above). We will seelater,that these curvesmotivate an interpretationwhich is of further significance. (Cf.47.8.)

47.6.Case (V)

47.6.1.Case(V):|g e < 3. In this case1 -^ g < 1+ ~ and

2^n l P

5- J < 1+q\"-

8 Theseinequalitiesexpress,as is easilyverified,

that theorientation of the inner triangle of Figure71is inverted,but that it isstill situated entirely within the outer(fundamental) triangle,as indicatedon Figure84. Thelatterisagain subdividedinto sevenareas,eachof whichcan becharacterizedby stating which two elementsetsareeffective in itin the senseof (47:5)in 47.2.3.Theonly difference betweenthe presentsituationand that onein Case(IV) (i.e.Figure71)is the behavior of area .Thelist is given belowFigure84.

Now we can draw the analogue of Figures54 and 72,indicatingfor eachpoint of the fundamental triangle the shaded areas4 which it dominates.This is done in Figure85,accordingto (47:5).

It is clearfrom Figure85 that no point of the area 5 is dominatedbyany point.6 HenceV must contain all of .

1 The lower triangle of Figure 83may degenerateto a point or even disappearalto-gether, cf.footnote 1 on p.409.

*Thesidesof the outer (fundamental) triangle aregiven by = f 1 + -^\\ those

of the inner triangle by' - - f 1 --jfj(Cf.Fig. 71). Thedifference of - ( I -f ~-))

and f 1 ~ ) is c .))

*This latter inequality is equivalent to e < 3.4 Excluding their boundaries.))6 Including its boundary.' * *8 I.e.by any a in E(CQ). It is easyto show that they aredominated by no a at all

they are the detachedimputations, by (45:D)in 45.2.4.Thepoints of the interior of the area dominate no other points either. I.e.they

dominate no a in #(e ). Again, it is easyto show that they dominate no a at allthey are the fully detachedimputations, cf. (45:C)in 45.2.4.

Thesestatements can alsobe verified directly, by using the definitions of 45.2.)))

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414 COMPOSITIONAND DECOMPOSITIONOF GAMES))

--ft1 ))

AD))

Figure 84.))

Area:))Effective two-element

setsS:))

(1,2),(1,8)(1,2), (2, 3)

(1,3) (2, 3)(2,3)(1,3)

(1,2)))

Figure 85.)))

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THREE-PERSONGAME)) 415))

47.6.2.Having found the part of V in , we proceedto determinetheremainderof V. SinceV is a solution,this remaindermust lie in the areawhich is undominated by the already known part of V f i.e.by . Con-siderationof Figure85 showsthat this undominated areaconsistspreciselyof the threetriangles@, , 0.1

It is clearfrom Figure85 that no point in any of the three trianglescan be dominated by a point in another one. Hencethe argumentof 47.4.2.shows, that our requirement of V must be precisely this:(44:E:c)of 44.7.3.must hold for the part of V in eachone of thesetriangles,taken for that triangle (in place ofthe entire fundamental triangle, i.e. v. Thetriangle I

and the curves))

Theconditionsin the triangles0,0, arethe sameas thosedescribedin Figures76,77 for the triangle T.Hencethe entiredeductionof 47.5.1.-47.5.4.may be repeatedliterally, andthe parts of V in 0,0, are curvesas shown on Figure81,characterizedby (47:6)in 47.5.5.))

Y))Figure 86.))

We can now obtain the general solution V for E(e^)(i.e.for the funda-mental triangle) by insertingsuchcurves into , 0, in Figure85. Theresult is shown in Figure86. Forfurther remarksconcerningthesesolu-tions cf. 47.8.,47.9.

47.7.Case(VI)))

47.7.eQ ^ 3. In this case)) <0< 1+))o))

and))

t 1 + ^ -2 Theseinequalitiesexpress,asiseasilyverified,

that the inner triangle of Figure84 has still the sameorientation, but thatit reachesthe boundariesof the outer (fundamental) triangle,and possiblybeyond,*as indicatedon Figure 87. The only differencebetweenthe presentsituation and that one in Case(V) (i.e.Figure84) is the disappearanceofthe areas@, , . The list is given belowFigure87.

Theanalogue of Figures54, 72 and 85 indicatingthe domination rela-tions, is containedin Figure88.

Theargument of 47.6.1.can be repeatedliterally, proving that V con-tains all of . Considerationof Figure88 showsthat leavesno partof the fundamental triangle undominated.4 HenceV is precisely . Forfurther remarksconcerningthis solution,cf. 47.9.

1The remainder of the fundamental triangle is dominated by the boundary ofwhich belongs to .

*This last inequality is equivalent to e < 3.8 When 6 > 3.4Theremainder of the fundamental triangle isdominated by the boundary of which

belongsto .)))

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416 COMPOSITIONAND DECOMPOSITIONOF GAMES))

47.8.Interpretation of the Result:TheCurves (OneDimensional Parts) in the Solution

47.8.1.Thesolutionsobtained in the discussionsof 47.2.-47.7.deservea brief interpretative analysis. It is quite conspicuousthat the repeated))

\\)),a'- -II-))

'- -())

v^_

IL))

Area:))

Figure 87.))

Effective two-elementsetsS:))

(1,2)))(1,3)))

(2,))

Figure 88.

appearanceof a smallnumberof qualitative featuresgoesfar in character-izing their structures insofar as they deviate from the types familiar in)))

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THREE-PERSONGAME 417

the solutionsof the essentialthree-persongame of the old theory. Thesefeatures are:Thecurves arbitrary within the restriction(47:6)of 47.5.5.which occuras soonas e > (and as long as e < 3); and the two-dimen-sional areas,which appear when e > f. We will now undertake theirinterpretation.

Considerfirst Case(IV): < eQ <f (in the \"normal\" zone). Let usconsiderthosesolutionsof the presentcasewhich extendthenon-discrimina-tory solution of the old theory (cf. 33.1.3.and (32:B)in 32.2.3.).Suchasolution is picturedon Figure82.

This figure showsthe threepoints which form the analogue of a solu-tion in the old theory. Taking,e.g.the lower point , oneverifies easilythat there))

i i . o 2 3 ,\" l - -V1- -57

= \" 1 +T a = a =2 V

1- T 2 3i.e.

Oil = 1 + Q, 2 = 3 = i-Thus thesethreepoints expressan arrangement where two players haveformed a coalition, obtainedits total proceeds(amounting to 1),and dividedthem evenly but the defeatedplayer hasnot beenreducedto his minimumvalue 1,becausehe retainedbeyondthat the total available excessCQ.

Now the curves,starting from thesepoints (in the fringe betweenthetwo triangles),expressthe situation where the total excesse is not leftin the indisputedpossessionof the defeatedplayer. By claiming any partof the excess,the victorious coalition exactsmore than the amount 1which it can actually get in the game i.e.it ceasesto beeffective. (Cf.the areas(2), 0, in the Figures71 and 72.) Therefore the conductof affairs of this coalition the distributionof the spoils within it is nolonger determinedby the realitiesof the game i.e.by the threats betweenthe partners but by the standard of behavior. This is expressedby thecurve, which is part of the solution. The possiblethreats between thepartners still restrictthis curve to a certain extent(cf. (47:6)in 47.5.5.),but beyondthat it is highly arbitrary. Itmust bere-emphasizedthat thisarbitrarinessis just an expressionof the multiplicity of stable standardsofbehavior but a definite standardof behavior, i.e.solution,meansa definitecurve, i.e.rule of conductin this situation.

47.8.2.Theseconsiderationssuggestthe followingtentative interpreta-tion:(47:A) In the presenceof a positive excessit may happen that a

coalition can obtain beyond its effective maximum also somefraction of the excess.This possibilityis then due entirelyto the standard of behavior and not to the physicalpossibilitiesof the game. The fraction of the excessthus obtainedmay varyfrom 0%to 100%and be left undeterminedby the standard ofbehavior. The latter will prescribe,however, uniquely, how)))

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418 COMPOSITIONAND DECOMPOSITIONOF GAMES

the fraction obtainedis to bedistributedbetweenthe membersof the coalition. This rule of division will dependon which o*the many possiblestable standards of behavior is chosen,andif the latteris varied, this rule will vary widely, although notquite unrestrictedly.

We have seenalready, that undeterminedcurves accordingto (47:6)occurin many solutions,and they will occuragain in the future. Theaboveinterpretationseemsto fit them in every case.

Theindefinitenessof the distributionof the excessbetweenthe victoriouscoalition and the defeatedplayer (in a given solution)is an instancehowcertainsocialadjustmentsmay be left open even within a specifiedsocialorder. Our curves expressthe further nuance that while suchan indefinitedistribution is decidedupon, some players can be tied to eachother bydefinite conventions. (We will seefurther instancesof this in the thirdremarkof 67.2.3,67.3.3.and in 62.6.2.)

47.9.Continuation :TheAreas (Two-dimensional Parts) in the Solution

47.9.1.Theinterpretation (47:A) in 47.8.2.couldbe tested by applyingit to the extensionof the discriminatorysolution of the oldtheory (cf.33.1.3.and (32:A) in 32.2.3.)as picturedon Figure83. Thiswould bring up someinstructive view-points, particularly with respectto the curve in the lowertriangle of Fig.83. However,we refrain from elaboratingthis caseanyfurther.

We turn, instead, to the Cases(V) and (VI),specifically when e >J(thesearethe \"toolarge\"excessesin the senseof 44.6.1.,45.2).Thesecasesarecharacterizedby the circumstance,that their solutionscontaintwo-dimensionalareas. Actually, two different situationsmay arise:

(a) Case(V), i.e.|< eQ <3. A solution V containsthe two-dimen-sionalarea0,but besidesalsocurvesas discussedin 47.8.(cf. Fig.86).

(b) Case(VI),i.e.eQ ^ 3. The unique solution V is the two-dimen-sionalarea , and nothing else(cf.Fig.88).

The emergenceof two-dimensionalareaswithin the solutionindicatesthat the standard of behavior fails tocontain rulesof distributionat leastwithin certainlimits. In the Cases(a), (b)theselimits arespecified. Inthe case(a) the curves of 47.8.appear outside of those limits, i.e.thestandard of behavior still sanctionscertaincoalitions in the Case(b) thisis no longerthe case.

47.9.2.So we see that the \"disorganizing\"effect of a \"toolarge11

excessi.e.gift from an outsidesource(cf. 44.6.1.)manifests itself in twosuccessivestages:In the Case(a) it is presentin a certaincentralarea,butdoesnot excludecertainconventional coalitions. In the Case(b) thestandard of behavior no longerallows coalitionsbut it setscertainlimitingprinciplesfor the distribution. .)))

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THREE-PERSONGAME 419We have seenthat.,thesesuccessivestagesof disorganizationarereached

at 60 = 1,3 respectively.1Theseconsiderationsseemto be quite instructive in a qualitative way

for the possibilitiesof standardsof behavior and organizations.It appearslikely that they will provideuseful guidancein the further developmentof the theory. But the readermust be cautioned against drawing farreachingconclusionsfrom the quantitative results:They apply to thethree-persongame with an excess,2 which is thus shown to bethe simplestmodel for their operation. But it must have becomeamply clearbynow that an increasein the number of participants will affect conditionsfundamentally.

1Note that |r|i f.2 Hencealsoto a decomposition six-persongame in the old theory, cf.46.12.)))

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SIMPLEGAMES

48.Winning and LosingCoalitionsand GamesWhere They Occur48.1.TheSecondTypeof 41.1.Decisionby Coalitions

48.1.1.The program formulated in 34.1.provided for far-reachinggeneralizationsof the gamescorrespondingto the 8 cornersof the cubeQ,introducedin 34.2.2.Thecorner77/7 (alsorepresentativeof 77,777,77)was taken up in 35.2.1.and providedthe sourcefor a generalization,lead-ing to the theory of compositionand decompositionto which all of ChapterIX was devoted. We now pass to the corner7 (alsorepresentativeof 7,77, 777),which we will treatin a similarfashion.

By generalizing the principle,of which a specialinstancecan be dis-cernedin this game,we will arrive at an extensiveclassof games,to becalledsimple. It will be seenthat a study of this classyieldsa body ofinformation which is of value for a deeperunderstandingof the generaltheory in the senseof 34.1.

48.1.2.Considerthe corner7 of Q, discussedin 35.1.As was broughtout in 35.1.1.,thisgamehas the followingconspicuousfeature:Theaim of theplayersis to form certaincoalitionsconsistingeitherof player4 and oneally,orof all threeotherplayerstogether. Any one of thesecoalitionsis winningin the full senseof the word. Any coalition which falls short of theseiscompletelydefeated.I.e.thequantitative element,the paymentsexpressedby the characteristicfunction, can be treatedas somethingsecondary theprimary aim in this game is to succeedin forming certaindecisivecoalitions.

This description suggestsstrongly that the number four of playersand the particular schemeof decisivecoalitionsarespecialand accidentaland that a moregeneralprinciplecan be extractedfrom this particulararrangement.

48.1.3.In carrying out this generalization,the following observationisuseful. In our above example,the decisivecoalitions the attainment ofwhich is the soleaim of the players werethese:(48:1) (1,4),(2,4),(3,4),(1,2,3).Now it is convenient to view not only theseas winning coalitions,but alsoall their (proper)supersets:(48:2) (1,2,4),(1,3,4),(2,3,4),(1,2,3,4).Thepoint isthat although the coalitions(48:2)contain participantswhosepresenceis not necessaryin orderto win, the coalitionis neverthelessa

420)))

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WINNING AND LOSINGCOALITIONS 421winning one i.e.the opponents are defeated.1 Theseopponents formthosecoalitionswhich arethe complementsof the seta in (48:1),(48:2),i.e.the sets

(48.3) (2,3),(1,3),(1,2),(4).( } (3), (2), (i), e-Thus (48:1),(48:2),contain the winning coalitions,and (48:3)containsthe defeatedones.

It is easily verified that every subset of 7 = (1,2,3,4)belongsto pre-ciselyoneof thesetwo classes:(48:1),(48:2),or (48:3).2

48.2.Winning and Losing Coalitions

48.2.1.Letus now considera setof n players:7 = (1, , n). Theschemeof 48.1.3.generalizesto subdividingthe system of all subsets of Iinto two classesW and L, such that the subsets of W will representthewinning coalitionsand the subsetsof L will representthe losingones. Theanaloguesof the propertiesformed in 48.1.3.can beformulated as follows:

Denotethe system of all subsets of / by 7.8 The mapping of everysubsetSof / on its complement(in /) :(48:4) S-+-Sis clearlya one-to-onemappingof I on itself. Now we have:

(48:A :a) Every coalition iseitherwinning or defeatedand not bothi.e.W and L arecomplementarysetsin 7.

(48:A:b) Complementation(in 7) carriesa winning coalition into alosingoneand viceversa i.e.the mapping(48:4)maps W andL on eachother.

(48:A:c) A coalition is winning, if part of it is winning i.e.W con-tains all supersetsof its elements.

(48:A :d) A coalition is losing,if it is part of one which is losing i.e.L containsall subsetsof its elements.

48.2.2.Beforewe discussthe conceptsof winning and losing in theirrelationship to the game,we may analyze the structureof conditions(48:A:a)-(48:A:d)somewhatfurther.

Thefirst conspicuousfact is that, although we needboth classesW andLto interpretthe game,theseclassesdetermineeachother. Indeedthey dothis in two ways:Given one of W or L (48:A:a)as well as (48:A:b)can beused to constructthe other. I.e.starting from one of thesesets,the otheroneis obtained in this way:

Accordingto (48:A :a):Takethe given setas a wholeand form its comple-ment (in J).

1I.e.the complements areflat in the senseof31.1.4.Cf.the discussion in 35.1.1.* (1,2,3,4)has 24 - 16subsets,of these8 are in (48:1),(48:2),and the remaining 8 in

(48:3).1As / has n elements,7 has 2 elements.)))

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422 SIMPLEGAMES

According to (48:A:b):Takeeachelementof the given setseparately,and replaceit by its complement(in 7).1

It shouldbe notedalsothat if the given set,W or L, possessesthe prop-erty (48:A:c)or (48:A:d)respectively,then the otherset obtained fromthe former by (48:A:a)or by (48:A:b)will possessthe other property(48:A:c)or (48:A:d).2

It follows from the above, th&t we can basethe entirestructure nowunderconsiderationon eitheroneof the two setsW and L. We must onlyrequirethat both transformation (48:A:a)and (48:A:b)lead from it tothesameset(which is then the otherone of W and L) and that it must satisfythe pertinent one of the two conditions(48:A:c)and (48:A:d)(theothercondition of (48:A:c)and (48:A:d)is then automatically taken careof,accordingto what we have seen).

Thus we have only two conditionsfor W or L:Firstthe equivalenceof(48:A:a)and (48:A:b)and second(48:A:c)or (48:A:d).

Theformer condition meansthis:Thenon-elementsof the set coincidewith the complements(in 7)of the elementsof the set. In other words:Oftwo complements(in 7)S, S,one and only onebelongsto the set.

Summing up:ThesetsTF( 7) arecharacterizedby theseproperties:

(48:W)(48:W:a) Of two complements(in 7) S, S, one and only one

belongsto W.

(48:W:b) W containsall supersetsof its elements.ThesetsL( I) arecharacterizedby theseproperties:

(48:L)(48:L:a) Of two complements (in 7) S, S, one and only one

belongsto L.(48:L:b) L containsall subsetsof its elements.

1 Thereaderwill note the remarkable structure of this condition :Thegiven setmustproduce the same result, irrespectively of whether complementation is applied to it as aunit, or to its elementsseparately.1This is actually true for (48:A:a)as well as for (48:A:b),and independent of thequestion whether (48:A:a)and (48:A:b)producethe sameset. Precisely:(48:B) Leta set M possessthe property (48:A:c)[(48:A:d)],then both setswhich

areobtained from it by (48:A:a)and (48:A:b) we do not assume that theyare identical possessthe other property (48:A:d)[(48:A:c)].

Proof:We must show that both transformations (48:A:a)and (48:A:b)carry (48:A:c)into (48:A:d)and viceversa.

Clearly (48:A:c)is equivalent to this:(48:A:c*) If Sis in M, and T is not, then SST isexcluded.

Again (48:A:d)is equivalent to this:

(48:A:d*) If Sis not in M, and T is, then S T is excluded.Now the transformation (48:A:a)interchanges \"being in M\" and \"not being in M \" witheach other. Hence it interchanges (48:A:c*)and (48:A:d*).The transformation(48:A:b)interchanges and - (this is brought about by individual complementation forthe elements S, T\\ besidesthe symbols S, T must be interchanged). Henceit, too,interchanges (48:A:c*)and (48:A:d*).)))

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CHARACTERIZATIONOF THESIMPLEGAMES 423

We restate:If W [L] fulfills (48:W) [(48:L)],then (48:A:a)and (48:A:b)yield the

samesetL [W]. W andL fulfill (48:A :a)-(48:A :d)and L [W] fulfills (48:L)[(48:W)].Conversely,if W, L fulfill (48:A:a)-(48:A:d),then they fulfill

separately(48:W), (48:L).49.Characterization of the SimpleGames

49.1.GeneralConceptsof Winning and Losing Coalitions

49.1.1.We now pass to the considerationof the connection betweenwinning and losingcoalitionsin the game itself.

Assume, therefore, that an n-persongame F is given. In all the con-siderationswhich follow, it is advantageousto restrictourselvesto the oldtheory in the senseof 30.1.1.or 42.4.1.Consequently,as pointedout in42.5.3.,we may assumeF to be zero or constant-sumas we desire.Forthepresentwe prefer to chooseF as a zero-sumgame.

Beyond this F is not restrictedand in particular no normalization isassumed.

49.1.2.Let us first analyze the conceptof a losingcoalition. Repeatingessentiallywhat was said in 35.1.1.,we may argueas follows:1The playert,when left to himself, obtains the amount v((i)). This is manifestly theworst thing that can ever happen to him, sincehe can protecthimselfagainst further losseswithout anyone else'shelp. Thus we may considerthe playeriwhen he getsthis amount v((i))to be completelydefeated.Acoalition S may be consideredas defeated,if it getsthe amount ] v((t)),

i in Ssincethen eachplayeri in it must necessarilygetv((i)).2 Thusthe criterionof defeat is

v(S) = v<)-tinS

In the terminology of 31.1.4.,this meansthat the coalition S is flat. (Cf.alsofootnote 3 on page296.)

We have obtaineda satisfactorydefinition of the systemLr 8 of all losing(defeated)coalitions:(49:L) Lr is the setof all flat setsS(&/).

It is now easy to say what a winning coalitionis. It is plausiblyone,the opponentsof which arelosing,i.e.the system Wr* of all winning coali-tions is this:

1Thedifference is that our present F is more general.1Sinceno player t needever acceptlessthan v((0),and those in the coalition Shave

together ^/ v ((*')) this is the only way in which they can split.))

1In orderto avoid confusion, we will use the symbols Wr, Lr instead of the W, L of48.2.2.The difference between this and the former is that 48.2.2.is a postulationaldiscussion of the properties which appeareddesirablefor the conceptsof \"winning\"and \"losing\" (describedby W, L) while we are now analyzing definite setsobtainedfrom a specificgame T.

The two viewpoints will bemerged in (49:E)of 49.3.3.)))

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424 SIMPLEGAMES

(49:W) W r is the setof all setsS(z/) for which -Sis flat.

It shouldbe conceptuallyclear,and is also immediately verified with

the help of 27.1.1.-2.,that the setsWr, I/r areinvariant under strategicequivalence.

49.1.3.We cannot expectthe above Wr, Lr to fulfill the conditions(48:A:a)-(48:A:d)(for W, L) of 48.2.1.Thegame in its presentgeneralityneednot beof thesimpletype referredto,where the only aim of all playersis to form certaindecisivecoalitionsand thereareno othermotives whichrequire a quantitative description.1 It will therefore be necessary torestrictin orderto expressthe property we have in mind. Indeed,thepreciseformulation of this restrictionis our immediateobjective.

Nevertheless,we beginby determininghow much of (48:A:a)-(48:A:d)holdstrue for the T in its presentgenerality. We give theanswerin severalstages.(49:A) W r, Lr always fulfill (48:A :b)-(48:A :d)

Proof:Ad (48:A:b):Immediateby comparing(49:L) and (49:W) in49.1.2.2

Ad (48:A:c),(48:A:d):Sincewe have (48:A:b),we can apply (48:B)in4S.2.2.8 and therefore (48:A:c)and (48:A:d)imply eachother.,

But (48:A:d)coincideswith (31:D:c)in 31.1.4.,considering(49:L).Thus the main difference betweenour presentWr, Lr and the setup of

48.2.lies in (48:A:a) i.e.in the questionwhether or not Wr and Lr arecomplements.We can decomposethis assertioninto two parts:(49:1)(49:1:a) TTr nLr = 0,4(49:1:b) JFr uLr = /.6

(49:l:a)leadsback to familiar concepts:(49:B)

(49:B:a) (49:l:a)holds if and only if T is essential.(49:B:b) If T is inessential,then W r = LT = /.6

1Our discussion of the four-person game has provided many illustrations of suchmotives, for which the end of36.1.2.provides a goodinstance. This situation is, indeed,the usual (general) one the classof gamesat which we are aiming now, is in a certainsensean extremecase,cf. the concluding observation of 49.3.3.

1Actually the conceptof \"winning

\" was basedon the conceptof \"losing\" by justthis operation of complementation.1It appearsnow why we separated(48:A:a)from (48:A:b)in 48.2.1:We have now(48:A:b),but not (48:A:a).

4 It may seemodd that this no coalition can at the same time be both winning andlosing must bestatedseparately. Themeaning of this condition will appearin (49:B)and footnote 6.

This states that every coalition i.e.every subset of / is definitely winning orlosing. This is, ofcourse,the ideaon the basisof which we wish to specializeT.

Thus a coalition can at the same time beboth winning and losing, when the game isinessential manifestly becausein this caseboth statesareirrelevant.)))

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CHARACTERIZATIONOF THESIMPLEGAMES 425

Proof:Ad (49:B:a):Thenegation of (49:l:a)is the existenceof an Ssuch that both S and S are flat. This amounts to inessentialityby(31:E:b)in31.1.4.

Ad (49:B:b):Wr = Z/r = / meansthat everySin 7isflat. Thisamountsto inessentialityby (31:E:c)in 31.1.4.

Beforepassingto (49:l:b)we note that Wr, I/r possessone propertywhich did not occurin (48:A:a)-(48:A:d).(49:C) Lrcontains the empty setand all one-elementsets.1

Proof:Thiscoincideswith (31:D:a),(31:D:b)in 31.1.4.(49:C)is really a new condition, i.e.it is not a consequenceof (48:A:a)-

(48:A:d);we will verify this in 49.2.below. Thus our plausiblediscussionof 48.2.overlookeda necessaryfeature of the Wr, Z/r- We must, therefore,make sure that the present conditionscontain everything. I.e.that theconditions(48:A:b)-(48:A:d)and (49:C),togetherwith the resultsof (49:B)on inessentiality,characterizethe Wr, LT completely. This will be shownin 49.3.below.

49.2.TheSpecialRole of One-elementSets49.2.1.We begin with the exampleannouncedabove:Two systems

TF, L which fulfill (48:A:a)-(48:A:d),2 but not (49:C). Actually, we candetermineall suchpairs.

(49:D) W, L fulfill (48:A:a)-(48:A:d),but not (49:C),if and only ifthey have the following form :W is the setof all Scontaining i ,Lis the setof all S not containing i , where IQ is an arbitrary butdefinite player.

Proof:Sufficiency:It is immediately verified that the W, L formed asindicated fulfill (48:A:a)-(48:A:d).(49:C) is violated, sincethe one ele-ment set (f ) belongsto TF, and not to L.

Necessity:Assume that W, L fulfill (48:A:a)-(48:A:d),but not (49:C).Let (to) be a one-elementset,which does not belong to L.3 Then (t' )belongsto W.

Every S containing t' has $2(i ), henceit belongsto W by (48:A:c).If S doesnot contain iQ, then S containsit;hence S belongsto W bythe above and S belongsto L by (48:A:b).

Finally W, L aredisjunctby (48:A:a),henceW is preciselythe set of theS containing t* andL ispreciselythe setof the S not containing t' .

49.2.2.Itmay be worth while to comment briefly upon this result.

1 It is clearly in the spirit of our entire analysis ofgamesthat a coalition ofoneplayeris to beconsideredasdefeated as this player has not succeededin finding partners for acoalition.

2 We mentioned originally (48:A:b) (48:A:d)only, but the abovestrengtheningrequires no extra effort.

8 If the empty setdoesnot belong to L, then no setcan belong to Lowing to (48:Aid) ;henceany (to) will do.)))

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426 SIMPLEGAMES

The W, L formed in (49:D) cannot be the Wr, Lr of any game sincethey violate (49:C).This may seemodd,since(49:D)appearsto convey avery clearidea of the kind of \"winning\" and \"losing\"describedby itsW, L. Indeed,they describethe situation where a coalition wins if theplayer io belongsto it, and losesif he doesnot. Why can no game be con-structedto fit this specification?

The reasonis that under the conditionsdescribed,\"winning\" wouldnot bea matter of forming coalitionsat all:1 The playeri is \"victorious\"without anybodyelse'shelp. Still worse,in our terminology this positionof to is no victory it is not the result of any strategicoperation,2 but afixedstategiven him by the rulesof the game.8 A game in which coalitionsinvolve no advantage is inessential,4 even if one player i shouldhave aconsiderablefixed advantage in it.

Thereaderwill understand,of course,that all this is just an additionalcomment on results which were already rigorously establishedabove (in(49:0),(49:D)).

49.3.Characterization of the Systems W, L of Actual Games49.3.1.We now turn to the secondsubjectmentionedat the end of

49.1.3.Let two systems W, L( 7) be given, which fulfill the conditions(48:A:b)-(48:A:d)and (49:C),and also(49:1:a).6 We wish to constructanessentialgame F with Wr = W, Lr = L. In doing this, we normalize Fwith 7 = 1.

ThesetsS in Lrarecharacterizedby their flatness, i.e.by v(S) = p,where p is the number of elementsof S.6 ThesetsS in Wr arecharacter-izedby the fact that S belongsto Lr , i.e.by v( S) = (n p), owingtothe above. Now v( S) = v(/S),hencewe may write for thisv(S) = n -p.

Hencewe have shown:ThedesiredrelationsWT = W, Lr = L areequivalent to this:

(49:2) Fora g-elementsetS,(q = 0,1, , n 1,n)

(49:2:a) v(5) = n - q

if and only if Sbelongsto W, and

(49:2:b) v(S) = -qif and only if S belongsto L.

1Theequivalent consideration wascarriedout in a specialcasein 35.1.4.1We always considerthis to be the same thing as forming appropriatecoalitions.*Cf.our treatment of the basicvalues a',6',c'in the three-persongame, in 22.3.4.

Theentire discussion of strategic equivalence,cf.27.1.1.,was made in the same spirit:advantages like this one can be removed by strategically equivalent transformations,while those which are really due to forming coalitions, cannot.

4Henceits WT, Lrarenot the desiredones,describedin (49:D),but thoseof (49:B:b).*We require (49:l:a)becausewe aim primarily at essentialgames(cf.(49:B)). Sub-

sequently we will make our discussion exhaustive aswill beseenin (49:E).'Recallthat aU v() - y - -1.)))

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Thus our task is to constructa game F (normalized and 7 = 1)with acharacteristicfunction v(S)which fulfills (49:2).

49.3.2.(49:2)determinesv(S) for the Sof W and L, so we needonlydefine it for thoseS which belongto neitherset. We try therethevalue 0.Accordingly we define:

H\"V

ff(Lr .Sl

rn ^1Sag-elementsetwith g = 0,1, ,n-l,n

q tor om L Jotherwise1

We first prove that v(/S) is a characteristicfunction, i.e.that it fulfills

(25:3:a)-(25:3:c)in 25.3.1.We prove theseconditionsin their equivalentform of (25:A) in 25.4.2.:

Casep = 1with = :Thisisv( I) =0,immediatesinceI is in W, because= -7is in L by (48:A:b),(49:C).Casep = 2 with = :This is v(S\\) + v(S2) == when Si,Sjarecomple-

ments. If both Si,S2 arenot in W, L, then v(Si) = v(S2) = 0. If one ofSi,S2 is in W or L, then the other is in L or W, respectively,by (48:A:b).Assume,by symmetry, Si in L, S2 in W. Let Si have q elements,henceS2 has n - q. Then v(Si)= -g,v(Si) = g.

Soat any ratev(Si)+ v(Si) = 0.Casep = 3 with g :This is v(Si)+ v(S2) + v(S8) ^ when Si,St,S,

arepairwisedisjunct with the sum /. If none of Si,S2, Sa is in W, thenv OSi),v(S2), v(S3) ^ O.2 If one of Si,S2, Sais in W, we may assumebysymmetry, that it is$3. Hence Sa = Siu S2 is in L by (48:A:b),and soSi,S2 arein L by (48:A :d).LetSihave q\\ elements,S2 have g2 elements,henceS8 has n gi g2. Then v(Si) = -cji,v(Si) = -g2, v(Si) = q\\ + g.

Soat any ratev(Si)+ v(S2) + v(S8) g 0.49.3.3.Thusv(S)belongsto a game T. We now establishthe remaining

assertions.v(S)(i.e.T) isnormalized and 7 == 1:Indeed,by (49:C)all v((i)) = 1.v(S) fulfills (49:2):Owing to (48:A:b)and v(-S)= -v(S),the two

parts of (49:2)go over into eachotherif we interchangeS and S. Weconsidertherefore only the secondhalf.

If Sis in L, then clearly v(S) = q. If Sis not in L, then v(S) = qwould necessitate= #,8 or q = 0. But this means that S is empty,contradicting(49:C).

Sothe game F possessesall desiredproperties.We arenow able to prove the followingexhaustivestatement:

(49:E) In orderthat two given systems W } L(/)be the TFr, Lrof a suitable game T, these requirementsare necessaryandsufficient:F inessential:W = L = /.T essential:(48:A:b)-(48:A:d),(49:0),(49:1:a).

1That the two first specifications do not conflict, is due to (49:l:a).Clearly v(S) if Sis not in W.

1As n - q T* -7,Scouldnot be in W', hencev(S) 0.)))

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428 SIMPLEGAMES

Proof:T inessential:Immediateby (49:B:b).T essential:The necessitywas establishedin (49:A), (49:B:a),(49:C).

Thesufficiencyis the contentof the constructionwhich we have carriedout.In concludingwe mention another interpretation of (49:2), Recalling

the inequalities(27:7)of 27.2.(also shown on Figure50),which specifylimitations for v(S),it appearsthat Wr is the setof thoseS for which v(S)reachesthe upper limiting value, and Lr the setof thoseS for which vCS)reachesthe lower limiting value.

49.4.Exact Definition of Simplicity

49.4.(49:E) permits us to give a rigorousdefinition of that classofgamesto which we alluded in 48.1.2.and 48.2.1.,and which was circum-scribed in more detail at the beginningof 49.1.3.:Where the only aim ofall playersis to form certaindecisivecoalitionsand where thereareno otherinvolved motives which requirea quantitative description.

By combining the part of (49:E) which refers to essentialgameswith

(49:1),it appears that the formal expressionof this idea is

(49:1:b) W r uLr = /.Indeed,this condition expressesthat any given coalition S belongseitherto the winning or to the losingcategory without any further qualification.

We define accordingly:An essentialgamewhich fulfills (49:l:b)iscalledsimple.

The conceptof simplicity is invariant under strategicequivalence,sincethe setsWr, Lrare.

49.6.SomeElementary Propertiesof Simplicity

49.5.1.Beforewe take up the detailedmathematical discussionof thisconcept,letus consideroncemore the closingremarkof 49.3.In the senseof that remark an essentialgame is simple,if v(S) liesfor every S on theboundary1of the areaassignedto it by the inequalities(27:7)in 27.2.

The variety of all essentialn-persongames (normalized,7 = 1) canbeviewed as a geometricalconfiguration of a certainnumber of dimensions,given in Figure65. More preciselythe inequalitiesreferred to define aconvex polyhedricdomain Qn in the linearspaceof the dimensionalityinquestion,and the pointsof this domain representall thesegames.2

49.5.2.E.g.:Forn = 3 the dimensionalityis zero,and the domain Qa single point. For n = 4 the dimensionalityis 3,and the domain Q4the cubeQ of 34.2.2.

Now the simple gamesare those for which we are on the boundaryof eachdefining inequality. With respectto the convex polyhedricdomainQn this means:Thesimplegamesarethe verticesof Qn , n = 3,4.

1Theboundary consistsof two points:the upper limiting value n p and the lowerone p, (y 1). v(S)must beoneof thesetwo, no matter which.'Thereaderwho is familiar with n-dimensional linear geometry, will note:SinceQnis defined by linear inequalities, it is a oolvhedron. The discussion of 27.6.allows toconcludethat it is convex.)))

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E.g.:Forn = 3 Q3 is a singlepoint, i.e.nothing but a vertex, so theessential3-persongame issimple.l Forn = 4 Q4 isthe cubeQ,so the simplegamesarethe vertices,i.e.cornersI-VIII.*

49.6.Simple Gamesand Their W, L. TheMinimal Winning Coalitions :Wm

49.6.1.Combining(49:E)with the definition of simplicity,we obtain:(49:F) In orderthat two given systems W, L(s/) be the Wr, Lr

of a suitable simplegame F, theserequirementsarenecessaryand sufficient: (48:A:a)-(48:A:d),(49:C).

That the S referred to in (49:2)exhaust all subsets of 7, is definitoryfor simplicity. Consequentlyit is for simplegamesand for thesealone,that knowledgeof Wr, Lr determinesv(S), provided that the game isnormalized and 7 = 1. I.e.,without the last proviso,that it determinesthe gameup to a strategicequivalence.

We restatethis:(49:G) In caseof simplicity,and only then, the game F is deter-

mined by its Wr, Lr up to a strategicequivalence.

Consequently,accordingto (49:F)and (49:G)a theory of simplegamesis coexistensionalwith the theory of those pairs of systems W, L whichfulfill (48:A:a)-(48:A:d),(49:C).

49.6.2.In studying the pairs TF, L describedabove,48.2.2.shouldberecalled,and particularly (48:W), (48:L) there and (49:2).Accordingto these,it is sufficient to name eitherW or L in orderto determinethepair W, L.

The conditions(48:A:a)-(48:A:d)are then to be replacedas follows:If W is used, by (48:W);if L is used,by (48:L).

As to (49:C),it refersto L directly. We can equallywell refer it to W,

by applying (48:A:b) then the setsmentioned in it must be replacedby theircomplements.

For the sakeof completeness,we restate(48:W) and (48:L),togetherwith the correspondingforms of (49:C).

ThesetsW( I) arecharacterizedby theseproperties:(49:W*)(49:W*:a) Of two complements(in /) S, S,one and only one

belongsto W.

(49:W*:b) W containsthe supersetsof its elements.(49:W*:c) W contains/ and all (n l)-elementsets.

*Cf.also(50:A)in 50.1.1.1As far as the corners7, V, VI, VII are concerned,this is no surprise:Our dis-

cussionstarted with thesein 48.1 and our conceptof simplicity obtained from them bygeneralization.

Thereappearanceof the corners77, 777, IV, VIIIismore puzzling: We treated themin 35.2.as the prototypes of decomposability. However, they aresimple too, as follows

easily from (50:A) and the beginning of 51.6.)))

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430 SIMPLEGAMES

ThesetsL ( I)arecharacterizedby theseproperties:(49:L*)(49:L*:a) Of two complements(in /) S, S, one and only one

belongsto L.(49:L*:b) L containsthe subsetsof its elements.(49:L*:c) L containsthe emptysetand all one-elementsets.

As pointedout above, wecouldbasethe theory on eitherW with (49:W*)or onL with (49:L*).

49.6.3.Sinceit is more in keepingwith the usual way of thinking aboutthesematters to specifythe winning rather than the losingcoalitions,weshalluse the first mentionedprocedure.

In this connectionwe observe that a certainsubset of W shares theimportanceof W. This is the setof thoseelementsS of W of which nopropersubset belongsto W. We call theseS the minimal elementsof W

(i.e.W r) and their setW m (i.e.TF?).The intuitive meaning of this conceptis clear:Theseminimal winning

coalitionsarethe really decisiveones,thosewinning coalitionsin which noparticipant can be spared. (Itwill be rememberedthat our discussionof48.1.3.beganwith the enumeration of thesecoalitionsfor the game we werethen considering.)

49.7.TheSolutions of Simple Games

49.7.1.The heuristic considerationswhich led us to the conceptofsimplegamesmake it plausible,that the discussionof gamesbelongingtothis category,may turn out easierthan that of (zero-sum)n-persongamesin general. Fora corroborationof this we must examinehow solutionsaredeterminedin a simplegame. Sincewe arenow consideringthe old formof the theory, 30.1.1.must be consulted.1 We beginwith the observationthat a considerablesimplification must be expectedfrom the fact that in asimplegameevery setis eithercertainly necessaryor certainly unnecessary(cf.31.1.2.).

49.7.2.In orderto establishthis assertionwe prove first:

(49:H) In any essentialgame T all setsS of Wr arecertainly neces-sary, and all setsSof Z/ r arecertainly unnecessary.

Proof:If S is in Z/r then it is flat, hencecertainlyunnecessaryby (31:F)in 31.1.5.If S is in W?, then S is flat (becauseit is in Lr) and S 7* Q(because is in Z/r, hencenot in Wr). So S is certainly necessaryby(31:G)in 31.1.5.

We can now fulfill our above promiseconcerningsimplegames indeed,this can be done in two different ways.

1In the terminology of the new form of the theory as introduced in 44.7.2.et sequ.this means:We are looking for solutions for #(0),i.e.the excessis being restricted to thevalue 0.

Thesigmficance.of this restriction will becomeclearerin the third remark of 51.6.)))

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MAJORITY GAMESAND THEMAIN SOLUTION 431(49:1) In any simplegame F all setsS of W r arecertainly necessary,

and all othersarecertainly unnecessary.

Proof:Combine (49:H) with the fact, that for a simplegame Lr ispreciselythe complementof Wr.

(49:J) In any simplegame F all setsS of W arecertainly necessary,and all othersarecertainly unnecessary.l

Proof:We can replacethe Wr of (49:1)by its subset W^, i.e.we cantransfer all S of WT W? from the certainly necessaryclassinto thecertainly unnecessary,owing to (31:C) in 31.1.3.Indeed,every in Wrpossessesa subset T in TFp.

Of these two criteria(49:1)and (49:J),the latter is more useful.Their importancewill be establishedby actual determination of solutionsin simple games.2 Indeed,this analysis of simple games permits thedeepestpenetration yet effected into the theory of games with manyparticipants.3

50.TheMajority Gamesand the MainSolution50.1.Examples of Simple Games:TheMajority Games

50.1.1.Beforegoing any further, it is appropriateto give someexamplesof simplegames,i.e.of the pairs W, L of (49:F)in 49.6.1.We know from49.6.2.that it sufficesto discussthe W as characterizedby (49:W*) there.

Let us therefore considersomepossibleways of introducing suchW i.e.possibledefinitions of a conceptof winning.

The principle of majority suggests itself as a particularly suitabledefinition of winning. Henceit is plausibleto define W as the systemof allthoseSwhich contain a majority of all players. Itwill benoticed,however,that we must excludeties indeed (49:W*:a)statesfor this W, that forevery S eitherS or S must contain the majority of all players, thusexcludingthat both may contain exactlyhalf. In other words:Thetotalnumber of participantsmust be odd.

So if n is odd, we may define W as the setof all S with > ^ elements.4

Thesimplegame which obtainsin this way,5 will becalledthe directmajoritygame.

1Comparison of (49:1)and (49:J)shows that the Sof Wr W aresimultaneouslycertainly necessaryand certainly unnecessary. This is another illustration for theremark at the end of footnote 1 on p.274.

1Cf.50.5.2.and 55.2.1Cf.55.2.-55.11.,and in particular the generalremarks of 54.4Sincethe smallest integer > ^ is - (n odd!),we may also say:S must have

j r elements.

Precisely:Theclassof strategically equivalent ones(of n participants).)))

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432 SIMPLEGAMES

Thesmallestn for which this can be done,1 is 3. We know that thereexistsonly one essentialthree-persongame,and that for this W it consistspreciselyof the 2 and 3-elementsets i.e.of the setswith >f elements.Sowe see:(50:A) The (unique) essentialthree-persongame is simple;it is

the directmajority game of threeparticipants.For the subsequent n which are eligible,n = 5, 7, , the direct

majority game is merelyonepossibilityamong many.50.1.2.Thedirectmajority gameis only available, when n is odd, and

yet simplegamesexistfor even n as well indeedour prototype of simplegames(cf.48.1.2.,48.1.3.)had n = 4.

However,the conceptof majority is easilyextendedto cover the caseofeven n as well. To this endwe introduceweighted majorities in thefollowingmanner:Let eachoneof the players1, , n be given a numerical weight,

say Wi,- , w n respectively. Define W as the setof all thoseS which

contain a majority of total weight. This means:n

(60:1) w. >i Wi,i in 8 t-1

orequivalently,

(50:2) % Wi> J) Wi.iinS iin -8

We must again take careto excludeties.However,owing to the greatergenerality of our presentsetup, it is betterto proceedimmediatelyto acompletediscussionof (49:W*).

50.1.3.Let us see,therefore, what restriction(49:W*) imposesuponthe MI, , w n.

Ad (49:W*:a):Sincewe can expressthat S belongsto W by (50:2),soS belongsto W when

(50:3) % Wi < % Witi in 8 i in -8

So (49:W*:a)means that always (50:2)or (50:3)holds, but never both.This meansclearly,that never

(50:4) Wi = w^i in 8 t in -S

orequivalently, that never

(50:5) Wi = i Wi.tinS -l

1I.e.which ia oddand for which a game can be essential.)))

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MAJORITY GAMESAND THEMAIN SOLUTION 433

Ad (49:W*:b):Usingthe definition of W in the form (50:1),this require-ment is clearlysatisfiedif all Wi ^ O.1

Ad (49:W*:c):Using again (50:1),it is clearthat / = (1, , n)belongs to W. For the general(n - l)-elementset S = / - (i' ), thecondition(50:1)statesthat))

Summing up:(50:B) The weights Wi,

- , w n can be used to define by (50:1)or (50:2)a W which satisfies(49:W*) if and only if they fulfill

the followingconditions:(50:B:a) Forall t = 1, , n

n

0** ^x i V^;SWi < ~y 2_t Wi.

(50:B:b) Forall S I))

tinS -lVerbally:A player has always non-negative weight, but never half of

the total weight or more;no combination of playershas preciselyhalf of thetotal weight.2

Thesimplegamewhich obtainsfrom this W 3 will be calledthe weighted

majority game (of n participantswith the weights w\\ 9 , w n). We will

alsodesignatethis game by the symbol[t0i, * * * , w n].Thus the directmajority game has the symbol[1, , 1],It will benoted that the four-person game representedby the corner

7 of Q, discussedin 48.1.2.,48.1.3.can be describedas a weightedmajoritygame. Indeed,the principleof winning found in 48.1.3.can be expressedby sayingthat players1,2,3have a common weight, while player4 has thedoubleweight. I.e.this game has the symbol[1,1,1,2].

60.2.Homogeneity

60.2.1.The introduction of majority gamesand their explanatorysymbols[wi, , w n] is a step in the directionof a quantitative (numeri-cal) classificationand characterizationof simplegames.Therearegoodreasonsto think that it would be mostdesirableto carry out sucha programfully: Simplicitywas defined in combinatorial,set-theoreticalterms andit is to be expectedthat a numerical characterizationwould make them

1This is, of course,a perfectly plausible condition; indeed,the surprising thing is thatwe arenot forcedto require u> > i.e.that wecanpermit a weight to vanish.

1Thefirst requirement obviates the difficulty of 49.2.,the secondexcludesties.1Precisely:Theclassof strategically equivalent ones.)))

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434 SIMPLEGAMES

easierto handle. Sucha characterizationusually facilitates a more exhaus-tive, quantitative understandingof the notion considered.Besides,in ourpresent problemwe areultimately searchingfor solutionsthat aredefinednumerically, and therefore it seemslikelythat a numerical characterizationwill correspondto them more directlythan a combinatorial one.

However,this first step is far from carrying out the transition.On the one hand, a simplegamemay possessmore than one symbol

[t0i, , w n] indeed,every simplegame that hasoneat all has infinitely

many. 1 On the other hand, we do not know whether all simplegamespossesssucha symbolat all.2

We begin by consideringthe first deficiency. Sincethe same simplegame may possessseveralsymbols[101, , w n], the natural procedureis to singleout from among them a particular one by some convenientprincipleof selection.It is desirableto specifyin thisprinciplesuchrequire-mentswhich increasethe significanceand usefulnessof the w\\ 9

- - , w n.First somepreliminary observations. Theconditions(50:1),(50:2)sug-

gestconsiderationof the difference

(50:6) aa = 2 % w<- %*><=% w %- 1*.

t in S i 1 \\ in S i in 8

This ds expresseshow much the coalition S outweighsits opponentshow much of a weighted majority it possesses.Theseare its immediateproperties:(50:C) as = -a_s

Proof:Usethe last form of (50:6)for as.(50:D:a) as > if and only if S belongsto W.

(50:D:b) as < if and only if S belongsto L.(50:D:c) as = is impossible.

Proof:Ad (50:D:a):Definitory.Ad (50:D:b):Immediateby (50:D:a)and (50:C).Ad (50:D:c):Immediateby (50:D:a),(50:D:b)sinceW, L exhaust all

S. It alsocoincideswith (50:B:b).60.2.2.Now it is natural to try to arrangethe weightsw\\, , w n so

that the amount of as which securesvictory be the same for eachwinningcoalition.It would beunreasonable,however, to requirethis actually forall Sof W: If Sbelongsto W, then its propersupersetsT dotoo,and they mayhave aT > as.z Sincesucha T containsparticipantswho arenot necessaryfor winning, it seemsnatural to disregardit. I.e.we requiretheconstancyof as only for thoseS of W which arenot propersupersetsof otherelements

1Obviously, sufficiently small changesof the u> will not disturb the validity of (60:1),particularly since(50:5)is excludedby (50:B:b).1We will seein 53.2.that certain simple gameshave none.

1Thus for T - / a S,a/ > asunless u>< for all i not in 8.)))

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MAJORITYGAMESAND THEMAIN SOLUTION 435

of W. In the terminology introducedin 49.6.3.:as is requiredto be con-stant for the minimal elementsof W i.e.the elementsof W m.

We define accordingly:

(50:E) The weights w\\ y , w n are homogeneous,if the as of(50:6)have a common value, to be denotedby a, for all S ofW1*.

Whenever (50:E)is valid we shall indicate this by writing [wi, , w n]h

insteadof [wi, , w n].Clearlya >0. A common positive factor affects none of the significant

propertiesof w\\, , w n , therefore we can use this in the caseof homo-geneityfor a final normalization: Makinga = 1.

We concludeby observing that the games mentionedat the end of50.1.3.arehomogeneousand normalized by a = 1. Thesearethe directmajority gamesof an odd number of participants [1, , l],.andthecorner7 of Q [1,1,1,2]which can accordinglybe written [1, , 1]*and [1,1,1,2]*.Indeed,the readerwill verify with easethat a3 = 1 for allSof W m in both instances.

50.3.A MoreDirectUseof the Conceptof Imputation in Forming Solutions

60.3.1.The homogeneouscaseintroduced above is closelyconnectedwith the ordinary economicconceptof imputation. We proposeto showthis now.

Moreprecisely:We defined in 30.1.1.a generalconceptof imputationsand based on it a conceptof solutions. In forming thesewe were led bythe sameprinciplesof judgment which areused in economics,and thereforesomerelationshipwith the ordinary economicconceptof imputation must beexpected.However, our considerationstook us rather far from thatconcept.This appliesespeciallyto the constructionswhich were necessarywhen we found that setsof imputations i.e.solutions and not singleimputationsmust be the subjectof our theory. It will now appear that forcertainsimplegames the connectionwith the ordinary economicconceptof imputation can beestablishedsomewhat more directly. Onemight saythat for the specialgamesin questionthe connectionbetweenthis primitiveconceptand our solutionscan be directly established.Actually it will

providea simplemethod to find a particularsolutionfor eachone of thosegames.

50.3.2.The two conceptsof solution,i.e.the two procedures,supporteachother quite effectively. The ordinary economicconceptprovidesauseful surmiseas to the form of a certainsolution. And then our mathe-matical theory may be used to determinethe solutionsin questionand tomake the requirementsof the ordinary approach complete.(Cf. 50.4.on the one hand, and 50.5.et sequ.on the other.)

Theseconsiderationsalsoserveanother end:They bringout the limita-tions of the ordinary approachwith greatclarity. Theordinary approach)))

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436 SIMPLEGAMES

functions in this form only for the simplegames,and even therenot alwaysand not entirely unaidedby our mathematical theory. Besides,it doesnotdiscloseall the solutions for the games to which it applies. (Furtherremarkson this subjectoccurthroughout the discussion,and particularlyin 50.8.2.)

In this connectionwe emphasizeagain that any game is a modelof apossiblesocial or economicorganization and any solution is a possiblestablestandardof behavior in it. And the gamesand solutionsnot coveredby the method referred to i.e.by the unimproved economicconceptofimputation will prove to be quite vital onesfor socialor economictheory.It will be seenthat the simplegameswhich can be treatedby this specialmethod are closelyconnectedwith the homogeneousweightedmajoritygamesof which they area generalization.

50.4.Discussionof This DirectApproach

60.4.1.Considera simplegame F which we assumein the reducedformwith 7 = 1,but which we do not yet restrictany further. Let us try todiscussit in the senseof the ordinary economicideaswithout making useofour systematictheory.

Clearly,in this game the soleaim of playersis to form a winning coalition,and oncea minimal coalition of this kind is formed, thereis no motive for itsparticipants to admit additionalmembers.Consequentlyone can assumethat the minimal winning coalitions the S of W m are the structuresthat will form. It is therefore plausibleto assume that a player's fatepresents only two significant alternatives:He eithersucceedsin joiningone of the desirablecoalitionsor he does not. In the lattercasehe isdefeated,hencehe obtains the amount 1. In the former casehe is suc-cessfuland accordingto ordinary ideasone ought to ascribeto this successa value. This value may vary from one player to another;for player iwe denoteit by 1+ z<so that x is the margin betweendefeat and successfor playeri.l

60.4.2.Letusnow formulate the requirementswhich must be imposedontheseXi, , xn in the courseof a conventional economicdiscussion.

First:By the very meaning of the z* necessarily(50:7) Xi ^ 0.

Second:If it happens that no minimal winning coalition contains acertainplayeri,then thereexistsfor him no alternative to the value 1,and so we neednot define any z for him. 2

1We assumehere that there is only one way of winning, i.e.that the margin x, is thesamewhichever (minimal winning) coalition the player succeedsin joining. This isplausible sincethere is only one kind of successin a simple game :the completeoneevery coalition being either fully defeatedor fully winning.

It will appearin 50.7.2.and 50.8.2.how far this standpoint carries. As far as itdoes,it can beadvantageously combined with our systematical theory.1For the really important simple gamessuch i do not exist i.e.every player belongsto someminimal winning coalition. Cf. the first observation in 51.7.1.and (51:0)in 51.7.3.)))

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MAJORITY GAMES AND THEMAIN SOLUTION 437

Third:If a minimal winning coalition S becomeseffective, then thedivision betweenthe playerswill be this:Eachplayerinot in S obtains 1,eachplayer t in S obtains -1+ x* Thesum of theseamounts must bezero. This means

0= (-1)+ (-!+*<)= -n+ 2) *,i not in S t in S i in S

i.e.(50:8) *t = n.))

tin))

In our system of notations this distributionis describedby the vector

a ={!,,}with the components

__ f 1 for t not in S,Qt\\ \\

( 1 -f Xi for i in S.We denotethis vector by a s. Our first conditionand the presentone

actually statejust that a 6' is an imputation in the senseof 30.1.1.50.4.3.Continuing the usual line of argument, \\ve shall now want to

determinethe x\\, , xn by means of the equations and inequalitiesof the threeabove remarks.In doing this, one more point must becon-sidered:We have stated in the third remark, that its S must be minimal

winning, i.e.belongto W m. However it may be askedwhether all S of W m

can be used.Indeed,the presentprocedureis nothing but the usual one to determine

the imputation of values to complementarygoodsby meansof their alterna-tive uses.1 Now thesealternative uses may be more numerous than thedifferent goodsunderconsideration i.e.Wm may have moreelementsthan n.2In such a situation onemight expectthat someof the usesareunprofitableand neednot be includedin the third remark. Indeed,we already madeuseof this principleby taking the S of W m only, and not all elementsof W,becausethe S of W W m (thenon-minimal winning coalitions)areclearlywasteful. Are we now sure that all S of W m must be consideredasequivalentsof profitable uses? They areclearlynot wasteful in the crudesenseof the S mentionedabove;no participantof an S inW m can be sparedwithout causingdefeat. But unprofitability can arisein lessdirectwaysthan this,asnumerouseconomicexamplesshow. Thusthe questionremainsunansweredas to which S of W m areto be used in the third remark.

It is clear,however, that if an S of W m is not includedthere,i.e.if(50:8) J) Xi = n

t in S1In this caseit would bemore suitable to say, services. Theobjectconsideredis the

total serviceofplayer i in cooperatingwithin a coalition which hejoins.1Cf.The fourth remark in 53.1.)))

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438 SIMPLEGAMES

fails to hold for it,then it must be definitely unprofitable. I.e.we musthave > in placeof = in (50:8):(60:9) 5) Xi > n.

tin S

Thus the questionarises:By what criteriaarewe to determinewhich Sof Wm fall underthe third remark i.e.for which must (50:8)hold. Denotetheir setby U(sW\.") Then (50:9)must hold for the S of W m - U. Sothe problemis to determineU.1

50.5.Connection with the GeneralTheory. Exact Formulation

60.5.1.Insteadof attempting a verbal description,let us settlethispoint by going back to our systematictheory. From the statementmadein 50.4.we carry over this much:Considera system of minimal winning

coalitions,i.e.a set U Wm and the z. Form the imputations))

as in 50.4.' 1for i not in S } . . . rr.. \\

when S is in U.1+ Xi for i in S]That these a 5, S in J7, areindeedimputations,is expressed,as we know,by the conditionsof 50.4.(50:7) Xi z 0,(50:8) x = n when S is in U.

tin S

Formthe setV of the a 5, S in U. We will decidewhether U and the Xi

aresatisfactory,by determiningwhether this V is a solution in the senseof 30.1.1.

II will be seenthat the result which is obtained in this manner can bestatedverbally and is perfectly reasonablefrom the ordinary economicpoint of view. But it may be questioned whether it could have beenunequivocally establishedby the usual procedures.This may serveas anillustration of how our mathematical theory can serve as a guide evenfor the purely verbal discussionsof the ordinary economicapproach,(cf.50.7.1.).

50.5.2.We proceedto investigatewhether V is a solution.*

Let us determinefirst, when a given imputation ft = {0i, , /3n }

is dominatedby a given a T, T in U. Sincethe gameis simple,the setS1It would be utterly mistaken to try to define Wm - U (and so 17) by means of

(50:9).This would not restrict the Xi, *, sufficiently and their determinationis the realobjective!)))

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MAJORITY GAMESAND THEMAIN SOLUTION 439

of 30.1.1.for this domination can be assumedto belongto W (oreven toW m, use (49:1)or (49:J) in 49.7.2.).Forevery i in S, af > ft ^ -1;for every i not in T9 a = -1:henceS T. Now T is in U c jp f 5 isin fF, therefore S S T7 yieldsS = T. So we see:The set S of 30.1.1.forthis domination must be our T. And T can be used there, sinceit iscertainly necessary,as it belongsto U W m 5 W, cf. above. Hencethe

domination a T H ft amounts to this:afr > ft* for i in T, i.e.

(50:10) ft, < -1+ z, for i in 71.Denotefor any imputation ft = {0i, , n ) the setof all i with

(50:11) ft -1+x.by 72( ft ). Then (50:10)statesthat 72(7)and T aredisjunct. An alterna-tive way of writing this is:(50:12) -72(7)2^.

We repeat:

(50:F) <*r *- is equivalent to (50:12).

From this we can infer:

(50:G) Let U* be the set of all 72( I) which possesssomesubsetbelongingto U.

Let U+be the setof all 72(s7) for which R doesnot belongto U*.

Then ft is undominated by any elementof V if and only if

7?(7)belongsto U+.

Proof:That ft is dominatedby someelementof V i.e.by some a r ,T in U means that (50:12)holdsfor some T in U. This is equivalent to

sayingthat -72( ) is in U*,i.e.that 72( ) isnot in U+.

HenceR( ft ) belongs to Lr+ if and only if is dominated by noelementof V.

60.5.3.Beforegoing any further we observe four simplepropertiesofthe set t/+of (50:G)(50:H:a) U* = U+ = W if U = W m

Proof:Assume U = W m. Then U* consistsof thosesetswhich possessasubset belongingto W m i.e.a minimal winning subset. HenceU* = W.Theoperationwhich leadsin (50:G)from U* tot/+is the combination of thetransformation (48:A:a)and (48:A:b)in 48.2.1.Now we noted already)))

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then, that thesetwo transformations compensateeachother,when appliedto W . Hence7* = W gives t/+ = W .(50:H:b) U*is a monotonic and U+ is an antimonotonic operation.

I.e.f/i s C/2impliesE7? s C7?and Uf => C7f.

Proof:It sufficesto recallthe definitions in (50:G),to seethat Ui s UzimpliesU* sU* and this in turn Uf 2 C/J.(50:H:c) All our U sWm have *7* cTF c f/+.

Proo/:Combine(50:H:a)and (50:H:b)(with U, W m in placeof Ui, C/2).(50:H:d) Both U*and [7+ contain all supersetsof their elements.

Proof:This is obvious for U*. Thepropertyunder considerationis thesameone which wasformulated in (48:A:c)in 48.2.1.(W taking the placeof our [/*,J7+.) Now the operationwhich leadsin (50:G)from U* to U+,is the combination of the transformations (48:A:a)and (48:A:b)in 48.2.1.(Cf. the proof of (50:H:a)).Application of (48:B)in 48.2.2.to thesetwotransformations shows that the property in question is conservedwhen

passingfrom U* to U+.50.5.4.Note that U*, U+ allow a simpleverbal interpretation. If we

knew only of the coalitionsbelongingto U that they arewinning, ofwhichcoalitionscould we then assertthat they are certainly winning, and ofwhich that they arenot certainly defeated?

Theformer is the casefor the coalitionswith subsetsin T,i.e.for thoseof U*. Thecertainly defeatedonesarethe complementsof these,i.e.thosenot in U+. HenceU* is the setof the first mentionedcoalitions,and {7+the setof the last mentioned ones.

Now the meaning of (50:H:a)-(50:H:c)becomesclear:For U = W m ,everything is unambiguous:The certainly winning coalitionsarepreciselythosewhich arenot certainly defeated,and they form the set W. As Udecreasesfrom W m , the gapwidens. Thefirst setdecreasesthrough subsetsof W, the secondone increasesthrough supersetsof W.

Theassertionof (50:H:d)is equally plausible.60.6.Reformulation of the Result

50.6.1.(50:G)of 50.5.2.allows us to state:(50:1) V is a solution if and only if R(ft) belongsto U+ precisely

when ft belongsto V.Sowe must only decidewhen (50:1)holds. Forthis purposewe consider

> >

an R in U+and determinethe ft for which R(ft) = R.Considerthe threepossibilities:

(50:13) (-1)+V (-!+*<)= 0,i not in A i in ft <)))

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MAJORITY GAMES AND THEMAIN SOLUTION 441

i.e.(50:14) Xi = n.

inB <>

If a ft with 12( ) = R exists,then we have

(50:15) = ft (-1)+ % (-1+ *,),i-1 i not in R i in ft

i.e. in (50:13),(50:14).So> in (50:13),(50:14)excludesthe existenceof any \"/? with R(~ft) = R. I.e.the setsR in C7+with > in (50:13),(50:14)neednot beconsideredfurther. Consideron the otherhand, an R in E7+

with < in (50:13),(50:14).Then thereareinfinitely many ways of choos-

ing 7 with ft = and ft ^ (\"I5**'***!.For all thesel I 1+ Xi for t m /c)

ft( ) necessarily2 ft. Henceit belongsto V by (50:H:d).SinceV is

finite, these ft cannot all belongto V. This is a contradiction.I.e.setsR in U+with < in (50:13),(50:14)must not exist.

60.6.2.It remainsfor us to considerthe setswhich arein U+ with = in(50:13),(50:14).According to the above, these must furnish preciselythe V of V.

If ft belongsto V, i.e.ft = a T, T in C7, then we have this situation:R ( ft ) is T plusthe setof thoset for which x> = 0. T belongsto Uc[7*c [/+

(for the second relation use (50:H:c)),hence R(ft) belongsto U+.Also

2 _ Xi =2) Xi = n -

i in R(0) i in T

Sowe have = in (50:13),(50:14).Hencethe ft of V areall takencareof.Conversely:Consideran R in U+with = in (50:13),(50:14).Addition

of all i for which x> = to R affects neitherthe fact that R belongsto [7+(by (50:H:d)),nor the equation (50:14).Sowe may assumethat R con-tains all thesei.

If now an imputation ft has R( ft ) = R, then ft ^ -1+ x> for i in fi.n

Always ft ^ 1. As ft = 0,this implies:))

(60:16) =j _}))

for i not infor t in R.)))

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442 SIMPLEGAMES

Conversely:(50:16)implies that ft is an imputation with R( ft ) = ft.

Henceour requirementin this casemust be that the ft of (50:16)be an

a r , T in U. This means,that T and R differ only in elementsi for which

Xi = 0. And this property is insensibleto our original modification of fi,the inclusion of all suchi into R.

Summing up:(50:J) V is a solution if and only if this is the case:Callan i indiffer-

ent when z t = O.1

Then we have

(50:8*) J) Xi = nin T

for the T of U and, of course,also for thosewhich differ fromtheseonly by indifferent elements.

And we must require

(50:9*) x t > ntin T

for all otherT of U+.

In making use of this result, one may chosethe set U c W m first, thenattempt to determinethe Xi from (50:8*)and finally verify whether thesex<

fulfill the inequalities

(50:7) Xi

and (50:9*).50.7.Interpretation of the Result

60.7.1.The result (50:J)permits the verbal statementpromised in

50.5.1.This is it:A solutionV is found by choosingarbitrarily the setU of thoseminimal

winning coalitions(i.e.(7cWm), which are to be consideredprofitable.The Xi must then satisfy the correspondingequations (50:8*).But afterthis, it must be verified that certain other coalitionsaredefinitely inprofit-ablein thesenseof (50:9*).This must berequirednot only for thosecoali-tionswhich areknown to be winning, (i.e.W), but for all thosewhich cannot

1Thesei constitute a slight complication which is further aggravated by the factthat we have no exampleof a game in which they actually occur. It may be that theynever exist;an indifferent i characterizesa player who belongs to someminimal winningcoalitions,but never receivesa share.

Theexcludedplayer in a discriminatory solution of the three-persongame is in thissituation (cf. (32:A) in 32.2.3.with c - 1). But that solution is an infinite set,whereasour V must be finite.

It would be of interest to decidethis existential question. At any rate we must atpresentprovide for the indifferent i to avoid lossofgenerality or rigour.)))

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MAJORITY GAMES AND THEMAIN SOLUTION 443

be establishedas definitely defeatedby the coalitionsof U alone (i.e.U+)excepting,of course,the coalitionsof U itself.1

The readermay now judgewhether the concluding remark of 50.5.1.is justified by this formulation.

50.7.2.The questionof finding the proper U for (50:1)is a rather deli-cateone. The antimonotony of U+ (cf.(50:H:b)in 50.5.3.)makes itselffelt now: Decreasing[7, i.e.the number of equations, increases[/+, thenumber of inequalities,and viceversa.

In particular,if we chooseU as largeas possible,i.e.U = W m, then theinequalities associatedwith U+ createno difficulties at all. Indeed:U = W\" implies U+= W by (50:H:a)in 50.5.3.A T of W certainlypossessesa subsetS which is minimal in W, i.e.belongsto U = W m. Nowif T differs from this S by more than indifferent elements,then we haveX{ > for somei in T S,hence ^ x > ^ x = n, i.e.(50:9*)asdesired.

in T i in 8Thus U = W m always yieldsa solution V, if its equations(50:8*)can

be solvedat all (with (50:7)).But as we pointed out in 50.4.3.,we have no right to expecta priori

that this will always be the case especiallysincetheremay be more equa-tions (50:8*)(i.e.elementsin TFm), than variablesXi.

The last objectionis not an absolute one;indeed it is easy to find asimplegame for which the number of theseequationsexceedsthe numberof variables and the solution neverthelessexists.2 On the other handthereexistsimplegamesfor which thoseequationshave no solutions. An

exampleof this is somewhat more hidden,8 but the phenomenon is probablyfairly general. When this occurs,one must investigatewhether a solutionV cannot be found by appropriate choicesU c W m. The difficulty anddelicacyof this questionhasbeencommentedupon alreadyat the beginningof this section.4

50.8.Connection with the HomogeneousMajority Games

60.8.1.We now restrictourselvesto the caseU = W m. I.e.we assumethat the full systemof equations

(50:17) % Xi = n for all S in W m,tinS

can be solvedwith

(50:7) Xi 0.1And those which differ from them only by indifferent elements.1This happens for the first time for n - 5, cf.the fifth remark in 53.1.1This happens for the first time for n - 6,cf.the fifth remark in 53.2.5.4 No instanceof a simple gamewith a solution V derived from U cWm is known, nor

is it established that none exists. The further-going question whether every simplegamepossessessolutions V of suitable U W m is equally open.

Theproblem seemsto be of some importance. It may be difficult to solveit. Itappearsto have somesimilarity with the solved questions mentioned in footnote 1 on

p. 154,but it has not beenpossible,sofar, to exploit this connection.)))

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444 SIMPLEGAMES

We saw that in this casethe setV of all a 5, S in W m, is a solution. In thissituation and only then, we call V a main simplesolution of the game.

There is a certainsimilarity between these requirementsand thosewhich characterizea homogeneousweighted majority game. Indeed,thelatteraredefined by

(50:18) *> = & for all S in Wtin S

where))

())

Wi; + a, a > (combine(50:D)(50:E)of 50.2.)and

(50:19) Wi 0.Actually, thereismore than similarity. Thus, if a systemof w iy fulfilling

(50:18),(50:19)is given, a system z< fulfilling (50:17),(50:7)obtains asfollows:The quantity b of (50:18)is positive.1 Multiplicationof all Wi

by a common positive factor leaveseverything unaffected, and by choosingthis factor as n/b we can replaceb in (50:18)by n. Now we can simplyputXi = Wi and (50:18),(50:19)become(50:17),(50:7).

If conversely a systemof Xi fulfilling (50:17),(50:7)is given, thereis anextra difficulty. We may put Wi ss x<.2 Then (50:7)becomes(50:19)

n

and (50:17)yields(50:18)with 6 = n, i.e.a = 2n ] w. But now the-iquestionariseswhether the last requirementa > is fulfilled i.e.whether

(50:20) Xi < 2n.-iSummingup:

(50:K) Every homogeneous,weighted majority game possessesamain simplesolution.

Conversely,if a (simple)game possessesa main simplesolu-tion, homogeneousweights for the game can be derivedfrom itif and only if (50:20)is fulfilled.

50.8.2.This connectionbetweenhomogeneousweightsand main simplesolutions is significant. But it must be stressedthat a homogeneous,weightedmajority gamewill in generalhave other solutionsbesidesthe

1Otherwiseall i occurring in the Sof Wm would have w> - by (60:18)and (50:19).Then (50:6)of 50.2.1.and (50:19)gives a.g for the 8of W t hencea 0, which isnot the case.

1Thei which belong to no minimal winning setcausea slight disturbance, sincetheyhave no Xi (cf. the secondremark in 50.4.),while we require their IP<. However, thecontingency is unimportant (cf.loc.cit.) and we can put theseWi as is easily con-cluded from the referencesof footnote 2 on p. 436.)))

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ENUMERATIONOF ALL SIMPLEGAMES 445

main simpleone.1 And a game with a main simplesolution may not fulfill

(50:20),i.e.thereneednot be < in

A <(50:21) % x{ = 2n.2-i >

Beyondall this, finally, we must not losesight of the main limitationof these considerations:Whether we take the conceptof \" ordinary \"

imputation in its narrower form of 50.8.1.(i.e.U = W m) or in its wideroriginal form of 50.6.,50.7.1.(i.e.C7sW m, cf. (50:1)in 50.6.2.),it is cer-tainly restrictedto simplegames. That it is necessaryto go beyondthese,and beyondthe specialsolutionsdescribedhere,and that this forces us tofall back completelyon the systematicaltheory of 30.1.1.,was pointedoutat the end of 50.3.

61.Methodsfor the Enumeration of All SimpleGames51.1.Preliminary Remarks

51.1.1.Beginningwith 50.1.1.we introducedspecificsimplegameswhich

permitted characterization by numerical criteriainstead of the originalsettheoreticalones(cf. the beginning of 50.2.1.).We saw,however, thatthesenumerical procedurescouldbe carriedout in severalways and thattherewas no certainty that all simplegamescouldbe accountedfor with

their help. Itis therefore desirableto devisecombinatorial (settheoretical)methodsthat producesystematicenumeration of all simplegames.

This is, indeed,indispensablein order to gain an insight into the possi-bilitiesof simplegamesand particularly to seehow far the above mentionednumerical procedurescarry us. It will appear that the decisiveexamplesof the non-obvious possibilitiesobtain only for relatively high numbers ofplayers,8 so that a mere verbal analysiscannot bevery effective.

51.1.2.We pointedout at the end of 49.6.3.that the enumeration of allsimplegamesis equivalent to the enumeration of their setsW, i.e.of allsetsW which fulfill (49:W*) in 49.6.2.We alsonoted there that it may beadvantageous to replacethe use of W (all winning coalitions),by Wm

(all minimal winning coalitions).Either procedureprovidesan enumeration of all simplegames. The

use of W is preferablefrom the conceptualstandpoint sinceW has the

1Themain simple solution of the essentialthree-person game ([1,1,1]*,cf.the end of50.2.)is the original solution of 29.1.2.,i.e.(32:B)of32.2.3.We know from 32.2.3.and33.1.that other solutions exist.

The main simple solution of corner 7 of Q ([1,1,1,2]*,cf. the end of 50.2.)is the

original solution of 35.1.3.We will discussthis game, together with the more generalone[1, , 1,n 2]*(n participants) in 55.and obtain all solutions.

All thesereferencesmake it clearthat the solutions other than the main simple onearequite significant, cf.33.1and 54.1.

s -occursfor the first time for n - 6, cf. the fourth remark of 53.2.4.> occursforthe first time for n 6 or 7, cf.the sixth remark of 53.2.6.

Both theseexamplesare quite interesting in their own right.8 n -6, 7 cf.53.2.)))

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446 SIMPLEGAMES

simplerdefinition and Wm was introducedindirectlywith the help of W.Fora practicalenumeration of all simplegames which is our presentaimthe useof Wm is preferablesinceWm isa smallersetthan W l and thereforemore readilydescribed.

We will give both proceduressuccessively. It will appear that thesediscussionsprovidea natural applicationof the conceptsof satisfactorinessand saturation introducedin 30.3.

51.2.TheSaturation Method :Enumeration by Meansof W

51.2.1.ThesetsW arecharacterizedby (49:W*) in 49.6.2.i.e.by theconditions(49:W*:a)-(49:W*:c)which constitute (49:W*).

Let us for a moment disregard (49:W*:c),and consider(49:W*:a),(49:W*:b).Thesetwo conditionsimply that no two elementsof W canbe disjunct.2 In other words:Denotethe negation of disjunctness i.e.of Sc\\T7*Qby ScRiT. Then (49:W*:a),(49:W*:b)imply (Hi-satis-factoriness.8 A more exhaustivestatementalong theselinesis this:(51:A) (49:W*:a),(49:W*:b)areequivalent to (Rrsaturation.a

Proof:(Ri-saturation of W meansthis:(51:1) Sbelongsto W, if and only if Sn T * for all T of W.

(49:W*:a),(49:W*:b)imply (51:1):Let W fulfill (49:W*:a),(49:W*:b).If S belongsto W, then we know that S n T 7* Q for all T of W. If Sdoesnot belongto W, then T = -Sbelongs to W by (49:W*:a),andS n T = 0.

(51:1)implies (49:W*:a),(49:W*:b):Let W fulfill (51:1).We prove(49:W*:a),(49:W*:b)in the reverseorder.

Ad (49:W*:b):If Smeetsthe criterionof (51:1),then every supersetof Sdoestoo. HenceW containsthe supersetsof its elements.

Ad (49:W*:a):Owing to the above, S is not in W if and only if nosubsetof Sis in W. I.e.when every T of W is not S,or again,whenfor every T of W, Sn T ^ . By (51:1)this means preciselythat S isin W.

Thus, at any rate,preciselyone of /S, Sbelongsto W.Now SfaiT is clearlysymmetric,hencewe canapply (30:G)in 30.3.5.41W, Laredisjunct sets. They have the samenumber of elementsowing to (48:A:b)

in 48.2.1.Togetherthey exhaust 7 which has 2n elements. HenceW as well asLhasexactly 2W~1elements.

Thenumber of elements in W m varies, but it is always considerably smaller. (Cf.the fourth remark in 53.1.)1Proof:LetS,Tbelong to W, Sn T -0. Then -Sa T,hence-Sbelongsto IF by(49:W*:b),thus violating (49:W*:a).1Cf.the definitions of 30.3.2.

4 It will be remembered that wealsoassumedin 30,3.5.the generalvalidity of s<JU?-i.e.in this caseof Sto\\S. This means S & Q soit fails for S .However, (49:W*:a),(49:W*:c)exclude form W, hencewe may useas domain D

in the senseof30.3.2.instead of/ (the system ofallsubsetsof/) equally well 7 ()(the

system of all non-empty subsetsof 7). This rids us of S -.)))

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ENUMERATIONOF ALL SIMPLEGAMES 447

51.2.2.Inorderto discuss(49:W*) on this basis,we must take (49:W*:c)also into account. This can be done in two ways. The first way will beuseful for a subsequentcomparison.

(51:B) W fulfills (49:W*) if and only if it is (Ri-saturated and con-tains neither nor any one-elementset.

Proof: (49:W*) is the conjunction of (49:W*:a),(49:W*:b)and(49:W*:c).The first two amount by (51:A) to (Ri-saturation. Taking(49:W*:a)for granted,(49:W*:c)may bestated thus:If S is/ oran (n -1)-elementset, then S is not in W. I.e.:Neither nor any one-elementsetis in W.

Thesecondway is more directlyuseful.Let V Q bethe systemof all sets of (49:W*:c)i.e.of 7 and all (n -1)-

elementsubsetsof I. Thenwe have:

(51:C) V is a subset of a W fulfilling (49:W*) if and only if V u VQ

is (Ri-satisfactory.

Proof:Wz. V and W fulfilling (49:W*) amount to this:W*V,W ful-fills (49:W*:a),(49:W*:b)i.e.W is (Ri-saturated by (51:A) W fulfills

(49:W* :c) i.e.W 2.VQ. Inother words:We arelookingfor an (Ri-saturatedW 2.V u VQ i.e.we are asking whether V u VQ can be extendedto an(Ri-saturated set.

Now we know that (30:G)of 30.3.5.applies,and hencethe considera-tions of the last part of 30.3.5.applytoo.1 Thisextensabilityis equivalentto the (Ri-satisfactorinessof V u VQ.

51.2.3.We rephrase(51:C)more explicitly:

(51:D) V is a subset of a W fulfilling (49:W*) if and only if it pos-sessestheseproperties:

(51:D:a) No two S,T of V aredisjunct.(51:D:b) V containsneither 2 nor any one-elementset.

Proof:We must express,accordingto (51:C),the (Ri-satisfactorinessofV u VQ. I.e.that no two S,T of V or VQ aredisjunct.

S,T areboth in V: This coincideswith (51:D:a).S,T areboth in VQ: Both have ^ n 1elements,hencethey cannot

be disjunct.8Of S,Toneis in V and the otherin W We may assumeby symmetry,

S as the former and T as thelatter. Soan S of V must not bedisjunctwith

/ or any (n l)-elementset. Thisis precisely(51:D:b).1Note that the domain !>-/-()(cf.footnote 4 on p.446)is finite.1For this cf.alsofootnote 4 on p.446.1We areusing that 2(n - 1)> n i.e.n > 2i.e.n ^ 3. This should have beenstated

explicitly at the beginning but it is a natural assumption, sincesimple games (i.e.setswith (49:W*)) exist only for n 3. (Cf.49:4,49:5.))))

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448 SIMPLEGAMES

(51:D)solvesthe questionof enumerating all W: Startingwith any V

which fulfills (51:D:a),(51.-Bib)1 we may increaseit gradually as long asthis can be donewithout violating (51:D:a),(51:D:b).When this processcannot be continued any further, than we have a V which is maximalamong the subsetsof the W (with (49:W*)) i.e.we then have sucha W.

In performing this gradual building up processin all possibleways,weobtain all W in question.

Thereadermay try this for n = 3 or n = 4. It will appear that theprocedureis quite cumbersomeeven for small n, although it is rigorousandexhaustive for all n.

51.3.Reasonsfor Passingfrom W to W. Difficulties of Using W~

51.3.1.Let us considerthe setsWm of 49.6.We wish to characterizetheseW m directlyand to find somesimpleprocess

to construct them all. In what follows, we will derive two different waysof characterization,both beingof the saturation type. Thefirst will bebymeansof an asymmetricrelation,while the secondwill be by a symmetricone. Thus it is the secondonewhich is suitedfor constructionpurposes,inanalogy with the constructionof W in 51.2.

We give neverthelessboth characterizationsbecausethe equivalenceisquite instructive:The first one is in some(technical)respectssimilarto thedefinition of a solution (cf. 30.3.3.and 30.3.7.),and therefore the transitionto the equivalent secondform is of interestsinceit points a way to solveproblemsof this type. We have mentionedbefore(in 30.3.7.)how desirablethe correspondingtransition for our conceptof solutionwould be.

61.3.2.Let W be a systemwhich containsall supersetsof its elements:e.g.fulfilling (49:W *:b). Then the systemof its minimal elementsW m deter-mines W: Indeed,it is clearthat W is the system of the supersetsof allelementsof W m.

Henceif a system V is given, and we arelookingfor a W with (49:W*)such that V = JFm, then this W must necessarilybe the system V' of thesupersetsof all elementsof V.

ConsequentlyV = Wm for a W with (49:W*) if and only if thesetworequirementsaremet by W = V. 2 We arenow goingto transform thischaracterizationof the V = W m into one of the saturation type.

Denote the assertion that neitherS n T = nor S D T, by S&iT.Then we have:(51:E) V = W for a W with (49:W*) if and only if V is (Hrsaturated

and containsneither nor any one-elementset.Proof:Accordingto the above,we must only investigatewhether W = V

has the desiredproperties:1In principle we may start with the empty set. Thereaderwill note that the exclu-

sion of from V (cf.above)doesnot affect the possibility of V - .1I.e.W - V is the only system which can possibly meet theserequirements, but even

it may fail.)))

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V = Wm :Let S be a minimal elementof this W. Then 82.T for someT of V. HenceT7 is in W, and so the minimality of SexcludesS D T. SoS = T i.e.Sbelongsto V.

Thus only the converseproperty must be discussed:Whether everySof V is really minimal in W. Any S ot V clearlybelongsto TF. Sotheminimality means that S D 7\", T'of W is impossible;i.e.that S D T 2 T,T7 of F is impossible. This impliesthe impossibilityof SD T, T of 7 andis impliedby it (put T7' = T). Sowe have this condition:

(51:2) Never S D T for S,T in F.TF fulfills (49:W*):We must consider(49:W*:a),(49:W*:b),(49:W*:c)

separately. We do this in a different order.Ad (49:W*:b):ClearlyW = F containsall supersetsof its elementsso

this is automatically fulfilled.Ad (49:W*:c):Take (49:W*:a)for granted. (Cf. below.) Then

(49:W*:c)may bestated thus:If S isI or a (n l)-elementset,then S isnot in TF. I.e.neither nor any one-elementsetis in TF; that is, no subsetof theseis in F. Sowe have this condition:

(51:3) Neither nor any one-elementsetis in F.Ad (49:W*:a):We considerthis in two parts:S', S'cannot both belongto W: I.e.if S,T belongto F, then we can-

not have S&S',T S'. Now the existenceof such an S' impliesS n T = and is impliedby it (put S'= S). Sowe have this condition:

(51:4) Never S n T = for S,T in F.Oneof S, S must belongto TF:Assume that neither of S, S belongs

to W. This meansthat no T of F has T cS or T c -S,the lattermeaningS n T = . I.e.no T of F has T = S or S D T or S n T = . Or again:S is not in F and no T of F fulfills the negation of S(R2r.1

I.e.S is not in F,but S(R8!Tfor all T of F.Now we have to expressthat this is impossible:i.e.:

(51:5) If S(R2r for all T of F,then S belongsto F.Thus (51:2)-(51:5)arethe criteriawe want.Now (51:2)and (51:4)can be stated togetherlikethis:S&zTfor all S,

TofF. I.e.:(51:6) S(R2T for all T of F,if S belongsto F.

(51:5)and (51:6)togetherexpresspreciselythe (Rj-saturation of F.Hencethis and (51:3)form the criterion and this is preciselywhat wewanted to prove.

(51:E) is of someinterestbecauseit is a perfect analogue of (51:B).Thus thesecharacterizationsof W and TFm differ only in thereplacementof

1This is indeedSz>TorSnT-Q.)))

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450 SIMPLEGAMES

S<RiT:not S n T = Qby

S(R2r:neither S n T = nor S D T.But as this replacesthe symmetric(Ri by the asymmetric(Ra, (51:E)cannotbe used in the way in which we used (51:B)or rather the underlying(51:A).

61.4.ChangedApproach : Enumeration by Meansof Wm

51.4.1.We now turn to the secondprocedure.This consistsin analyz-ing the following question:Given a system F,what does it mean for a W

fulfilling (49:W*) that VsWm ?Themeaning of the V W m is this:Every S of V is a minimal element

of W. I.e.such an S must belongto W but its proper subsetsmust notbelongto W. As W fulfills (49:W*:b),i.e.containsthe supersetsof all itselements,it sufficesto statethis for the maximal propersubsetsof S only;i.e.for the S (i),i in S. As W fulfills (49:W*:a)we may say instead ofS- (i) not belongingto W, that -(S- (0) = (-AS)u (i) is in W. Sowe see:(51:F) V c W m (W with (49:W*))means preciselythis:Forevery

S of V, S belongsto W ; and for every i of this S,(-S)u (i)belongsto W.

We now prove:

(s51:G) V is a subset of the W m of a W fulfilling (49:W*) if andonly if it possessestheseproperties:

(51:G:a) No two S,T of V aredisjunct.(51:G:b) No two S,T of V have SD T.(51:G:c) For S, T of F, S u T = / implies that S n 71 is a one-

elementset.(51:G:d) Neither nor any one-elementset, nor 7 must belong

to V.

Proof:Let Vi be the setof all (-5)u (i),Sin V, i in S. Then V c JF\" 1

means by (51:F),that 7u 7i W. This is possiblefor someW with

(49:W*)accordingto (51:D),if V u Vi fulfills (51:D:a),(51:D:b).Let us therefore formulate (51:D:a),(51:D:b)for V u Vi.Ad (51:D:a):S,T7 areboth in V: Thiscoincideswith (51:G:a).S,Tareboth in Vii I.e.S = (-S')u(i),r = (-T')u(j),8't 7\" in V, tin

fl',j'inl\".ThedisjunctnessofS,T meansthese:-S',-T'disjunct,i.e.S'u T'= 7;

(i), (j) disjunct, i.e.i 7* j\\ S',(j) disjunct,i.e.j in S';-T7', (i)disjunct,i.e.i in T'.

Summingup:S'u T' = 7;i,j two different elementsof both S'and T'i.e.of S'n I\".)))

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ENUMERATIONOF ALL SIMPLEGAMES 451Now we must state that this is impossible. I.e.if S'u T' = 7, then

S'n Tr cannot possesstwo different elements.As S'n T'cannot be emptyby (51:G:a),this meansthat it must be a one-elementset.

Thusprecisely(51:G:c)obtains(S',T'in placeof its S,T).Of S,T one is in V and the other in Fr.We may assumeby symmetry,

that S is the former and T the latter. So T = (-T1) u (j), T' in V, j

in T'. The disjunctnessS, ( T') u (j) means:S, T1 aredisjunct, i.e.S T'\\ Sy (j) aredisjunct,i.e.j not in S.

Summing up:SST',j an elementof T'not in S.Now we must state that this is impossible. I.e.not S cT'. Thus

precisely(51:G:b)obtains. (T7',S in placeof its S,T.)Ad (51:D:b):Neither nor any one-elementsetmust belongto V nor

to V\\. The lattermeansthat neither must be a ( S) u (i),S in F,i in S.Only a one-elementsetcouldbe such a ( S) u (i) and this would mean:-S= 0,i.e.S = /.

Summing up:Neither nor any one-elementset,nor I must belongto V. This coincideswith (51:G:d).

Thus we have obtained preciselythe conditions(51:G:a)-(51:G:d)asdesired.

(51:G) solves the problemof enumerating all W m in perfect analogyto the solutionby (51:D)of the correspondingproblemfor the W: Startingwith any V which fulfills (51:G:a)-(51:G:d)1 we increaseit gradually aslong as this can be done without violating (51:G:a)-(51:G:d).When this

processcannot becontinuedany further, we have a V which is maximalamong the subsetsof the Wm of a W with (49:W*) i.e.we have sucha W m.

In performing sucha gradualprocessof buildingup in all possibleways,we obtain all W m in question.

51.4.2.Our last remarks show that the practicalenumeration of allsimplegamescan be basedon (51:G) and we will, indeed,undertake it in52. But someotherconsiderationsarebettercarriedout at first.

We now proposeto analyze the assertionthat (51:G)isa conditionof thesaturation type a little more closely.

Observefirst, that as (51:G:b)refers to two arbitrary S,T of F,wecan interchangethesein it. I.e.we can replace(51:G:b)by this:(51:G:b*) No two S,T of F have S D T or S cT.

Denotethe assertionthat S, T fulfill (51:G:a),(51:G:b*),(51:G:c)i.e.that neither S n T = nor S D T7

, nor ScT, nor S u T = / withoutS n T beinga one-elementset by S(RjT.

Then (51:G)simply statesthat F is (Rs-satisfactory, togetherwith

(51:G:d).Now let the domain D be the system 7 of thosesubsets of /which fulfill (51:G:d)i.e.neither0,nor a one-elementset,nor I. Thenthe last remarksof 51.4.1.show that the W arethe maximal (Rs-satisfactorysubsetsof 7.

1In principle we may start with the empty set.)))

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452 SIMPLEGAMES

S<R8T is clearly symmetric.1 Hencewe may apply (30:G) of 30.3.5.This gives:(51:H) V = W m for a W with (49:W*) if and only if V is (R 3-saturated

(in 7).Comparing(51:E)with (51:H)showsthat we have succeededin passing

from theasymmetric(Ra to the symmetric (Rs fulfilling the promisemadein

footnote 1p.271.51.4.3.It is quite instructive to compare(R* (in 51.3.2.)with our (R 8

*

S(R2T:neitherS n T = 0,nor S a T.S(R3T:neitherS n T = , nor S => T, nor S cT,

nor Su T = 7 without S n T7 beinga one-elementset.

Meresymmetrization of (R 2 (cf.30.3.2.)would give the threefirst partsof (Rs, but not the last one. This last part is the essentialachievement of(51:G)and (51:H) and not connectedin any obvious way with the threeothers.

One can infer from this how reconditethe operationsmust be by whichthe program of 30.3.7.might becarriedout if this proves to be feasibleat all

61.5.Simplicity and Decomposition

61.5.1.Let us considerthe connectionsbetweenthe conceptof a simplegameand that of decomposition.

Assume,therefore, that F is a decomposablegame with the constituentsA, H (J, K complementsin /). Then we must answer this question:What doesit mean for A, H that F is simple?

We beginby determiningthe setsW, L. Sincewe must considerthemfor all three gamesF, A, H, it is necessaryto indicatethis dependence.We write therefore Wr, L?\\ W^ LA; Wu y Z>H.

It shouldbe added that we assumeneitheressentialitynor any normal-ization for the gamesF, A, H. It is convenient, however, to assumethemall in a zero-sumform.2

(51:1) S = R u T (R J, T fi K) belongsto W? [Lr] if and only ifR belongsto TTA [LA] and T to WH [LH].

Proof:Replacementof Sby its complement(in /),/ S,3 replacesR,T by their respectivecomplements(in /, K). This transformation inter-

1And StovS holds in 7:Sn S = occursonly for S - , S=> S never, S U S - /only for S / henceneither of thesehappen for an Sof7.

2Thereaderwho recallsthe discussionsof46.10.may want to know at this point howthe question of the excesses(in r, A, H the e , , ^ loc.cit.)is to behandled. This ques-tion will be clarified in the discussion of 51.6.

8 It is preferableto write the complement in this way, instead of the usual -S,-ft,T, sincewe arecomplementing in different sets.)))

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ENUMERATION OF ALL SIMPLEGAMES 453

changesWr, W y Wn with Z/r, I/A, Z/H. Henceour statementconcerningthe W impliesthat oneconcerningthe L and viceversa. We aregoing toprove the latter.

That S belongsto Z/r is expressedby

(51:7) v(S) = v((0)t in 8

sinceA, H arethe constituentsof T, we have v(S) = v(ft) + v(T). Hencewe can write (51:7)thus:

(51:8) v(fl) + v(T) = S v<) + 2 v 0).t in ft t in 7

That ft belongsto LA and T to LH is expressedby

(51:9) vGR) = S v((0),tin ft

(51:10) v(T) - v((t)).iinT

Theassertionwhich we must prove, then, is the equivalenceof (51:7)and(51:9),(51:10).

Clearly (51:9),(51:10)imply (51:7);the reverseimplication can bedrawn sincealways

v(fl) ^ X y (W)>i in ft

v(T) ^ J) V (W)in 71

(cf.(31:2)in 31.1.4.).51.5.2.We arenow able to prove:

(51:J) F is simpleif and only if this is true:Of the two constituentsA, H one is simple,and the otheris inessential.

Proof:Thecondition is necessary:Simplicityof F meansthis:(51:11) Forany S / one and only one of thesetwo statementsis

true:(51:ll:a) S is in W r.(51:ll:b) S is in Lr.Put S = flu T (Rz J, TsK)and apply (51:1)to (51:11).Then thisresults:

(51:12) Forany two R J, TfiK one and only one of thesetwostatementsis true:

(51:12:a) R is in W* and T is in W tt .(51:12:b) R is in LA and T is in LH .)))

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454 SIMPLEGAMES

Now put R = , T = K. Then R belongsto LA and'77 belongsto WVHencefor (51:12:a)W& and LA have a commonelement:R, and for (51:12:b)WH and LH have a common element:T. By (49:E) in 49.3.3.(appliedtoA, H instead of its r) the former implies that A is inessential,and thelatter,that H is inessential.

Sowe see:(51:13) If F is simple,then eitherA or H is inessential.

Thecondition is sufficient: We assume,by symmetry, that H is inessen-tial. Then (49:E)in 49.3.3.(appliedto H in placeof its F) showsthat everyT K belongsto both WH and Z/H. Hencewe can now reformulate thecharacterization (51:12)of the simplicityof F.(51:14) Forany R J one and only one of thesetwo statementsis

true:(51:14:a) R is in W*.

(51:14:b) R is in LA.This is preciselythe statementof the simplicityof A. Sowe see:

(51:15) If H [A] is inessential,then the simplicityof F is equivalentto that one of A [H].

(51:13),(51:15)togethercompletethe proof.

61.6.Inessentiality, Simplicity and Composition. Treatment of the Excess51.6.It is worth while to compare(51:J)with (46:A:c)of 46.1.1.

Therewe found that a decomposablegame is inessential if and only ifits two constituentsare i.e.inessentialityis hereditaryunder composition.This is not true for simplicity,which as we know, is the simplestform ofessentiality:By (51:J) a decomposablegameis not simpleif its two con-stituents are. (51:J)shows that a simplegameA remains simpleundercompositionif and only if it is combinedwith an inessentialgameH i.e.with a setof \" dummies\" (cf. footnote 1 on p.340).

In this connectionfour further remarksareappropriate:First:If the simplegame F obtains as describedabove from the con-

stituent (simple)gameA by an additionof \"dummies\"(i.e.of the inessen-tial game H),then the solutionsof F aredirectlyobtainablefrom thoseof A.

Indeed,this is describedin detail in 46.9.l

Second:We stated at the beginningof 49.7.that we use the old form ofthe theory for simplegames.It is therefore worth noting that the type ofcompositionto which we were led (cf.the above remark) is preciselytheone for which the old form of the theory is hereditary. (Cf. the end of46.9.or (46:M)in the first remark in 46.10.4.)

1Thisis, of course,what common senseleadsone to expectanyhow. Thesurprisingturns of the theory of decomposition cf. in particular the resum6 in 46.11.show,however, that it is unsafe to losesight of the exactresults. In this case46,9.providesthefirm ground.)))

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ENUMERATIONOF ALL SIMPLEGAMES 455

Third:In this connectionit becomescleareralsowhy we had to refrainfrom consideringotherexcessesthan zero i.e.the new form of the theoryin the senseof 44.7. for the theory of simplegames.

Indeed:If we had beenable to carry this out successfully,then theresults of 46.6.and 46.8.would enableus to deal with all compositionsofsimplegames.Now we have seenthat a compositionof simplegamesis nota simplegame. In otherwords: A theory of simplegameswith generalexcesswould indirectlyembracenon-simplegamesas well. It is thereforenot surprisingthat we couldnot proceedin generality.1

Fourth:In the light of the analysisof 46.10.the above remarksconcern-ing the excessassumethe following significance:They show that the con-ceptof simplicitydoesnot stand the generaloperation of imbedding.2This showsthat the methodicalprincipleconsideredin 46.10.5.cannot beappliedunder all conditions.

51.7.A Criterion of Decomposability in Termsof W m

51.7.1.In 51.5.we discussedwhen a decomposablegame F is simple.We now tacklethe converseproblem:To decidewhen a simplegame isdecomposable.

Let a simplegameF be given. Itwill appearthat the followingconceptis of importance:An i of / is significant if and only if it belongsto someSof Wm .* Denotethe setof all significant elementsof / i.e.the sum of allS in W by 7 .

We now proceedin severalsuccessivesteps:(51:K) If F is simpleand decomposable,and if thesimpleconstituent

is A (cf. (51:J)and the use of notations of 51.5.)then F and A

have the same W m.Proof:According to (51:1)the S = R u T (RzJ, TsK) of W r

obtain by taking any R of W& and any T of WH- H is inessential(by(51:J)),hencethe T of WH aresimplyall T K (cf.the proof of (51:J)).Consequentlythis S = R u T is minimal i.e.it belongs to Wf if itsR, T areminimal. This meansthat R belongsto TFj and that T = 0,i.e.S = R.

Thus W and TFj coincide,i.e.F and A have the same W m.(51:L) With the sameassumptionsas in (51:K),necessarilyJ 27 .

Proof:F and A have the sameW m (by (51:K)),hencethe samesignificantelements therefore thoseof F, which form the set7 , areall among the par-ticipants of A, which form the setJ.

1In a certain sensethis may beviewed as an application of the methodical principlereferred to in footnote 3 on p.270.

1Unlessit is merely an addition of \"dummies\" asdiscussedabove.1Thus a player i is significant if there existsa minimal winning coalition to which he

belongs;i.e.if there existsa conceivableessentialservicehe may render.It will beseenthat the oppositeof this is a \"dummy

\" (cf.the end of 51.7.3.).All this refers,of course,to simple games.)))

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456 SIMPLEGAMES

(51:M) Assume only that F is simple.Then 7o is a splittingset,1the Jo-constituentA beingsimple,and the (/ 7 )-constituentH inessential(cf.(51:J)).

Proof:Consideran S = R u T, R 7 , T I - 7 . Then:

(51:16) Sis in W if and only if R is in W.

Indeed:If R is in W } then SZL R is too. Conversely:Let S be in W.

Then a minimal TmW with T Sexists. SoT is in TTm , every i of T isin 7 . HenceT s7 . Thus T7 S n 7 = -R, and therefore fl is in W

along with T.(51:17) TisinL.

Indeed:ReplaceS by T(sI - 7 ); this replacesour 72, T by Q, T.As is in L,(51:16)permitsto infer the samefor T.

We now prove:

(51:18) v(S)- v(ft)+v(r).Considerthe S of L and of TV separately:

Sis in L:/2, T c5arealso in L. Hence

v(S) = ^ v((t)) = X v((0) + S v((t)) = v() + v(D,t in S t in A t in T

i.e.(51:18).S in W: By (51:16),(51:17)fl is in IFand T in L. Hence))

t not in S

v() = - 2; v((o>- - 2) v(^) - 2)t not in ft t not in S i in T

v(T) = 2) v((0),iin T

and so

v(S) = v(R) + v(T).i.e.(51:18)

(51:18)is preciselythe statementthat /o is a splitting set. ForallTzI-h (51:17)gives

v(D - 2) v((t)),in T

hencethe 7 7o constituentH is inessential.Consequentlythe 7o-con-stituent A must besimpleby (51:J).

Thus the proof is completed.'Inthe senseof 43.1.)))

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SIMPLEGAMES FOR SMALLn 457

51.7.2.We are now able to describethe decomposibilityof a simplegame F completely i.e.we can name its decompositionpartition Up in thesenseof 43.3.(51:N) With the sameassumptionsas in (51:M):Thedecomposition

partition n r consistsof the set7o and of the one-elementsets(i) for all i in 7 7 .

Proof:All (t),i in / 7 , belongto n r :By (51:M)7 7 is a splittingsetof F with an inessentialconstituentH. Henceevery (i),i in 7 7o, is asplitting set of H (use,e.g.(43:J)in 43.4.1.)and so of F (use (43:D) in43.3.1.).Beinga one-elementset, (i) is necessarilyminimal. Henceitbelongsto IIr.

7o belongsto lip:7 is a splittingsetby (51:M). If J is a splittingsetT , then (51:L)appliesto J or to 7 J,henceeitherJ 27 or 7 J 27o,7 n J = . Both excludeJ c7 . Thus 7 is minimal. Henceit belongstollr.

No further J belongsto IIr :Any otherJ of IIr must be disjunct with

7 and with all (i),i in 7 7 , (use(43:F) in 43.3.2.).As the sum of thesesetsis 7, this would necessitateJ but is not an elementof n r(cf. the beginningof 43.3.2).

Thus the proof is completed.51.7.3.Combinationof (43:K)in 43.4.1.with (51:N)gives:1

(51:0) A simplegame F is indecomposableif and only if 7 = 7,i.e.if and only if all its participantsaresignificant.

We concludeby proving:(51:P) A simplegameF possessespreciselyone/-constituentwhich

is simpleand indecomposable:That with J = 7 .Proof:The7 -constituentcan be formed and is simpleby (51:M).Now considera simple/-constituent.Thenit has,by (51:K),the same

W m and the samesignificant elementsas F itself, hencethe latterformthe set 7 . So the indecomposabilityof the /-constituentis by (51:0)equivalent to / = 7 .

We callthe 7 -constituentA of F its kernel. All otherparticipantsi.e.those of 7 7 are \"dummies.\"(Cf. (51:M) or (51:N), and thelast part of 43.4.2.).Henceall that matters in the gameF takesplaceinits kernelA ; to seethis, it sufficesto apply the first remark in 51.6.

52.theSimpleGamesfor Smalln52.1.Program:n 1,2 Play No Role. Disposalof n = 3

52.1.Our nextobjectiveis the enumerationof all simplegamesfor thesmallervalues of n. We proposeto push this casuisticanalysisso far as isnecessaryto producethe examplesreferred to in 50.2.(cf. footnote 2 on

*Ormore directly of (43:K)in 43.4.1.with (51:L),(51:M).)))

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468 SIMPLEGAMES

p.434),50.7.2.,(cf.footnotes 2,3,4 on p.443),50.8.2.(cf. footnotes 1,2, onp.445).

Sinceevery simplegame is essential,we needonly considergameswith

n 3.Forn = 3 the situation is this:The(unique)essentialthree-persongame

is simpleand it has the symbol[1,1,1]*.1

Sowe can assumefrom now on that n ^ 4.

52.2.Proceduref or n ^ 4:<TheTwo-elementSetsand TheirRolein Classifying the Wm

52.2.1.Let an n ^ 4 be given. We wish to enumerateall simplegameswith this n. In orderto do this it is convenient to introducea principleoffurther classification of thesegameswhich is very effective for the smallervalues of n.

The enumeration in questionamounts to the enumeration of the setsW m for which we have various characterizationsavailable e.g.that one of(51:G)in 51.4.1.

Considerthe smallestsetswhich may belong to W m. Since(51:G:d)loc.cit.excludesthe empty set and the one-elementsetsfrom W m, thismeans consideringthe two-elementsets in W m. Thesesetspossessthefollowingproperty:(52:A) A two-elementsetbelongsto W m if and only if it belongsto

JT.2

Proof:The forward implication is obvious. Now assume conversely,that the two-elementsetS belongsto W. The propersubsets of S areempty or one-elementsets,hencenot in W. Therefore Sbelongsto W m.

We proposeto classifyaccordingto two-elementsetsin W m.52.2.2.ConceivablyWm may contain no two-elementsetsat all. We

denotethis possibilityby the symbolCo.Thenextalternative is that W m containspreciselyone two-elementset.

Bya permutation of the players1, , n we can makethis setto be (1,2).We denotethis possibilityby the symbolCi.

Further, W m may contain two or more two-elementsets.Considertwo of these. By (51:G:a)loc.cit.they must have a common element.By a permutation of the players 1, , n we can make the commonelementto be1,and the two otherelementsof thesetwo sets2 and3.

SeW m contains(1,2)and (1,3).We denotethe possibilitythat W m containsno further two-elementsets

by the symbolC*52.2.3.Now assume that W m doescontain further two-elementsets.

Assumefurthermore that not all of them contain 1.Considertherefore a two-elementset of W m not containing 1. By

(51:G:a)loc.cit.it must have common elementswith (1,2)and (1,3)1beingexcluded,thesemust be2 and 3 sothesetmust be(2,3).

1Cf. (50:A)in 50.1.1.and the last remark of 50.2.2.*I.e.a non-minimal set in W must have at leastthree elements.)))

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Thus (1,2),(1,3),(2,3)belongto Wm. (To this extentwe have perfectsymmetry in 1,2,3.)

Now considerany othertwo-element setwhich may belongto W m. Itcannot contain all threeof 1,2,3;by a permutation of theseplayerswe canarrangeit so that the setin questionfails to contain 1. Now it must havecommon elementswith (1,2)and (1,3)1being excluded,thesemust be2 and 3 so the setmust be (2,3),but we assumedit to be different from(2,3)(among others).

Thus W m containsthe two-elementsets(1,2),(1,3),(2?3),and no others.We denotethis possibilityby the symbolC*.

52.2.4.The remaining alternative is that Wm contains other two-elementsetsbesides(1,2),(1,3),but that they all contain 1.

Bya permutation of the players4, , n we can maketheseplayerstobe 4, , k + 1with a k = 3, , n - 1.

Thus W m containsthe two-elementsets(1,2),(1,3),(1,4), (1,*+ 1),and no others. We denotethis possibilityby the symbolC*.

52.2.5.It is convenient to bracketthe casesC , Ci, C*of 52.2.2.and theCk, k = 3, , n 1of 52.2.4.together:We then have the cases

Ck, k = 0,1, , n - 1.InthecaseCjknowTFm contaijasthetwo-elementsets(l,2),,(l,fc+ l),

and no others. By an additionalpermutation of the players1, , n 1

we can replacethesesetsby (l,n), , (k,n).Itis in this form that wearegoingto usethe caseC*,k =0,1 , n 1.

Now Ck containsthe two-element sets(l,n), , (fc,n), and no others.BesidestheseC* the only alternative is C* of 52.2.3.which we will not

transform.

52.3.Decomposability of the CasesC*, C*_i, C*_i

52.3.1.Of all thesealternatives three can be disposedof immediately:C*,Cn-2,<?n-i. We discussthesein a different order.

Ad C*:Consideran S 7. If S contains two or more of 1,2,3say (atleast) 1,2,then Sa(1,2).(1,2)belongsto W, henceS does too. If Scontains one or fewer of 1,2,3,say (at most) 1,then Sfi (2,3).(2,3)belongsto W, -(2,3)to L, henceS to L too. Sowesee:W consistspreciselyof thoseS which contain two or more of 1,2,3.HenceW m consistspreciselyof the sets(1,2),(1,3),(2,3).2 So(1,2,3)is the 7 of 51.7.for this game.

In otherwords:Thekernelof the game under considerationis a three-persongame with the participants 1,2,3,its W m consistingagain of (1,2),(1,3),(2,3).As mentionedbefore for the last time in 52.1.this game hasthe symbol [1,1,1]*.The remaining n 3 players, 4, , n are\" dummies.\"

*Namely (!'?'}'\"'*i)'cf.28.1.1.\\n, 1,2, , n I/1Thesewere the two-element setsof Wm by definition but wehave now shown that

they exhaust Wmcompletely.)))

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460 SIMPLEGAMES

Sowe see:CaseC* is representedby preciselyone game:Thethree-persongame

[1,1,1]*,with the necessarynumber (n 3) of \"dummies.\"52.3.2.CaseCn-i:Consideran Ss/.Assume first, that n belongs

to S. If Shas no further elements,then it is the one-elementset(n),andso in L. If Shas further elements,say t = 1, , n 1,then Sa (t, n) .Now this (i,n) belongsto W, henceS doestoo. In otherwords:if n is in S,then Sbelongsto W, exceptwhen S = (n). Applying this to S gives:If n is not in /S, then Sbelongsto W, when S doesnot, i.e.if and only if-S= (n),i.e.S = (1, , n - 1).

HenceW consistspreciselyof theseS:All setscontaining n, exceptthesmallest one (n); no set not containing n, exceptthe largest one,(1, , n 1). One verifies easily that this IF indeed fulfills therequirements(49:W*). Also that this game can bedescribedas a weightedmajority game, all players 1, , n 1 having a common weight,while player n has the n 2 fold weight. I.e.this gamehas the symbol[1, , 1,n - 2].

TFm obtains immediately from IT. It consists preciselyof theseS:(1,n), , (n 1,n) and (1, , n I).1 It is now easy to verifythat this gameis homogeneousand normalized by a = 1. I.e.that as = 1(cf.50.2.)for all theSof this Wm. Hencewe can write [1, , 1,n 2]h.

Sowe see:CaseCw-i is representedby preciselyone game:The n-person game

[1,- - , 1,n -2],.62.3.3.Ad Cn-2:Consideran S /. Assume first, that n belongsto S.

IfShas no further elementsotherthan possiblyn 1,then S (n 1,n).Now (n 1,n) is not in JPm, hencenot in W (by (52:A) in 52.2.1.)-SoSis inL along with (n 1,n). UShas further elements,otherthan n 1,say i = l, , n 2, then Ss (t, n). Now this (i, n) belongsto WhenceSdoestoo. Sowe see:If n is in S,then Sbelongsto W, exceptwhenS = (n) or (n 1,n). Applying this to S gives:If n is not in S thenSbelongsto W when Sdoesnot, i.e.if and only if S = (n) or (n 1,n),i.e.S = (1, , n - 1)or (1, - , n -2).

HenceW consistspreciselyof thesesetsS:All setscontaining n, except(n),and (n 1,n); no setnot containing n, except(1, , n 1)and(1, , n 2). One verifies easily that this indeedfulfills the require-ment (49:W*).

Wm obtains immediately from W. It consists preciselyof theseS:(1,n), f (n - 2,n), and (1, , n -2). So(1, , n - 2,n)is the /o of 51.7.for this game.

1Thus the two-element setsin Wm are (1,n), , (n 1,n), as it should be bydefinition. Thenew fact is that the only further element of W m is (1, , n 1).

Note that this last set is not a two-element setonly becauseof n ^ 4.1Thus the two-element setsin Wm are (1,n), , (n 2, n) f as it should be by

definition. Thenew fact is that the only further element of Wm is (1, , n 2).For n 4 this last set is alsoa two-element set, thereby falsifying the classof the

game. (ItbecomesC*instead of C_2, i.e.Ct.)Hencethis class(C_j)is void, unlessn 5.)))

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SIMPLEGAMESFOR SMALLn 461In other words: The kernel of the game under considerationis an

(n l)-persongame with the participants 1, , n 2, n, its Wm

consistingagain of (1,n), , (n 2, n), (1, , n 2). Thusthis is the caseCn-2for n 1players the analogue of the caseCn_i for nplayers(replacingn by n 1!)discussedabove. Henceit has the symbol[1, , 1,n 3]*. The remaining playern 1is a \"dummy.\"

Sowe see:CaseCn-2 is representedby preciselyone game:1 The (n l)-person

game[1, , 1,n 3]* with one dummy.

62.4.The Simple GamesOther than [1, , 1,I - 2]* (with Dummies) :TheCasesCk, k -0, 1, , n - 3

52.4.The results of 52.3.deserve to be consideredsomewhat furtherand to be reformulated. We saw that for every I ^ 4 the homogeneousweightedmajority game of I players[1, , 1,I 2]*can be formed.2We can even form it for I = 3:Then it is the directmajority game of threeparticipants[1,1,1]*.Sowe will use it for all / ^ 3.

If n ^ 4 then we can obtain a simplen-persongame by forming this[1, , 1,/ 2]*for any / = 3, , n and adding to it the necessarynumber of \" dummies/'

Theresult of 52.3.was this:This game with I = 3,n, and (for n ^ 5)n 1exhauststhe casesC*9 Cn-i,Cn-2.

Theodd thing about this result is that thesevalues of I do not exhaustthe full set of its possibilitiesI = 3, , n (cf. above). That is tosay, they do this for n = 4, 5, but not for n ^ 6. There remain the/ = 4, , n 2 for n ^ 6. What is their significance?

Theansweris this:Considerthe game [1, , 1,I 2]* (I players)with n I \"dummies.\"Assume only I = 3, , n and n ^ 4. TheW m consistsof (1,1), , (I - 1,I) and (1, , 1- I).8 Hencewehave caseC*when I = 3 and caseCz_iwhen I = 4, , n.4

Thuswehave in thesegamesspecimensfrom the casesC*,Cs, , Cn-i.Theresultof 52.3.can now be formulated likethis:The casesC*,C_j,Cn-iareexhaustedby the pertinentonesamong thesegames.6

We restatethis conclusion:

(52:B) We wish to enumerateall simple n-person games n ^ 4.The game [1, , 1,I 2]* (I players) with the necessarynumber (n I) of \"dummies\" is a simplen-persongame forall Z = 3, 4, , n. Itscaseis C*, C3, , C*-i,respec-

1 For n 6 it is void for n 4. Cf.footnote 2 on p.460.1Cf.Case _!above,with I in placeof n.3 We take players 1,- , I as the participants of the kernel [1, , 1,1 2]*and

players/ -f 1, , n as \"dummies.\" This differs from the arrangement in caseC.iof52.3.where / n 1 and player n 1was \"dummy\" by an interchange of playersn 1 and n.

4 For / - 3, C*replacesCisince(1, ,/- 1)is in this cosea two-element set.1HenceCiis void for n ^ 4,sinceit occurson the secondlist, but not on the first one.

Cf, 52.3.)))

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482 SIMPLEGAMES

lively. All other simple n-person games(if any) are in thecasesCo,Ci, , Cn-s.1

52.5.Disposalof n = 4,5

52.6.1.We will discussthe values n = 4, 5 fully and somecharacteristicinstancesin n = 6,7.

n = 4 is easily settled. By (52:B) above, we needonly investigateCo,Ci for this n. In thesecasesW m containsg 1two-elementsets.How-ever this is impossible:Sincethe complementof a two-elementsetis a two-elementset,theremust be the samenumber of two-elementsetsin W and inL. I.e.half of the total number,which is 6. SoW contains3 two-elementsetsand the sameis true for W m.2

Thus the only simplegamesfor n = 4 arethoseof (52:B).We statethis as follows:

(52:C) Disregardinggames which obtain by adding dummies tosimplegamesof < four persons,8 thereexistspreciselyone simplefour-persongame:[1,1,1,2]*.

62.6.2.Consider next n = 5. By (52:B)above we must investigateI = 0,1,2.In contrast to the n = 4 case,all of theserepresentconcretepossibilities.

Co:No two-elementset is in W m i.e.in W. So they areall in L andtheir complements,the three-elementsets,areall in W. Thus W consistsof all setsof ^ threeelements,and W m of all setsof threeelements.Hencethis is the directmajority game [1,1,1,1,1]A.

Ci:(1,2)is the only two-elementset in W m i.e.in W. Passingto thecomplements:(3,4,5)is the only three-elementsetin L i.e.the others arein W. Thus W consistspreciselyof thesesets:(1,2),all three-elementsetsbut (3,4,5),all four- and five-element sets.It is easy to verify that thisfulfills (49:W*) and also that its Wm consistsof the followingsets:

(1,2),(a,6,c),where a = 1,2,and 6, c = any two of 3,4,5.Now one shows without difficulty, that this game has the symbol

[2,2,1,1,1]*.d:(1,2),(1,3)arethe only two-elementsetsin W m, i.e.in W Passingto the complements:(3,4,5),(2,4,5)arethe only three-elementsetsin!/ i.e.the othersarein W. Thus W consistspreciselyof thesesets:(1,2),(1,3),all three-elementsetsbut (2,4,5),(3,4,5),all four- and five-element sets.It iseasyto verify that this W fulfills (49:W*) and also that its W m consistsof the followingsets:

(1,2),(1,3),(2,3,4),(2,3,5),(1,4,5).Now one showswithout difficulty that this gamehas the symbol[3,2,2,1,1]*.

1All those caseswhich we succeededin exhausting so far were void or containedpreciselyonegame. This is, however, not generally true. Of.the first remark in 53.2.1.

1Owing to (52:A) in 52.2.1.This will beusedin what follows continuously withoutfurther reference*.'I.e.to the unique simple three-persongame [1,1,1]*.)))

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Hencethe simplegamesfor n = 5 arethesethree,and thoseof (52:B).We stateas follows:

(52:D) Disregardinggameswhich obtainby addingdummiesto sim-plegamesof < five persons,1thereexistpreciselyfour simplefive-persongames:[1,1,1,1,1]*,[1,1,1,2,2]*,'[1,1,2,2,3]*,'[1,1,1,1,3]*.53.TheNew Possibilitiesof SimpleGamesfor n ^ 6

53.1.TheRegularities Observedfor n < 6

53.1.Beforewe go further, letus draw someconclusionsfrom the abovelists.

First:All simple games which we have obtained so far, possessedasymbol, [wi, , Wn]h, i.e.they were homogeneousweightedmajoritygames. Having verified this for n = 4, 5, the questionariseswhether it isalways true. As stated in footnote 3 on p.443, this is not so;the first

counter-examplecomesfor n = 6.Second:Sofar every classC* which containedany gameat all, contained

only one. This too fails from n = 6 on. (Cf. the first remark in 53.2.1.)Third:Onemight think a priori, that thereis greatfreedom in choosing

the weightsfor a homogeneousweightedmajority game. Our lists show,however, that the possibilitiesarevery limited:Oneeachfor n = 3,4, andfour for n = 5.8 We emphasizethat sinceour lists areexhaustive,this isa rigorously establishedobjectivefact and not a more or less arbitrarypeculiarityof our procedure.

Fourth :We can verify the statement of footnote 1on p.446 that while

the number of elementsin W is determinedby n (it is 2n~l)> that one oiW m may vary for simplegamesof the same n. This phenomenon beginsfor n = 5.

Forn = 3: W has 4 elements,W m in the unique instancehas 3. Foin = 4:W has 8 elements,W m in the unique instance4. Forn = 5:Whas16elements,W m in the four instances10,7,5,5,respectively.

Fifth: We can verify the statement of footnote 2 on p.443, that the

equations(50:8)of 50.4.3.,50.6.2.(with U = W m) may be more numerousthan their variables,and neverthelesspossessa solution i.e.a system ol

imputations in the ordinary sense. The former means that Wm has > r

elements,the latteris certainly the casefor homogeneousweighted majorit}games((50:K) in 50.8.1.).

We saw above that for n = 3, 4 Wm necessarilyhas n elements,bul

for n = 5, it may have 10or 7 elementsas well. And all thesegamesanhomogeneousweightedmajority games.4

1 I.e.to [1,1,IK and to [1,1,1,2]*.*We permute the players of thesegames (belonging to Oiand C2) in order to have ai

increasing arrangement of weights.3 Disregarding permutations of the players!4 Thus we have the first counter-examples for n 5:[1,1,1,1,1]*(the direct majority

game) and [1,1,1,2,2]*.)))

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464 SIMPLEGAMES

Fora simplegame,where thesesolutionsdo not exist,cf.the fifth remarkin 53.2.5.

5S.2.TheSix Main Counter-examples(for n 6, 7)

53.2.1.We now passto n = 6,7. A completeexhaustionof thesecases,even of n = 6,would be rather voluminous. We forego it for this reason.We will only give somecharacteristicinstancesof simplegamesin n = 6, 7which illustrate certainphenomena which begin as mentionedbefore atthesen.

First:We mentionedin the secondremark of 53.1.that for n = 6, acaseCk may contain severalgames. Indeed,it is not difficult to verify thatthe two homogeneousweightedmajority games

[1,1,1,2,2,4],,[1,1,1,3,3,4],,(cf. footnote 2 on p.463)aredifferent from eachotherand belongboth toC,.

63.2.2.Second:We mentioned in the first remarkof 53.1.that for n = 6a simplegame existswhich is not a homogeneousweighted majority game,i.e.one which doesnot possessany symbol[u>i, , i0n]*. By (50:K) in

50.8.1.this is necessarilythe casewhen thereexistsno main simplesolution;i.e.no systemof imputations in the ordinary sense. (Cf.the fifth remarkin

53.1.)Sucha game existsindeed,and it is even possibleto differentiate further:

It is possibleto find onewhich is neverthelessa weightedmajority game(without homogeneity!),i.e.which possessesa symbol[wi, , w n], andit is alsopossibleto find one which doesnot even have that property.

We beginwith the first mentionedalternative.Put n = 6:Define W as the systemof all thosesets8 I = (1, , 6)

which eithercontain a majority of all players (i.e.have ^ 4 elements),orwhich contain exactlyhalf of all players(i.e.have 3 elements),but a major-ity of all the players1,2,3(i.e. 2 of these). In otherwords:Theplayers1,2,3form a privilegedgroupas againstthe players4,5,6 but their privilegeis ratherlimited:Normally the overall majority wins;only in caseof a tiedoesthe majority of the privilegedgroupdecide.

It is easy to verify that this W satisfies(49:W*). Thegameis clearlyaweightedmajority one:It suffices to give the membersof the privilegedgroup (1,2,3)someexcessweight over thoseof the others(4,5,6),whichmust beinsufficient to overridean overall majority. Any symbol

[w, w, w, 1,1,1]with 1<w < 3 will do.1

1w > 1is necessary,e.g.for 8- (1,2,4)to defeat-S- (3,5,6)(i.e.2u> + 1 > w + 2).w < 3 is necessary,e.g.for S - (3,4,5,6)to defeat-S- (1,2)(i.e.w + 3 > 2u>).)))

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NEW POSSIBILITIESOF SIMPLEGAMES 465

Wm is quicklydetermined;it consistsof thesesets:(Si): (1,2,3)))

(fli)))(S'/): (a,&,/0 wherea,b = any two of 1,2,3,

h = 4 or 5 or 6(Si\:") (o,4,5,6) where a = 1or 2 or 3.l))

Theequations (50:8)of 50.4.3.,50.6.2.(with U = W m) which deter-mine a main simplesolution in the senseof 50.8.1.are:

/ET/\\. /. I / 1 /- A\\&i). X\\ -f\" X} -j- X$ = 0,(Ei): xa + xb + xh = 6, where 0,6= any two of 1,2,3,

h = 4 or 5 or 6(E'if

) : xa + z4 + z5 + c = 6, where a = 1or 2 or 3

Theseequations(Ei)cannot besolved.2 Indeed,(EC)with a = 1,6= 2and /i = 4,5,6shows that x4 = x b

= x& ; (EC')with a = 1,2,3showsthatx\\ = 2 = x8 ; now (J) gives 3^i = 6, Xi = 2;hence(EC)gives 4 + x4 = 6,x4 = 2;and then (#\"')gives 2 + 6 = 6 a contradiction.

It shouldbe noted that the ordinary economicaspectof this occurrencewould bethis:(Si')(i.e.(EC))showsthat the servicesof players4,5;6 can besubstituted for each other hencethey are of the same value. (Si\(i.e.(E{\showsthe same for 1,2,3.Now comparisonof (Si) and (Si')showsthat one player of the group 1,2,3can be substituted for one playerof the group4,5,6 and comparisonof (S'/)and (Si\")showsthat one playerof the former groupcan be substitutedfor two playersof the latter. Henceno substitution ratebetweenthesetwo groupscanbe defined at all. Thenatural way out would be to declaresomeof the setsof W m enumeratedin

(Si) to be \"no profitable uses\"of the players' services. In the senseof50.4.3.this amounts to choosingU c W m (Cf.also50.7.1.and footnote 4 onp.443). Whether in this game a U cW m can have the requiredproperties(cf.50.7.1.)couldbe decidedby a simplebut somewhat lengthy combina-torial discussion,which has not yet been carriedout. The existenceofsuch a V is highly improbable,becauseit can beshown that it would have

mathematically unlikely characteristicsif it existed.This gameis alsovery peculiarin another respect:It is possibleto prove

that there existsno solution V which contains only a finite number ofimputations and which possessesthe full symmetry of the game itself;i.e.invariance under all permutationsof the players 1,2,3and under allpermutationsof the players4,5,6.We do not discussthis ratherlengthyproof at this place.1 Thus the type of solution which onewould term thenatural onedoesnot exist.

1Thus Wm has 1 + 9 + 3 - 13elements.1They are 13equations in 6variables, but this in itself is not necessarilyan obstacle,

as the fifth remark in 53.1.shows.8 Whether any finite solution V exists at all, is not known. We suspectthat even

this question will be anwered in the negative.)))

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466 SIMPLEGAMES

This is an indication of how extremelycareful one must be in termingextraordinarysolutions\"unnatural/'or in trying to excludethem.

63.2.3.Third:Let us now considerthe secondalternative referred toin the secondremarkabove:A simplegame for n = 6,which is no majoritygameat all i.e.which has no symbol [u>i, , w*]. This alternativeitself canbe subdividedfurther:It is possibleto find a game such that it

possessesa main simplesolution(cf.above) and it is also possibleto find

one that has no main simplesolution.Considerthe first case:Put n == 6. Define W as the systemof all thosesetsS(fi7 = (1, ,6))

which eithercontain a majority of all players(i.e.have ^ 4 elements),orwhich contain exactlyhalf (i.e.have 3 elements),but an even number of theplayers1,2,3(i.e.have or 2 of these). Comparingthis with the examplein the secondremark above, this observationmust be made:The players1,2,3still form a groupof specialsignificance,but it would be misleadingtocall their significance a privilege sincetheir absencefrom the tying (i.e.three-element)setS is just as advantageousas their strong representation(presenceof preciselytwo of them),and the presenceof all of them just asdisastrousas their weakrepresentation(presenceof preciselyoneof them).Theybringabouta decisionnot by their presencein Sbut by an arithmeticalrelation:1

It is easy to verify that this W fulfills (49:W*) in 49.6.2.2

Let us now determineW m. SinceW containsall ^ four-element sets,no ^ five-elementsetcan be in Wm. Considernow a four-element setin W.

If the number of players1,2,3in it is even, remove from it a player4 or 5or 6.1 If the number of players1,2,3in it is odd, remove from it a player1or 2 or 3.4 At any ratea three-elementsubset with an even number of

players1,2,3obtains i.e.one in W. Sono four-element setcan be in Wm.HenceW m consistsof the three-elementsetsin W. Theseare:

(Si): (4,5,6)(S\:") (<*>>b,h) where a,6= any two of 1,2,3;

h = 4 or 5 or 6.6

1Note alsothat the group 4,5,6has a similar significance:SinceSmust have threeelements (in order that thesecriteria becomeoperative), the statement that an evennumber of 1,2,3is in Sis equivalent to the statement that an odd number of 4,5,6is in S.

This lends further emphasis if any beneeded to our frequently made observationconcerning the great complexity of the possibleforms of socialorganization, and theextremewealth of attendant phenomena.1Note in particular that always oneof Sand Sbelongsto W: This is evident if oneof the two has ^ 4 elements (and so the other 2s 2). Otherwise both S and Shave3elements. Henceoneof them contains an even number of players 1,2,3and the otheran odd one.

1This is possible,as 1,2,3areonly 3 players.4This is possible,as 4,5,6are only 3 players.1Thus W has 1 + 9 -10elements.)))

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NEW POSSIBILITIESOF SIMPLEGAMES 467

If this game had a symbol [wi, , w n]9 then therewould be

Wi > Wi for all Sin W.i in 8 t not in 8

Apply this to the setsof Wm enumeratedin (Si). This gives in particular:Wi + Wi + W* > Wi + Wi + Wj,u>i + Wj + w > w* + Wi + MS,u>i + w + MS > Wi + wi + toe,Wj + W + Wi > Wi + W>5 + W*.

Adding theseinequalitiesgives:2(wi + w 2 + u>8 + W* + W* + WB) > 2(wi + ti?2 + W* + w 4 + Ws + ti> 6),

a contradiction.The equations (50:8)of 50.4.3.,50.6.2.(with U = Wm} which deter-

mine a main simplesolution on the other hand are:

EQ: x4 + x* + x< = 6,Ey): xa + xb + xh = 6, where a,6 = any two of 1,2,3;

h = 4 or 5 or 6.They areobviously solved by x\\ = = xe = 2.1

In the ordinary economicterminology one would have to say that thestructuraldifferencebetweenthe groupsof players1,2,3and 4,5,6cannot beexpressedby weights and majorities,and that as far as values areconcerned,thereis no difference.

63.2.4.Fourth:Note that the above exampleis alsosuited to establishthe difference betweenthe homogeneousweighted majority principleandthe existenceof a main simplesolution,as discussedin 50.8.2.Indeed,it is an instance of = in (50:21)loc.cit.:SinceZi = = x = 2 (cf.above),so

Xi = 12= 2n.t-i

53.2.5.Fifth: Now consider the second casedescribed in the thirdremark above:A simplegame for n = 6, for which neither a symbol

[Wl,' , Wn]

nor a main simplesolution exists.Comparedwith the two previous examplesgiven in the secondand

third remarkabove this oneis basedon lesstransparent principles.Thisis not surprisingsincenow all our simplifying criteriaareto be unfulfilled.

This is theexample:Put n = 6. Define W as the systemof all thosesetsS( / = (1, ,6))

which contain eithera majority of all players(i.e.have ^ 4 elements),or1It is easilyseenthat this is their only solution.)))

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468 SIMPLEGAMES

which contain exactly half (i.e.have 3) elements,and fulfill the followingfurther condition:EitherScontainsplayer1,but it isnot (1,3,4)or (1,5,6)*or Sis (2,3,4)or (2,5,6).2'8

It is easy to verify that this W satisfies(49:W*)in 49.6.2.W m can be determined without serious difficulties. It turns out to

consistof thesesets:(Si): (1,2,6) where 6 = 3 or 4 or 5 or 6(S'z'): (l,a,6) where a = 3 or 4, b = 5 or 6 4

(Si\"): (2,p,g) where p = 3,q = 4, or p = 5, q = 6.4(SJO: 0,4,5,6)'

If this gamehad a symbol[w\\, , w n], then therewould be

2) Wi > Wi for all Sin W.i in 8 i in -8

Apply this to the setsof W m , enumeratedin (S3). This gives in particular:))

W2 + W* + W4 > Wi + Wf> + Wt,w* + MS + w 6 > Wi + Ws + W4.

Adding thesefour inequalitiesgives:2(wi + Wi + W* + WA + ws + w 6) > 2(wi + wz + w 3 + W* + w& + u> 6),

a contradiction.The equations (50:8)of 50.4.3.,50.0.2.(with U = W m) which deter-

mine a main simplesolution on the otherhand are:(E't): x\\ + xi+ xb

= 6, where 6 = 3 or 4 or 5 or 6,(E't): Xi + xa + xb

= 6, where a = 3 or 4, b = 5 or 6,(#i\"): X<L + xp + x q

= 6, where p = 3,q = 4, or p = 5,= 6,

(#'/): 0:3+ 0:4+ 0:5+ 0:6= 6.Theseequations (1?3) cannot be solved.8 Indeed(/?')showsthat o:3 = x4

and xs = x, hence (E'3\") gives x2 + 2x8 = 6, x2 + 2x6 = 6, therefore1I.e.it is (l,a,b)with a 2, 6 3 or 4 or 5 or 6;or with a 3 or 4, b 5 or 6.1Thecomplements of the previously excludedsets(1,5,6)and (1,3,4).1If this last exception concerning (1,3,4),(1,5,6)and (2,3,4),(2,5,6) wereomitted,

then W would bedefined by this principle:Theplayer 1is privileged normally the over-all majority wins, but ties aredecidedby player 1.

It is easyto verify that this is simply the game [2,1,1,1,1,1]*.I.e.this caseis evensimpler, than our in some ways, analogous example in the secondremark abovesincethe privilege existing herehas a numerical value in the conventional sense.

Thus the complicating exception concerning (1,3,4),(1,5,6)and (2,3,4),(2,5,6) isdecisivein bringing forth the real characterof our example.

4 Note that a,6vary independently of eachother, while p,q do not!Thus PFw has4+4+2+ l - 11elements.

1They are 10equations in 6 variables,cf.footnote 2 on p. 465.)))

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NEW POSSIBILITIESOF SIMPLEGAMES 469

x9 =B, and so 3 = a?4 = z6 = * Now (E*v) &ves *x* = 6 ^a = f

whence (#','),(#',\")yieldxi+ 3 = 6, z2 + 3 = 6,i.e.xl = x2 = 3. Finally(JEJ) becomes3 + 3 + f = 6, a contradiction.

As to the interpretationof this insolubility, essentiallythe same com-mentsarein orderasat the correspondingpoint of the secondremarkabove.

63.2.6.Sixth:We have already referred to the difference betweenthehomogeneousweighted majority principle,and the existenceof a mainsimplesolution,as discussedin 50.8.2.This wasdonein the fourth remarkabove,where an examplefor = in (50:21)loc.cit.wasgiven. We will nowgive an examplefor > in (50:21)loc.cit.

Sincewe found that for n ^ 5 all simple games were homogeneousweightedmajority games,we must now assumen <t 6. We do not knowwhether an exampleof the desiredkind existsfor n = 6 the one whichwill be given has n = 7.

Put n = 7. Define W as the systemof all thosesetsS( / = (1, ,7))which contain any one of the 7 followingthree-elementsets:1

(S4): (1,2,4),(2,3,5),(3,4,6),(4,5,7),(5,6,1),(6,7,2),(7,1,3)The principleembodiedin this definition canbe illustrated in various

ways.This is one:The7 setsof (SOobtain from the first one (1,2,4) by

cyclicpermutation. I.e.by increasingall its elementsby any one of thenumbers0,1,2,3,4,5,6but all threeby the sameone providedthat thenumbers8,9,10,11,12,13areidentified with 1,2,3,4,5,6respectively.2

In otherwords:They obtain from the setmarked x x x on Figure89,by any one of the 7 rotationswhich this figure allows.

Another illustration:Figure90 shows the players 1, , 7 in anarrangementin which it is feasibleto mark 7 setsof (SOdirectly. They areindicatedby the 6 straight linesand the circleO.1

The verification, that this W fulfills (49:W*) is not difficult, but weprefer to leave it to the readerif he is interestedin this typeof combinatorics.W m consistsobviously of the 7 sets of (SO-

It is easy to show along the linesgiven in the third and fifth remarksabove that this isnot a weightedmajority game. We omit thisdiscussion.

The equations (50:8)of 50.4.3.,50.6.2.(with U = W\") which deter-mine a main simplesolution on the otherhand are:(#0: xa + xb + xc = 7, where (o,6,c)runs over the 7 setsof (SO-

They areobviously solvedby zi = = Xi = i.41Thus Wm has 7 elements.f In the terminology of number theory :Reducedmodulo 7.1Thereaderwho is familiar with projectivegeometry will note that Figure 90is the

picture o!the so-called7 point plane geometry. The seven setsin question are itsstraight lines, eachonecontaining 3 points, and the circleO alsorating assuch.

Oneshould add that other projective geometriesdo not seemto be suited for ourpresent purpose.

4Itis easilyseenthat this is their only solution.)))

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470)) SIMPLEGAMES))

Now we can establishthat > holdsin (50:21)in 50.8.2.Indeed:))

x* = V- > 14- 2n.i-lAs the gamesdiscussedin the second,third and fifth remarks,this one

too correspondsto an organizational principlethat deservescloserstudy.In this game the setsof W m , i.e.the decisivewinning coalitionsarealwaysminorities (three-elementsets).Nevertheless,no playerhasany advantageover any other:Figure89 and its discussionshow that any cyclicpermuta-tion of the players1, , 7 i.e.any rotation of the circleof Figure89leavesthe structureof the game unaffected. Any playercan be carriedinthis manner into any other player'splace.1 Thus the structureof the game))

Figure 89.)) Figure 90.))

is determinednot by the individual propertiesof the players2 all are,aswe saw,in exactlythesameposition but by the relation among the players.Itis,indeed,the understandingreachedamong 3 playerswho arecorrelatedby (/S4) 8 which decidesabout victory or defeat.

64.Determinationof All Solutionsin SuitableGames

54.1.Reasonsto ConsiderOther Solutions than the Main Solution in Simple Games

64.1.1.Our discussionof simplegamesthus far placedmost emphasisupon the specialkind of solutionsdiscussedin 50.5.1.-50.7.2.and particu-larly on the main simplesolution of 50.8.1.On the basisof what we havelearnedin theprevioussections especiallyfrom the examplesof 53.2.thisapproachdoesnot appearto do justiceto all aspectsof our problem.

1Thegame is neverthelessnot fair in the senseof 28.2.1.,sincee.g.the two three-elementsets(1,2,4)and (1,3,4)act differently: Theformer belongsto W, the latter to L.(Soin the reducedform of the game, with y -1,the v(S)of the former is 4, and that ofthe latter is -3.)

1Which the rules of the game might give them.'Thereexist in this game no significant relations between any two players:It is

possibleto carry any two given playersinto any two given onesby a suitable permutation(of all players 1, , 7) which leavesthe game invariant.)))

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ALL SOLUTIONSIN SUITABLEGAMES 471

Tobeginwith, we have seenthat we cannot expectall simplegamestohave solutionsof the type mentioned.Already for n 6 a wealth of newpossibilitiesemerged.This is significant, since6 is a sizeablenumberfrom the point of view of combinatorics,but a small one when viewed inthe contextof socialorganization.

But further, even when thesesolutionsexist,indeedeven for the homog-eneousweightedmajority games,they donot tellthe whole story. Forthemost primitive specimenof that class,the essentialthree-persongamewhich as we know has the symbol [1,1,1]*,there existmany solutions.And our discussionin 33.showedthat they areall essentialfor our under-standingof the characteristicsand the implicationsof our theory actuallysomefundamental interpretationswere first obtainedat that point.

64.1.2.Consequentlyit is important to determineall solutions of asimplegame and, as long as we arenot able to do this for all simplegames,to do it for as many simplegamesas possible.Inparticular this shouldbe donefor at leastone simplegame at eachvalue of n. Suchresultswouldprovidesome information about the structural possibilitiesand principlesof classificationof solutionsfor n*participants.

It is true that this information would be equallywelcomeif it couldbeobtainedfor otherthan simplegames. Howeverthe simplegamespossessa manifest advantage over all others when solutionsareto bedeterminedsystematically:Forsimplegames the so-calledpreliminary conditionsof30.1.1.causeno difficulties (cf.31.1.2.),sincethereevery setS is certainlynecessaryorcertainly unnecessary(cf.49.7.).

It is equallytrue that the determinationswhich we envisagewould onlyprovide information concerninga few isolated cases. But they wouldneverthelesscover all n i.e.enableus to vary n at will. This is bound tolead to essentialinsights.

54.2.Enumeration of ThoseGamesfor Which All Solutions Are Known

64.2.1.Let us take inventory of the casesfor which we already knowall solutionsof a game. Therearethree:

(a) All inessentialgames(cf.(31:P) in 31.2.3.,complementedby (31:1)in 31.2.1.).

(b) The essentialthree-persongameboth in the old theory (excesszero)and in the new one (generalexcess).(Cf.32.2.3.for the formerand the analysisof 47.2.1.-47.7.for the latter.)

(c) All decomposablegames provided that all solutionsof the con-stituents areknown. (Cf. (46:1)in 46.6.)

Clearlywe canuse the device(c)to combinethe gamesprovidedby (a)and (b) thus obtaining games for which all solutionsare known.1 Inthis processof buildingup (a) furnishes only

\" dummies\" (cf.the end of43.4.2.),hencewe may well dispense with it, sincewe want structural

1This can alsobe expressedin the following way:A given game F is the compositeof its indecomposableconstituents, accordingto

the definition of the decomposition partition at the end of 43.3.and (43:E)eod. We)))

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472 SIMPLEGAMES

information. Thus we areleft with thosegameswhich areobtained byiteratedapplicationof (c)to (b). In this way we can obtain gameswhicharethe compositeof essentialthree-persongames.1

64.2.2.This gives n = 3fc-persongamesfor which we know all solutions.Sincek is arbitrary,we can maken arbitrarily great. Tothis extentthingsare satisfactory. Howeverthe fact remainsthat such an n-persongameis just a polymer of the essentialthree-persongame the playersform in

reality setsof 3 which the rulesof the game fail to link to eachother. Itis true that our results concerningthe solutionsof decomposablegamesshow that a linkageof thesesets of playersis neverthelessprovidedfor inthe typical solution i.e.by the typical standard of behavior. But natu-rally we want to seehow the ordinary type of linkage,explicitlysetby therulesof the game,affects the organization of the players i.e.the solutionsor standards. And we want this for greatnumbersof players.

Consequentlywe must look for further n-persongamesfor which it ispossibleto determineall solutions.

64.3.Reasonsto Considerthe Simple Game[1, - , 1,n 2]*

64.3.1.As pointedout above,we aregoing to look for thesespecimensamong the simplegames.2 Now it turns out that thereis a certainsimplegame for every n ^ 3,for which this determinationcan be carriedout.This is theonly n-persongame,of a generaln, for which we succeededthusfar in the generaldetermination.This obviously gives it a position ofspecialinterest. We will alsoseethat it permitsinterestinginterpretationsin severalrespects.

The game in questionhas already occurredin 52.3.and in (52:B)of52.4.It is the homogeneousweighted majority game [1, , 1,n 2]*,(n players).know from (43:L)in 43.4.2.that the setsof participants into which this partition sub-divides them, aresetsof 1or ^3elements.

Thesimplest possibility is therefore that they areall one-elementsets. Accordingto (43:J)in 43.4.1.this means that the game is inessential i.e.it takes us back to thecase(a) above.

Thenext simplest possibility is that they are all one-or three-element sets. Theseare exactly the gameswhich we can form accordingto (c) from (a) and (b). I.e.it isfor thesethat we know all solutions.

This is satisfactory sinceit shows that -a classification basedon the sizesof theindecomposableconstituents (i.e.of the elements of the decomposition partition, cf.(43:L)in 43.4.2.)is a natural one:Ourprogressin obtaining all solutions follows preciselythe lines drawn by it.

Italsostresseshow limited theseresults are:Itis indeed a very specialoccurrencewhen a game is decomposableat all. (Remember the denning equations of (41:6)or(41:7)in 41.3.2.,accordingto the criterion at the end of 42.5.2.!)Thetypical n-persongame is indecomposableand cannot bereachedby means of (c).1By application ofstrategicequivalence we can assume them all to bein the reducedform. But denoting their y by 71, , 7* respectively,we cannot expectto makethem all equal to 1by a changeof unit (unless A; 1). Indeed,their ratios71: :y k

are unaffected by changesof unit.*For this reasonwe use the old theory, i.e.excesszero. Cf. the third and fourth

remarks in 51.6.)))

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54.3.2.As discussedin 52.3.in this game the minimal winning coalitionsaretheseS:(1,n), , (n - 1,n) and (1, , n - 1). I.e.playernwins as soonas he finds any ally at all, but if he remainscompletelyisolatedthen he loses.1 This result invites someremarks:

First:Thestatementof this rule suggestsstronglythat playern is in aprivilegedposition:Heneedsonly one ally to win, while the others needeach other without exception.Actually the situation is this:Player nneeds a coalition of two, the others togetherneedone of n 1,henceaprivilege existsonly if n 1> 2, i.e.n ^ 4.

For n = 3 there is, indeed, no difference between the three players:We have then the game [1,1,1]*,the unique essentialthree-persongamewhich is obviously symmetric.

Second:Theprivilege of player n is as extensiveas a privilege can be:We requiredthat n must find at leastone ally in orderto win and it wouldnot have beenpossibleto requireless.2 It is impossibleto specifythat ncan win without an ally, i.e.to declarethat the one-elementset (n) to bewinning this is incompatiblewith the essentialityof the game. (Thiswasdiscussedextensivelyin 49.2.)

65.TheSimpleGame[1, , 1,n - 2]h

55.1.Preliminary Remarks

66.1.Thedeterminationof all solutionsof the gamewhich we discussedabove will show that they fall into a complexarray of classes,exhibitingwidely varying characteristics.These createan opportunity for theinterpretations we have alluded to previously. We will discusssome ofthem, while further discussionsalong the same line will probablyfollow insubsequentinvestigations.

Theexactderivation of this completelist of solutionswill be given in thesectionswhich follow (55.2.-55.11.).This derivation is of not inconsider-able complexity. We are giving it in full for the same reasonsas theanalogousone concerningthe solutionsof decomposablegamesin ChapterIX.:Theproof itself is a convenient and natural vehicle for certaininterpre-tations. It presentsat several stagesan opportunityto bringout verballythe emergingstructural features of the organizationsunder consideration.In fact this circumstancewill be even more pronouncedin the proofsof thischapterthan in thoseof Chapter IX.

55.2.Domination. TheChief Player. Cases(I) and (II)

66.2.1.After thesepreliminarieswe proceedto the systematicinvestiga-tion of the game[1, , 1,n 2]* (n players). Assume that it is in thereducedform, normalized by y = 1.

1 As every one-elementset must.1We statedabovethat player n is not at all privileged in this game when n 3

and now we state that he is asprivileged as he possibly can be! Yet n -3 is no excep-tion from this statement:Sincethere exists only one essential three-persongame, the)))

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474 SIMPLEGAMES

We beginwith an immediate observation on domination :))

(55:A) For = {i, , an }, = {j9i, ' , A.},a H ft

if and only if either

(55:1) n > Pn and a< > ft for somei = 1, , n 1,or

(55:2) oti > ft for all i = 1, , n - 1.Proo/:Thiscoincideswith (49:J)in 49.7.2.,sinceW m consistsof the sets

(1,n), , (n - 1,n) and (1, , n - 1).n n

Note that a = ft = permit us to infer from (55:2)the validity-i -iof

(55:3) ctn < ftn.

Hence:

(55:B) a ^ ft necessitate an ^ ft*.i

Proof:By symmetry we needonly considera s-< /3 . We sawthat thisimplies(55:1)or (55:3),henceat any ratean 5^ft.

These two results, simple as they are, deserve some interpretativecomment.

We discussedin 54.3.that the player n has a privilegedrolein thisgame.1 Heis in a situation which is comparableto that of a monopolist,with the inescapablelimitation (cf. the secondremark loc.cit.)that hemust find at leastoneally. I.e.a generalcoalition of all othersagainsthim but nothing lessthan that can defeat him. We will call him thechief player in this game.2

55.2.2.Thesecircumstancesarebrought out clearly in (55:1)and (55:2).Onemay say that (55:1)is the directform of domination by the chief playerand an arbitrary ally (any playeri = 1, , n 1)while (55:2)may betermeda stateof generalcooperationagainsthim. (55:1),(55:3)or (55:B)show that in a domination the chief player is certainly affected:Advan-tageouslyin (55:1)(the directform of domination with the chief player),adverselyin case(55:2)(thegeneralcooperationagainst the chief player).Any otherplayercan be unaffected, left aside,in a domination.8

position in which a player finds himself there may aswell becalledthe bestpossibleonesinceit is the only onethere is.

1Exceptthe casen 3, about which more will be said later.1As to the casen 3, the end of the first remark in 54.3.should bekept in mind.

> >1I.e.It may happen for an i -1, , n - 1 that H ft and a< - 0<. This is

actually only possiblewhen n 4,cf.again the observations concerning n - 3.)))

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THESIMPLEGAME [1, , 1,n -2]* 475

55.2.3.Now considera solution V of this game.1 Form

Max-*. an = w,mVMin~> . an = tf. 2

, in V

Clearly~~ 1^ tf ^ w.

The meaning of #, o> is plain:They representthe.worst and the bestpossibleoutcomefor the chief player,within the solution V.

We distinguishtwo possibilities:(I) = w,

(II) w < w.

65.3.Disposalof Case(I)

.55.3.1.Considerthe case(I). This means that for all a in V

(55:4) an = J>,

i.e.that the chief player obtains the same amount under all conditionswithin the solution. In other words: (I) expressesthat the chief playeris segregatedin the game in the senseof 33.1.Consideringthe centralroleof the chief player it is not unreasonablethat the first alternativedistinctionin our discussionshouldproceedalong this line.8

55.3.2.Let us now discussV in case(I).(55:C) V is preciselythe setof all a fulfilling (55:4).

1In the senseof the old theory, cf.footnote 2 on p.472.1That these quantities can be formed, i.e.that the maximum and minimum existand

areassumed can beascertainedin the sameway as in footnote 1on p.384. Cf.in partic-ular (*) loc.cit.

*Thereferenceto 33.1.re-emphasizesthat this procedureis analogous to that one ofthe essentialthree-persongame.

This will appeareven more natural if it is recalledthat the essentialthree-persongame is a specialcaseof the one we considernow pertaining to n 3. (Cf.e.g.theend of the first remark in 54.3.)

Closerconsideration of the casen 3 shows, however, that this analogy suffersfrom a rather unsatisfactory limitation: In this casethe game is really symmetric, and soany oneof the three players couldhave beencalledthe chief player. (Cf.alsofootnote 2on p.474.) In 33.1.the segregation in question was indeedapplicableto any one of thethree players,and now we have arbitrarily restrictedit to player n!

Yet there is no way so far to apply this to the other players too if we want ourdiscussion to coverall n ^ 3 (and not only n 3):For n 4 the chief player and hisrole are unique.

The only sensein which this situation can be accepted temporarily is that ofkeeping in mind that case(II)must in fine turn out to bea compositeone.

Thus for n -3 comparison with the classification of 32.2.3.which is analyzed in33.1.shows this: Our case(I)is one of the possibilities of (32:A)there:discriminationagainst player 3. Our case(II),on the other hand, coversthe other two possibilities of(32:A): discrimination against players 1,2together with (32:B),the non-discriminatorysolution. So (II)is really an aggregateof 3 possibilities when n 3.

This schemewill, indeed, generalizefor all n. Cf. (e) in the fourth remark of55.12.5.)))

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476 SIMPLEGAMES

>

Proof:We know alreadythat all a of V fulfill (55:4). If, conversely,a

ft fulfills (55:4),then every a of V has an = 0n, hence(55:B)excludesct H ft . Henceft belongsto V-

Thus V is determinedeasily enough, but we must now answer theconversequestion:Given an w ^ 1,is the V defined by (55:4)(i.e.by(55:C))a solution? I.e.doesit fulfill (30:5:a),(30:5:b)of 30.1:1.?

Now (55:B)and (55:4)excludea *- ft for a , ft in V, hence(30:5:a)is automatically satisfied. Therefore we need only investigate (30:5:b)of30.1.1.I.e.we must securethis property:

(55:5) If ftn ^ w, then a H ft for some a with an = w.

More explicitly:We must determine what limitations (55:5)imposesupon w.

Theftn 9* & of (55:5)can be classified:

(55:6) ft n > u,(55:7) ftn < .

We show first:

(55:D) In the case(55:6)condition(55:5)is automatically fulfilled.

Proof:Assume ft n > w, i.e.ft n = & + , * >0. Define))

by a< = fti H--= for t = 1, , n 1,and an = ft n = co. a is71 \"\"* 1

an imputation of the desiredkind with a H ft by (55:2).Thus only the case(55:7)remains. Concerningthis casewe have:

(55:E) Forw = 1,(55:7)is impossible.

Proof:ftn ^ 1,hencenot ft n < w = 1.Thepossibilityw > 1is somewhat deeper.l

(55:F) Assume o> > 1 and case(55:7).Then condition (55:5)is equivalent to w < n 2 -- r-

1<i 1 means that the chief player is not only segregatedbut alsodiscriminatedagainst (by V) in the worst possibleway. (Cf.33.1.)

Thus w 1 gives a solution outright, while w > 1 necessitatesthe moredetailedanalysis of (55:F). This is not surprising: An extreme form of discriminationis a more elementary proposition and requires lessdelicateadjustments than an inter-mediate one.)))

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THESIMPLEGAME [1, , 1,n -2]> 477

Proof:Assume ft n < w. Foran a with = o>, (55:3)of 55.2.1.is

excluded,i.e.domination a H ft must operatethrough (55:1)(and not(55:2)!)in (55:A). Sincean > n, this condition amounts merely to

(55:8) > fa for somei = 1, , n 1./ Thus (55:5)requires the existenceof an imputation a with an =

and (55:8).Considerfirst (55:8)for a fixed i = 1, , n 1. Then this con-

dition and an = o> can be met by an imputation a if and only if ft andadd up with n 2 addends 1 to < 0. I.e.ft + <o (n 2) <0,ft < n 2 W. Consequently(55:8)is unfulfillable for all i = 1, ,n_r- 1,if and only if

(55:9) ft n - 2 - J> for all i = 1, , n - 1.

(55:5)expressesthat this shouldhappen for no ft with ft n < o>. I.e.no imputation ft could have (55:9)togetherwith 1^ ft n < w. 1 Thismeans that n 1addends n 2 co and oneaddend 1must add up

to > 0. I.e.(n - l)(n - 2 - w) - 1>0,n - 2 - J> > ? p and sopn 1))

< n 2--r> as desired.n 1))

Combining (55:E),(55:F) and recalling(55:D) and the statementsmade concerning(55:5)and (55:6),(55:7),we can summarize as follows:

(55:G) Let w be any number with))

n))

Form the setV of all a with

n

Thesearepreciselyall solutionsV in the case(I).1We areassuming that (55:9)implies ^ 1 for i 1, , n 1.This means

n 2 w gt 1, n 1. Indeed w > n 1 must be excludedsinceit makes(55:4)unfulfillable by imputations: and n 1 addends 1 would then add up to>0.

Thereforethe hypotheses of (55:F)imply this: & n 1.1In pursuance with the parallelism with the discussion of the specialcasen 3 in

33.1.referred to in footnote 3 on p.4Z5 wenote that this correspondsto the cloc.cit.For n 3our n 2 r becomesthe J occurring there.)))

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478)) SIMPLEGAMES))

Thefirst values of the quantity n 2---r are:))

n) 3) 4) 5) 6)

u 2 1) 1 n*) 51A7)

1127*)

19IB)n 2 n-1) 2 -0.5) 3 1'6r) ' -6.70

4) 5- 3'8))

Figure 91.65.3.3.Theinterpretationof this result is not difficult:This standard of behavior (solution)is based on the exclusionof the

chief player from the game. This makes the distribution between theother players quite indefinite i.e.any imputation which gives the chiefplayer the \"assigned\"amount J> belongsto the solution. Theupper limit

of the \"assigned\"amount o>, n 2 -- = could also be motivated

following the linesof 33.1.2.,but we will not considerthis question.

55.4.Case(II):Determination of V

66.4.1.We now pass to the considerablymore difficult case(II). (Cf.the last part of footnote 3 on p.475.) We then have))

This suggeststhe following decompositionof V into three pairwisedisjunctsets:

V,setof all a in V with an = w,

V,setof all a in V with an =a>,

V*, setof all a in V with o> < an < o>.

By the very nature of co, o> (cf. the beginningof 55.2.3)V, V cannot beempty while we cannot makesuchan assertionconcerningV*.1

66.4.2.We beginby investigating V.))

(55:H) If belongsto V and ft to V u V*, then <x<

t = 1, , n - 1.))

for all))

Proof:Otherwise ft > c^ for a suitable i=!,,n 1. Now

n = w, 0n > , so jSn > nj hence ft H a by (55:1),which is impossible,sincea , ft belongto V.

1V* ia actually empty in the caseconsideredpreceding (55:V).)))

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THESIMPLEGAME [1, - , 1,n - 2]h 479

Form

cti = Min^. t for i = 1, - , n I.1

Now (55:H)gives immediately:

(55:1) If ~0 belongsto V u V*, theng ^ fa for all i = 1, n - I.2

We prove further

n-l(66J) & + o> 0.'))

n-1Proof:Assume that V a> + o? <0. Then we can choose7 > g for))

i = 1, , n 1,7n = w with 2) ? = 0, forming the imputationi-i>

7 |<v , ...\"v > 4l7l> > 7n|.V is not empty, choosea from V. Then by (55:1)fa ^ a< < 7< for

all i = 1, , n - 1,henceby (55:2)7 H ft . As belongsto V,

this excludes7 from V.Hencethere existsan a in V with a - 7 . If a belongsto V,

then an = a> = 7n , hencea ** 7 contradicts(55:B).So a must belongto S/u V*.\" Now by (55:1)a, ^ a, < 7, for all t = 1, , n - 1.

^ *But both (55:1)and (55:2)in (55:A) require sincea H 7 that a, > 7,for at leastone i = l, ,n l. Thus we have a contradiction.

Now the determinationof V can be completed:(55:K) V has preciselyone element:))

1That thesequantities can be formed, i.e.that theseminima exist and are assumedcan be ascertainedin the same way as in footnote 1 on p. 384. Cf.in particular (*)

loc.cit. What is stated there concerning V, is equally true for V, Jhe intersection ofV

with the closedset of the a with a* #.1Note that this cannot be assertedfor the ft of V since0,may exceedthe minimum

value g,. Cf.,however, (55:L).1Cf.however, (55:12)below.4 Note that by their definitions all g< -1,(i - 1, , n -1)and 2 -1,

henceall our 7* & 1,(t - 1, , n 1,n).)))

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480 SIMPLEGAMES

Proof:Let a = \\cn, , a_i,<*} be an elementof V. Then

(55:10) \"'--' f r' \" 1.

by the definition of these quantities. Now ]? ou = and by (55:J),-in-l

oti + w ^ 0,hence> is excludedfrom all inequalitiesof (55:10).-i \"\"

Thus

(55:11) I a< = y for * = *' ' ' ' ' n ~ *'} a* = w,

i.e.{ai, , an-i,a} = {!, , an-i,a>).

So V can have no elementother than {ai, , an_i, wj. SinceV

is not empty, this is its unique element.55.4.3.Note that as a = {i, , an_i, w} belongs to V f it is

necessarilyan imputation. Sowe can strengthen (55:J)ton-l

(55:12) a< + = 0.-iWe canalsostrengthen(55:1):

(55:L) If \"^ belongsto V, then<* ^ fa for all i = 1, , n 1.

Proo/:For )3 in V u V* this has beenstatedin (55:1),for ft in V

(55:K)yieldseven ft = a,-.We concludethis part of the analysisby proving:

(55:M) = -1.Proof:Assume w > 1,i.e.o>= 1+ , c > 0. Define))

by ft = a< + ^ ^ for {=!,-,n 1, and ^ = w - e = 1.is an imputation (cf. (55:12)above). n < w, or equally(55:L),excludes

ft from V.)))

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THESIMPLEGAME [1, , lf n - 2k 481

Hencethere exists an a in V with a H ft . By (55:L)on g a< < fafor all i = 1, - , n - 1. But both (55:1)and (55:2)in (55:A) require-since a H ft that on > fa for at leastone i = 1, , n 1. Thuswe have a contradiction.

Note that now (55:12)becomes

(55:N)*

a, = 1.i-iTheessentialresultsof this analysisare (55:K),(55:L),(55:M). They

can be summarized as follows:1The worst possibleoutcome for the chief player is completedefeat

(value 1).Thereis one and only one arrangement i.e.imputation(in V) which doesthis, and for all otherplayersthis is the best possibleone(in V).

This arrangement (in V) is the stateof completecooperationagainstthe chief player.1

The readerwill note that while this verbal formulation is not at allcomplicated,it could only be establishedby a mathematical, not by averbal, procedure.

55.5.Case(II):Determination of V

56.5.1.We arenow able to investigate V.>

(55:0) Consideran imputation = {0i, , n } with & ^ a,>

for some i = 1, , n I and /3n ^ o>. Then belongsto V.

* . *Proof:Assume that doesnot belongto V- Then thereexistsan a

in V with a H 0. Hence(55:1)or (55:2)of (55:A) must hold. As

a is in V f n ^ w ^ *, and this excludes(55:1).By (55:L)a< ^ a, for alli = 1, , n 1,hencea ^ a ^ fa for at least onet = 1, , n 1,and this excludes(55:2).Sowe have a contradictionin both cases.

(55:P) ai ^ n - 2 - J> for i = 1, , n - 1.Proof:Assume that < < n - 2 - o> for a suitablei = 1, , n - 1,

i.e.that -(n - 2) + a< + w < 0.1All this applies,of course,to the case(II)only.1This expressionwas alsoused in a related, but somewhat different sense in the

last part of 55.2.)))

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482 SIMPLEGAMES

Then we can chooseft r ife 1(j = 1, , n 1,j ^ i} i.e.n 2 valuesn

of j) fa < a,-, ft n > w with ] ft = 0,forming the imputationj-i7 = I0i, , ft>}.

This meetsthe requirementsof (55:0),henceit belongsto V. But thisnecessitatesn ^ o> by the definition of that quantity contradicting

n > ^.Now put

(55:13) a* = Min.i n-i i-1

Then (55:P)statesthis:(55:14) a* ^ n - 2 - w. 2

Denotethe setof all i(= 1, , n 1)with

(55:15) a = a*

by S*. By its nature this setmust have thesetwo properties:(55:Q) S+e(1, , n - 1), S* is not empty.

65.5.2.We continue:(55:R) a* = n - 2 - a>.

(55:S) V consistsof theseelements:a where i rijns over all S*,>

and where a* = {a\\, , a*n_i, a*n|with

a, = a* for j = f,for j = n,

1 otherwise.

Proof of (55:R)and (55:S):We beginby consideringan elementft of \\f.

If ft < g for all i = 1, , n 1,then (55:2)gives a H /3 since

a = {gi, an-i,w). As a belongsto V by (55:K),so a , areboth in V hencethis is impossible. So(55:16) ft ^ g ^ a* for some i = 1, , n 1.

Necessarily(55:17) ft ^ -1 for all j = 1, , n - 1,;*i,and since is in V, so

(55:18) ft> = .1 This time the minimum is formed with respectto a finite domain!1Cf.however, (55:R)below.)))

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n

Now % Pi = and by (55:14)-(n- 2) + a* + ^ 0,hence> isy-i

excludedfrom all inequalitiesof (55:16),(55:17).Henceg< = a*, i.e.ibelongsto S*. And

for j = i,for j = n,otherwise,

i.e.ft = a * as defined above.Sowe see:

(55:19) Every ft of V is necessarilyan a* with i in S*.

Now V is not empty, hencean a i in V (i in S+)exists. Consequently\"-*

this a* is an imputation, hence2}aj = 0,i.e. (n 2) + a* + w = 0.;-iThis is equivalent to (55:R).

Considerfinally any i of S+. Since(55:R)is true,we have

-(n - 2) + a* + a> = 0.n ^

Hence a} = 0,i.e.a* is an imputation. But } = =*, n = J>,y-i

hence(55:0)guaranteesthat a'belongsto V- And sincea*n =a>, a i is

even in V. I.e.

(55:20) Every a i with i in S* is an imputation and belongsto V.

(55:19)and (55:20)togetherestablish (55:S). (55:R) was demon-stratedabove. Thus the proof is completed.

55.5.3.Theessentialresultsof this analysisare(55:R),(55:S)togetherwith the introductionof the setS*. It is again possibleto give a verbalsummary.1

The best possibleoutcome for the chief playerassignsto him a certainvalue o>. In orderto achieve this he needspreciselyone ally who can beselectedat will from a certainsetS+of players. This setconsistsof thoseamong the players 1, , n 1who areleastfavojed in the stateofcompletecooperationagainstthe chief player,referred to at the endof 55.4.

Thus the arrangementswhich the players1, , n 1makebetweenthemselves,when they combine to defeat the chief player completely,determinehis conduct in thosecaseswhere he achievescompletesuccess.This \" interaction \" between fundamentally different situations is worth

1 All this applies,of course,to the case(II)only.)))

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484 SIMPLEGAMES

noting.1 Itisalsoof interestthat the natural alliesof the chiefplayer,whenhe aims at completesuccess,arethe leastfavored membersof a potentialabsoluteoppositionagainsthim. 2

Theconcludingremark of 55.4.concerningthe contrastof formulationand proof appliesagain.

55.6.Case(II):a and S+

55.6.We determinedin 55.4.,55.5.the two parts V, V of V.1 It istherefore time to turn to the last remaining part of V:V*.

Let a be the setof all a with a = g, = a* for all i in S+. Then wehave:(55:T) VuV'sa.

Proof:Consideran a in V u V*. We must prove a = g< for all i in S+.Now at g a* for all i = 1, , n 1by (55:L).Hencewe need

only excludeen < g, when i is in S*.Fori in S+ form the a. * of (55:S).It belongsto V, so a\\ = o>; a

belongs to V u V*, so an < w. Henceai> an. Now a, < a< means\"~

> **i= (ft;> i> hence a'H a by (55:1)and this is impossible,since

>

a *, a both belongto V.(55:U) VfiG if and only if S+ is a one-elementset or a* = 1;

otherwiseV and a aredisjunct.

Proof:Consideran a in V- Then a = a* (from (55:S)),i in S+.>

Comparingthe definitions of a * and & makesit clearthat this belongsto Q,if and only if S+has a unique elementi or , = 1.

1In 4.3.3.we insisted on the influence exercisedby the \"virtual\" existenceof animputation i.e.of its belonging to a certain standard of behavior (solution) on allother imputations of the same standard. Almost all solutions of n ^ 3 persongameswhich we found can be used to illustrate this principle. A specificreferenceto it wasmade at an early stageof the discussion, in 25.2.2.The present instance, however, isparticularly striking.

1Politicalsituations to illustrate this principle arewell known and in connection withthem its generalvalidity is frequently asserted. It is difficult to deny, however, that thecasewhich can be made purely verbally for this principle is no better than that whichcouldbemade for a number of other conflicting ones.

The point is that for the particular game i.e.socialstructure we consider atpresent, this and no other principle is valid. To establish it a mathematical proof ofsomecomplexity was needed. All purely verbal plausibility arguments would havebeeninconclusive and ambiguous.

The set S* is still unknown, although restricted by (55:Q). The numberstin, , -iare alsounknown, but restricted by (55:N).They determine a* (theirminimum), w, are given by (55:M),(55:R). Thedetermination of theseunknownswill beattended to later. Cf.(55:0')(i.e.(55:L'),(55:N')and (55:P')).

Nevertheless,the form of V and of V has beenfound, and the remaining uncertain-tiesareof a lessfundamental character.)))

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Theverbal meaning of (55:T),(55:U)is this:Eachplayerof the leastfavored group (S*,cf. the end of 55.5.)reacheshis optimum 1 in everyapportionment in which the chief player is not fully successful(i.e.inV u V*). When the chief player is fully defeated(i.e.in V), this is eventrue for all players1, , n 1(cf.the end of 55.4.).When the chiefplayeris fully successfulthen this is true for one and only one player,whomay be any memberof the leastfavored group (S+,cf. the end of 55.5.).

65.7.Cases(II')and (II\.") Disposalof Case(II')55.7.1.Considerthe caseS+ = (1, , n 1),to becalledcase(IP).

In this caseg = a* for all i= 1, , n 1,so(55:N)gives (n l)a* = 1,i.e.,a* =--> and (55:R)gives o> = n 2--=- If a belongsto Ct

71 ~~\"~ 1 71 1then at = gt = a* = for i = 1, , n 1. Hencean =

1,))

i.e.a = | Tt ; -t l|- By (55:T)this is equally true for

all 7 in V u V*.This a is clearlythe unique elementa of V by (55:K),henceV* is

empty. HenceV = V u Vi and now (55:K),(55:S)give

(55:V) V consistsof theseelements:))

where i = 1, , n 1and where

7'= (a\\,-

with))

1n - 1

TJ 2)

for j = i,for j = n,

otherwise.)

71 & -n 1))

(55:V) determinesthe only possiblesolutionV in the case(II'). Thisdoesnot necessarilyimply, however, that this V is eithera solutionor incase(!!')Indeed,if it failed to meetany one of thesetwo requirements,then we would only have shown although in a rather indirectway that

1 His individual optimum within the given standard i.e.solution V. For the

player i( 1, , n 1)this optimum (maximum) is < owing to (55:L) althoughHi wasoriginally defined ashis pessimum (minimum) in the part V of V*)))

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486 SIMPLEGAMES

no solution in the case(IF) exists. We will prove, therefore, that bothrequirementsare met.1

66.7.2.(55:W) TheV of (55:V) is the unique solution in the case(II').

Proof:We need only show that this V is a solution in the case(IF)the uniquenessthen followsfrom the above, i.e.from (55:V).

Case(IF) is easilyestablished:Clearly,for this V

^ = 1, o> = n 2))

S* = (1, , n - 1).It remains for us to prove that V is a solution,i.e.to verify (30:5:c)

in 30.1.1.To this end we must determinethe imputations ft which areundominated by elementsof V.

> >

For a H (55:1)is excludedsinceoj= 1. So this domination

can operatethrough (55:2)alone,henceit amounts to aj > ft, i.e.ft <-r71 - 1

for t = 1, , n 1.For a*H ft , k = 1,' , n 1,(55:1)is excludedwhen i 9* k

and (55:2)is excludedsinceaf = 1for i ^ k. So this domination canoperatethrough (55:1)with i = k alone,henceit amounts to a* > ft for

j = fc, n, i.e.pk <-r>ft. < n - 2 - -))

Hence is undominated by elementsof V if and only if this is true:;>-..holds for somei = 1, , n 1and it holds even for all711))

thesei in casethat n < n 2))

n -))

Thus /3n < n 2--r necessitates8\\. , |8n_i ^-r- Alson 1 tt 1

A - -*ft, ^ 1. Hence2) ft = yields= for all these^ relations,i.e.ft = a .-iOn the otherhand ft, ^ n 2--r necessitatesft ^-r for one

n 1 n 1*'(=1, ,n 1) and ft ^ 1for the othern 2 values of j. Hence

n _^ _^

JJ ft = again yields = for all these^ relations,i.e.ft = a *.y-i

1Cf.this situation with (55:G),where the case(I)was settled. No such secondaryconsiderationswereneededthere,because(65:G)wasabwonecessaryand sufficient.)))

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Sothe undominated by V are aand a',,a *~l ; i.e.preciselythe elementsof V, as desired.

65.7.3.This solution is of importancebecauseit is a finite set as weshallseeit is the only solution with that property. If the generalcoalitionagainstthe chief playeris formed, the n 1participantssharein it equally

as describedby a . If the chief player finds an ally, he gives him the

sameamount as a and retains the remainder as describedby

7', , ?-i.All this is perfectly reasonableand non-discriminatory.1 Neverthelessthis is not the only possiblesolution we found another one in 55.3.(cf.(55:G))and more will emergein the sectionswhich follow.

55.8.Case(II\"):a and V. Domination

65.8.1.Consider next the caseS* 9* (1, , n 1),to be calledcase(II\.

Using(55:Q)we may also formulate this as follows:

(55:X) S*c(1, , n - 1), S*not empty.We can also say: Cases(II')and (II\")arecharacterizedrespectively

by the absenceand by the presenceof discriminationwithin the possiblegeneralcoalition against the chief player.

As we enterupon the discussionof case(II\")the following remark isindicated:

The argumentation of 55.4.-55.7.was mathematical,but the (inter-mediate)results which were obtainedthereallowed simpleverbal formula-tions. I.e.it was possible to work into the mathematical deductionrelatively frequent interruptions, giving verbal illustrationsof the stagesattained successively.

This situation changesnow, insofar as a longer mathematical deductionis neededto carry us to the next point (in 55.12.)where a verbal interpreta-tion is again appropriate.

55.8.2.We now proceedto give this deduction.Write V = a n V (thepart of V in a). By (55:T),(55:11)V = V u V*

or V = V u V* u V = V, accordingto whether the condition of (55:U)isnot or is satisfied.

(55:Y) Thecondition(30:5:c)holdsfor V in ft.

Proof:Replace(30:5:c)by the equivalent (30:5:a),(30:5:b)in 30.1.1.Ad (30:5:a):SinceV V, elementsof V cannot dominateeachother

becausethe sameis true for V.1The specialcasesn 3, 4 of this solution are familiar: For n 3 it is the non-

discriminatory solution of the essentialthree-persongame; for n 4 it was discussedin35.1.)))

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488 SIMPLEGAMES

Ad (30:5:b):Let ft in Ct not be in V. Then we must find an a in Vwith a H ft.

> >

To begin with, ft is even not in V. Hencean a in V with a H ft

exists. This a must be in V if it is not in S?(cf. the remarkspreceding(55:Y)),and this would establishour statement. Soweneedonly to excludea from V.

_ > *Assume that a is in V f i.e.(by 55:S) a = a*, k in S+. We have

a*H ft. (55:1)is excludedwhen i ** k, and (55:2)is excludedsinceak.= 1 for i j k (i = 1, , n 1). So this domination can onlyoperatethrough (55:1)with i = fc, henceit impliesaj >0*,i.e.ftk < * = a*.However,this is impossible,sinceft belongsto Ct.

56.8.3.Thus our task is now to find all solutions(i.e.all setsfulfilling

(30:5:c)in 30.1.1.)for Ct. This necessitatesdetermining the nature ofdomination in Ct.

(55:Z) For a, ft in Ct, a H ft is equivalent to this:an > ft n anda > fti for someiin(I, , n 1) S+.

Proof:For a H ft (55:1)is excludedwhen i is in S*,and (S&:2)isexcludedsinceak = 0*(=g* = <O for all k of S*.

Sothis domination can only operatethrough (55:1)with i in (1, ,n 1) S*. And this meansan > ft n and a, > ft, as asserted.

We have replacedthe setof all imputations by Ct and the conceptofdomination describedin (55:A) by that onedescribedin (55:Z). Otherwisethe problemof finding all solutionshas remainedthe same. Theprogressis that the conceptof domination in (55:Z)can be workedmore easilythanthat onein (55:A) as will beseenin what follows.

56.9.Case(II\"):Determination of V'

55.9.1.Let p be the number of elementsin S+.Then we have:

(55:AX

) 1:gp n - 2.

Proof:Immediateby (55:X).

.1))(55:B'))) -1))

Proof: 1g a* is evident. Next a< = a* for i in 5*,a< > a+ for iin (ly , n 1) S+, and by (55:A') neither set is empty. So)))

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n-l))

. > (n - l)a*,and hence(55:N)gives 1> (n - l)a*,a* <__ ^- n 1

as desired.An a in a has p fixed components:the (= o = a*), t in S*;and

n p variable ones:the a, i in (1, , n) fif*. Theseare subjectto the conditions

(55:21) a, -1 for t in (1,- - , n) - S*9

n

and ] a,- = 0,i.e.-i(55:22) a,- -pa,.

In (1,...,*)-The lower limits in (55:21)add up to less than the sum prescribedin

(55:22),i.e. (n - p) < -pa*. Indeed,this meansa* < n \"\"\" P = -- 1.p p

And by (55:A')p<n-l,so--l> 5 - 1= 1_,and (55:B')P 71 \"\"* 1 71 1guaranteesa* < ^--r

Sowe see:(55:C') Thedomain a is (n p l)-dimensional.

65.9.2.We now proceedto a closeranalysisof V and of a.1Put

(55:23) co* = n-p-l-pa*.By (55:R)we can write

(55:24) * - (p - !)(*+ 1).(55:D') a>*= w if and only if S*is a one-elementset(i.e.p = 1)or

a* = 1i.e.if and only if the condition of (55:11)isunfulfilled;otherwisew* < co.

Proof:Sincep ^ 1,a* ^ 1by (55:A'), (55:B'),this is immediatefrom (55:24).(55:E;) Max- an = w*.

a in a*The lemmas (55:D')-(55:P')which follow are the analytical equivalent of the

graphical deduction of 47.5.2.-47.5.4.Thetechnical background differs, but the analo-giesbetween the two proofs are nevertheless very marked the interested readermayfollow them up stepby step.

(55*C')shows that a graphical discussion would have to take placein a (n p 1)dimensional space(by (55:A') this is 1,g n -2). This is the reasonwhy we usean

analytical one. (Thegraphical proof referred to above took placein a plane, i.e.itrequired 2 dimensions.))))

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490 SIMPLEGAMES))

(55:F') This maximum is assumedfor preciselyone in <J:

with))

* = {r,))

(<*

= a* for i in S+,a>* for i = n,

1 otherwise.1

Proof of (55:E')and (55:F):It is clearfrom the definition of a that for

the a of (i the variable componentan assumesits maximum when the othervariable components at , i in (!,,n 1) S* assume theirminima. Theseminima are 1. Sofor this maximum))

(g-1))= a* for i in S*,

foriin(1, , n - 1)- S*.))

n-1Now an = ^/ = pa* + (n 1 p)=n p 1 pa*. By

(55:23)this meansan = a>*.This provesall our assertions.

(55:G') ^* belongsto V.Proof:a * belongsto a,for any a of a (55:E'),(55:F')give

a* :S:(=a,*).

So (55:Z)excludesa H a *, and therefore (55:Y) necessitatesthat a*belongto V.

55.9.3.After these preparations the decisive part of the deductionfollows:

(55:H') If a, ft belongto V, then an = ft n implies a = ft .Proof:Considertwo a , ft in V with dn = n.Put 7, = Min (at, 0,)(i= 1, , n 1,n) and assume first that

n n

5) Ti < 0,say 5) 7. = , >0.

Put 6 = {81, , 6n-i,6n|where

(7for i in S*,

7. + _1^ for i in (1, , n - 1,n) - S*.n p

1Comparison of this definition with (55:D;) shows that this a * is an a , i in 5 i.e.that it belongsto V if and only if the condition of (55:11)is fulfilled.

Sincea *belongsto a this is in agreement with the result of (55:11).)))

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This 6 is clearly an imputation, and as i in/S* gives 5l =7i=ai==/3l =g<=a# ,

so 5 belongs to ft. We have 6,, > y n = an = ft*, and for t in

(1, , n 1) S*,5i > 7 = a or ft, hence5 H a or 6 H .Since a , belongto V, this excludes5 from V. Hencethere exists

an ij in V with rj H 5 .Now by (55:Z)ryn > 5n and ry t > 5t for an i in (1, , n 1) S+.

>

A fortiori rj n > dn > y n = an = ft, ? > 5; > 7< = at or ft. Thus t? S-\" aor 17 H . As a , , 77 all belongto V'f this is a contradiction.

n

Consequently^ 7 < is impossible,sot-i

n

(55:25) 7. ^ 0.i-1

n n

Now 7i g ai; 7i ^ ft and 2}ca = ^ j8 t- = 0. Hence(55:25)yields= for=i =i

> >

all these^ relations,i.e.7 = a = ft. This proves a = as desired.>

(55:!') Thevalues of the an for all a in V makeup preciselytheinterval))

Proof:Foran a in V,an ^ 1 is evident, and an g o>*followsfrom

(55:E').Hencewe needonly excludethe existenceof a yi in

-1 tfi ^ co*,

such that a 7^ 2/1 for all a in V.Thereexistcertainly elementsa of V with an ^ y\\\\ Indeeda * belongs

to V by (55:G'),and a* = co* ^ y l8 Form

Min... an = 2/2,1

a in V with an ^ 2/i

1In this caseit is not necessaryto form the exactminimum, but the procedurewhich

achievesthis is somewhat longer than the one used below. That this minimum can beformed, i.e.that it existsand is assumed, can beascertainedin the sameway as in foot-note 1 on p.384. Cf.in particular (*) loc.cit. What is stated there for V is equallytrue for the analogous set V in a and for the intersection of V with the closedset of

the a with an ^2/i.Becauseof this needfor closurewe must use the condition a* ^ ^i and not n > y\\

although we are really aiming at the latter. But the two will be seento be equivalentin the caseunder consideration. (Cf.(55:26)below.))))

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492 SIMPLEGAMES

and choosean a + in V with a+ ^ y\\ for which this minimum is assumed:a+ = t/ 2. By (55:H')this a + is unique.

So2/2 ^ 2/i, and sincenecessarilya+ 5^ y it so t/ 8 ^ 2/1, i.e.(55:26) 2/1 < 2/2-

It follows from the definition of 2/2 that

(55:27) y\\ g an < 2/2 for no a in V.Now put t/i = 2/2 ~-

> > and form the imputation

= {01, * * , fti-l, fti}

with ft, = +- = 2/2 - = 2/i ft = a+ = a< = a* for i in S*,ft =*

a+ ^---f or t'

in (i . . . n _ i) _ 5 Clearly belongston 1 p

Ct and n = y\\ excludesft from V. Hencethereexistsa y in V with

7^7.By (55:Z)thismeans7n> n and %>frforaniin(1, , n 1) S+.Now 7n > 0n = 2/i necessitatesby (55:27)yn ^ ^2- Tn = 2/2 would

imply 7 = a + (by (55:H;), cf. above). Hence7 = at < ft for theabove i in (1, , n 1) S+, and not 7, > ft as required. Hencey* > 2/2-

Thus 7n >2/2=<*n and y t >ft >at for the above iin(1, ,n 1) S^.So 7 H a +, and as 7 , a + both belongto V r this is a contradiction.

56.9.4.By (55:1'),(55:H')we see:Forevery y in

-1 V ^\"*

thereexistsa unique a in V with an = y. Denotethis a by))

Clearly n (2/) =2/ and a t (t/) = = * for i in S*. Sothe functions which

matterarethe at (t/) for i in (1, , n 1) S*.Combiningthis with (55:1')gives:

(55:J') V consistsof theseelements:

7(2/)where y runs over the interval)))

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and where a (y) = [ai(y)9 , an_i(y), <xn(y)}with

= a* for i in S*,y for i = n,a suitable function of y (and i) for i in

(1, , n - 1)- S,.55.9.5.And to conclude:

(55:K') The functions on(y), t in (1, , n - 1)- <S,of (55:J')fulfill the followingconditions:

(55:K':a) Thedomain of at(2/) is the interval

-1 y ^ co*.(55:K':b) i/!g y 2 implies at(i/i) ^ a,(yi).1(55:K':c) a.(-l)= a,.(55:K':d) .(*)= -1.(55:K':e) a,(y) = -pa*- i/.2*8

tin (l,-,n-l)-S*1I.e.a,(t/) is an antimonotonic function of y.1From theserelations the continuity of all functions (t/), i in (1, , n 1) *

follows. Indeed,we can evenprove more, the so-calledLipschitz condition:

(55:28) \\<x*(yi) a,(yO|^ |yi yi|.

Proof:This relation is symmetric in yi, y* hence we may assume y\\ ^ y*. Nowapplication of (55:K':e)to y y\\ and y =

2/2 and subtraction give

tin (l,-,n-l)-S*By (55:K':b)all theseaddendsa(j/0 a(2/j)are ^ 0,hencethey arealso^ than theirsum y* y\\. Thus

Theseinequalities make it alsoclearthat the middle term is \\a t (y t) a(t/i)|andthat the last term is \\yt yi|. Hencewe have

as desired.Thereaderwill note that we never assumed any continuity we proved it! This is

quite interesting from the technical mathematical point ofview.8 Note that (55:K':c),(55:K':d)do not conflict with (55:K':e).Indeed:

For y - -1(55:K':e)gives ] a<(-l)- ~pa# 4- 1, hencetm(l,-,n-l)-S*

n-1(55:K':c)requires ^/ a \"*

\"\"P a* \"^\" J S < == 1 agreeing with (55:N).im(lf ,n-l)-5* -l

For y - w* (55:K':e)gives J) a<(w*)- ~pa# - *, hence))

(55:K':d)requires (n p 1)- ~p*-w*,w*-n~p~l--pa*, agreeing with

(55:23).)))

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494 SIMPLEGAMES

Proof:Ad (55:K':a):Containedin (55:J').Ad (55:K':b):Assume the opposite:y\\ ^ t/ 2 and a(t/i) < a(y2) (for a

suitable i in (1, , n - 1) S*). This excludesy\\ = !/2, so y\\ < y 2.Then a (7/2) H a (t/i), which is impossiblesincea (y\\), a (y2) both belongtoV.

Ad (55:K':c):This is a restatementof the fact that a belongsto V,indeedit belongsto V. (Cf. (55:K),(55:M).)

Ad (55:K':d):This is a restatementof the fact that a * belongsto V

^ n

Ad (55:K':e):a (y) is an imputation, hence^ 4 (y) = 0.

By (55:J') this means that a (2/) + Pa* + 2/= 0>

i.e.that (?/) = pa* t/> as desired.tin (l,-,n-l)-S>

55.10.Disposalof Case(II\55.10.1.The resultsobtained in 55.S.-55.9.contain a completedescrip-

tion of the solution V. Indeed:As we saw at the beginning of 55.8.2.V = V u V, although the addend V may be omitted (becauseit is cV)if and only if the condition of (55:11)is satisfied. V is describedin (55:S),V in (55:J')- Thesecharacterizationsmakeuse of the parameters

a.: (i = 1, , n - 1),or*, S,w, *,at(t/)(iin (1, , n - 1)-S,,-1g y g *),

which aresubjectto the restrictionsstated in (55:N);(55:13),(55:15)in

55.5.1.;(55:R);(55:23),(55:24)in 55.9.2.;(55:K').Sincethis material is dispersedover seven sections,it is convenient to

restatethe completeresult in one place:

(55:L')(55:L':a) S* is a set c(1, , n 1),not empty. Let p be the

number of elementsof S*,so that 1 g p g n - 2.n-l

(55:L':b) gi, , gn-iarenumbers ^ -1,with ^ at = 1.

(55:L':c) Forall i in S,a> = a^, for all i in (1, , n - 1)- S*

(55:L':d) Put = n - 2 - a*, a>* = /i - p - 1 ~ pa^, so thatw w* = (p l)(a* + 1).)))

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(55:L':e) ai(y) is defined for t in (1, , n 1) S+,

-1 y *.Thesefunctions satisfy the conditions(55:K':a)-(55:K':e).V consistsof theseelements:

(a) a (y) where y runs over theinterval 1 y ^ w*, andwhere))

with

&i = a* for i in S+,y for i = n,the av(y) of (55:L':e)for i in(1, -

f n- 1)-S,.(b) a * where i runs over all S*and where

with

for j = i,for j = n,otherwise.

Remark:If p = 1 (S* a one-elementset) or a* = 1,then eo = w*

and the a * of (b) coincidewith a (y) of (a) for y = w*. If this is not the

case i.e.p *z 2, a* > 1,then a> > a>* and the a * of (b) aredisjunctfrom the a (y) of (a).

The readerwill verify with little difficulty that all thesestatementsarenothing but reformulations of the resultsreferred to above.

65.10.2.(55:L')must be followed by similar considerationsas (55:V).We must investigate whether all V obtainedfrom (55:L')aresolutionsandin the case(II\.")Thoseof them which meet both these requirementsform the completesystemof all solutionsin the case(II\.")We will provethat all V of (55:L')meet theserequirements.

(55:M') The V of (55:L')arepreciselyall solutionsin the case(II\.Proof:We need only show that every V of (55:L/)is a solution in the

case(II\") that theseV arepreciselyall such solutionsthen follows from

(55:L').Case(II\")is easilyestablished:Clearlyfor this V, w = 1, and)))

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496 SIMPLEGAMES

(in the senseof their definitions given in 55.2.-55.5.)are preciselythequantitiesdesignatedin (55:L/)by thesesymbols,1hence

S,c(1, , n - 1)by (55:L':a).

It remains for us to prove that V is a solution. In the present casewe will do this by proving that V fulfills (30:5:a),(30:5:b)in 30.1.1.

Ad (30:5:a):Assume a H ft for a , ft in V. We must distinguishto which cases(a), (b) of (55:L/) a , ft belong. Therearefour possiblecombinations:

a, ft in (a):I.e.a = a (2/1), ft = a (1/2) and so a (t/i) H a (t/2).Now (55:1)is excludedwhen i is in S* and (55:2)is excludedsince

(yO = <*(y) = * for i in S*. So this domination can operatethrough (55:1)with i in (1, , n - 1)- S* only. By (55:L':e)thismeans an (t/i) > ctn(yd, y\\ > 2/2, and a(s/i) > a,(t/2) for a suitable i in

(1, , n - 1)-S*,contradicting(55:K':b).>

a in (a), ft in (b): I.e.a = a (i/), = a * (i in S*), and so> *

(j/) Ha*. Now (55:1)is excluded,since otn (y) = y ^ w* ^ a> = aj,,and (55:2)is excluded,since <(i/) = a\\ = g = a*. So we have a con-tradiction.

a in (b), ft in (a):I.e.a = a* (i in S*), ft = a (y), and so

a * H a (#). Now a\\ = a(v) = at = a^ and for j j* i,n, a}= -1^ a,(j/),i.e.a}g a,(j/) for all j = 1, , n 1. This excludesboth (55:1),(55:2),and gives a contradiction.

a , in (b): I.e.a = a *, = a * (i, A in S*),and so a * H a *.Nowai= a* = w, thus contradicting(55:B)

Ad (30:5:b):Assume that ft is undominated by the elementsof V.We wish toprove that this impliesthat ft belongsto V which establishes(30:5:b).

Assumefirst that ft n ^ w. If ft < a, = a* for all i = 1, - , n 1,then a( 1)H ft , contradictingour assumption. Henceft ^ g forsomet*=l, , n 1. Now the argument used in the proof of (55:R)showsthat necessarilyi in S* and == a *. Therefore belongsto Vin this case.

l w obtains from (b), 31, , gn-ifrom (a) with y - -1,and then a*, > from(55:L':c).)))

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Assume next that n < o>. If ft < g = a* for some i in S*,then

clearly a * H contradictingour assumption. So ft ^ g = a* for allt in S+.

n n 1Now % ft = gives ft, = - ft ^ n -p - 1-pa* = *, i.e.-i i-i-1 ft> a)*. Put y -ft..Assumethat ft ^ a(y) for all i in (1, , n 1) S*. Then we

have clearly ft 2> a(2/) for all i = 1, , n. (Fori in S* and f = nn n

we have even = , cf. above.) Hence^ ft = i(t/) = necessitates-i t-ithat we have = in all these ^ relations.So = a (y). Therefore/3belongsto V in this subcasetoo.

Thereremainsthe possibilitythat ft < a<(y) for a suitablei in (1, ,n 1) S+. A sufficiently small increaseof y (from y = to somey > j8 n) will not affect this relation ft < <*i(y).1 Forthis new y we have

>, >

y >ft, a,-(y)> ft, and therefore a (y) s* ft contradictingour assumption.Thus all possibilitiesareaccountedfor.

55.11.Reformulation of the CompleteResult

55.11.1.Thesethreecases(I), (IF),(II\") into which we subdividedour problem have beencompletelysettledby (55:G),(55:W), (55:M;)respectively. Let us now seeto what extentthesethreeclassesof solutionsarerelatedto eachother.

Among the undetermined parametersoccurringin (55:L/) i.e.in(55:M'),describingcase (II\") is the set S*. According to (55:L':a)this is any setc(1, , n 1)with the exceptionof (1, , n 1)and 0. This raisesthe questionwhether it is not possibleto find someinterpretationfor theseexcludedcasesS+ = (1, , n IJandS*=also.

For S* = (1, , n 1) the answer is easy. If we use this S*(disregarding(55:L':a)to this extent),then we obtain (usingall otherparts

of (55:L')):p = n 1 by (55:l/:a),gi = = g-i= a* =n _ t

by (55:L':b),(55:L':c),= n - 2 - jpz-y <>* = -1,by (55:L':d).Thereis no occasionto introduce the functions a(y) of (55:L':e),since(1, , n 1) S+is empty. Inasmuchas the interval 1^ y ^ w*

plays a role(in (a) of (55:L':e)),it must be noted that it shrinks to thepoint y = 1 (sincew* = 1). Now comparisonwith (55:V) disclosesthat under theseconditions(55:L/)coincideswith (55:V).

1cu (y) is continuous! Cf.footnote 2 on p. 493.)))

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498 SIMPLEGAMES

Sowe have:(55:N') If we includein (55:L':a)S= (1, , n - 1)(hencep =

n 1) also,then (55:L')enumeratesall solutionsin the cases(IF)and (II\:")Case(IF)correspondsto 5* = (1, , n - 1)and case(II\")to S+ * (1, , n - 1).

55.11.2.After this resultone might feel inclinedto correlatethe remain-ing exceptionS* = with the remaining case(I). However,inspectionof[(55:L')with S* = and comparisonwith (55:G)show that this is notpossible at leastnot in this direct way.

Indeed:Use of (55:L)with S+ = Q (hencep = 0) gives an empty (b),so a V coincidingwith (a) i.e.V is the set of all

1^ y ^ w*, with suitablefunctions i(t/), , a_i(y). Disregarding

othermaladjustments1 we note:In this arrangement the an of an a in Vdeterminesits !,-,an-i;while in (55:G)an was constantand

arbitrary !2Summing up:

(55:0') All solutionsV are enumeratedby (55:G) Case(I) and(55:N') Cases(IF) and (II\.")(55:N')coincideswith (55:L'),when (55:L/:a)is widenedto includeall S+ (1, , n 1)with S* 7* . The exclusionof S+ = Q is necessary;thischoicewould producea V which isnot the solution of (55:G),andindeedis no solution at all.

65.11.3.We concludewith the followingobservations:

(55:F)(55:P':a) Incase(IF),i.e.S* = (1, , n - 1),p = n - 1,we have:

o>* = 1,i.e.the interval 1 ^ y ^ w* of (55:L':e)shrinksto

a point. Also a+ = -(55:P':b) In Case(II\,")i.e.S*c(1, , n - 1),p <n - 1,we have

w* > 1,i.e.the interval -1g y <; o>*of (55:L':e)doesnot. 1))

shrink to a point. Also))n))

1 Owing to p - (55:23)now gives w - a>* - -( + 1),hencewe may have))

> w, and soMax-* . M&\\_ l ^y -w*, although it should bew!))

For S+ * , (55:L':b),(55:L':c)gave Min.^j n-1gi - ; for S - they

give Min^j , n-ig< > a* althou gl1tn ^ former was the definition of a !TheV of (55:L')with 8- is thus not a set from our list ofsolutions, henceit is

no solution at all. It would have beeneasyto verify this directly.)))

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THESIMPLEGAME [1, , 1,n -2]* 499

Proof:Ad (55:P':a):We proved thesestatementsimmediatelypreceding(55:N').

Ad (55:P':b):We saw in the proof of (55:B')that a* < 2-ZJP,hence

w * + l = n ~p-pa^ > 0,w* > -1.a* < _ wasstated in (55:B').55.12.Interpretation of the Result

55.12.1.We can now beginto interpret this result. Itis hardly possibleto do this in an exhaustive way for two reasons.Firstthe final resultcontainedin (55:0'),i.e.in (55:G),(55:K'),(55:L') is rather involved,hencea precisestatementmust necessarilybe mathematical and not verbal.Any verbal formulation would fail to do justiceto someof the numerousnuancesexpressedby the mathematical result. Secondwe still lack thenecessaryexperienceand perspectivefor a really thoroughgoing interpreta-tion of a situation like the presentone. The game which we considerhereis a characteristicn-persongame in somesignificant ways, as we setforth in 54.1.2.and 54.3.But our successin determiningall of its solutionsis still an isolatedoccurrence(thecaseof 54.2.1.notwithstanding). It will

takemany more discussionslike this one before one can attempt reallyexhaustiveinterpretationsof characteristicn-persongames.

It is neverthelessuseful to do a certain amount of interpreting without

any claim of completeness.We have seenin several previous instancesthat such interpretationsgive valuable guidancefor the further progressof the theory. Besides,this proceduredoesthrow some light on thesignificance of our rathercomplicatedmathematical result.

Sincewe do not try to be complete,the interpretation is best madein the form of severalremarks.

55.12.2.First:Thesolution of Case(I) describedin (55:G)is an infinite

setof imputations. Thesameis true for the solutionsof (II\,")describedin (55:L')(cf. (55:N')) sincethe y mentioned therevaries over an entireinterval which doesnot shrink to a point. (Cf. (55:P':b).)On the otherhand the solution of Case(IF) is a finite setof imputationsas was alreadyobservedat theend of 55.7.l Thissolution alsohas the attractive propertyof sharingthe full symmetry of the game i.e.invariance under all permuta-tions of the players1, , n 1.

Thus it is in severalways the simplestsolution of our game. Heuristicdiscussionsof its specialcasesn = 3,4 (in 22.,35.1.respectively)led to thissolutionand it is easy to extendthem to the generaln.2 It takesthe full

machineryof our formal theory to find the other solutions.Itwill besufficientlyclearto the readerby now that theseothersolutions

can in no way be disregarded.Besides,the existenceand the uniqueness1 Thereadermay compareto the same effect (55:P':a),(55:P':b).*The (heuristic) argument would run as follows: The chief player needsan ally to

win, with any such ally he obtains n - 2. Thus if he wishes to retain the amount w (thiscorrespondsto the o> of our exactdeductions) he canconcedeeachally n 2 w. If)))

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500 SIMPLEGAMES

of a finite solution is a favorable contingency in thepresentgame,but by nomeansgeneral.1

66.12.3.Second:Theabove solution correspondedto thelargestpossibleS+:(1, , n 1). Theother extremeis the solution which we associ-atedwith S* = Q (cf. preceding(55:0')).This is the solution in Case(I)describedin (55:G).Like that one in the precedingremark it possessesthe full symmetry of the game. Indeedthesetwo the Cases(I)and (IF)arethe only oneswith this symmetry.2

On the otherhand thissolution is infinite. As wesawin 55.3.it expressesthe organizational principlethat the chief player is segregatedin the gamein the senseof 33.1.Inspectionof (55:G)disclosesthat this standard ofbehavior i.e.solution offers absolutelyno principleof division among theotherplayers i.e.all imputations where the chief player receivesthe pre-scribedamount belong to it. This is perfectly reasonableby commonsense:Thechief playerbeingexcluded,the other playerscan only combinewith eaol^otherunanimously. All quantitative checksin their relation-ships (i.e.the possibilityof siding with the chief player) being forbidden,there is no telling what the outcomeof their bargaining with eachotherwill be.

66.12.4.Third:The remainingsolutionsarethosein Case(II\"),describedin (55:L')(cf.(55:N')),i.e.those with S* ^ 0, (1, , n - 1). Theyform a morecomplicatedgroup than the two solutionsdealt with above.Indeed,they took up a considerablepart and the most involved one ofour mathematical deductions.Theirinterpretation,too,is more difficultand complicated.We will indicatethe main pointsonly.

We describedin (55:L')in detail how in all imputationsof a standardbehavior i.e.a solution of thiscategorythe playersof (1, , n 1) S*

his n 1potential allies together can make more than that, i.e.if(n - l)(n - 2 - ) < 1,

then his chancesof finding an ally are destroyed and this is the only limit to hisexactions.

Thus w is only limited by (n l)(n 2 ) ^ 1,i.e.w n 2 2-

Sow - n - 2 :n I

Sothe chiefplayer obtains n 2 r if he succeedsin forming a coalition, and

of course 1ifhe doesnot. For the other playersthe corresponding amounts are rit i

and -1.The readercan now verify that this is just the solution arrived at in (55:V), i.e.

Case(II').1As to the uncertainty concerning the existencecf. the end of the secondremark in

53.2.2.An instance where the uniqueness fails is analyzed in 38.3.1.1Any other solution belongsto Case(II\") and sohas an S* * , (1, , n - 1).

Hencean appropriate permutation of the players 1, , n - 1 will carry an elementof S* into oneoutside,thus changing 5* and with it the solution under consideration.)))

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THESIMPLEGAME [1, , 1,n - 2]* 501arecausallylinked to the chief player. I.e.how the respectiveamountswhich they get areuniquely determinedby the amount assignedto the chiefplayer. This connection was expressedby definite functions.1 Thesefunctions couldbechosenin differentways, thus yieldingdifferentstandardsof behavior i.e.solutions but a definite standard meant a definite choiceof thesefunctions. Thus the uncorrelatednessoftheplayers1, ,n 1,so prominent in the secondremark, is now gone. Thereis obviously somekind of indefinite bargaining going on betweenthe chief playerand thoseof (1, , n 1) S*,2 but the relationshipof the latterplayers toeachother is completelydeterminedby the standard.

It is worth while to emphasizeoncemore this difference betweenthesituation describedin the secondremark and in the presentone i.e.betweenthe Cases(I) and (II\.")In the former casetherewas bargaining betweenall playersexceptthe chief player with absolutelyno rulesor correlationsto cover it, 8 so that the standard of behavior had to makeno provision inthis respect.Now we have bargaining betweenthe chief playerand someof the others,but this time the standardmust provide definite correlationsand rules for the opponentsof the chief player. Accordingly there is amultiplicity of possiblestandards.

Thequalitative typesof indefinitenessarising in the Cases(I)and (II\,as discussedabove, are a more general form of that onewhich we investi-gated in 47.8.,47.9. The remarks made there'about the 2-dimensional(area) and one-dimensional (curve) parts of those solutionsare indeedapplicableto our presentCases(I) and (II\,")respectively.

While it is possibleto motivate this difference by verbal arguments ofsome plausibility,they are all far from convincing. The mathematicaldeductionalone,suchas we gave it, gives the real reason and its relative

complicationshowshow difficult it must be to translate it into ordinarylanguage. This is another instanceof a resultwhich can beexpressed,but

scarcelydemonstrated,verbally.66.12.6.Fourth:Thesituation of the remaining players those in S+

has alsoits interestingaspects.Inspectionof (55:L/) showsthat in every imputation of our solution

eitherall theseplayersget the amount a* , or one of them getsa% and theothersthe amounts -1. From this one infers immediately:

(a) If S* is a one-elementset then the player in S* getsalways thesameamount:a*

(b) If a* = -1then eachplayer of S* always getsthe sameamount:-1.(c) If neither of (a) or (b) is the case i.e.if the condition of (55:U)

(alsoreferred to in (55:D'))is fulfilled then eachplayer in S+

1Thea<(y), t in (1, , n 1) 5*.1Thiscorrespondsto the variability of y in (55:L':e).Of.also(55:P':b).1Exceptfor the assignment to the chief player who is segregated.)))

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502 SIMPLEGAMES

always gets one of the two different amounts a* and 1,andneithercan be omitted.1

From thesewe can draw the followinginterpretativeconclusions:

(d) In the two cases(a) and (b),but not in (c),the playersof S+ aresegregatedin the senseof 33.1.

(e) Thecase(a)where S* is a one-elementset:S+ = (i),i = 1, ,n 1,expressesthe segregationof the player i alone. Thevaluea* which is then assignedto him, is limited by (55:B'):

(55:29) -1Sa,<^~This is a satisfactory complementto the segregationof the chiefplayer, Case(I), describedin the secondremark.2 The value w

which was then assignedto the chief playerwas limited by (55:G):(55:30) -lgw<n-2- ^

(f) If S* is not a one-elementset,then thereis within the cases(a), (b)only the possibility(b):a* = 1.In other words:If more than one playeris to be segregated,then their setmust notcontain the chief player, nor all otherplayers, and the segregatedplayersmust all be assignedthe value:

(55:31) a* = -1.(g) We concludefrom (e), (f) that thosesetsof playerswhich can be

segregatedarepreciselythe setsof L 3 the defeatedsets.(h) If only oneplayeris segregated,then (e)showsthat he neednot be

discriminatedagainstin an absolutelydisadvantageousway. I.e.hemay beassignedmore than 1. (55:29), (55:30)alsostatethe upperlimit of what this assignmentcan be:It is clearlythe sameamountwhich this player would get in the finite solutionof Case(I), dis-cussedin the first remark.4 It is very satisfactorythat this extendsthe result of 33.1.2.from n = 3 to all n.

(i) If, on the other hand, more than one player is segregated5 then(55:31)showsthat therecan be no concessions:They must all begiven the absoluteminimum 1.

1I.e.both occurin appropriate imputations of the solution.1This resolvesthe difficulty pointed out in footnote 3 on p.475.8 This is bestverified by recalling the enumeration of the elementsof W and soofL

in the caseC_i in 52.3.4 n 2 ; for the chief player, for the others. Theassignment must be

n i n ilessthan theseamounts.

*I.e.the number p of elements in # is 2. Sincep n 2 (cf.(55:L:a))this canhappen only when n 2 ^ 2 i.e.n :4. This is the reasonwhy the phenomena of(i) and (j) werenot observedin the discussion of n 3.)))

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THESIMPLEGAME [1, , 1,n - 2]* 503

(j) This assertionmust be qualified to the following extent:If S+ hasmore than one element,the a* of (55:29)are still all possibleindeed (55:L')with (55:B')allows for them explicitly. But thesituation of the players in S+ is then describedby (c),and can nolongerbe termedsegregation:They may join coalitionsand therebyimprove their status.

It is clearthat theseremarks, particularly (g), (h), (i), invite furthercomment. Howeverwe will now restrictourselvesto these indicationsand return to the subjectat another occasion.

55.12.6.Fifth: We found a great number of solutions,characterizedby numerousparameters,someof which were even functions which couldbe chosenwith considerablefreedom. The main classification,however,was rather simple:It was affected by the set S* (1, , n I).1The pairs S*, S* exhaust obviously all partitions of 7 = (1, , n)into two sets.Possiblythis is the first indicationof a generalprinciple.In a simplegame a partition into two complementsseemsto decideevery-thing, sinceone of them is necessarilywinning and the othernecessarilydefeated.In generalgames partitions into more setsmay be equallyimportant. At any ratethe role of S* in the presentspecialcasegives thefirst idea of what may be a generalclassifying principlein all games.

We arenot in a position,as yet, to give this surmisea more preciseform.

1We use as in the secondremark S+ to symbolize the case(I),the discussionpreceding (55:0')notwithstanding.)))

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CHAPTER XIGENERAL NON-ZERO-SUMGAMES

56.Extensionof the Theory56.1.Formulation of the Problem

66.1.1.Our considerationshave reachedthe stageat which it is possibleto drop the zero-sum restrictionfor games. We have already relaxedthiscondition onceto the extentof consideringconstant-sumgames with asum different from zero. But this was not a really significant extensionof the zero-sumcasesincethesegameswere relatedto it by the isomorphismof strategicequivalence(cf.42.1.and 42.2.).We now proposeto go thewhole way and abandonall restrictionsconcerningthe sum.

We pointed out before that the zero-sum restrictionweakensthe con-nection betweengamesand economicproblemsquite considerably.1 Spe-cifically, it emphasizesthe problemof apportionmentto the detriment ofproblemsof \" productivity*' proper (cf.4.2.1.,particularly footnote 2 onp.34;also5.2.1.). Thisisespeciallyclearin thecaseof the one-persongame:behavior in this situation is manifestly a matter of productionalone,with

no conceivableimputation (apportionment)betweenplayers. And indeedthe one-persongame offers no problemat all in the zero-sumcase,and aperfectly goodmaximum problemin the non-zero-sumcase(cf. 12.2.1.).

Accordinglyour presentprogram of extendingthe theory to all non-zero-sum gamesmust be expectedto bring us into closercontactwith questionsof the familiar economictype. In the discussionswhich follow, the readerwill soonobservea changein the trend of the illustrative examplesand ofthe interpretations:we shall begin to deal with questions of bilateralmonopoly, oligopoly,markets,etc.

66.1.2.Completeabandonmentof the zero-sumrestrictionfor our gamesmeans, as was pointed out in 42.1.,that the functions 5Cjk(ri, , r n)which characterizedit in the senseof 11.2.3.arenow entirely unrestricted.I.e.,that the requirement))

(56:1) Kk (r ly- , r n) s*-l

1It should benoted that zero-sum gamesnot only coverthe type ofgamesplayed forentertainment (cf.5.2.1.),but alsothat many of them describequite adequatelyrelation-ships of a definitely socialnature. Thereaderwho has progressedup to this point andrecallsthe interpretations which we have made in numerous caseswill be fully aware ofthe validity of this statement.

Thus the distinction betweenzero-sum and non-zero-sum gamesreflectsto a certainextent the distinction between purely socialand social-economicquestions. (Thenextsentencein the text expressesthe sameidea.)

04)))

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EXTENSION OF THETHEORY 505

of 11.4.and 25.1.3.is droppedwith nothing to take its place. Accordinglywe proceedon this basisfrom now on.

This changenecessitatesa completereconsiderationof our theorywith all the attendant conceptson which it is based. Characteristicfunc-tions,domination, solutions, all theseconceptsareno longer defined when(56:1)is dropped.We emphasizethe fact that the problemwhich ariseshereis a conceptualone,and not merely technicalas wereall thosetreatedin ChaptersVI-X,on the basisof our theory of the zero-sum games.1

56.1.3.The prospectof having to start all over again would be verydiscouraging:we have already spent considerableeffort on theseconceptsand the theory basedon them. Furthermorewe face a conceptualproblemand the qualitative principleson which our theory wasbaseddo not seemto carry beyond the zero-sumcase. Thus this final generalization thepassagefrom the zero-sum to the non-zero-sumcase would seem tonullify all our past efforts. We must find therefore a way to avoid thisdifficulty.

At this point onemight recallthe comparablesituation which arosein42.2.Thereour transition from the zero-sumcaseto the constant-sumcasethreatened on a narrower scale with similar consequences.They wereavoidedby an appropriateuseof the isomorphismsof strategicequivalence,as effected in 42.3.and 42.4.

The usefulnessof this particular device was, however, exhaustedbythe applicationreferredto:strategicequivalencesextendthe family of allzero-gamespreciselyto the family of all constant-sumgamesand no further.(This shouldbeclearfrom the considerationsof 42.2.2.,42.2.3.or 42.3.1.)

Sowe must find someotherprocedureto link the theory of the non-zero-sum gamesto the establishedtheory of the zero-sumgames.

56.2.TheFictitious Player. TheZero-sum Extension r66.2.1.Beforegoing further we have to clarify a point of terminology.

Thegameswhich we shall now considerarethosewhere asstated in 56.1.2.condition (56:1)is droppedwithout anything elsetaking its place. We

talked of theseas non-zero-sumgames,but it is important to realize thatthis expressionis meant in the neutral sense, i.e.that we do not wish toexcludethosegamesfor which (56:1)happensto be true. It is thereforepreferableto use a lessnegative name for thesegames. Accordingly weshallcall thegameswith entirely unrestricted3C*(Ti, ,T)generalgames.*

We have formulated the programof linking the theory of the generalgamesin someway to the theory of the zero-sumgames.It will actuallybepossibleto do more:any given generalgamecan bere-interpretedas azero-sumgame.

1Among thesetechnical problems was one which we preferredto treat by a methodinvolving a certain conceptualgeneralization: the caseof the constant-sum games,whichwill be referred to further below in the text.

1This is in agreement with 12.1.2.)))

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606 GENERAL NON-ZERO-SUMGAMES

This may seemparadoxicalsincethe generalgamesform a much moreextensivefamily than the zero-sum games.However,our procedurewill

beto interpret an n-persongeneralgame as an n + 1-personzero-sumgame.Thus the restrictioncausedby the passagefrom generalgamesto zero-sumgameswill be compensatedfor indeed made possible by the extensiondue to the increasein the number of participants.1

66.2.2.Theprocedureby which a given generaln-persongameisre-inter-pretedasan n + 1-personzero-sumgame isa very simpleand natural one.

It consistsof introducing a fictitious n + 1-stplayerwho isassumedto losethe amount which the totality of the other n real players winsand viceversa. Hemust, of course,have no directinfluence on the courseof the game.

Let us expressthis mathematically :Considerthe generaln-persongameF of the players 1, , n, with the functions 3C*(ri, , r n) (k =1, , n) in the senseof 11.2.3.We introducethe fictitiousplayer n + 1by defining

n

(56:2) 3Cn+1 (r,, , r w) = - 3C*(nf , r).*iThevariablesri, , r n arecontrolledby the real players1, , n,

respectively. They representtheir influence on the courseof the game.Sinceit is intended that the fictitious player have no influence on thecourseof the game,a variable r n+i which he controls,was not introduced.2

In this way we obtain a zero-sumn + 1-persongame,the zero-sumextensionof F, to be denotedby T.

66.3.QuestionsConcerning the Characterof P56.3.1.In stating that we have re-interpretedthe generaln-persongame

F as the zero-sumn + 1-persongameT, we imply prima facie that theentire theory of T has validity for T. This assertion requires,of course,closestscrutiny.

We shall now undertake this investigation. It must be understoodthat this cannot be a purely mathematical analysis,like the analysesin theprecedingchapterswhich were basedon a definite theory. We areanalyz-ing oncemorethe foundations of a proposedtheory. Consequentlytheanalysismust in the main be in the nature of plausibilityarguments, evenif intermixedwith subsidiarymathematical considerations.Thesituationis exactlythesameas in thoseearlierinstanceswhere we madeour decisions

1This may serveas a further illustration of the principle statedrepeatedly that anyincreasein the number of participants necessarilyentails a generalization and complica-tion of the structural possibilities of the game.

The formalism of 11.2.3.provided a variable rk for every player k. (In ordertoreplaceit for the present casewe must replaceits n by our n H- 1.) Henceonemightinsist on the appearanceof the variable rn+i of the fictitious player n + 1.

This requirement is easyto meet. It suffices to introduce a variable T+Iwith onlyonepossiblevalue (i.e.to put B +i -1,loc.cit.). Actually one could even use anydomain ofT*+I(i.e.any /3 +i) aslong asall 3C*On, , T, rn+i) areindependent ofTH+I,sothat they arereally functions 3C*(n, , r)asused in the text.)))

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EXTENSIONOF THETHEORY 507

concerningthe theoriesof the zero-sum two-, three-,n-person games.(Cf. 14.1.-14.5.,17.1.-17.9.for the zero-sum two-persongame;Chap.Vfor the zero-sum three-persongame;29.,30.1.,30.2.for the zero-sumn-person game. For the generaln-person game i.e.the relationshipbetweenthe theoriesof T and T the equivalent sectionsbeginwith 56.2.,and go on to 56.12.)

Theresult of our analysiswill be that it is not the entiretheory of Fas a zero-sumn + 1-persongame,in the senseof 30.1.1.which appliestoF,but only a part of it which we shalldetermine.In other words,we shallfind that not the system of all solutions for P, but only a certainsub-systemproduceswhat will beinterpretedas the solutionsof F.

66.3.2.The fictitious player was introducedas a mathematical deviceto make the sum of the amounts obtainedby the playersequal to zero. Itis therefore absolutelyessentialthat he shouldhave no influence whateveron the courseof the game. This principlewas duly observedin the defini-tion of T as given in 56.2.2.We must neverthelessput to ourselvesthequestion whether the fictitious player is absolutely excludedfrom alltransactionsconnectedwith the game.

This caveat is not at all superfluous. As soon as T involves threeormore persons1 the game is ruled by coalitions,as we observedat an earlystageof our analysis. A participationof the fictitious player in any coali-tion which is likelyto involve the payment of compensationsbetweentheparticipants would be completelycontrary to the spirit in which he isintroduced. Specifically:the fictitious player is no player at all, but onlya formal devicefor a formal purpose. As long as he takesno part in thegamein any director indirectform, this is permissible. But as soonas hebegins to interfere, his introduction into the game i.e.the passagefromF to T ceasesto be legitimate.That is, T cannot then be regardedas anequivalent, or a re-interpretationof F, sincethe real playersof T, 1, , n,may have to provide against dangersor may profit by possibilitieswhich

certainlydo not existin F.56.3.3.One might think that this objectionis void due to the way in

which the fictitious playerwas introduced. Indeed,the amounts

3Ci, * * , 3Cn ,

which the real players 1, , n obtain at the end of the play, do notdependon any variable which he controls2 i.e.he hasno moves in the play.Howcan he then be a desirablepartner in a coalition?

It may appearat first that this argument has somemerit. The condi-tionsdescribedmakeit seemthat any coalition of real playersis just as well

1I.e.when n -f 1 ^ 3 which means n ^ 2. Thus only the general one-persongameis free from the objectionswhich follow. This is in harmony with the fact which we haveemphasized repeatedly, that the general n-persongame is a pure maximum problem onlywhen n 1.

n

2 Nor doesthe amount 3Cn_n 2^ 3Cfc which he obtains.)))

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508 GENERAL NON-ZERO-SUMGAMES

off without the fictitious playeras with him. Ishe anything but a dummy?If this wereso,the theory of T couldbe appliedwithout any further quali-fications to P. Howeverthis is not the case.

It is true that the fictitious player, having no moves to influence thecourseof the game,is not a desirablepartner for any coalition. I.e.noplayeror groupof playerswill pay a (positive)compensationfor hiscooper-ation. Howeverhe himself may have an interestin finding allies. Theamount which hegetsat the end of the play 3Cn+i(ri, , r) dependson the moves of the otherplayers on n, , r n and it may be worthhis while to pay one ormore among the playersa (positive)compensationfor ceasingto cooperatewith the others. It is important not to misunder-stand this:As long as F is played, i.e.as long as the fictitious player isreally a formalistic fiction, no such thing will happen;but if the gamereally playedis T, i.e.if the fictitious playerbehavesas a real playerwouldin hisposition,then his offerof compensationsto the othersmust beexpected.

56.3.4.As soon as the fictitious player begins to offer compensationsto otherplayersfor cooperationwith him which, as wesawabove,amountsto their non-cooperationwith others he is an influenceto be reckonedwith.Heoffers to join coalitionsand to pay a price for this privilege and hiswillingnessto pay is fully as goodas a directinfluenceon the game exercisedby ability to makesignificant moves.

Thus the fictitious player gets into the gamein spite of his inabilitytoinfluence its coursedirectlyby moves of his own. Indeedit is just thisimpotencewhich determineshis policyof offering compensationsto others,and thus setsthe above mechanism into motion.

Fora betterunderstandingof the situation it may be helpful to give aspecificexample.

66.4.Limitations of the Useof f

66.4.1.Considera generaltwo-persongame in which eachof the players1,2if left to himself can securefor himself only the amount 1,while thetwo togethercansecurethe amount 1. It is easy to specifydefinite rulesfor a game to bring this about.1 A particularly simple combinatorialarrangementwhich doesit, is as follows:2

Eachplayer will, by a personalmove, chooseoneof the numbers 1,2.Eachone makeshis choiceuninformed about the choiceof theotherplayer.

After this the paymentswill be made as follows:if both players havechosenthe number 1,eachgets the amount i,otherwiseeachgets theamount l.a

1Thus it will beseenin 60.2.,61.2.,61.3.that the bilateral monopoly correspondstojust this.

This construction should be compared with the one used in defining the simplemajority game of three personsin 21.1.,with which it has a certain similarity.1With the notations of all 11.2.3.:0, 2 2 and)))

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EXTENSIONOFTHETHEORY)) 509))

It is easy to verify that this gamepossessesthe desiredproperties.Let us now considerthe fictitious player3 and form the gameasdefined

in 56.2.2.,with its characteristicfunction v(S), Ss(1,2,3).Accordingtowhat we said above))

-1,))

Obviously))v(0) - 0,))

and by the generalpropertiesof the characteristicfunction (of a zero-sumgame)

*3))= -v((l,2))= -1,v((l,3)) = -v((2))= 1,v((2,3))= -v((l))= 1,

v((l,2,3))= -v(0)= 0.))

Summingup:))

(56:3))) v(S) =))

-11

when S has)) elements.))

This formula (56:3)is preciselythe (29:1)of 29.1.2.;i.e.Pis theessentialzero-sum three-persongame in its reducedform, with 7 = 1. Thus itcoincidesequallywith the simplemajority gameof threepersons,which wasdiscussedin 21.1

Now we learnedpreviously from the heuristicdiscussionsof 21.-23.thatthis gameis nothing but a competitionof all playersfor coalitions. Indeed,this is immediatelyobvious,consideringthe nature of the simplemajoritygameof threepersons(cf.21.2.1.).Hencea fictitious playerwill certainlyshowa strong tendency to enterinto coalitions.In fact the gameT is, asfar as the characteristicfunction is concerned,completelysymmetricwith

respectto its three players. Consequentlythe two realplayers 1,2playexactly the same roleas the fictitious player 3,and so thereis no reasonwhy theirability to entercoalitionsshouldbe at all different from his.2

56.4.2.We canalsorevert to theargument usedin the last part of 56.3.3.and apply it to this game:If the fictitious player3 in T behavesas a realonewould,hehasevery reasonto try to prevent theformation of a coupleof theplayers1,2,sincehe losesthe amount 1if this coupleis formed, and wins

1Ofcourseall thesegames coincideonly as far as their characteristic functions areconcerned,but the entire theory of30.1.1.is basedon the characteristicfunctions alone.

To avoid misunderstandings we re-emphasizethis: The rules of the game f, fully

expressedby the JC,arenot at all symmetric with respectto the players1,2,3;3Ct dependson n, rs but not on n. It is only the characteristicfunction v(), SS(1,2,3),which issymmetric in 1,2,3.But weknow that v(S)alonematters. (Cf.footnote 1,above.))))

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510 GENERAL NON-ZERO-SUMGAMES

theamount 2if it is not formed.1 Hencehe will offer player1or player2 acompensationfor disruptingthis couple,i.e.for choosingTI or TJ, respec-tively, equal to 2 instead of 1. This compensationcan bedeterminedbytheconsiderationsof 22.,23.,and turns out to be$.2 Thereadermay verifythis for himself, togetherwith the fact that thisprocedureleadsto the knownresultsconcerningthe simplemajority game of threepersons.

56.4.3.Theexampleof 56.4.1.gives substanceto the objectionformu-latedin 56.3.3.and 56.3.4.Thus the fictitious player n + 1can influencethe game T not directly through personalmoves but indirectlyby offeringcompensationsand thereby modifying the conditionsand the outcomeof the competitionfor coalitions.As pointedout at the end of 56.3.3.,thisdoesnot mean that any suchthing happensin T, i.e.as long as the fictitiousplayer is a mereformalistic fiction. It doeshappen in T if the theory of30.1.1.is appliedto it literally, i.e.if the fictitious player is permitted tobehave (in offering compensations)as if he were a real one. In otherwords,thfi considerationsof the last paragraphsdo not mean that we wantto attribute to the fictitious player abilitiesconflicting with the spirit inwhich he wasintroduced. They servedonly to show that an uncompromis-ing applicationof our original theory to T brings us into such a conflict.Hencewe must concludethat the zero-sumgameF cannot beconsideredanunqualified equivalent of the generalgameF.

What arewe then to do? In orderto answerthis question,it is best toreturn to the analysisof the specificexampleof 56.4.1.wherje the difficultywasexpressedfully.

56.5.TheTwo PossibleProcedures

56.5.1.Onemight try to escapefrom our presentdifficulty by observingthat it wasbroughtabout in 56.4.1.by the exclusiveuseof the characteristicfunction. Indeedthe game T there coincidedwith the simple majoritygameof three persons where the mechanism of coalitionformation isbeyond doubt only to the extentthat they had the samecharacteristicfunctions, but not the same3C*(cf. in particularfootnotes1and 2 on p.509)Thus a possibleexpedientmight be to abandon the claim that the charac-teristicfunction alone matters,and to base the theory on the 3C*themselves.

At closerinspection,however, this suggestionappears to be entirelywithout merit at leastfor the problemunder consideration.

First:Abandonment of the characteristicfunction v(S) in favor of theunderlying 3C*would depriveus of all means to handle the problem.For

1By footnote 3 on p.508and by (56:2):

, r.) - -K,(rlf ) - .(,,,r t )))

f This is the compensation which brings up the player 1or 2 (who joined the fictitiousplayer 3)from the loss-1,to a gain i which is what he would obtain in a couple,i.e.in acoalition of the players 1 and 2. It alsobrings the fictitious player'sgain from 2 downto 1,which is indeed what it should be.)))

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EXTENSION OF THETHEORY 511zero-sumgameswe possessno generaltheory other than that of 30.1.1.,basedon v(S)exclusively. Thus the adoptionof this program would makeour passage from the generalgame F to the zero-sum game T entirelyuseless,sinceit would renderus just as incapableof handling zero-sumgamesas, originally, generalgames.Hencethis sacrifice of our entireexistingtheory wouldbereasonableonly if it werequitecertainthat despiteits adequacy in all other respectsthere was no otheravenue of escape.However,neitherof thesetwo conditionsis fulfilled.

Second:Theretrogressionfrom the characteristicfunction to the under-lying 3C*doesnot meetthe objectionsof the previousparagraphs. Indeed,at the end of 56.3.2.as well as in 56.4.2.we did operatein a manner whichtook the 3Cjt into account. We establishedthe necessityfor the fictitiousplayer in T to offer compensationsin a directmanner, a necessitywhichwas in no way dependentupon a replacementof T by a different gamewith

the samecharacteristicfunction. l

Third:It will appearfrom the discussionwhich follows that it is notnecessaryto sacrificethe theory based on the characteristicfunction, butratherthat the objectionscan be met by a simplerestrictionof its scope.

66.6.2.Reconsiderationof 56.S.2.-56.4.2.shows that we were notjustified in placingthe blamefor our presentdifficulties,with regardto thebehavior of the fictitious player,entirely upon the theory of 30.1.1.

Theconsiderationsof 56.3.2.-S6.3.4.and 56.4.2.were entirely heuristic.Thisisparticularlyimportant in the case56.4.2.where the undesirableresultwasobtainedin a definite way for a specificinstance. Indeed,the treatmentof 56.4.2.referredto the \" preliminary\" heuristicdiscussionof the essentialzero-sumthree-persongame in 21.-23.,and not to its exacttheory in 32.

What happenedthere in 56.4.2.as well as in 56.4.1.can bedescribedin the terminology of the exacttheory as follows:the generaltwo-persongameF of 56.4.1.led to a zero-sum three-persongame T which coincideswith the simple majority game of three persons.The exacttheory of30.1.1.providedvarious solutionsfor this game,which wereclassifiedandanalyzedin 33.1.Now the considerationsof 56.4.1.and 56.4.2.amountedto selectinga particularonefrom among thesesolutions:the non-discrimina-tory solutionof 33.1.3.

Consequentlywe must ask ourselves:Was it reasonableto selectjustthis the non-discriminatory solution? Is it not possiblethat anotheroneamong the solutions i.e.a discriminatoryonein the senseof 33.1.3.is free from the objectionswhich hold us up?

56.6.TheDiscriminatory Solutions

66.6.1.If we had approachedthe essentialzero-sumthree-persongamei.e.the simplemajority game of threepersons from any otherangle,andif it had beennecessaryto selecta particularonefrom among its solutions,

1We made repeateduse of such replacementsin 56.4.1.,but not in the subsequentargument of 56.4.2.!)))

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512 GENERALNON-ZERO-SUMGAMES

therewould have beena strongpresumptionin favor of the non-discrimina-tory one. This solution i.e.the standard of behavior which it represents

gives all threeplayersequal possibilitiesto competefor coalitions,andin the absenceof any definite motive for discrimination one is tempted totreatit as the most \"natural\" solution of this game.1

However,in our presentsituation thereis every reasonto discriminate:In the game P,players1,2arereal players,the original participantsof r,while player 3 is, as repeatedly emphasized,just a formalistic fiction.Throughout the discussionof the precedingparagraphswe have stressedthat this playershouldnot competefor a coalition, and that he shouldnotbe treatedlike the others. In other words,if we expectto be able at alltoapply the theory of 30.1.1.to this situation, then thereis an absolutenecessityof discriminatingagainstthe fictitious player3, i.e.to chooseoneof those solutionswhich were termeddiscriminatory in 33.1.,the excludedplayerbeingthe fictitious player3.

We saw, loc.cit.,that thesediscriminatory solutionsarecharacterizedby the fact that the excludedplayer whom the solution, i.e.the standardofbehavior, disqualifiesfrom competingfor a coalition is assigneda fixedamount c in all imputations of the solution. Itappearedin 33.1.2.that thisamount neednot bethe minimum at which the excludedplayercan maintainhimself alone, i.e.not necessarilyc = 1. Actually c could be chosenfrom a certaininterval: 1^ c <i.

66.6.2.At this point it may be useful to interrupt the discussionfor amoment in orderto comment briefly upon the discriminatory solution whichexcludesthe fictitious player in the worst possiblesituation, i.e.withc = 1. According to 33.1.1.this solution consists of preciselythoseimputationsin which the fictitious player3 gets 1,and eachone of the tworeal playersgets ^ 1.

As pointedout loc.cit.this meansthat the solution i.e.the standardof behavior restrictsin no way the division of the proceedsbetweenthetwo realplayers. The reason given there is now valid in a much morefundamental way: the bargaining of the players 1,2has becomeentirelyunrestricted,not only becausethe acceptedstandard of behavior excludestheinterferenceof player3 which was the only normative influence in therelationshipof players1,2 but alsofor the still betterreasonthat player3doesnot exist. It is easy to seethat this removes the threat that player1or 2 will forsake cooperationwith the other if his \"fair share\" is notconcededby hispartnerand that insteadhe will cooperatewith player3 andobtain a compensationfrom that source.

66.7.Alternative Possibilities66.7.1.Let us now continue the discussionwhere it left off at the end of

66.6.1.1Ofcoursethe other solutions arejust asgoodin the rigorous senseof30.1.1.,but the

abovestatement is nevertheless reasonableprime facie.)))

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EXTENSION OF THETHEORY 513It may seemquestionablewhether we shouldinsist upon c = 1,or

allow the entire variability 1^ c < . Thefirst alternative is the moreplausibleprima fade. Indeed,c > 1meansthat the realplayersdo notexploitthe fictitious playerto the utmost of their possibilities,i.e.that theydonot endeavor to gain as much (asa totality) asfeasible. One might viewsucha self-denial as a compensationpaid to the fictitious player by virtueof the acceptedstable standard of behavior. And sincewe have decidedto excludeany participationof the fictitious playerin the interplay of coali-tions and compensations,thereis somejustification in forbidding this.

It must be conceded,however, that this argument is not altogethercogent. A (positive) compensationpaid by the fictitious player is aqualitatively different thing from one paid to him. The former is a patentabsurdity,sincethe fictitious playerdoesnot existand will therefore notpay compensations. The latter,on the other hand, is not absurd at all.Itmerelyexpressesa self-denialin exploitinga possiblecollectiveadvantage,and wehave had several instancesshowingthat a stablestandardofbehaviorcan requiresuchconduct.1 It is not a priorievident that sucha self-denialis out of the questionin the presentsituation.2 To excludeit would meanthat a stablestandardof behavior in the presenceof completeinformation

necessarilyentails attainment of the maximum collective benefit. Thereaderwho is familiar with the existingsociologicalliterature will knowthat the discussionof this point is far from concluded.

We shall neverthelesssucceedin settlingthis questionwithin the frame-work of our theory by showingthat cmust berestrictedto itsminimum value.3

56.7.2.Forthe moment, however, we must developboth alternativesconcurrently.

Tothis endwereturn to the generaln-persongame r,and the correspond-ing zero-sum (n + l)-persongame T. We arenow able to formulate therelevant conceptsrigorously.

(56:A:a) Denotethe setof all solutionsV of T by 12.

(56:A:b) Given a number c, denotethe systemof thosesolutionsV

of T for which every imputation a = {i, , an , <*n+i} ofV has an.fi = c,4 by Qe.

1This is, of course,just another way to expressthe possibility of 33.1.2.Another

instance, in a zero-sum four-person game, is given in (38:F)of 38.3.2.Still anotherobtains for all decomposablegames in 46.11.(In this last instance the self-denialisexercisedby the players of A when < 0, and by thoseof H when & > 0,cf.loc.cit.)

We emphasize that such self-denial is exercisedunder the pressureof the acceptedstandard of behavior, although the players areassumed as always in our theory to beinformed fully about the possibilities of the game.

1However if it occurred, it would be regarded normally as an inefficient thoughstable form of socialorganization.

8 I.e.the self-denial in question doesnot occurand the maximum socialbenefit isalways obtained. This result is not assweeping as it may seem,sincewe areassuming anumerical and unrestrictedly transferable utility, aswell ascompleteinformation.

4I.e.where the fictitious player gets the same amount c in all imputations of the

solution.)))

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514 GENERAL NON-ZERO-SUMGAMES

(56:A:c) Denotethe sum of all setsQc by ft'.(56:A:d) Denotethe ft c of c = v((n+ 1))= -v((l, , n)) by

ft\". 1

In connectionwith (56:A:c)we notethis:Forsomec the set ft c is empty. Thesec may obviously be omitted

when ft' is formed. Thus an+i ^ v((n+ 1)) v((l, , n)) necessi-tatesc <z v((l, , rc)),otherwiseft c is empty. Again))

= - a.g -))

necessitatesc ^ ^ v((fc)), otherwiseft c is empty. So c is subjecttofc-i

the restrictionn

(56:4) -v((l, , n)) S c S -))

Jb-l

Actually it is usually even more restricted.2

The ft\" of (56:A:d)belongsto the minimum c of (56:4).

56.8.TheNew Setup

56.8.1.Our discussionof S6.3.2.-56.4.3.showedthat the solutionsof ft

are certainly not all significant for T. The analysis of 56.6.1.restrictedthosesolutionsfurther, but it left the question unansweredwhether thesystemof all significant solutionsis ft' or 12\".

Thus the systems12'and 0\" correspondto the two alternativesreferredto.

We now proceedto differentiate between12'and ft\".

Considerthe imputations

(56:5) *= {i, , n, n+i}

of the gameT. Among the components i, , n, an+i the n firstones,01, , an expressrealities:the amounts which the realplayers,1, , n respectively,are to obtain from this imputation. The lastcomponent,a+i on the other hand, expressesa fictitious operation:theamount attributed to the fictitious playern + 1. Further,this component

1I.e.where the fictitious player gets, in all imputations of the solution, only thatamount which he couldobtain for himself even in opposition to all others. This meansasweknow that the real players obtain together the maximum collectivebenefit.

1Thus in the essentialzero-sum three-persongame, (56:4)gives-1 c 2,

while weknow from 32.2.2.that the exactdomain ofc (with non-empty ft) is

-1 c < J.)))

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EXTENSION OF THETHEORY 515an+i is not only fictitious in the interpretationof P, but it is also mathe-matically unnecessary, i.e.it is determinedif the ai, , an areknown.

Indeed(sincethe sum of all componentsof the imputation a must bezero)

n

(56:6) an+i = - J) ak .))

Consequentlyit may be preferableto expressa by specifying its compon-ents i, , an only, always rememberingthat an+i can be obtained ifdesired from (56:6).Thus we shallwrite

(56:7) 7= {{at, -,.}}.We observethat thisnotation isnot intendedto supersedethe originalone,i.e.we wish to befree to useboth (56:5)and (56:7),whichever may be moresuitable. Indeed,it is in orderto avoid misunderstandingswhich mightensuefrom this double notation, that we areusing the doublebrackets{ { }}in (56:7)insteadof the simpleones { |in (56'.5).1

56.8.2.Theimputation a in its form (56:5)was subjectto thezero-sumrestriction,and also to the restriction

(56:8) a, ^ v((t)) for i = 1, , n, n + 1.We must express(56:8)for (56:7)(with (56:6)).

Now for t = 1, , n (56:8)is unaffected by the transition from

(56:5)to (56:7),but for t = n + 1we must make use of (56:6).So itbecomes

a* ^ -v((n+ 1))= v((l, , n)).-i1 Of coursewe couldhave done this all along, i.e.for the original zero-sum n-person

games. Herean imputation

a (ai, , a*)

is determined if only its components ,-,i ?* i* aregiven (for any fixed t ), since))

In conformity with this we have observedalready in (31:1)in 31.2.1.that the imputa-tions of the (essential)zero-sum n-person game form an (n l)-dimensional, and not ann-dimensional, manifold.

However, there was no particular advantage to be gained by getting rid of an vand there was no way to decidewhich a f should beeliminated, if any. In the graphical

discussion of the essential xero-sum three-persongame we actually made an effort tokeepall a, in the picture. (Cf.32.1.2.)

Thesituation now is altogether different, considering the specialrole of <x+i. Theelimination of <x+i will beessentialfor our subsequent deductions.)))

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516 GENERAL NON-ZERO-SUMGAMES

Thus (56:8)goesover into this:(56:9) a< v((t)) for i = 1, , n;

(56:10) < g v((l, , n)).t-i56.0.Reconsiderationof the CaseWhere T isa Zero-sum Game

56.9.1.Let us stop for a moment to interpret theserestrictions.(56:9)is not new. It expressesagain what we had already for the

zero-sumgames,namely that nobodywould acceptin any caselessthan hecan get for himself in oppositionto all others. (56:10),however, appearsfor the first time. Itsmeaning becomestransparent if we considerthequantity v((l, , n)) more closely.

v((l, , n)) is the value of the game for the compositeplayercomprisingall realplayers 1, , n, and playing against the fictitiousplayer n + 1. The amount which this compositeplayer getsat the endof a play is,of course))

Hecontrols the variablesTI, , r n , i.e.all the variableswhich occurin this expression.Thus in the zero-sum two-persongame the real playerscontrol all moves the fictitious player having no influence on the courseof the game.

In comparingthis with the zero-sumtwo-personschemedescribedinn

14.1.1.our 2) 3C* correspondsto 5C there, all our variablesTI, - , r n*-l

to the one variable TI there,while no domain of variability in our presentset-upcorrespondsto the variable r 2 there.

It is intuitively clearthat the value of sucha game (for the first player)obtains by maximizing with respectto all variables (sincethey are allcontrolledby him). This is))

(56:11) Maxrj.....% 3C*(n, , r n)k-lin our setup, the correspondingexpressionin the schemeof 14.1.1.being

(56:12) MaxTi OC(ri,r 2) (r* is really absent).

Of coursethe systematictheory of 14.,17.givesthe sameresult:Vi, v 8

in 14.4.1.areequal to eachotherand to (56:12),sincethe operationMinTjis void. Sothe gameis strictly determinedand has the value (56:12)in)))

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EXTENSION OF THETHEORY 517the senseof 14.4.2.and 14.5.Consequentlythe generaltheory of 17.yieldsnecessarilythe samevalue.

Sowe see:))

(56:13) v((l, - - ,n)) - MaxTi.....% X*(n, , r n).*-iConsequently(56:10)expressesthis:No imputation shouldoffer all (real)players togethermore than the totality can expectin the most favorablecase,i.e.assumingcompleteco-operationand the bestpossiblestrategy.1*2

Summing up:(56:B) The imputations of (56:7)are subject to the following

restrictions:(56:B:a) No real playermust be offered lessthan he can obtain for

himself even in oppositionto all otherplayers(cf. (56:9)).(56:B:b) All real playerstogethermust not be offeredmore than the

totality can expectin the most favorable case,i.e.assumingcompleteco-operationand the best possible strategy (cf.(56:10)and (56:13)).

This formulation makes the common-sensemeaning of our restrictions(56:9),(56:10)(i.e.(56:B:a),(56:B:b))quite clear:A violation of (56:9)(i.e.of (56:B:a))meansthat one of the (real)playersreceivesan offerwhichis more unfavorable than what can beenforcedagainsthim. A violation of(56:10)(i.e.of (56:B:b))meansthat thetotality of all (real)playersreceivesan offer which is more favorable than it couldever expectto achieve. Itseemsreasonableto considertheseas preciselythe conditionsunder which

playerswho actrationally will refuse to considera distributionscheme(animputation) becauseit is manifestly absurd.

56.9.2.Beforeproceedingany further we must oncemore retraceour

steps and compareour presentsetup with the previousone,in the caseswhere both apply.

Specifically:Assume that we are applying the procedureof the pastsectionsto an n-persongame T which is already zero-sum.Accordinglywe form for this gamethe zero-sumn + 1-persongame T as describedin

56.2.2.,and then proceedas in 56.8.2.1Note that the conceptof a beststrategy for the totality of all real players is clearly

defined: if there is complete co-operation,then the totality facesa pure maximum

problem.1If the game in its original form i.e.beforethe normalization of 12.1.1.and 11.2.3.is performed contains chancemoves, then the \"most favorable case\"referredto abovemust not be taken to include thesetoo. I.e.only co-operationand optimal choiceofstrategiesis to be assumed,while the chancemoves must be accountedfor by formingexpectationvalues. Indeed,it is in this way that wepassedin 11.2.3.from the

9*(T, Ti, , Tn)

(TO representing the influence of all chancemoves) to the JC*(n, , r) which we areusing now.)))

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518 GENERAL NON-ZERO-SUMGAMES

It is important not to misunderstandthe meaning of this operation.Obviously the operationsof 56.2.2.and 56.8.2.areentirely unnecessaryifF itself is a zero-sumgame;we possessa theory which disposesof this case.But if a more generaltheory, valid for all games,is to be constructedonthis basis, then we must demand that it agreewith the (morespecial)oldtheory as far as the lattergoes. I.e.in the domain of the old theory, wherethe new theory is superfluous,the new must agreewith the old.1

56.9.3.That F is a zero-sumn-persongame meansn

^(n, , r n) m 0,*-l

i.e.OCn+iCn, , r n) m 0. Thus v(S) is not affected if the fictitiousplayern + 1 is added to (orremoved from) the set*S. I.e.:(56:14) v(S) = v(Su (n + 1)) for 8 (1, , n).ThespecialcasesS = , (1, , n) give

(56:15) v((n + 1)) = 0,(56:16) v((l, , n + 1)) = 0.

(56:14),(56:15)togethershow that the gameT is decomposablewith

the splitting sets(1, , n) and (n + 1). Its(1, , n) constituentis the original game F, while the fictitious player n + 1 is a dummy.2

(Forthe decompositioncf. the end of 42.5.2.as well as 43.1.For thedummiescf. footnote 1 on p.340,and the end of 43.4.2.)

Now we can observe:56.9.4.First:SinceT obtains from F by the addition of a dummy, the

solutions of F and T (in the old theory) correspondto eachother, theonly difference beingthat the latter takes careof the dummy (thefictitiousplayer n + 1) also,assigninghim the amount v((n 4- 1)),i.e.zero. (Cf.46.9.1.or (46:M)in 46.10.4.)

Our proposednew theory would obtain the solutions for F from the(oldtheory) solutionsof P. Hencethe above considerationprovesthat allthenew solutionsto be obtainedfor F will be among the oldones. Further-morewe seethat in this casewe can indeedmust take the entiresystemQ of (56:A:a)in 56.7.2.It must be noted,however, that in this caseallsolutionsof Q automatically assignthe fictitious player n + 1 the amountv((n + 1)). I.e.here fl = Oc with c = v((n + 1)), i.e.fl * ft\". (Cf.

1This isa well known methodological principle of mathematical generalization.*Thereadershould recallthat the fictitious player is not, in general,a dummy in the

game f . This may sound paradoxical,but it was establishedin 56.3.for the very specialcaseof the generaltwo-person gamesr. Indeed,it is just becausethe rules of the game fdo not in general assign him the role of a dummy that we must restrict the solution Vof F to those which do restrict him to such a role. This is the meaning of the discussionsof 56.3.2.-56.6.2.

We shall determine in 57.5.3.which propertiesof r are necessaryand sufficient inorderthat the fictitious player bea dummy.)))

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EXTENSION OF THETHEORY 519(56:A:b)and (56:A:d),ioc.cit.)Consequentlyany sets we might definebetweenft and 0\" in particular,both 12'and 8\" of (56:A:c)and (56:A:d),Ioc.cit. coincidewith Qand areequallyacceptablefor our purpose.

In other words:the choicebetweenQ'and 8\" which is still aheadof usis of no significancein this case. Both alternatives herearein agreementwith theoldtheory;indeed,thereis no needhereto abandonthe old theoryat all.1

56.9.5.Second:The imputations for a zero-sum n-persongame weredefined in the old theory in this way:

(56:C:a) a = (i, -,};(56:C:b) <* v((t)) for i = 1, , n;

n

(56:C:c) <= 0.-i

Our newarrangement of (56:7)in 56.8.1.differsfrom this. Herewehave

(56:C:a*) \"^ - ({ -,}};and by (56:9),(56:10)and (56:16),(56:C:b*) < ^ v((t)) for i = 1, , n;

n

(56:C:c*) a, 0.i-lWe alreadyknow from the precedingremark that therecan be no real

difference in the present casebetween the old theory and the new one.2It is neverthelessuseful to seedirectlythat (from the point of view of theold theory) the two procedures(56:C:a)-(56:C:c)and (56:C:a*)-(56:C:c*)contain really no discrepancy.

The only difference between these two arrangementslies in (56:C:c)and (56:C:c*).Recalling the definitions of 44.7.2.,we seethat the differ-encebetween(56:C:a)-(56:C:c)and (56:C:a*)-(56:C:c*)can alsobe statedin this way:Thefirst amounts to consideringsolutionsfor E(G); the second,to consideringsolutionsfor F(0).Now we have noted in 46.8.1.that liesin the \"normal\" zone of the game T, and by (45:0:b)in 45.6.1.,E(0)and

F(0)have the samesolutions. Thus we have a perfect agreement.Thesetwo remarksmade systematicuse of the theory of composition

and decompositionof Chapter IX, in order to analyze the influenceof our

contemplatednew procedureon the zero-sum gamesr. This procedureconsistedmainly in the passagefrom T to T, which as we sawamounted tothe additionof a dummy to F. This is considerablymore specialthan thegeneralcompositionsdealt with Ioc.cit. The specificresults used could

1Thenecessityof restricting ft was deducedin 56.5.-S6.6.by considering a non-serosum game r.

1Orrather any new onebuilt along the lines contemplated we have not yet made thedecisionbetween ft' and ft\".)))

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520 GENERAL NON-ZERO-SUMGAMES

accordinglyhave beenobtainedwith lesseffort than by the use of the farmore generaltheoremsreferredto. We shall not enter into this subjectfurther sincethegeneralresultsof ChapterIXareavailable in any case,andbecausethe above treatmentprojectsour presentconsiderationsmoreclearlyupon theirproperbackground.

66.10.Analysis of the Conceptof Domination

56.10.1.We now return to the generaln-person game F, its zero-sum extensionF, and the new treatment of imputations as introducedin 56.8.

Certainlyall solutionsof T in generalcannot be used todefine a satis-factory conceptof solutionsfor F. This was establishedby the consider-ation of a specialcase i.e.by casuisticprocedure in S6.5.-56.6.Let usnow approachthis problemsystematically;i.e.applyto thegamethe formaldefinition of a solution as given in 30.1.1.and try to determinein full

generalitywhich of its featuresareunsatisfactory and requiremodification.In doingthis we shall use the conceptof imputation (of T) in the new

arrangement(56:7)in 56.8.1.Theimportant point about this arrangementis that it stressesdb initio the primary importanceof the realplayersin P,i.e.directsour attention more to F than to T. This doesnot impair, ofcourse,the fact that we apply the formal theory of 30.1.1.to the zero-sumn + 1-persongameT, and not to the generaln-persongameF (whichwould not bepossible).

Theconceptsof 30.1.1.areall basedon that of domination. We there-fore beginby expressingthe meaning of domination as defined loc.cit.forimputations (of T) with the new arrangementof (56:7)in 56.8.1.

Considertwo imputations

7 = {{a,, .a.}}, 7 = {{/Si, -,&)}.Domination))

means that thereexistsa non-empty set S S (1, n, n + 1) which is

effective for a , i.e.(56:17) < V (S)'

inSsuch that

(56:18) at > ft for all i in 8.We wish to expressthis in terms of the ,,ft with i = 1, , n alone.It is therefore necessaryto distinguishbetweentwo possibilities:

56.10.2.First:Sdoesnot contain n + 1. Then

(66:19) Ss(1, ,n), Snot empty.)))

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EXTENSION OF THETHEORY 521Theconditions(56:17),(56:18)above neednot be reformulated sincetheyinvolve only the a, ft with i = 1, , n. BesidesS (1, , n)in the v(S)of (56:17).

Second:Sdoescontain n + 1. Put T = S - (n + 1).Then

(56:20) T (1,- , n), T may beempty.

The conditions(56:17),(56:18)above must be reformulated sincetheyinvolve an+i, fti+i.

It is natural to form S in (1, , n, n + 1),i.e.as (1, , n,n + 1)- <S; and -T in (1, , n); i.e.as (1, , n) -T. Thesetwo setsareclearly equal,but it is neverthelessuseful to have symbolsforboth. We denotethe first by _LSand the secondby T.

n+lSince^ a< = 0,so))

iinS tin 5 in -Tv(S) = -v(JLS)= -v(-T).

Hence(56:17)becomes

(56:21) ,-fcv(-T).in -71

This involves only the a,with i = 1, , n. Besides Ts(1, , n)in the v( 5P) of (56:21).Next (56:18)becomes

(56:22) >& for all t in T,and

<*n+l > 0n+l'

This last inequality meansthat

(56:23) f i < t ft--i -i(56:22),(56:23)also involve only the a, ft with t = 1, , n.

Summing up:

(56:D) a H means that thereexistseither(56:D:a) an Swith (56:19)and (56:17),(56:18);

or(56:D:b) a T with (56:20)and (56:21),(56:22),(56:23).Note that thesecriteriainvolve only setsS,T, -Tfi(1, f n) and the

cti, ft with i 1, , n. I.e.they refer only to the original game Tand to the realplayers1, , n.)))

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522 GENERAL NON-ZERO-SUMGAMES

56.10.3.Thecriterion(56:D)of domination was obtained by a literalapplicationof the original definition of 30.1.1.,the applicationbeingmadedirectly to T and then translated in terms of T. This rigorousoperationhaving beencarriedout, let us now examinethe result from the point ofview of interpretation;i.e.let us seewhether the conditionsof (56:D)producea reasonabledefinition of domination for the presentcase.

According to (56:D) domination holds in two cases(56:D:a)and(56:D:b).

(56:D:a)is merely a restatementof the original definition of 30.1.1.1Itexpressesthat thereexistsa groupof (real)players(thesetSof (56:19)),eachof whom prefershis individual situation in a to that in ft (this is(56:18)),and who know that they areableas a group,i.e.as an alliance,toenforcethis preferenceof theirs (this is (56:17)).

(56:D:b)on the otherhand is, when viewed in terms of T and of thereal players alone,somethingentirely new. It requiresagain that thereexista group of (real)players (theset T of (56:20))eachof whom prefershis own individual situation in a to that in ft (this is (56:22)).Theability of this group to enforce the preferencein question(i.e.(56:17))isnot required. Insteadwe have the condition that the real players leftout of this group must not be able to block the preferredimputation in

question,that is insofar as it affects them (this is (56:21)).2

Finally there is the peculiarcondition that the totality of all (real)players i.e.societyas a whole must be worseoff under the (preferred)

regimea than under the (rejected)regimeft (this is (56:23)).56.10.4.This strangealternative (56:D:b)was, of course,obtained by

treating the fictitious player n + 1as a real entity. If we refrain from

1 Applied, however, to the generalgame F, for which that theory was not intended !1The (real) players left out, i.e.those of 7\\ could block the preferred imputation

a if they couldget for themselves separatelymore than a assigns to them together;i.e.if))

in -T

(Note that we had to excludeequality here,sincethat would not block a .) Thenegationof this is indeed:

(56:21) J) <fcv(-T).tin -T

Thismay becomparedwith the expressionof the ability of the original group T toenforceits preference,i.e.(56:17) J) , v(T).

iinT

Itshould be noted that neither of (56:17),(56:21)implies the other:It is perfectly

possiblethat the group T can enforcea as far as it affectsthe members of T,and that at)))

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EXTENSION OF THETHEORY 523

doingthat and try to appraisematters in termsof realitiesi.e.of the realplayers then it becomesvery difficult to interpret (56:D:b).The bestone cansay of it is that it seemsto assumethe effective operationof aninfluencewhich is definitely setto injure societyas a whole (i.e.the totalityof all real players). Specificallyin this casedomination is assertedwhenall playersof a certaingroup (of real players)prefer their individual situa-

tion in a. to that in , if the remaining (real)playerscannot blockthisarrangement,and if it is definitely injurious to societyas a whole.

In comparingthis domination (56:D:b)to the ordinary one (56:D:a)the followingdifferencesareparticularly conspicuous:First,that in (56:D:a)the ability to enforce one'spreferenceis essential,while in (56:D:b)theessentialpoint is the ability of the others to block it. Second,that in(56:D:a)the active grouphad to be a non-empty set,whereasin (56:D:b)it couldalso be an empty set (cf. (56:19)and (56:20)).Third, the anti-socialviewpoint figures in (56:D:b),but not at all in (56:D:a).

Thereaderwill have noticedby now that (56:D:b)is of a rather irra-tional character,but neverthelessnot altogetherunfamiliar. It would beeasyto enlargeupon the imagesand allegoriesof which (56:D:b)is an exactformalization. Thereis no need to dwell further upon this subjecthere.What mattersis that we have every reasonto seein the alternative (56:D:b)the generalcausefor thosedifficultiesfor which a specialcasewas analyzedin 56.5.-56.6.Clearly(56:D:b)is not an immediately plausibleapproachto the conceptof domination in the sensein which (56:D:a)is.

We shall therefore try to resolveour difficultiesby the simpleexpedientof rejecting(56:D:b)altogether.

66.11.Rigorous Discussion

56.11.1.We have decidedto redefine domination by rejecting(56:D:b)and retaining (56:D:a)in (56:D)of 56.10.2.This new conceptof domina-tion can be stated in two ways which both seemto deserveconsideration.

First:As pointedout at the beginning of 56.10.3.,(56:D:a)amounts toa repetitionof the correspondingdefinition of 30.1.1.Theonly difference))

the same time the group Tcan block a asfar asit affectsthe members of T. Ontheother hand it isalsopossiblethat neither group can enforceor prevent anything.

n

However, if Tisa zero-sum game, and if we require (asin the old theory) 2/ ~0,-ithen (56:17)and (56:21)areequivalent. Indeed,in this case

v(T) + v(-T) - v((l, , n)) -0,so

2) - - 2) , v(-T) - -v(T),iin-r iinT

from which the equivalence follows as asserted.)))

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524 GENERAL NON-ZERO-SUMGAMES

is that then F was a zero-sum n-persongame,while now it is a generaln-persongame!

Thus our presentproceduremeans that we extend to the present casethe definition of domination in 30.1.1.unchanged,irrespectiveof the factthat the gameis no longerrequiredto be of zero sum.1

Second:Let us now view the restriction to (56:D:a)from the stand-point of T rather than of F. Our original discussionin 56.10.yieldedthe two cases(56:D:a)and (56:D:b)dependingupon the followingdisjunc-tion. In the senseof 30.1.1.the domination in T had to bebasedon asetS.Now (56:D:a)obtains when n + 1does not belong to S,while (56:D:b)obtainswhen it does.Hencethe restrictionto (56:D:a)amounts to requir-ingthat thesetS must not contain n + 1.

We repeat:Our new conceptof domination means, in terms of T,that in the definition of domination in 30.1.1.we add to the conditions(30:4:a)-(30:4:c)imposedupon the set/S, the further condition that S mustnot contain a specifiedelement,namely n + 1.

This can alsobeconstruedas a restrictionon the conceptof effectivityloc.cit.:We regarda setSas effective only if it doesnot contain n -f 1.(Of course,the original condition(30:3)loc.cit.is alsorequired.)

66.11*2.We now proceedto study the new conceptof solution for P,i.e.for F, basedupon the new conceptof domination, introducedin 56.11.1.In analyzing it we shall rely upon the game T and the form (56:5)of impu-tations (ratherthan upon the game F and the form (56:7)of imputations)and the definition of domination as formulated in the secondremark of66.11.1.

We obtain our result by proving four successivelemmas:(66:E) If V is a solution for T in the new sense,then every

a = {i, -,,an+i)

of V has aw+1 = v((n + 1)).Proof:Assume the opposite.Necessarily an+i ^ v((n + 1)), hence

therewouldexistan a = (i, , aw, nfi} in V withan+i > v((n + 1)).Put an+i - v((n+ !))+,>0. Define 7 = |0i, , 0,P*+i\\ with

Pi = cti + ^ for i = 1, , n;n+i = <*+!- t = v((n + 1)).

1It may seempeculiar that it took us so long to reachthis simple principle, in factweneedthe further considerations of 56.11.2.beforeweacceptit finally. However, theact of taking over the definition of 30.1.1.without any alternatives, in spite of theextremely wide generalization which is now performed, requires most careful attention.Thedetailedinductive approachgiven in theseparagraphs seemedto bebestsuited forthis purpose.)))

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EXTENSIONOF THETHEORY 525n

Since ft = -ft+i= -v((n+ 1)) = v((l, , n)) and ft > < fort-it = 1, , n, sothe useof S = (1, , n) establishes ft H a .l As- _a belongsto Vf ft cannot belongto it. Hencethereexistsa y in V

with 7 H ft . Now considerthe set S which enforcesthis domination.SinceS doesnot contain n + 1,we have S (1, , n). As ft > a<

> > _for t = 1, , n, 7 H implies 7 H a . But 7 , a areboth in VJhencewe have a contradiction.

(56:F) If V is a solutionfor T in the new sense,then it is so alsoin the old sense.

Proof:We must showthat (30:5:a),(30:5:b)of 30.1.1.,with dominationin the new sensehere imply the samewith domination in the old sense.Now domination in thenew senseimpliesdomination in theoldsense;henceour assertionconcerning(30:5:b)is immediate.Soonly (30:5:a)requirescloserinspection.

Assume therefore that (30:5:a)is invalid in the old sense,i.e.that for

two a. , ft in V, a H ft in theoldsense. Let Sbe thesetwhich enforcesthis domination. By (56:E) an+i = ft*+i (= v((n+ 1))), h$nce n + 1cannotbelongto S. Consequentlya H ft in the new sense,i.e.(30:5:a)fails in the new sensetoo. This completesthe proof.(56:G) If V is a solutionfor T in the old sense,and if every))

of \\f has an+i = v((n+ 1)),then V is alsoa solutionin the newsense.

Proof:We must show that (30:5:a),(30:5:b)of 30.1.1.with dominationin the old sensehere imply the samewith domination in the new sense.Now domination in theoldsenseis impliedby domination in thenew sense;hencethis time our assertionconcerning(30:5:a)is immediate.So only(30:5:b)requirescloserinspection.

Considertherefore an a = {i, , an , <*n+i} not in V- As (30:5:b)> _

holds in the old sense,thereexistsa ft = {0i, , n, n+i} in V with

ft H a in the oldsense. Let S bethesetwhich enforcesthis domination.

Necessarilyan+i ^ v((n+ 1)),and since ft belongsto V by assumption,ft n+i = v((n + 1)).Henceft n+\\ g an+i and so n + 1cannot belongto S.Consequentlyft H a in the new sense,i.e.(30:5:b)holdsin the new sensetoo. This completesthe proof.

1This domination aswell asall others in this proof are in the new sense.)))

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526 GENERAL NON-ZERO-SUMGAMES

(56:H) S?is a solutionof T in the new sense,if and only if it belongsto the system Q\" of (56:A:d)in 56.7.2.

Proof:The forward implication results from (56:E) and (56:F),theinverseimplication from (56:G),

56.11.3.In interpretingthe resultof (56:H)we must rememberthat thediscussionoriginatedfrom the necessityof restrictingthe system 8 of allsolutionsof T for thepurposesof the theory of F. We sawin 56.7.that theplausibleresult of this restrictionshouldbe the set Q' or ft\" (orpossiblysomesetbetweenthe two). Thereafter our effort wasdirectedto makinga decisionbetween these two possibilities. Furthermorewe concludedin 56.10.-56.11.1.that a modification of the conceptof domination in Tmight answer our problem.Now the statementof (56:H) is that thismodification of the conceptof domination leads preciselyto the set Q\".By theseconcurrentresults the decisionis clearlyindicated. We acceptQ\" as the systemof all solutionsfor F.

56.12.TheNew Definition of a Solution

56.12.We reformulate this togetherwith referencesto the main resultson which the decisionwasbased:(56:1)(56:I:a) Fora generalw-person gameF, a solution is any solution

(in the original senseof 30.1.1.)of its zero-sumextension,thezero-sumn + 1-persongame T, for which all

a = {i, -,,,<*n+i}in V have

(56:24) an+1 = v((n+ 1)).Thesesolutionsform preciselythe set Q\" of (56:A:d)in 56.7.2.

(56:I:b) Using the form (56:7),a ={ {ai; fc- , an } } for these

imputations (i.e.emphasizingF and its playersratherthan T),transforms (56:24)above into

(56:25) <= v((l, , n)).-i

This is clearlya strengthenedform of (56:10)in 56.8.2.(56:I:c) In the specialcasein which F is itself a zero-sumgame,our

new conceptof solutions(for F) coincideswith the old one,i.e.the unmodified applicationof 30.1.1.(Cf. the first remarkin 56.9.4.)Thus it is no longernecessary to distinguishbetweenthe old theory and the new one. (Cf.alsofootnote 1on p.518.))))

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THECHARACTERISTICFUNCTION 527

(56:I:d) Fora generaln-persongameF, the solutionscan also beobtained by applying the definitions of 30.1.1.(which wereintended for zero-sumgamesonly) to F directlyand withoutany modification. Theconceptof imputationsfor F must thenbeused in the form (56:7). (Cf.the first remark in 56.11.1.)

(56:I:e) Thevalidity of (56:I:d)meansthat nothing must beaddedto the characterizationof the imputations in the form (56:7)as given in 56.8.2.However,by (56:I:b)the equation(56:25)will then automatically hold in eachsolution \\7. Hencewemay, if we wish, add (56:25),i.e.strengthen (56:10)of56.8.2.to (56:25).*

(56:I:f) Therestrictionimposedin (56:I:a)upon the solutionsof Fcan also be expressedby modifying the conceptof dominationfor F but then allowingall solutionsin the modifiedsense. Thismodification consistsof imposingupon effective sets (in thesenseof 30.1.1.)the further requirementthat they must notcontain n + 1. (Cf. the secondremarkof 56.11.1.)

57.TheCharacteristicFunction and RelatedTopics

57.1.TheCharacteristicFunction :TheExtended and the Restricted Forms

57.1.We arenow in the possessionof a theory which appliesto all gamesand like the theory of 30.1.1.for zero-sum games,of which it is an exten-sion is based exclusivelyupon the characteristicfunction. I.e.thefunctions OC*(ri, , r n), k = 1, , n of 11.2.3.,which actually definethegame,do not affect the theory directly,but only through the character-istic function v(S).2

Thereis, however, a difference between the use of the characteristicfunction v(S)for a zero-sumgame,and for a generalgame. Fora zero-sumn-person gameF the characteristicfunction v(/S) is defined for all setsS (1, , n) and for only these. (Cf. 25.1.)Fora generaln-persongame F we had to form its zero-sumextension,the zero-sum (n + l)-persongame F, and the characteristicfunction v(S) was actually formed like thecharacteristicfunction (in the old sense)of F. (This is the v(S) which

1This permissibility of restricting

n

(56:10) v((l, ,n))i-ito

n

(56.25) 2} .-v((l, , n))t-iis analogous to (but more general than) the equivalenceof E(Q)and F(0) referred to in

the secondremark of 56.9.5.1Ofcoursev(S)is defined with the help of the 3C*(n, , r). Cf.25.1.3.and 58.1.)))

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528 GENERALNON-ZERO-SUMGAMES

figured in all of our recentdiscussions,particularly throughout 56.4.1.,56.5.1.,56.7.2.,56.8.2.,56.9.1.,56.9.3.-56.10.3.,56.11.2.-56.12.)Accordinglyv(S)is now defined for all setsS (1,- , n, n + 1)and for only these.We may, however, if we wish, considerv(S)for the setsS (1, , n)only. When this is done,we shall speakof the restrictedcharacteristicfunc-tion; while v(S)in its original domain, embracingall S S(1, , n, n + 1)is the extendedcharacteristicfunction.

From this we concludein the specialcaseof a zero-sum game:Herethe characteristicfunction of the old theory is the restrictedcharacteristicfunction of the new one.1

Returning to the generalgames,we seethat the characteristicfunctionis the basisof our entirepresent theory. Among the equivalent formula-tionsof that theory, (56:I:a)in 56.12.usesthe extendedcharacteristicfunc-tion, while (56:I:d)uses the restrictedone.

Consequentlyour next objective is necessarilythe determination ofthe nature of thesecharacteristicfunctions, and of their relationshipto eachother.

57.2.Fundamental Properties

57.2.1.Considera general n-persongame F, and its two characteristicfunctions, as defined above:The restrictedone,v(S)defined for all subsetsSoil= (1, , n) and the extendedone,v(S)definedfor all subsetsSofJ = (1,- - - , n, n + !).

In what follows we must distinguishbetween two possibilitiesin ournotations for S, as in the secondremark in 56.10.2.ForSsJ=(1, , n, n + 1) we can form S in 7, i.e.as / S,while for S / =(1, , n) we can also form S in /, i.e.as / S* We denoteagainthe first setby XS and the secondone by S.

We proposeto determinethe essentialpropertiesof both characteristicfunctions of the generaln-persongame just as we did in 25.3.and 26.,forthe characteristicfunction of the zero-sum n-persongame.

Consider the extended characteristicfunction first. Sinceit is thecharacteristicfunction in the old sensefor the zero-sum (n + l)-persongame T, it must have the properties(25:3:a)-(25:3:c)formulated in 25.3.1.only with I = (1, , n + 1) in placeof the / = (1, , n) there.In this way we obtain:

(57:1:a) v(0) - 0,1 All thesedistinctions and definitions cannot and do not affect the rigorously estab-

lished fact that for all zero-sum games the two theories are equivalent to eachother.(Cf.(56:I:c)in 56.12.)

*We denote them by the sameletter, v, sincethey have the samevalue wherever botharedefined.

1Formed for the sameS (of course8 /), thesetwo setsareclearlydifferent. Loc.cit. we claimed that they are equal, but there we formed them for two different setsSandT.)))

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THECHARACTERISTICFUNCTION 529

(57:1:b) v(S) = -v(S),(57:1:c) v(8 u T) v(5) + v(T) if 5 n T = 0.

OS,TfiJ).Considernext the restrictedcharacteristicfunction. We obtain con-

ditionsfor it from (57:l:a)-(57:l:c),by restrictingourselvesto subsetsof J.This is immediately feasiblefor (57:l:a),(57:l:c),but it is impossiblefor(57:1:b).* In this way we obtain:(57:2:a) v(0) = 0,(57:2:c) v(Su T) v(fl) + v(T) if S n T = 0.

(5,Ts7)Note that we cannot replace(57:l:b)by somethingequivalent for 8.Indeed,all we can do with S,is to put T = S in (57:1:c).This gives

(57:2:b) v(-S)g v(7) - v(S).Even if v(7) = 0,which neednot be the case,(57:2:b)becomesmerely

(57:2:b*) v(-S)^ -v(S),but not the equivalent of (25:3:b)in 25.3.1.

v(-S)= -v(S).(57:l:a)-(57:l:c)as well as (57:2:a),(57:2:c)are, by virtue of their

derivation, only necessarypropertiesof the (extendedor restricted)char-acteristicfunctions. We must now seewhether they aresufficient as well.

67.2.2.If T werean arbitrary zero-sum(n + l)-persongame,then wecouldconcludefrom the resultof 26.2.that any v(S)which fulfills (57:l:a)-(57:l:c)is the (old sense)characteristicfunction of a suitable T, i.e.theextendedcharacteristicfunction of a suitable generaln-persongameF.In otherwords:This would prove that the conditions(57:l:a)-(57:l:c)arenecessary and sufficient that they contain a completemathematicalcharacterizationof the characteristicfunctions of all possible generaln-persongamesF.

However,T is not at all arbitrary. As we saw in 56.2.2.,the (fictitious)player n + I has no influence on the courseof the game i.e.he has nopersonal moves; the 3C*(ri, , rn , rn+i) do not really depend on hisvariable rn+i. Furthermore,it is clearfrom 56.2.2.that this is the onlyrestrictionto which T must be subjected:If in a zero-sumn + 1-persongameF the player n + 1has no influence on the courseof the game,thenwe can view T as the zero-sum extensionof a generaln-persongameF

playedby the remaining players1, , n.2

1S, S cannot be both /(!, , n) sinceone of them must necessarilycontain n -f 1.

*I.e.we can treat n -f 1 as if he were a fictitious player as far_as the rules of the

game areconcerned. We know, of course,that there are solutions V for f which bringout the fact that he is a real player (those in but not in Q\", cf. (56:A:a)-(56:A:d)in56.7.2.and (56:1a) in 56.12.;recallalso56.3.2.,56.3.4.).)))

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530 GENERAL NON-ZERO-SUMGAMES

Consequentlythe following questionarises:(57:l:a)-(57:l:c)areneces-sary and sufficient conditionsfor the characteristicfunctions in the oldsenseof all zero-sumn + 1-persongames.Howmust they bestrengthenedso as to do the same thing for the (oldsense)characteristicfunctions of allthose zero-sumn + 1-persongames in which the player n + 1 has noinfluence on the courseof the game?

Answering this question would amount to giving a completemathe-matical characterization for the extendedcharacteristicfunctions of allgeneraln + 1-persongames.But then the problem of doing the samefor the restrictedcharacteristicfunctions would still remain.

It will be seenthat by attacking the last problem first, a somewhatmore advantageousarrangement obtains:the first problemcan besolvedin a few lineswith the help of the latterone. However,our approachwill

bedominatedby the aboveconsiderations.

67.3.Determination of All CharacteristicFunctions

57.3.1.We proceedto prove that the necessaryconditions (57:2:a),(57:2:c)arealso sufficient: Forany numerical setfunction v(S)which ful-fills (57:2:a),(57:2:c)thereexistsa generaln-persongame F of which thisv(/S) is the restrictedcharacteristicfunction.1

In orderto avoid confusion it is betterto denotethe given numericalset-function which fulfills (57:2:a>,(57:2:c)by v (S). With its help weshall define a certaingeneraln-persongameF, and denotethe restrictedcharacteristicfunction of this F by v(/S). Itwill then be necessaryto provethat v(S) = vo(S).

Let therefore a numerical set-function v (S) which fulfills (57:2:a),(57:2:c)be given. We define the generaln-persongame F as follows:2

Eachplayer k = 1, , n will, by a personalmove, choosea subsetSk of / which containsfc. Eachonemakeshis choiceindependentlyof thechoiceof the otherplayers.

After this the paymentsaremadeas follows:Any setS of playersfor which

(57:3) Sk = S for every A; belongingto Sis calleda ring. Any two ringswith a common elementareidentical.Inotherwords:Thetotality of all rings(which actually have formed in a play)is a systemof pairwisedisjunctsubsetsof /.

Eachplayer who is containedin none of the rings thus defined formsby himself a (one-element)setwhich is calleda soloset. Thus the totalityof all ringsand solosets(whichactually have formed in a play)isa decompo-sition of 7, i.e.a system of pairwisedisjunct subsets of / with the sum I.Denotethesesetsby Ci, , Cp and the respectivenumbers of theirelementsby n\\, , nP.

1Theconstruction which follows has much in common with that of 26.11Thereadershould now comparethe detailswith those in 26.1.2.)))

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THECHARACTERISTICFUNCTION 631Now considera player k. Hebelongs to preciselyone of thesesets

Ci, , Cp, say to C,. Then the playerk getsthe amount

(57:4) I v,(C,).n q

This completesthe descriptionof the game F. F is clearlya generaln-persongame,and it is clearwhat its zero-sumextensionT is. We empha-sizein particular that in F the fictitious playern + 1getsthe amount

(57:5) - Vo(C fl).'-iWe arenow going to show that F has thedesiredrestrictedcharacteristic

function v (/S).57.3.2.Denotethe restrictedcharacteristicfunction of F by v(S).

Rememberthat (57:2:a),(57:2:c)hold for v(S) becauseit is a restrictedcharacteristicfunction, and for v (S)by hypothesis.

If Sis empty, then v(S) = v Q(S)by (57:2:a).Sowe may assumethatS is not empty. In this casea coalition of all playersbelongingto S cangovern the choicesof its Sk so as with certainty to make S a ring. Itsufficesfor every k in Sto choosehis Sk = S. Whatever the otherplayers(in S) do, Swill thus be one of the sets(ringsorsolosets)C\\ 9 9 Cp

say Cq. Eachk in C,= Sgetsthe amount (57:4),hencetheentirecoalitiongetsthe amount v (S). Consequently

(57:6) v(S) ^ v (5).Now considerthe complement S. A coalition of all playersk belong-

ing to S can govern the choicesof its k so as with certainty to make Sa sum of rings and solosets.If S is empty, then this is automaticallytrue, sincethen 5 = /. If S is not empty, then it suffices for everyk in S to choosehis Sk = S. Hence S is a ring, and therefore Sis a sum of rings and solosets.

Thus S is the sum of someamong the setsCi, , Cp say of))

(!', , r' aresomeamong the numbers 1, , p). Each k in Cq

(q = s' = i'y 9 r') getsthe amount (57:4),hencethe n q playersin Cq

togetherget the amount v (C),and so all players of S togetherget ther

amount v (CV) Sincethe CV, , CV are pairwisedisjunct setss-l

1The n q players in C9 get together the amount vo(C)by (57:4);henceall playersp

1, , n, i.e.all players in Ci, , Cp , get together the amount 2^vo(C). Now-i(57:5)ensues.)))

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632 GENERAL NON-ZERO-SUMGAMES

r

with the sum 5,repeatedapplicationof (57:2:c)gives v (C,')^ v (S).-iI.e.:Whatever the playersof S do, togetherthey get an amount ^ v (S).Consequently

(57:7) v(S) g vo(S).Now (57:6),(57:7)togethergive

(57:8) v(S) = voGS),

as desired.57.3.3.Let us now considerthe extendedcharacteristicfunctions.

Herewe know that the conditions(57:l:a)-(57:l:c)are necessary. Weshall prove that they arealsosufficient: That for any numerical setfunctionv(S)which fulfills (57:l:a)-(57:l:c)thereexistsa generaln-persongameFof which this v(S)is the extendedcharacteristicfunction.

In orderto avoid confusion, it is again betterto denotethegiven numer-ical set function which fulfills (57:l:a)-(57:l:c)by Vo(S). The extendedcharacteristicfunction of the generaln-persongame F which we shallusewill bedenotedby v(/S).

Let therefore a numerical set function v (S) which fulfills (57:l:a)-(57:l:c)begiven.Considerit for a moment for thesets5 / = (1, , n)only, then it fulfills (57:2:a),(57:2:c).Henceour constructionof 57.3.1.,57.3.2.can beappliedto this vo(/S). Soa generaln-persongameF obtains,such that its restrictedcharacteristicfunction has always v(S) = voOS)1and so its extendedcharacteristicfunction has v(S) = v ($) for S /.I.e.,if we revert to the natural domain of theseS,2 then we have:(57:9) v(S) = vo(S) if n + 1is not in S.

Now let n + 1be in S. Then it is not in S. Hence(57:9)givesv(JL/S) = vo(J_S). (57:1:a)-(57:l:c)holdfor v(S)becauseit isan extendedcharacteristicfunction, and for v (/S) by hypothesis. Therefore(57:l:b)gives v(J_/S) = v(/S), v (J_S)= v (/S). All theseequations combineto

(57:10) v(5) = v (fl) if n + 1is in S.Now (57:9),(57:10)togethergive

(57:11) v(S) = v (S)unrestrictedly,as desired.

67.3.4.Tosum up:We have obtainedcompletemathematical character-izations of both the restrictedand the extendedcharacteristicfunctionsv(S) of all possiblegeneraln-persongamesF. The former aredescribedby (57:2:a),(57:2:c),and the latterby (57:l:a)-(57:l:c).

1The \"always\" in this caserefers,of course,only to the SS/.1Which in this caseconsistsof all Sfi /.)))

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THECHARACTERISTICFUNCTION 533

We follow therefore the comparableprocedureof 26.2.,and call thefunctions which satisfy theseconditionsrestrictedcharacteristicfunctions orextendedcharacteristicJunctions, respectively even when thev areviewedin themselves,without referenceto any game.

67.4.Removable Setsof Players67.4.1.The result which we obtained for extendedcharacteristicfunc-

tions canalsobestated as follows:Every characteristicfunction (in the oldsense)of any zero-sum(n + l)-persongame is alsothe extendedcharacter-istic function of a suitable generaln-person game.1 Rememberingthediscussionof 57.2.2.,this means:Every characteristicfunction of anyzero-sum n + 1-persongame is also the characteristicfunction of asuitable zero-sumn + 1-persongamein which the player n + 1has noinfluence on the courseof the play.

Let us replacein this statementn + I by n, obtaining the equivalentfor zero-sumn-persongames and the roleof the player n. In ordertoformulate this result,it is convenient to define:

(57:A) Leta zero-sumn-persongame PandasetSsI = (1, , n)be given. Then we call S removable for F, if it is possibletofind another zero-sumn-persongameI\", which has the samecharacteristicfunction as F but in which no playerbelongingto Shas an influence upon the courseof the game.

Usingthis definition, our assertionbecomesthat the setS = (n) is remov-able. Given any playerk = !, , n, we can interchangethe rolesof theplayersk and n, hencethe setS = (k) is alsoremovable.Sowe see:(57:B) Every one-elementset>Sis removable in every game F.

Now it shouldbenoted that accordingto our theory the entirestrategyof coalitionsand compensationsin A gamedependsonly on its characteristicfunction. Consequentlythe two games F and F' of (57:A) areexactlyalike from that point of view.

Hence(57:B)can beinterpretedas follows:Theroleof any one playerin

any zero-sumn-persongame insofar as the strategicpossibilitiesof coali-tions and compensationsareconcernedcan be duplicatedexactly in anarrangementwhich depriveshim of all directinfluence upon the courseofthe game. Herewe mean his \"role\"in the mostextendedsense:includinghis relationshipto all otherplayers,and his influence on their relationshipsto eachother.

In otherwords:We describedin S6.3.2.-56.3.4.a mechanismby whicha player who has no directinfluence upon the courseof the gamecanneverthelessinfluence the negotiationsfor coalitionsand compensations.We have now shown in (57:B),that this mechanismis perfectly adequate

'Indeedthe conditions (57:l:a)-(57:l:c)coincidewith (25:3:a)-(25:3:c)of 25.3.1.with 7 - (1, , n, n + 1)in placeof its / - (1, , n).)))

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534 GENERALNON-ZERO-SUMGAMES

to describethe influence which any player in any game couldhave in thisrespect.This statement must be taken absolutely literally: Our resultguaranteesthat all conceivabledetailsand nuanceswill be reproduced.

57.4.2.By (57:B)every playerk = !, , n isremovableindividuallyi.e.the one-elementsetS = (k) is but this doesnot mean that all these

playersareremovable simultaneously i.e.that the setS = / = (1, , n)

is. Indeedwe have:

(57:C) The set S = / is removable if and only if the game T isinessential.

Proof:That no player k = 1, , n has an influence on the courseof the game F',meansthat all functions 3C[(Ti, , r n) areindependentof all their variablesTI, , rn i.e.that they areconstants

(57:12) aci(n,- ,rn) = <**.

From this

(57:13) v(S)= % <*k for all SzLkin 8

Conversely,if (57:13)is required,it canbe securedby (57:12).Hence(57:13)is the characteristicfunction of a game T for which such

a I*exists and (57:13)is preciselythe definition of inessentiality.Forn = 1,2every game F is inessential,hencetherethe setS = /

and with it every set is removable.1 For n ^ 3 there existessentialgames,and therefore S = I is in generalnot removable.

Therefore this questionarises:(57:D) Which arethe removable sets for an essentialgame F?(57:B),(57:C)contain a partialanswer:Theone-elementsetsareremovable,the n-elementset(S = /) is not. Where is the dividing line?

57.4.3.Theupper extremeis reachedwhen all (n l)-elementsetsand with them all setsexcept/ are removable. We call such a gameextreme. It is worth while to visualize what this property entails:Thestrategicsituation in such a game is equivalent to that where only oneplayerhas an influence on the courseof the game,and the roleof all othersconsistsmerely in trying to influencehisdecisions.Themeansof influenc-ing him is,of course,offering him compensations;the motive is to induce

1Themain result concerning zero-sum two-person games, according to which eachgame of this type has a definite value for eachplayer (say v, v cf. the discussion of17.8.,17.9.),means just this: It statesthat the game is equivalent to the fixed paymentsv, v to the two players and this is an arrangement where neither of them can influenceanything.

In every essentialgame, on the other hand, there existsthe interplay of negotiationsfor coalitions and compensations and this excludesthe simultaneous removability ofall players.)))

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THECHARACTERISTICFUNCTION 535

him to makedecisionswhich arefavorable to the playeror playerswho makethe offer.

Now we can prove:(57:E) For n = 3:The essentialzero-sum three-persongame is

extreme.(57:F) Forn = 4:Thereexistextremeas well as non-extremeessen-

tial zero-sumfour-persongames.More in detail:(57:E*) For the essentialzero-sum three-persongame,all two-

elementsetsareremovable.(57:F*) For an essentialzero-sum four-person game all three-

elementsetsareremovable, or all but one.1-2

Theproofsof thesestatementspresentno seriousdifficulties,but we do notproposeto give them here.

The results (57:B),(57:0),(57:E),(57:F) show that a generaltheoryof removable setsand extremegamesis not likely to bevery simple.Itwill be consideredsystematicallyin a subsequentpublication.

57.5.StrategicEquivalence. Zero-sum and Constant-sum Games

57.5.1.We have exhaustedthe usefulnessof the zero-sumextensionPof the generaln-persongame F, and therefore from now on we shalldiscussthe theory of generaln-persongames without referring to that concept.Consequently,hereafter we shalluseonly thegameF itself and its restrictedcharacteristicfunction, unlessexplicitlystated to the contrary. Forthisreasonthe qualification \" restricted\" will bedropped,and we shall speaksimply of the characteristicfunction of F. This is also in harmony with

our precedingterminology for zero-sumn-persongames,sincenow the oldand the new use of the conceptof a characteristicfunction areconcordant.(Cf.the remarksnext to the end of 57.1.)

Consideringthese arrangements,the definition of the conceptof asolutionmust be that describedin (56:I:d)of 56.12.Imputationsarebestdefined as describedin (56:I:b)and in the last part of (56:I:e)id. Itseemsworth while to restatethis latterdefinition explicitly:

An imputation is a vector

(57:14) 7 - {{!, -,)}the components!,,beingsubjectto the conditions

(57:15) a, v((i)) for i = 1, , n;1Every two-element set is a subset of two three-elementsets(remember that n 4),

and by the above at leastone of theseis removable. Henceevery two-element set isremovable in any event.

1The parts of the cubeQof 34.2.2.which correspondto thesevarious alternativescanbe explicitly determined.)))

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536 GENERAL NON-ZERO-SUMGAMES

(57:16) en - v(/).1

We cannow extendthe conceptof strategicequivalenceto the presentsetup. This will bedoneexactlyas in 42.2.and 42.3.1.,i.e.in analogy to27.1.1.:

Given a generaln-persongame T with the functions 3Cfc(ri, , rn)and a setof constantsaj, , J we define a new gameT'with the func-tions3C*(ri, ' ' * , rn) by

(57:17) 3Ci(r!, , rn) = 3C*(ri, , rn) + aj.From this we conclude,exactlyas before,that the characteristicfunctions

and v'(S) of thesetwo gamesareconnectedby))

(57:18) v'(S) = v(S)+ aj.kinS

We call two such games,as well as their characteristicfunctions, stra-tegicallyequivalent.

Sincewe arefree of all zero-sumrestrictions,the constantsa\", , aJJ

areunrestricted,just as in (42:B)in 42.2.2.We notethat this strategicequivalenceinducesan isomorphismof the

imputationsof F and I\" exactlyas in the two previousinstancesreferredtoabove. Specificallythe considerationsand conclusionsof 31.3.3.and of42.4.2.carry over to the presentcaseunchanged,so that it seemsunneces-sary to reformulate them explicitly.

67.5.2.Thedomain of all characteristicfunctions (of all generaln-persongames)was characterizedby the conditions(57:2:a),(57:2:c),which werestate:(57:2:a) v(0) = 0,(57:2:c) v(S iTT) ^ v(S)+ v(T) for S n T = 0.Among thesethe characteristicfunctions of zero-sumgamesand of constant-sum gamesform two specialclasses.The former are characterizedby(25:3:a)-(25:3:c)of 25.3.1.(Cf. 26.2.)I.e.we must add to our (57:2:a),(57:2:c)(which coincidewith the (25:3:a),(25:3:c)mentioned)the furthercondition

(57:19) v(-S)= -v(S).Thelatterarecharacterizedby (42:6:a)-(42:6:c)in 42.3.2.(cf.id.).I.e.wemustadd to our (57:2:a),(57:2:c)(which coincidewith the (42:6:a),(42:6:c)

1As was pointed out loc.cit. we couldhave used equivalently

n

X < *v(7).i-lIndeedthis is the original form of this condition. However, we prefer (57:16).)))

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THECHARACTERISTICFUNCTION 537

mentioned)the further condition

(57:20) v(fi) +v(-S)- v(J).Sincethe zero-sumgamesarea specialcaseof the constant-sumgames,

(57:20)must bea consequenceof (57:19),always assuming(57:2:a),(57:2:c).This is indeedso;we canactually prove somewhatmore,namely:(57:G) (57:19)is equivalent to the conjunction of (57:20)with

v(/) = 0.Proof:1 Assuming v(/) = 0, (57:19)and (57:20)are clearly the same

assertion.Henceit sufficesto showthat (57:19)impliesv(7) = 0. Indeed,(57:2:a),(57:19)give v(I) = v(-0)= -v(0)= 0.

Note that (57:20)is the assertionthat equality holds in (57:2:c)whenS u T = I.2 Thus the v(S) of constant-sum games are characterizedby thepropertythat the mergerof two distinct coalitionsSand T producesno further profit if togetherthey contain all players.

For the v(S) of zero-sumgamesthe further requirement v(7) =0must beadded.

To conclude,we emphasizethat the extraconditions(57:19)or (57:20)do not mean that any gamewith such a characteristicfunction is neces-sarily a zero-sumor a constant-sumgame. They imply only that such acharacteristicfunction must belong amongothers to at leastonezero-sum or constant-sumgame. It can happen that a gamewithout beingzero-sum (or constant-sum)itself has such a characteristicfunction, i.e.the characteristicfunction of a zero-sum (or constant-sum) game. Inthis caseit will behavefrom the point of view of the strategy of coalitions,and the compensationslike a zero-sum(or constant-sum)gamewithout

actually beingone.67.6.3.We arenow in the position to settlea question which was in

the foreground several times in our discussions.The analysisof 56.3.2.-56.4.3.was concernedalready with the fact that the fictitious player in

spite of his unreality is not ipsofacto a dummy. I.e.not onein thesenseof the extendedcharacteristicfunction and the decompositiontheory of thezero-sumextensionI\\ 8 This subjectcameup again at the beginningof56.9.3.,wherewe noted that he is a dummy for zero-sumgamesF.

Thequestionwhich we will answernow is accordinglythis:ForwhichgeneralgamesT is the fictitious playera dummy? 4 We prove:(57:H) The fictitious player is a dummy if and only if T has the

samecharacteristicfunction as a constant-sum game i.e.if(57:20)is fulfilled.

1Essentially this argument was made in 42.3.2.'IndeedSU T -I and the usual hypothesis of (57:2:c)Sn T - mean that

T - ~8'8 We had to excludehim from the game by explicitly restricting the solutions from

to \".4Theargument of the first remark in 56.9.4.,shows then that for thesegames and

ft\" coincide i.e.the restriction of the solutions of f is unnecessary.)))

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638 GENERAL NON-ZERO-SUMGAMES

Proof:As observedat the end of 43.4.2.a player is a dummy, if andonly if he forms (as a one-elementset) a constituent of the game. Wemust apply this to the fictitious player n + 1 in the zero-sum gameT.That (n + 1)is a constituent,meansobviously that

(57:21)v(fl)+v((n + l))-v(Su(n+ l)) for all Sfi(l, ,n).Now we have

v((n+ 1)) = -v(7),v(Su(n + l)) = -v(Su(n+ 1))= -v(-S).

Hence(57:21)becomes

v(fl) -v(7) = -v(-S),i.e.(57:22) v(S)+ v(-5)= v(7).

And this is preciselythe condition (57:20).

58.Interpretationof the CharacteristicFunction

68.1.Analysis of the Definition

58.1.We have arrived at a formulation of the theory of the generaln-persongame,and found that the conceptof the characteristicfunctionis justas fundamental in it as it wasin theprecedingtheory of the zero-sumn-persongame. It is therefore appropriate to survey the meaning of thisconceptoncemore,putting its mathematical definition into an explicitform and addingsomeinterpretativeremarks.

Consideraccordinglya generaln-persongameF, describedby the func-tionsJC*(ri, , r n) (k = 1, , n) in the senseof 11.2.3.Thevaluev(S) of the characteristicfunction for a setS 7 = (1, , n) obtainsby forming this quantity for the zero-sumn + 1-persongameT the zero-sum extensionof F.1 Hencewe can expressit by meansof the definitoryformulae of 25.1.3.:

(58:1) v(S) = Max-Min->K(7,7) = Min->Max-K(T,7),where we have:

{ is a vector with the components T,))

1We restrict ourselvesto the S 7 (1, , n) i.e.to the restrictedcharacteristicfunction. The use of all / - (1, , n + 1),i.e.of the extendedcharacteristicfunction, is contrary to our presentstandpoint. (Cf.the beginning of57.5.1.

))))

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INTERPRETATION 539

17 is a vector with the componentsrj r-a

t?r- ^ 0, J) ir- = 1;T-a

TS is the aggregateof the variables r*, k in S;r~5 is the aggregateof thevariablesT*, fc in S;1 and finally

(58:2) K(7,7)= OC(rV-*)W-<,rV-a

where

(58:3) 3C(r5, r~5) = JC*(n, , rw) .2fcinS

58.2.TheDesireto Make a Gain vs. That to Inflict a Loss>58.2.1.K( , 77 ) isobviously the expectationvalue of a play of the game

T for the coalition S,if the coalition S usesthe mixed strategy and the>

opposingcoalition S 8 uses the mixed strategy 17 . Hence(58:1)definesv(S),the value of a play for the coalition S under the assumptionthat the

coalition S wants to maximize the expectationvalue K( , t\\ ), while theopposingcoalition S wants to minimize it, and they choosetheir respec-tive (mixed)strategies , T? accordingly.

Now this principleis certainly correctin the zero-sum n + 1-persongame T,4 but we are really dealing with the general n-persongame F

1 SdenotesI S. Sincewe aredealing with f , we should have formed Swhichis I S. (Cf. the beginning of 57.2.1.)However, this is immaterial, becausenovariable TW+ I exists. (Cf.the end of 56.2.2.)

1We useonly the original 3C*,A; = !, , n, i.e.the W n +i of (56:2)in 56.2.2.))

(58:4) OC nr l(T,, -fcl

doesnot occurhere. This is, of course,due to the fact that S / =(!,,n).It must be remembered that formula (58:3)above is the first formula of (25:2)in25.1.3.Thesecondformula of (25:2)loc.cit.gives

(58:5) OC(rV-s) s ~ S 3C*0-i.- , r).fcin IS

(Note that we, must now definitely use S / S for the S loc.cit.,sincewe aredealing with T. Cf.alsofootnote 1 above.) Sincen + 1is not in S,it is in _US;hencethe sum ^ of (58:5)doescontain the JCM i of (58:4), However, (58:4)guarantees,

kin ISas it must, the identity of the right-hand sidesof (58:3)and (58:5).

3Theobservations of footnote 1 above apply again.4I.e.if we view S / Sas really representing 1.5 / S.)))

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540 GENERAL NON-ZERO-SUMGAMES

P is merely a \"working hypothesis\"!And in T the desireof the coalitionS to harm its opponent,the coalition S,isby no meansobvious. Indeed,

the natural wish of the coalition S shouldbe not so much to decreasethe expectationvalue K( , t? ) of the coalition S as to increaseits own

expectationvalue K'( , i\\ ). Thesetwo principleswould be identical>

if every decreaseof K( , i? ) wereequivalent to an increaseof K'( , i\\ ).This is of coursethe casewhen F is a zero-sumgame,1 but it neednotat all beso for a generalgame F.

I.e.in a generalgame F the advantage of one group of players neednot besynonymouswith the disadvantageof the others. In such a gamemoves or ratherchangesin strategy may existwhich areadvantageousto both groups. In other words, there may existan opportunity forgenuine increasesof productivity, simultaneouslyin all sectorsof society.

58.2.2.Indeed,this is more than a merepossibility the situations towhich it refers constituteoneof the majorsubjectswith which economicand socialtheory must deal. Hencethe questionarises:Doesour approachnot disregardthis aspectaltogether? Did we not losethis cooperativeside of socialrelationshipsbecauseof the greatemphasiswhich we placedon theiropposite,antagonistic,side?

We think that this is not so. It is difficult to presenta completecase,sincethe validity of a theory is ultima analysionly establishedby successin the applications and we have made no applicationsin our discussionthus far. We will suggesttherefore only the main points which seemto

1This is sobecausewhen T is zero-sum then

(58:6) K(7,7) + K'(T,7) -0.This is clearby common sense;a formal proof obtains in this way :Clearly

(58:7) K'(7,T) - 5) rc'(r,'-a )*r*iT-*

where

(58:8) 3C'(T5, T-3) s 3Cfc (Ti, . . . , Tn).*in -S

(Note that this is not the 2} 3C*(n, , r) which occursin (58:5)).Now coin-Jbin _LS

parison of (58:2)with (58:7)shows that (58:6)is equivalent to

(58:9) 3C(r5, r~s) + 3C'(r5, T~3) a 0,and (58:3),(58:5)imply that (58:9)amounts to

n

% JC*(n, - - - , r.) - 0,

i.e.the zero-sum condition for T.)))

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INTERPRETATION 541*

support our procedure,and then refer to the applicationswhich provide adefinite corroboration.

58.3.Discussion

58.3.1.The following considerationsdeserve particular attention inthis connection:

First:Inflicting losseson the adversarymay not bedirectlyprofitablein a general(i.e.not necessarilyzero-sum)game,but it is the way to exertpressureon him. Hemay be inducedby such threatsto pay a compen-sation, to adjust his strategy in a desiredway, etc. Henceit is not a limineunreasonablethat this category of strategicpossibilitiesshouldbe takeninto account;and our procedurein forming the characteristicfunction, asanalyzed above, might be the proper one to do just that. It must beadmitted, however, that this is not a justification of our procedure itmerely preparestheground for the realjustification which consistsof successin applications.

Second:A further considerationpointing in the same directionis this.We have seenthat in our theory all solutionscorrespondto attainmentof the maximum collectiveprofit by the totality of all players.1 Whenthis maximum is reached,any further gain of one groupof playersmust becompensatedby an at leastequallossof the others. True,therecouldbeovercompensation:i.e.one groupmight obtain a gain by inflicting a greaterloss on the others. However,we have assumedcompleteinformation forall players,and a perfect interplay of threats,counterthreatsand compensa-tions among them.2 Henceone may assumethat such possibilitieswill beeffective only as threats,and that the correspondingactionswill beobviatedalways by negotiationsand compensations.By this we do not mean thatthese threats are \" bluffs \" which arenever \"called.\"Sincethereexistscompleteinformation for all players,therecan never be any doubt. Butwhen an action is threatenedby which one party gainslessthan the otherone loses,then thereexistsipsofacto the possibilityof avoiding it by com-pensationsin a way which is advantageousto both sides.8 And when thishappensit is again true that one sidegainsexactlywhat the otherloses.

If this argument is acceptedas generallyvalid, then our difficultiesdisappear.

58.3.2.Third:It may be said that the argumentation of the two pre-cedingremarksis too sketchyand that it doesnot justify our theory in theexactform in which we proposeto useit. Thisis true,but our very detailed

1Cf.the end of 56.7.1.,particularly footnote 3 on p.513.1Our entire attitude towards coalitions and compensations was basedon this, already

in the theory of the zero-sum games.1We do not proposeto determine here the amount of the compensation i.e.thenature of the compromise. This is the task of the exacttheory which we possessalready.It will be the main subjectin eachapplication. (Cf. the various interpretations in61.-63.) At this point we want only to show that actionswhich would leadto a lossforthe totality ofall players,can beavoidedby the mechanism describedabove.)))

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542 GENERAL NON-ZERO-SUMGAMES4

motivation of that theory, as given in 56.2.2.-57.1.meetsthe latterrequire-ment. If the readerreconsidersthosesectionsin the light of the two pre-cedingremarks,then he will seethat the detailedjustification in the desiredsensewas their subject. Indeed,the possibilityof the objectionnowunder considerationwas our reasonfor making the discussionof our theoryso detailed,and avoiding plausibleshortcuts.1

Fourth: In spite of all this, the readermay feel that we have over-emphasizedthe roleof threats,compensations,etc.,and that this may be aone-sidednessof our approach which is likely to vitiate the results in

applications.Thebest answerto this is, as repeatedlypointedout before,the examination of thoseapplications.

We shall therefore considerdefinite applicationswhich correspondtofamiliar economicproblems.Their study will disclosethat our theoryleadsto resultswhich are,up to a certain point, in satisfactoryagreementwith the usual common-senseviews on thesematters. This is the caseaslong as the two following conditionsarefulfilled: First that the setup issimple enough to allow a purely verbal analysis,not making use of anymathematical apparatus. Secondthat those factors which are inseparablefrom our theory,but often excludedin the ordinary,verbal approach coali-tions and compensations have not comeessentiallyinto play. This situ-ation will be found to existin the application of 61.2.2.-61.4.Indeed,that exampleprovidesthe decisivecorroborationof our procedure.

Beyond this point, where the first condition is still satisfied,but notthe second,we shall find discrepanciesjust in the direction and to theextent to which the difference in standpoint justifies it. This will beparticularlyclearin the applicationsof 61.5.2.,61.6.3.and 62.6.

Finally, as even the first conditionfails, becausethe problemis no longerelementary, we gradually reachground where the theoreticalprocedurenecessarilytakesover the leading role from the ordinary, purely verbalone.2

59.GeneralConsiderations59.1.Discussionof the Program

69.1.1.We can now proceedto the applicationsof our theory of thegeneraln-persongame. The best way of starting such applicationsis asystematicdiscussionof all generaln.-persongamesfor small values of n.It will appearthat we can carry this out in absolutecompletenessfor thesamen as for the zero-sumgames:Forthe n ^ 3. Thediscussionfor thegreatervalues,i.e.for n ^ 4, is necessarilyat leastas difficult as it was for

1A possibleone would have beento define the characteristicfunction as in 58.1.andto comeout then with a flat generalization of the theory for zero-sum games,i.e.with(56:1:d)in 56.12.

*This gradual transfer of the emphasis from corroboration of the theory by thereliablecommon-senseresults in the simple case,to overriding any un theoretical approachby the theory in the complicatedones,is,ofcourse,quite characteristicin the formation ofscientific theories.)))

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GENERALCONSIDERATIONS 643

the zero-sum gameswhere we couldonly disposeof specialcasesof variouskinds.

We proposeto do considerablyless in the way of analyzing gameswithn ^ 4 this time. We can afford to be considerablybriefer now than wewere in discussingthe zero-sum games:Thedetaileddiscussiontherewasnecessaryin order to reassureourselvesof the proprietyof our procedure,and of the generalideas and methodicalprinciplesunderlying it. At thestagewhich we have reachednow, the generalsetup of the theory appearsto be justified, and we want only to gain assuranceconcerningthe onegeneralizing step carriedout in this chapter.Forthis purposea lessexten-sive analysisof applicationsshouldsuffice.

Further,it will be possiblealready to connectthe generalgameswith

n g 3 with sometypical economicproblems(bilateralmonopoly, duopolyversusmonopoly, etc.)which allow judgmentof the appropriatenessof ourtheory in the senseindicatedbefore.

Moredetailedinvestigations of generalgameswith n ^ 4 will beunder-taken in subsequentpublications.

69.1.2.The systematicapplicationof our new theory is best introducedby a generaldiscussion,similar to that of 31.It will not be necessary,however, to carry out the equivalent considerationsin detail;we must onlyanalyze to what extentthe resultsobtainedtherecarry over to the presentsituation,or what modificationsarerequired.

We neednot discussagain the roleof strategicequivalence,as expoundedin 31.3.,sincethis subjecthas already beendealt with satisfactorily in57.5.1.On the other hand, we shall take up certainmatters originatingelsewherethan in 31.:reducedforms, inequalitieswhich hold for the char-acteristicfunction, inessentialityand essentiality(cf. 27.1.-27.5.);further,the absolutevalues |T|i,|T|2 (cf. 45.3.),and finally someremarksconcerningthe theory of decompositionof Chapter IX.

69.2.TheReducedForms. TheInequalities

69.2.1.The conceptof strategicequivalence,as introduced in 57.5.1.can be used to define reducedforms for all characteristicfunctions, alongthe linesof 27.1.

Given a characteristicfunction v(S)its generalstrategicallyequivalenttransformation is given by (57:18)in 57.5.1.,i.e.by))

(59:1) v'GS) = vOS)+))

kmS))

This is precisely(27:2)in 27.1.1.,but the a?, - , <* arenow completelyunrestricted,while they were subjectloc.cit.to the condition (27:1):

n

] <x = 0. Hencethe J, , ojarenow n independentparameters,)))

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644 GENERAL NON-ZERO-SUMGAMES

while they representedonly n 1 independent parameters formerly(cf.27.L3.).1

It would be erroneousto assume, however, that this leads to morerestrictivepossibilitiesof normalization than we found in 27.1.4.Indeed,we desiredloc.cit.to obtain a particular v'(S) to be denotedby v(S)which fulfills the n 1conditions(27:3):(59:2) v((l)) = v((2))= = v((n)).Yet, the characteristicfunctions consideredat that time belongedto zero-sum games;hencewe had automatically

(59:3) v((l, , n)) = 0.In imposingthis as a normalizing requirement,we now have n conditions:(59:2)and (59:3).Sowe obtain

(59:4) v(/) + J = 0,(59:5) v((l)) + a? = v((2))+&==v((n))+ a.(59:4)expresses(59:3);(59:5)expresses(59:2).Theseequations cor-respondto (27:1*),(27:2*)loc.cit.,and it is easy to verify that they aresolvedby preciselyone systemof aj, , a:))

(59:6) a{= -v((A)) + v((*))- v(J)))

Sowe can say:(59:A) We call a characteristicfunction v(S)reducedif and only if it

satisfies (59:2),(59:3).3 Then every characteristicfunctionv(S) is in strategicequivalence with precisely one reducedvGS). This v(/S) is given by the formulae (59:1)and (59:6),andwe call it the reducedform of v(S).

59.2.2.Another possiblerequirementfor the n parametersa?, , aj|consistsin requiringfor v'(S) to be denotedby V(S) the n conditions))

(59:7) *((!))= *2))= = ?((n)) = 0.1Our present standpoint in this respectis similar to that which we took for the

constant-sum games in 42.2.2.f Proof:Denotethe joint value of n terms in (59:5)by 0. Then (59:5)amounts to

n

aj - -v((fc))+ ft and so (59:4)becomesv(7) - v((fc)) -f n/3 -0, i.e.))

8 This is preciselythe definition of27.1.4.)))

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GENERAL CONSIDERATIONS 545

This means

(59:8) v((l)) + a\\ = v((2))+ $...v((n))+ 2 = 0,i.e.(59:9) cfk = -v((fc)).Sowe can say:

(59:B) We call a characteristicfunction v(S)zeroreducedif and onlyif it satisfies(59:7).Then every characteristicfunction v(/S)is in strategicequivalence with preciselyone zero-reducedv(/S).This v(S) is given by the formulae (59:1)and (59:9),and wecall it the zero-reducedform of v(S).

59.2.3.Let us considerthe reducedcharacteristicfunction v(S). Wedenotethe joint value of the n terms in (59:2)by 7, i.e.(59:10) -7 = v((l)) = v((2))= = v((n)).Hence y = v((fc)) + and so (59:6)gives

n

(59:11) 7 =^ jv(/)-

Jb-1

If we use the zero-reducedform v(S) of the same v(S) then we haven n

v(/) = v(J) + ak , henceby (59:9)v(/) = v(7) - % v((fc)),i.e.using*-i *-i(59:11)(59:12) ny = ?(/).

Returning to the reducedform v(/S),we seethat someequalitiesand allinequalitiesof 27.2.arestill valid.

To beginwith, (59:10)can be stated as follows:

(59:13) v(S) 7 for every one-elementsetS.This coincideswith (27:5*)loc.cit.,while (27:5**)id.fails, sincewe sawin 57.2.1.that the equivalent of (25:3:b)in 25.3.1.is now missing,and thiswasrequiredto derive (27:5**)from (27:5*)there.

Repeatedapplicationof (57:2:c)in 57.2.1.to the sets(1), , (n)gives by (59:13)-ny ^ 0,i.e.:(59:14) 7^0.This coincideswith (27:6)in 27.2.

Considernextan arbitrary subsetS of I. Let p be the number of itselements:S = (*i, , kp). Repeatedapplicationof (57:2:c)in 57.2.1.)))

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646 GENERAL NON-ZERO-SUMGAMES

to the sets(fci), , (k p) gives by (59:13)v(S) ^ -P7-

Apply this to /S which has n p elements.Owing to (57:2:b)in 57.2.1.and (59:3),we have

v(-S)^ -v(S)1hencethe precedinginequalitynow becomes

v(S) (n -p)y.

Combiningthesetwo inequalitiesgives:(59:15) py ^ v(/S) ^ (n p)y for every p-elementsetS.This coincideswith (27:7)in 27.2.

(59:13)and v() = (i.e.(57:2:a)in 57.2.1.)can also be formulatedas follows:

(59:16) Forp = 0,1 we have = in the first relation of (59:15).This coincideswith (27:7*)in 27.2.v(7) = (i.e.(59:3))can also beformulated as follows:(59:17) Forp = n we have = in the secondrelation of (59:15).This coincideswith (27:7**)loc.cit.,exceptthat p = n 1is missing,forthe samereasonfor which the equivalent of (27:5**)id.is missing(cf.theremark following our (59:13)).

59.3.Various Topics59.3.1.Theseinequalitiescan now be treatedin the sameway as in

27.3.1.Therearetwo alternatives, basedon (59:14):Firstcase:7 = 0. Then (59:15)gives v(S) = for all S. This is

preciselythe inessentialcasediscussedin 27.3.1.,with all the attributesenumeratedthere. Considering(59:A), the inessentialgamesarepreciselythosewhich areequivalent to the gamewith v(S) = 0, the gamewhich isperfectly\"vacuous.\"

Secondcase:y >0. By a changeof unit we couldmake 7 = 1,with

the consequencespointed out in 27.3.2.And just as there,we refrainfrom doingthis immediately. Forthe same reasonsas pointed out there,thestrategy of coalitionsis decisivein sucha game. We call a gamein thiscaseessential.

Thecriteria(27:B),(27:C),(27:D)of 27.4.forinessentialityandessenti-n

ality areagain valid:In (27:B) v((fc)) must be replacedby*-i1Note that in our present application this inequality replacesthis missing equality

(25:3:b)in 25.3.1.,which was used in 27.2.)))

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GENERALCONSIDERATIONS 547

- v(7),))

while (27:C),(27:D) are completelyunaffected. Indeed,it is easy toverify that the proofs given therecarry over to the presentcase,their basesbeingprovidedin 59.2.1.

We leave it to the readerto apply the considerationsof 27.5. for theessentialcase,with the normalization 7 = 1 to the presentsituation.

59.3.2.We can now pass to the considerationswhich correspondtothoseof 31.

Theremarksof 31.1.1.-31.1.3.concerningthe structureof the conceptofdomination and certainly necessaryand certainly unnecessary setscan berepeatedwithout any change. The conceptsof convexity and of flatnesscan be introducedas in 31.1.4.Theconclusionsof 31.1.4.-31.1.5.arealsounaffected, exceptfor (31:E:b)in 31.1.4.and (31:G)in 31.1.5.,as well as(31:H)id.for p = n - 1. Thesearethe only oneswhere (25:3:b)of25.3.1.(cf.57.2.1.)is used.

Finally, the remark at the end of 31.1.5.must be modified. Owing towhat wesaidabove, the value p = n 1isjust asdubiousas thoseincludedin (31:8)loc.cit. I.e.the p for which the necessityof S is in doubt, arerestrictedto p 7* 0,1,n, i.e.to the interval

(59:18) 2 g p g n - 1.Thus this interval beginsto play a rolewhen n ^ 3, not only, as loc.cit.,when n ^ 4.1

Considernext the results of 31.2.The readerwho consultsthat sec-tion will have no difficulty in verifying the following: (31:1),(31:J),(31:K)areunaffected. In (31:L)the constructionof ft with the help of a can

becarriedout without any change;the first assertion,ft H a , cannot bemaintained,sinceit uses that part of (31:H) in 31.1.5.which is no longer

valid; the secondassertion,not a H ft , is unaffected. This weakeningof (31:L)removes (31:M). (31:N)remainstrue becauseit uses the intactpart of (31:L)only. (31:0),(31:P)areunaffected.

59.3.3.To conclude,letus considersomeof theconceptsof ChapterIX.We defined therethe two numbers|F|i,|T|ithe former in 45.1.,the latter

in 45.2.3.,and we discussedtheir propertiesin 45.3.Both definitions i.e.the pertinent considerationsof 45.1.,45.2.

carry over literally. Thereare,however, essentialchangesin 45.3.:In(45:F) only the secondpart of the proof is valid, but not the first part,sincethat and that alone makesuse of (25:3:b)in 25.3.1.(cf.57.2.1.).

1This is in agreement with the connection of general n-person gamesand zero-sumn -f 1-persongames,which was prominent throughout 56.2-56.12.)))

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548 GENERAL NON-ZERO-SUMGAMES

Specifically:We still have

(59:19) |r|t s?L=_?|r|If

and so we can evaluate |F|2 in terms of |F|i;but we do not have

(59:20) |r|i (n-))

nor can weevaluate |F|iin termsof |F|2at all. Indeed,weshallseein 60.2.1.that

(59:21) |F|!>0, |r|,=occursfor certaingames.

In consequenceof this, the remarksof 45.3.3.-45.3.4.becomepointless.Thesameholdsfor 45.3.1.,i.e.its result (45:E)fails in so far as it concerns|F|j.It is true for |r|ibut this is merely a restatementof the definitions.Consideringthis, and (59:19),(59:21)above, we seethat (45:E) must beweakenedas follows:

(59:C) If T is inessential,then |r|i= 0,|F|2 = 0.If T is essential,then |r|i> 0,|r|2 ^ 0.

The theory of compositionand decomposition,which is the main objectof ChapterIX, can be extendedin its essentialparts to our presentset-up.Thedifferencebetweenthe behavior of |F|iand |F|2 discussedabove,neces-sitatessomeminor changes,but theseareeasily applied.Of coursethetheoryof excessesand of solutionsin the setsE(eo)andF(e) (cf.there)mustbeextendedto the presentcase but this, too, entailsno realdifficulties.

A detailedanalysisof this subjectwould lengthen our expositionbeyondthe limits that we setourselvesin 59.1.1.Furthermore,the interpretativevalue of the results would not differ materially from what was alreadyobtainedin ChapterIX, when consideringzero-sumgames.

60.TheSolutionsof All GeneralGameswith n ^ 3

60.1.TheCasen -160.1.We proceedto the systematicdiscussionof all generaln-person

gameswith n ^ 3,as announcedin 59.1.1.Considerfirst n = 1. This casehas already beenconsidered(and, for

practicalpurposes,settled)in 12.2.In particular,we pointedout in 12.2.1.that in this (and only in this) casewe dealwith a pure maximum problem.It is neverthelessdesirableto verify that our generaltheory producesin this(trivial) specialcasethe common-senseresult.2 We apply therefore thegeneraltheory in completemathematical rigor.

1 (59:20)and (59:19)expressthe two parts of (45:F),respectively.*Thisbrings us backto the fourth remark in 58.3.2.)))

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SOLUTIONSFOR n S 3 549

A generalgame F with n = 1is necessarilyinessential:This is clearbyconsideringthe characteristicfunction v(/S) of its reducedform, sincethen (59:16)and (59:17)in 59.2.3.give (for p = l=n)-7= 0,i.e.7 = 0.We may alsouse without reducing any one criterion(27:B),(27:C),(27:D)of 27.4.(cf.59.3.1.).E.g.(27:C)loc.cit.is clearlysatisfied,with

ai = v((l)). Note that this is v(7), i.e.by (56:13)in 56,9.1.(with thenotation of 12.2.1.)MaxT3C(r). We restatethis:(60:1) ai = v((l)) = v(7) = MaxT3C(r).SinceT is inessential,we can apply (31:0)or (31:P) in 31.2.3.(cf.59.3.2.).This gives:

(60:A) T possessespreciselyone solution,the one-elementset(a)where))

with the ai of (60:1).This is obviously the \" common-sense\" result of 12.2.1.as it shouldbe.

60.2.TheCasen - 2

60.2.1.Considernextn = 2. Themain fact is that a generalgamewith

n = 2 neednot be inessential thus differing from thezero-sumgameswith

n = 2. (Thelatterareinessentialby the first remark in 27.5.2.)Indeed:Thecharacteristicfunction v(S)of its reducedform is completely

determinedby (59:16)and (59:17)in 59.2.3.It is

I(60:2) v(S) = < -7 when S hasI

Now oneverifies immediatelythat a v(S) of (60:2)fulfills the conditions(57:2:a),(57:2:c)of 57.2.1.,i.e.that it is the characteristicfunction of asuitableT (cf.57.3.4.),if and only if 7 ^ 0. This is preciselythe condition(59:14)in 59.2.3.Sowe see:the 7 ^ of (59:14)in 59.2.3.arepreciselythe possibilitiesin (60:2).

Thus 7 > 0, i.e.essentiality, is among the possibilities,as asserted.In the caseof essentialitywe may further normalize 7 = 1,thereby com-pletely determining(60:2).Thus thereexistsonly one type of essentialgeneraltwo-persongames.

Notethat while |r|i= 2?may thus be > 0,thereis always (for n = 2)|r|i= 0. It sufficesto prove this for the reducedform, i.e.for (60:2).

Indeed:Recallingthe definitions of 45.2.1.and 45.2.3.we see that

a a* {{ai,aj}) is detachedwhen a\\ 9 a ^ 7, a\\ + 2 ^ 0,and that theminimum of the correspondinge = a\\ + a2 is O.1 Hence|F|j= asdesired.

1 It is assumede.g.for ai - a* 0.)))

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550 GENERAL NON-ZERO-SUMGAMES

Summingup:Forn = 2 a zero-sumgamemust be inessential,a generalgameneednot be. Accordingly the former must have |F|i= 0;the lattermay have |F|i> too. But both have always |F|2 = 0.

We leave it to the readerto interpretthis result in the light of previousdiscussions,and particularly of 45.3.4.

60.2.2.Thesolutionsfor a generalgame F with n = 2 areeasilydeter-mined.

By the valid part of (31:H) in 31.1.5.(cf. the pertinent observationsin 59.3.2.)all setsS / with 0,1or n elementsarecertainly unnecessarybut sincen = 2,theseexhaust all subsets. Hencewe may determinethesolutionsof T as if domination never held. Consequentlya solution issimply defined by the property that no imputation can be outside of it.I.e.thereexistspreciselyone solution:the setof all imputations.

Thegeneralimputation is given in this caseas a ={ (ai,a2J },subject

to the conditions(57:15),(57:16)in 57.5.1.,which now become:))

(60:3) a!(60:4) ai + 2 = v((l,2)) = v(/).We restatethe result:

(60:B) F possessespreciselyonesolution,the setof all imputations.Thesearethe))

a = {{ai,a2|}with the ai,a2 of (60:3),(60:4).

Note that (60:3),(60:4)determinea unique pair i, 2 (i.e.a ) if andonly if

(60:5) v((l)) + v((2))= v((l,2)).

By the criteriaof 27.4.this expressesprecisely the inessentiality of F.This resultis,as it shouldbe,in harmony with (31:P)in 31.2.3.(cf.59.3.2.).

Otherwise

(60:6) v((l)) + v((2))< v((l,2)),

and thereexistinfinitely many i, 2, i.e.a . Thisis the caseof essential-ity for F.

Theinterpretationof theseresultswill be given in 61.2.-61.4.60.3.TheCasen = 3

60.3.1.Considerfinally n = 3. These games include the essentialzero-sumthree-persongamefor which |F|i> and |F|2 > (cf.45.3.3.).Sowe see:

Forn = 3 a zero-sumgameas well as a generalgamemay be essential,and both |F|i> and |F|2 > atepossibilities.)))

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SOLUTIONSFOR n 3 551Thecasewhere T is inessentialis taken careof by (31:O)or (31:P)in31.2.3.(cf.59.3.2.).We assumetherefore that F is essential.Use the reducedform of F in the normalization 7 = 1. Then we

can describeits characteristicfunction v(S) with the help of (59:16)and(59:17)in 59.2.3.as follows:))

(60:7) v(S) = { -1 when S has

and

(60:8) v((2,3))= en, v((l,3)) = a2, v((l,2))= a3 when Shas2 elements.

And it is verified immediately that a v(*S) of (60:7),(60:8)fulfills the con-ditions (57:2:a),(57:2:c)of 57.2.1.,i.e.that it is the characteristicfunctionof a suitable F (cf.57.3.4.),if and only if

(60:9) -2g ai,a 2, a3 ^ 1.Note that this F can be chosenzero-sum,i.e.that (25:3:b)of 25.3.1.

holdsif and only if

(60:10) ai = a2 = <i8 = 1.In otherwords:The domain (60:9)representsall generalgames,while itsupper boundary point (60:10)representsthe (unique) zero-sum game ofour case.

60.3.2.Let us now determine tffie solutionsof this (essential)generalthree-persongame.

The generalimputation is given in this caseas a = {{i,a2, 8}),subjectto the conditions(57:15),(57:16)in 57.5.1.,which now become:

(60:11) i -1, S -1, ^ -1,(60:12) ai + a2 + 3 = 0.

Theseconditionsarepreciselythose of 32.1.1.for i, a2, s (cf. (32:2),(32:3)there),i.e.those used in the theory of the essentialzero-sum three-

person game. They agreealso, apart from the factor 1+ -$'with theoconditionsof 47.2.2.for a 1, a2, a3 (cf. (47:2*),(47:3*)there),i.e.with thoseused in the theory of the essentialzero-sum three-persongame with excess.Consequentlywe can use the graphicalrepresentationdescribedin 32.1.2.,in particular in Figure52. We obtain the domain of the a as the funda-mental triangle in 32.1.2.in Figure53. It is alsosimilar to that in 47.2.2.in Figure70.

We expressthe relationshipof domination in this graphicalrepresenta-> >

tion. Concerningthe setS of 30.1.1.for a domination a H ft , the follow-)))

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552 GENERAL NON-ZERO-SUMGAMES

ing can besaid. By the valid part of (31:H) in 31.1.5.(cf.the pertinentobservationsin 59.3.2.)all setsS / with 0,1or n elementsarecertainlyunnecessary but sincen = 3,this restrictsour analysis to two-elementsetsS.

Put therefore S = (t, j).1 Thendomination meansthat

on + ai ^ v((t, j)) = ak and a, > ft, a, > ft.

By (60:12)the first condition may be written ctk ^ ak.We restatethis:Domination

a Hmeansthat

f either i > fa, 2 > ftz and a8 S #3;(60:13) j or ai > 0i, a3 > ft* and 2 ^ j;

1or 2 > ft, 3 > 03 and on ^ ai.2Thecircumstancesdescribedin (60:13)can now beadded to the picture

of the fundamental triangle. Thesimilarity is now more with 47.than with

32. Theoperationcorrespondsto the transition from Figure70 to Figures71,72, or to Figures84,85,or to Figures87,88. Indeed,the difference asagainst Figures71,84, 87 (which all describethe sameoperation,in thesuccessiveCases(IV),(V), (VI))is only this:

Thesix lines))

(60:14)

which form the configuration there,arenow replacedby the six lines

(60:15)))

respectively. Hencethe secondtriangle (formed by the three last lines)which appears in the fundamental triangle(formed by the threefirst lines)neednot beplacedsymmetrically with respectto the latter,as it is in thethreefigures mentioned.

11,j,k a permutation of 1,2,3.1This is quite similar to (47:5)in 47.2.3.,exceptthat we have there 1 r- in place

60of all three 01,az, at. Thereis alsothe changeof scaleby the factor 1-f -r referred tooafter (60:11),(60:12).

The relation to (32:4)in 32.1.3.is the sameas for (47:5)in 47.2.3.,cf. footnote 2on p. 406.)))

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60.3.3.It is convenient to distinguishtwo cases,accordingto whetherthe

(60:16) ai ^ -di, a2 ^ -a2, c*8 S -assidesof the threelast linesof (60:15)(wherethe threedomination relationsof (60:13)arevalid) intersectin a common area,or not. Owingto (60:12)the former meansthat

(60:17:a) ai + a2 + a8 >0,while the latter means that

(60:17:b) ai + a2 + a3 ^ 0.We call thesecases(a)and (b),respectively.

Case(a): We have the conditionsof Figures71,72, exceptthat theinner triangle neednot be placedsymmetrically with respectto the funda-mental triangle, as it is there. If this is bornein mind, then the discussionof Case(IV),as given in 47.4.-47.S.can berepeatedliterally. The solutionsaretherefore, with the samequalification, thosedepictedin Figures82, 83.

We note that if an a< = 1,then the correspondingsides of the innerand the fundamental triangle coincide(cf. (60:15)),and the correspondingcurve disappears.1

Case(b): We have essentiallythe conditionsof Figures84, 85 ofwhich thoseof Figures87, 88 arebut a variant with the sameproviso forasymmetry as in Case(a)above.

We redraw the arrangement of Figure84, the fundamental trianglebeingmarkedby / and the inner triangle by \\: Figure92. The arrange-ment has severalvariants, becausethe inner triangle can stick out fromthe fundamental triangle in various ways.2 Figures92-95depict thesevariants.8-4

If thesecircumstancesarebornein mind, then the discussionof Case(V)as given in 47.6.can be repeatedliterally.5 Thesolutionsaretherefore,

1Thus in the zero-sum case,where ai =03= a 1,none of these curves occurin accordwith the result of 32.

2 By (60:9) 2 ^ a, ^ 1.This means, as the readermay easily verify for himself,that eachsideof the \" inner\" triangle must passbetween the corresponding sideof thefundamental triangle and its oppositevertex. OurFigures 92-95exhaust all possibilitieswithin this restriction.

*Theonly ones which can occur in a zero-sum game, i.e.for ai a* at 1,arethose which can be symmetric: Figures 92, 95. Of these, Figure 92correspondstoFigure 84,and Figure 95correspondsto Figure 87.

4The Figures 92-95differ from eachother by the successivedisappearanceof theareas , , . Besidesone or more of the areas and , , may degenerateto alinear interval or evento a point. It is sometimes not quite easyto distinguish betweenthe \" disappearance\"mentioned above, and this \"degeneration.\" A rule which allowsdifferentiation between the four casescorresponding to the Figures 92-95and in which

this difficulty doesnot present itself, is this: Figures 92-95correspondrespectively tothe caseswhere the \"inner\" triangle meets 0, 1,2, 3 sidesof the fundamental triangle.(Meeting a vertex counts asmeeting both sidesto which it belongs.)

*Thediscussion of Case(VI) in 47.7,may alsobeconsideredas such a repetitionunder much simpler conditions.)))

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554)) GENERAL NON-ZERO-SUMGAMES))

with the necessaryqualifications of asymmetryand the possibledisappear-anceordegenerationof someareas - (cf. Figures92-95and footnote4 on p.553),thosedepictedin Figure86.))

Figure 92.)) Figure 93.))

Figure 94.)) Figure 95.))

60.4.Comparison with the Zero-sum Games

60.4.1.We have determinedall solutionsof the generaln-persongameswith n = 3 in a rigorousway, but we have not yet made an attempt toanalyze the meaning of our results. We therefore pass now to thisanalysis.

Letus begin with someremarksof a rather formal nature. We haveseenthat the smallestn for which a generalgame can beessentialis n = 2,while for the zero-sumgamesthe correspondingnumber was n = 3. Wehave also seen that there exists (assumingreductionand normalization7 = 1) preciselyone essential generalgame for n = 2, whereas for thezero-sumgamesthe same thing was true for n = 3. Again the essentialgeneralgamesfor n = 3 (under the sameassumptionsas above) form athreeparametermanifold, while for the zero-sumgames this was true forn = 4. All this indicatesan analogy betweengeneraln-persongamesandzero-sumn + 1-persongames.Of course,we know the reason:Thezero-sum extensionsof the generaln-persongamesarezero-sumn + 1-person)))

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ECONOMIC INTERPRETATION FOR n = 1,2 555

gamesand we saw that every zero-sum n + 1-persongame can be obtainedin this way.1

60.4.2.It must be remembered,however, that while the zero-sumn + 1-persongamesareexhaustedby this procedure,their solutionsarenot the solutionsof a generaln-persongame form only a subset of thoseof its zero-sumextension(cf. e.g.(56:I:a)in 56.12.).

Thus our determinationof all solutionsof \"all general three-persongamesmeans only that we know some,but not all solutionsof all zero-sum four-persongames.Indeed,the voluminous and yet incompletediscussionofChapterVIIshowsthat determiningall solutionsof all zero-sumfour-persongamesis a task of considerablygreatersize. Our results concerning thegeneralthree-persongames imply, however, this much:Thereexistsolu-tions for every zero-sumfour-persongame. (ThecasuisticdiscussionofChapterVIIdid not reveal this.)

61.EconomicInterpretationof the Resultsfor n = 1,261.1.TheCasen -1

61.1.We now cometo the main objectiveof our presentanalysis:Theinterpretationof our resultsfor n = 1,2,3.

Considerfirst n = 1:What matters in this casewas alreadystated orreferredto in 60.1.Our resultwas,as it had to be,a repetitionof the simplemaximum principle which characterizesthis case and this caseonly,which therefore describesthe \" RobinsonCrusoe\"or completelyplannedcommunistic economy.

61.2.TheCasen = 2. TheTwo-personMarket

61.2.1.Considernextn = 2:Our resultfor this case,obtainedin 60.2.2.can bestated verbally as follows:

Thereexistspreciselyone solution. It consistsof all those imputationswhere each player gets individually at leastthat amount which he cansecurefor himself, while the two gettogetherpreciselythemaximum amountwhich they can securetogether.

Herethe \" amount which a playercan get for himself\" must be under-stood to be the amount which he can get for himself, irrespectiveof whathis opponentdoes,even assumingthat his opponentis guidedby the desireto inflict a lossratherthan to achieve a gain.2

In examining the solution we find the opportunity to fulfill the promisecontained in the fourth remarkin 58.3.2.:We must seewhether our abovedefinition of the \" amount which a playercan getfor himself\" basedon a

hypotheticaldesireof the opponentto inflict a lossrather than to achieve

1Precisely: It is strategically equivalent to one which is so obtainable. (Cf. the

beginning of 57.4.1.)8 Cf.the detaileddiscussion at the end of 58.2.1.and in 58.3.Theamount which the

player k cansecurefor himself is, of course,v((fc)).)))

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556 GENERAL NON-ZERO-SUMGAMES

a gain leads to common-senseresults.1 In order to comparethe resultof our theory in this way with \"common-sense/'it is desirableto presentthegeneraltwo-persongame in a form which is easilyaccessibleto ordinaryintuition. Sucha form is readilyfound by consideringsomefundamentaleconomicrelationshipswhich can existbetweentwo persons.

61.2.2.Accordinglywe considerthe situation of two personsin a market,a sellerand a buyer. We wish to analyze one transactiononly and it will

appear that this is equivalent to the generaltwo-persongame. It isobviously also equivalent to the simplest form of the classicaleconomicproblemof bilateral monopoly.

The two participants are 1,2:the seller1 and the buyer 2. Thetransactionwhich we consideris the saleof one unit A of a certaincommod-ity by 1to 2. Denotethe value of the possessionof A to 1by u and for 2by v. I.e.,u representsthe best alternative use of A for the seller,while v

is the value to the buyer, after the sale.In order that sucha transactionhave any sense,the value of A for the

buyer must exceedthat onefor the seller. I.e.we must have

(61:1) u<v.It is convenient to use the stateof the buyer when no saleoccurs i.e.

his original financial position as the zero of his utility. 2

Let us now describethis as a game. In doingthis it is best to omit A

from the picture altogetherand to deal instead with the value connectedwith its transfer or its alternative uses. We may then formulate the rulesof the game as follows.

1offers 2 a \"price\"p, which 2 may \"accept\"or \"decline.\"In thefirst case1,2, get the amounts p, v p. In the secondcasethey gettheamounts u, O.3

The common-senseresult is that the pricep will have some valuebetweenthe limits setby the alternative valuations of the two participants,

1Thereaderwill understand that we do not ascribethis desireto the opponent. Itis only that our theory can be formulated as if he had this desire. What matters is notthis possibleformulation, but the results of the theory.

Indeed,this \"malevolent\" behavior of the opponent determines only some, butnot all features of the solution :It gives the lower limit of what eachplayer must obtainindividually, but what both get together can only bedescribedby the oppositehypothesisof perfectcooperation. (Cf.above.)

This is just a specialcaseof the general fact, that only the entire, rigorous theory isa reliable guide under all conditions, while the verbal illustrations of its parts are oflimited applicability and may conflict with eachother.

All this can bebrought out evenbetter by the detaileddiscussion of58.3.1We arepurposely disregarding the possibility of describing a saleas an exchangeof

goodsfor goods. Our theory forcesus, for reasonswhich we have statedrepeatedly, touse an unrestrictedly transferable numerical utility, which we may as well describeinterms of money.

We shall deviate from this standpoint only in Chapter XII.1We leaveit to the readerto formulate this in terms of our original combinatorial

definition of games.)))

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ECONOMICINTERPRETATIONFOR n 1,2 557

e.that51:2) u g p ^ v.

Vhere p will actually be between the limits of (61:2)dependson factorsottaken into accountin this description.Indeed,this rule of the gamerovides for one bid only, which must be acceptedor declined this islearly the final bid of the transaction. It may have beenprecededbyegotiating,bargaining,higgling, contractingand recontracting,aboutrhich we said nothing. Consequentlya satisfactory theory of this highly

implified modelshoulcHeavethe entireinterval (61:2)availablefor p.61.3.Discussionof the Two-personMarket and Its CharacteristicFunction

61.3.1.Before going any further we add two remarks concerningthisescriptionof the game,which is our modelfor the economicset-upunderonsideration.

First:It would be possibleto use more elaboratemodelsallowing forreater(but limited)numbersof alternative bids, etc.

Thereis a prima facie evidencefor consideringsuch variants, sinceallxisting marketsaregovernedby more or lesselaboraterules for successiveids by all participants,which appearto be essentialfor the understandingf their character.Besides,we did investigate in detail the game of'okerin 19. This gameis basedon the interplay of the bids of all partici-ants and we saw loc.cit.that the sequenceand arrangementof thesebidsas of decisiveimportancefor its structureand theory. (Cf.in particularle descriptivepart 19.1.-19.3.,the variants discussedin 19.11.-19.14.andleconcludingsummaryof 19.16.)

A closerinspectionshows, however, that in our presentsetup theseetailsdo not becomedecisive.The situation is altogetherdifferent fromokerwhich is a zero-sumgameand whereany lossof oneplayer is a gain>r the otherone.1 Specifically,the readermay discussany more compli-Etted market(but with only two participants!)in the sameway as we shall

it for our simpleversionin 61.3.3.Hewill find the samecharacteristicmctionas we obtain (61:5),(61:6)of 61.3.3.Indeed,thedeductionsgiventiere apply mutatis mutandis in any market (of two participants!):The^aderwho carriesout this comparisonwill observethat all that matters1 thoseproofs2 is that the seller(or the buyer) may, if he wishes,insistbsolutely on the particularpricementioned there,irrespectiveof theounter-offers he may getand the numberof successivebids required.8

Theseelaborationsleadessentially to the sameresultsas our simplelodel.We refrain thereforefrom consideringthem.

1 Thisappliesdirectly to Poker as a two-persongame, as consideredin 19.If moretan two personsparticipate, then our treatment by means of coalitionsbrings about theime situation.

2The significant one is the proof of (61:5)in 61.3.3.3 Returning to our previous remarks concerning Poker:.Thereadermay verify for

imself how a corresponding simple overall policy would not work there-dueto the)))

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558 GENERAL NON-ZERO-SUMGAMES

61.3.2.Second:On the otherhand our modelcould also be simplifiedfurther. Indeed,the mechanism of compensationsbetween(co-operating)players,which we assumedin all parts of our theoriesis perfectly adequateto replacebids of prices. I.e.it is not necessary to introduceoffering,acceptingor decliningof pricesas part of the rules of the game. Themechanismof compensationsis fully able to takecareof this, includingthepreliminary negotiating,bargaining,higgling, contractingand recontracting.

Sucha simplified game couldbe describedas follows:Both players 1,2may chooseto exchangeor not. If eitheronechoosesnot to exchange,then1,2get the amounts u, 0. If both choseto exchangethen they get theamounts w', u\" where u'y u\" aretwo arbitrary but fixed quantities with

the sum t;. 1

In other words:The rules of the game may provide for an arbitrary\" price\" p = u' (then v p u\,") which the players cannot influencethey will neverthelessbringaboutany otherpricethey desireby appropriatecompensations.

Thus it appears that the arrangementchosenin 61.2.2.is neitherthesimplestnor the most completeone. We areusingit becauseit seemsto bebest suited to bringout the essentialtraits of the situation without unneces-sary details.

61.3.3.The \" common-sense\" result of 61.2.2.amounts in the terminol-ogy of imputationsto this:Thereexistspreciselyonesolution and this is thesetof all imputations))

with

(61:3) ai ^ u, <*2 ^ 0,(61:4) ai + a2 = v.

Comparingthis with the applicationof our theory in 60.2.2.,we seethat agreementobtains when (61:3),(61:4)coincidewith (60:3),(60:4)there. This meansthat we must have))

(61:5) v((l)) =u, v((2))=0,(61:6) v((l,2))=.

It is easilyverified that (61:5),(61:6)areindeedtrue. Forthe sakeofcompletenesswe do this for both arrangementsof 61.2.2.and 61.3.1.,penalties which the rules of that game inflict upon any prohibitive, excessive,or in anyother simple way uniform schemeof bidding.

One could, of course,incorporate similar provisions into the rules governing amarket. Indeed,there are certain traditional forms of transactions which are possiblyof this type, such as options. But it doesnot seem advisable to include them in thisfirst, elementary survey of the problem.

1Thecharacteristicfunctions ofboth arrangements (that of61.2.2.and the oneabove)will bedetermined in 61.3.3.and they will be found to be identical.)))

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ECONOMICINTERPRETATION FOR n - 1,2 559

61.3.2.,the first beingdealt with in the text,and the variants requiredforthe secondalternative in brackets[ ].

Ad (61:5):Player 1can make sure to obtain u by offering the pricep = u [by choosingnot to exchange].Player2 can makesurethat player1obtains u by decliningevery price [by choosingnot to exchange].Hence))

Replacementof p - u by p = v [the same conduct of both players]yieldsin the sameway that v((2))= 0.

Ad (61:6):Thetwo playerstogethergeteitheru or v the latterarisingfrom p + (v p) [from u' + u\"}. By (61:1)v is preferable;hence

v((l,2))= v.

61.4.Justification of the Standpoint of 68.61.4.Thecoincidenceof the values of the characteristicfunction v(S)

with the u, 0,v as observedin 61.3.3.may appear fairly trivial. Thereis,however, one significant point about it:Itwasobtainedwith our definitionof the characteristicfunction to which the criticismsof 58.3.and 61.2.apply. I.e.it is dependentupon eachplayerascribingto hisopponent ina certainpart of the theory but not in all of it the desireto inflict a lossrather than to achieve a gain.

It is important to realize that this dependenceis really significant,i.e.that modificationof this assumptionwould alterthe result,and thereforefalsify it, sincethe resultwas seento becorrect.Thisisbestdonewith thearrangementof 61.2.2.

Indeed,assume that player 2 would under certainconditionspreferto make a profit for himself rather than to inflict a loss upon player 1.Assume that theseconditionsexist,e.g.when player1offers a certainpricePO > u but < v. In this caseplayer2 obtains v p if he accepts,andif he declines.Hencehe gains by accepting. On the otherhand player1obtains p<> if player2 acceptsand u if he declines.Henceplayer2 inflictsa loss (uponplayer 1)by declining.Consequentlyour presentassumptionconcerningthe intentions of player2 meansthat he will accept.

Thus under theseconditionsplayer 1 can count upon obtaining theamount p . This conflicts with our previous result accordingto which theentirepriceinterval (61:2)shouldbe permissible,and we sawin 61.2.2.thatit is the latterresult that must be consideredthe natural one.

Summing up:Thediscussionof the generaltwo-persongame which wecarriedout in 61.2.-61.4.has shown that the generaltwo-persongame iscrucialwith regard to the decisionwhether the characteristicfunctionshouldbe formed as used in our theory. Thesetup wassimpleenough asto allow a ''common-sense\"predictionof the result and any changeinthe procedureof forming the characteristicfunction would have alteredthe theoreticalresult significantly. In this way we have obtainedby theapplicationof the theory, a corroborationin the senseof the fourth remarkin 58.3.)))

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560 GENERAL NON-ZERO-SUMGAMES

61.6.Divisible Goods. The\"Marginal Pairs\"

61.5.1.Thediscussionof 61.2.-61.4.referredto a very elementarycasebut it neverthelesssufficedfor the task of \" corroboration\" which we hadsetfor ourselves.Besides,by interpretingoneessentialgeneraltwo-persongame,all were interpreted,sinceall of them arestrategicallyequivalentto onereducedform (which couldbe normalized to y = 1).

Sofar everything is satisfactory. But it is still desirableto verify thatour theory can do equal justiceto somewhatlesstrivial economicsetups.Forthispurposewe will first extendthe descriptionof the two-personmarketsomewhat. It will be seenthat this yieldsnothing really new. Thenweshall turn to the generalthree-persongames. Therewe will find genuinelynew corroborationsand opportunitiesfor more fundamental interpretations.

61.5.2.Let us return to the situation describedin 61.2.2:the seller1and the buyer 2 in a market. We allow now for transactions involvingany or all of s (indivisibleand mutually substitutable)units AI, , A t

of a commodity.1 Denotethe value of the possessionof tf( = 0,1, , s)of theseunits for 1by u t and for 2 by vt. Thus the quantities

(61:7) U Q= 0, MI, , M.,

(61:8) V Q= 0, t;i,---,.,

describethe variable utilities of theseunits to eachparticipant. As in61.2.2.we use for the buyer his original positionas the zero of his utility.

Thereis no need to repeatthe considerationsof 61.2.2.,61.3.1.,61.3.2.concerningthe rulesof the game which modelsthis setup.

It is easy to see,what its characteristicfunction must be. Sinceeachplayercan blockall sales,2 it followsas in 61.3.3.that

(61:9) v((l)) = 11., v((2))=0.Sincethe two players togethercan determinethe number of units to betransferredand sincewith a transfer of t units they obtain togetheru,-t + vt ,therefore

(61:10) v((l,2)) = Max,..o,i.....(*.-,+ v>).

This v(S) is a characteristicfunction, henceit must fulfill the inequali-ties(57:2:a),(57:2:c)of 57.2.1.Considering(61:9),(61:10),the only onewhich is not immediately obvious is

(61:11) v((l,2)) v((l)).This obtains, by observingthat the left-hand side is ^ u9 + v<>

= u, by(61:10)(uset = 0), and the right-hand sideis = u t.

1We could alsoallow for continuous divisibility, but this would make no materialdifference.

2 1by offering an inacceptibly high price,2by declining every price.)))

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ECONOMICINTERPRETATION FOR n 1,2)) 561))

61.5.3.Considernow the t for which the maximum in (61:10)isassumed,say t = t Q. It is characterizedby w._t o + v

to ^ u,-.t + vt for all t. Thisneedonly bestated for the t ^ fa and wecan stateit for the t ^ fa separately.We may write theseinequalitiesas follows:

(61:12) u.-t9 - u._t ^ vt - vt9

for t > fa,(61:13) u.-t- u..t vt - vt for t < fa.))

Specialize(61:12)to t = fa + 1 (exceptwhen fa s in which case(61:12)is vacuous):(61:14) u,-t - u.-* > vt+l - vt ,\\ / I

Bi

flto-t-i *

Q7

and (61:13)to t = t Q 1 (exceptwhen to = in which case(61:13)isvacuous) :(61:15) ti._i +i - i/._!< v t - vt-i.0^ 00Note that (0:12),(0:13)(without the specializationt = fa 1 that led to(6:14),(6:15))can be written as follows))

(61:16))) +>))

- ) ^ (,- ';-))) > (,,))

(61:17))) for))

In generalwe can say that (61:14),(61:15)is necessaryonly, while

(61:16),(61:17)is necessaryand sufficient. However,we may now profit-ably introducethe assumptionof decreasingutility that is, that the utilityof eachadditional unit decreases,as the total holding increases,for bothparticipants 1,2. As a formula))

(61:18)(61:19)))

U\\

t'l))

This impliesr))

(61:20)))

V))

>>))

Mi >V\\ >)) > V, ?'-!))

for t >))

- W.-,) ^ (/O))

for I <))

encenow (61:14),(61:15)imply (61:16),(01:17).Consequently(61:14),)))

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562 GENERAL NON-ZERO-SUMGAMES

(61:15)too arenecessaryand sufficient. Combining(61:14),(61:15)with

part of (61:18),(61:19)we may alsowrite:

Eachoneof

(61:21) U'-'*~\"\"/r1' Vt

> 7 \"'-1 ,is greaterthan eachoneof))

According to the usual ideas,the maximizing t = to is the number ofunits actually transferred. We have shown that it is characterizedby(61:21),and the readerwill verify that (61:21)is preciselyBohm-Bawerk'sdefinition of the \" marginal pairs.\"2

Sowe see:(61:A) The sizeof the transaction,i.e.the number to of units trans-

ferred, is determined in accordwith Bohm-Bawerk'scriterionof the \" marginal pairs.\"

To this extentwe may say that the ordinary common-senseresult hasbeenreproducedby our theory.

It may be noted,to conclude,that the casewhen this game is inessentialhas a simplemeaning. Inessentialitymeanshere))

i.e.by (61:9)equality in (61:11).Considering(61:9),(61:10)this meansthat the maximum in the latteris assumedat t = 0,i.e.that /o = 0. Sowe see:(61:B) Our game is inessentialif and only if no transferstake place

in it i e.when t Q= O.3

61.6.ThePrice. Discussion

61.6.1.Let us now pass to the determinationof the pricein this set-up.In order to provide an interpretationin this respectwe must considermorecloselythe (unique)solution of our game,as providedby the considerationsof 60.2.2.

Mathematicallythe present set-up is no more generalthan the earlierone analyzed in 61.2.-61.4.: both representessentialgeneraltwo-persongames,and we know that thereexistsonly one suchgame. Nevertheless,that set-upwas only a specialcaseof our presentone:Correspondingto8 = 1. This differencewill be felt as we now passto the interpretation.

1Comparing the first term of the first line with the secondterm of the secondline is(61:14);comparing similarly secondand first is (61:15).Comparing first and first is aninequality from (61:18);secondand second,one from (61:19).1E.von Bohm-Bawerk: Positive TheoriedesKapitals, 4th Edit. Jena1921,p.266ff.

8 Note that in our earlierarrangement of61.2.2.we forcedthe occurrenceof a transferby requiring (61:1).Our presentset-upleavesboth possibilities open.)))

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Comparisonof (61:5),(61:6)in (61.3.3.)with (61:9),(61:10)in 61.5.2.shows that the mathematical identity of these two setups rests uponsubstituting the u, v of the former accordingto

(61:22) u = u.y v = Max,_,i......(u,-.t + vt).The (unique)solutionconsists,therefore, of all imputations

a = {{!,a2}}fulfilling (61:3),(61:4)in 61.3.3.In terms of 2 this means

(61:23) ^ 2 ^ v - u.1

Let us now formulate this in terms of the ordinary conceptof pricesinsteadof the imputationswhich arethe meansof expressionof our theory.2Since,as we concludedin 61.5.3.,t Q units will have been transferred to thebuyer 2,theremust be

(61:24) vt9- Up = 2,

if the pricepaid wasp per unit. Consequently(61:23)means,in terms ofp, that

(61:25) T(U.-u.-t ) ^ p ^}\\*to to

This canalsobe written as<o 'o

(61:26) f V (u.^.1- w,_ t-) ^ p g 1V fe -^0.to ^-/ ^o ^W-l J-l

61.6.2.Now the limits in (61:26)arenot at all those which the Bohm-Bawerk theory provides. According to that theory, the pricemust liebetweenthe utilitiesof the two marginal pairs named in (61:21)of 61.5.3.,i.e.in the interval

(61:27) K^-^oUp (*<-<o--<o- - ^-i))

This canalsobe written as

(61:28)Max (w,-e+i - w._v vt,+i- vf|) ^ p^ Min (ti,_ <o

tt._ o_i, v<o

t;<o_i)

In orderto comparethis interval with (61:26),it is convenient* to forma further interval

(61:29) u8-to+ i - tt._i a ^ p ^ w o- -i.

1We could baseour discussion equally well on on, but the presentprocedureis bettersuited to berepeatedin the caseof the three-personmarket.

1It may be worth re-emphasizing: This is interpretation, and not the theory itself!3Note that by (61:22)u -w,, v = w<t + t><

p.)))

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The two last inequalitiesof (61:20)in 61.5.3.(with t = 0) yield that thelower limit of (61:29)is ^ that of (61:26),and that the upperlimit of (61:29)is ^ that of (61:26).Hencethe interval (61:29)is containedin the interval(61:26).Again, (61:29)obviously containsthe interval (61:27),i.e.(61:28).Summing up:Theintervals (61:26),(61:29),(61:28)contain eachother,inthis order.

Sowe see:(61:C) Thepricep per unit is limited to the interval (61:26)only,

while Bohm-Bawerk'stheory restrictsit to thenarrower interval(61:28).

61.6.3.The two results (61:A) and (61:C)give a precisepicture of therelation of our theory, in the presentapplication,to the ordinary commonsensestandpoint.1 They show that there is completeagreementcon-cerningwhat will happenin fact i.e.the number of units transferred buta divergenceas to the conditionsunder which it will takeplace i.e.thepriceper unit. Specifically,our theory provideda widerinterval for thatpricethan the ordinary viewpoint.

That the divergenceshouldcomeat this point and in this directionisreadilyunderstandable. Our theory is essentiallydependentupon assum-ing (among otherthings) a completemechanism of compensationsamongthe players. This amounts to possiblepaymentsof varying premiumsorrebatesin connection with the various units transferred. Now the narrowpriceinterval of the ordinary standpoint (defined by Bohm-Bawerk's\"marginal pairs\")is notoriously dependentupon the existenceof a uniqueprice equallyvalid for all transfers which occur. Sincewe areactuallyallowing premiumsand rebates,as indicated above, the unique priceisobliterated.Our pricep per unit is merely an average price indeed itwas defined as such by (61:24)in 61.6.1.and it is therefore quite naturalthat we obtained a wider interval than the one defined by

\" marginalpairs.\"

Toconclude,we observethat suchabnormalitiesin the formation of thepricestructurearealso quite in agreementwith the fact that the marketunder considerationis a bilaterallymonopolisticone.

62.EconomicInterpretationof the Resultsfor n = 3 :SpecialCase

62.1.TheCasen 3, SpecialCase. TheThree-personMarket

62.1.1.Considerfinally n = 3. We proposeto obtain an interpretationin the samesenseas was outlined in 61.2.1.This will be doneby extendingthe modelof 61.2.2.dealingwith two personsin a market to one dealingwith threepersons.

1We took Bohm-Bawerk's treatment as representative for that standpoint. Indeed,the views ofmost other writers on this subjectsinceCarl Menger areessentially the sameas his.)))

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As we have pointedout before,the first mentioned discussioncouldnotfail to beexhaustive,sincethereexistsonly one essentialgeneraltwo-persongame. On the other hand we know that the essentialgeneral three-persongamesform a 3 parameterfamily and their detailed discussionin 60.3.2.forced us to distinguish numerous alternatives.1 Accordingly severalmodelswould be required to accountfor all possibilitiesof the essentialgeneralthree-persongame. We shall restrictourselvesto the discussionof one typicalclass. An exhaustive discussionwould be somewhat lengthyand would not contributeproportionally to our understandingof the theory

but it would not presentany additionaldifficulties.62.1.2.We consider accordinglythe situation of three-personsin a

market, one sellerand two buyers. Thediscussionof two sellersand onebuyer would lead to the same mathematical setup and to correspondingconclusions.Forthe sakeof definitenesswe discussthe first form of theproblemand leave it to the readerto carry out the paralleldiscussionofthe secondform.

Thethreeparticipantsare1,2,3the seller1,the (prospective)buyers2,3. We shall consider successivelythe specialarrangement of 61.2.2.and the more generaloneof 61.5.2.In contrastto what wefound there,thelatterwill now provide a real generalization of the former.

Let us begin with the setup of 61.2.2.:The transactionwhich we con-sider is the saleof one (indivisible)unit A of a certain commodity by 1toeither2 or 3. Denotethe value of the possessionof A for 1 by u, for 2 by v,and for 3 by w.

In orderthat thesetransactionsshouldmakesensefor all participants,the value of A for eachbuyer must exceedthat for the seller. Also, unlessthe two buyers 2,3happen to be in exactlyequal positions,oneof themmust be strongerthan the other i.e.able to derivea greaterutility fromthe possessionof A. We may assumethat in this casethe strongerbuyeris 3. Theseassumptionsmean that we have

(62:1) u < v ^ w.

As in 61.2.2.and 61.5.2.weusefor eachbuyerhis originalpositionas the zeroof his utility.

As in 61.5.,there is no need to repeatthe considerationsof 61.2.2.,61.3.concerningthe rulesof the gamewhich modelsthis setup.

It is easy to seewhat its characteristicfunction must be:Sinceeachbuyer can blocksalesto him, and the selleras well as both buyerstogethercan blockall sales(cf. 61.5.2.),it followsas in 61.3.3.that

(62:2) v((l)) = u, v((2))= v((3))= 0,(62:3) v((l,2)) = v, v((l,3)) = w, v((2,3))= 0,(62:4) v((l,2,3))= w*

1The two main cases(a) and (b), the latter being subdivided into the four subcasesrepresentedby the Figures 92-95.

1Ofcoursethis makes useof w < v 5* w.)))

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566 GENERAL NON-ZERO-SUMGAMES

Thisv(S)isa characteristicfunction, henceit must fulfill theinequalitiesof (57:2:a),(57:2:c)in 57.2.1.The verification can be carriedout with

little trouble,and is left to the reader.By the nature of things the gameto which v(S)belongsis not constant

sum,1henceit is a fortiori essential.

62.2.Preliminary Discussion

62.2.We can now apply the results obtained in 60.3.concerningtheessentialgeneralthree-persongame,to obtain all solutionsfor our presentproblem.We shall again comparethe mathematical result with what theapplicationof ordinary common sensemethodsgives.

Theagreementwill turn out to bebetterthan in 61.5.2.-61.6.3.up to acertainpoint specificallythe limits to bederivedfor the pricewill bethesame with both methods. This is probablyascribableto the fact that wearedealingnow with oneunit only, just as in 61.2.2.When we pass to *

units, in 63.1.-63.6.,the complicationsof 61.5.2.-61.6.3.will reappear.Beyond the point referred to, however, there will be a qualitative

discrepancybetweenour theory and the ordinary view point. It will beseenthat this isdue to the possibilityof forming coalitions.Thispossibilitybecomesa reality for the first time for threeparticipants, and it must beexpectedthat our theory will do it full justice while the ordinaryapproachusually neglectsit. Thus the divergenceof the two procedureswill alsoturn out to bea legitimateonefrom the point of view of our theory.

62.3.TheSolutions :First Subcase62.3.1.We proceedto the applicationof 60.3.1.,60.3.2.to the v(S) of

(62:2)-(62:4)above.Theimputations in this setuparethe))

with

(62:5) ai ^ w, at 0, a, 0,(62:6) ai + a2 + as = w.

In orderto apply 60.3.1.,60.3.2.,it is necessary to bring this to itsreducedform, and then to normalize y = 1.

The first operationcorrespondsto the replacementof our ai,aj, 8 bythe aj,aj,a of

(62:7) ai = a* + aismentionedin 57.5.1.and discussedin 31.3.2.and in 42.4.2.Thea?,aj,aj obtain as describedin the discussionwhich leadsto (59:A) in 59.2.1.

1Proof:(57:20)in 57.5.2.is violated, e.g.by

v((2,3 - u < w -v()))

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Specifically,

,AO Q\\ ' w + 2u , w u , w u(62:8)a{= i --g> a = a2 --g

* o = 8 --g

Thecorrespondingchangeson v(S)aregiven by (59:1)in 59.2.1.; they carry(62:2)-(62:4)into

(62:9)))

(62:10)v'((l,2)) = -*-*v'((i,3))))3 ' v \\\\*t**JJ Q

>

v'((2,3))- -?l^p),(62:11) v'((l,2,3))= 0.

Thus 7 => and so the secondoperationconsistsof dividing every-o

thing by this quantity. Insteadof doingthis, we prefer to apply 60.3.1.,60.3.2.directly, inserting everywhere (where 7 = 1 was assumed) the

proportionalityfactor -1o

Comparisonwith (60:8)in 60.3.1.showsthat

2(w u) w u 3v 2w ufll = 1 , az , ag _

The six linesof (60:15)in 60.3.2.,which describethe triangle from whichwe derivedour solutions,becomenow:))

(62:12)v ;))

62.3.2.We can now discussthis configuration in the senseof 60.3.3.Clearly

i + 2 + 3 = v w rg 0,hencewe have (60:17:b)loc.cit. i.e.we have the Case(b) id.,and itremains to decidewhich one of its four subcases,representedby Fig-

1Theprocedureis analogous to that used in the discussion of the essentialzero-sumthree-persongame with excessin 47.,in particular in 47.2.2.and 47.3.2.(Case(III)),47.4.2.(a certain phaseof Case(IV)).

*The 1 in (60:15)loc.cit. stands for 7, so we must multiply it by the propor-w u

tionality factor r mentioned above.u

8 The ai, ai, at in (60:15)loc.cit.,which reappearhere, include already ther * w -ufactor r))

, w u) , w u)At')

n? u2)

**1 ') a2 o j a3) 3 ')

, 2(w - u)) , w u) ,) 3t> 2w u)ai o ;) a2 3)a

3)3)))

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568)) GENERAL NON-ZERO-SUMGAMES))

ures92-95,is present. Therefore we proceedfrom hereon by graphicalrepresentation.

Forthis representationweuse,asbefore,theplaneof Figure52. Repre-sentingthe sixlinesof (62:12)as thoseof (60:15)in 60.3.2.wererepresented))

-u)))

, w u.< 3-Figure 96.)) Figure 97.))

V: Theline /and the curve))

Figure 98.

by Figures92-95,we obtain Figure96. The qualitative features of this

figure follow from the followingconsiderations:))

(62:A:a))) The secondaj-linegoesthrough the intersectionof thefirst ajj- and c^-lines. Indeed:))

2(w -u))) w u)) w u)) = 0.))

(62:A:b)(62:A:c)))

The two a-linesareidentical.Theseconda^-line is to the left of the first one. Indeed:

It has a greatera'3-value, since3v 2w u . w u ^ ~))

Comparisonof this figure with Figures92-95showsthat it is a (rotatedand) degenerateform of Figure94 -,

1Thearea is degeneratedto a point(the upper vertex of the fundamental triangleA), the areas , also

1For this and the remarks which follow, cf. footnote 4 on p. 553.)))

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degenerated,but to two linear intervals (the upper and the lower partof the left side of the fundamental triangle A), while the areas , arestill undegenerated(the trapezonand the smallertriangle, into which thefundamental triangle A is divided on our figure). This dispositionofthe five areasof Figure94 is shown in Figure97. The generalsolution Vnow obtains,asstated at the endof 60.3.3.,by fitting the pictureof Figure86into the situation describedby Figure97. Figure98 showsthe result.1

62.4.TheSolutions :GeneralForm

62.4.Before we go any further we note that Figure97 is of generalvalidity, assuming

(62:13) u < v ^ w,

but the picture it gives refers qualitatively to

(62:14) v < w.

When

(62:15) v = w,

then thearea in Figure97 i.e.the upper interval on the left sideof thefundamental triangle degeneratesto a point. (Cf. (62:A:c)in 62.3.2.)Hencein this caseFigure98 assumesthe appearanceof Figure99.))

V: Theline \\and the curve

V: Thecurve))

Figure 99. Figure 100.This discussioncan be renderedquite symmetricwith respectto the

players2,3 the two buyers by:Assuming (62:14)or (62:15),we may replace(62:13)by the weaker

condition

(62:16) u < v, w.

Let us therefore assume(62:16)only and not (62:13)with (62:14),(62:15).Thismeansthat eachbuyerderivesa higher utility from the possessionof A

than the seller,but it doesnot placethe buyerswith respectto eachother.(Cf. the discussionin the first part of 62.1.2.)

1Thecurve in Figure 98 is like those in Figure 86,subject to the restriction statedthere:(47:6)in 47.5.5.)))

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570 GENERAL NON-ZERO-SUMGAMES

Now (62:16)leavesthreepossibilitiesopen:(62:14),(62:15),and

(62:17) v > w.

The solutionsof (62:14),(62:15)were given by Figures98,99. (62:17)obtainsfrom (62:14)by interchangingthe two players2,3 the two buyers

and t>, w. This means that Figure98 must be reflectedon its verticalmiddleline (after interchangingv, w). This is shown in Figure100.

Summingup:(62:B) Assuming (62:16),the generalsolutionV is given by Figures

98,99,100for v <,=, > w, respectively.

62.6.Algebraical Form of the Result

62.5.1.The result expressedby Figure98 can be statedalgebraicallyas follows:1

Thesolution V consistsof the upper part of the left side of the funda-mental triangle,and the curve ~.

Thefirst part of V is characterizedby

' w ~~ u _ 3v ~\" 2w u , __ w u3 3 ~~ ~~ 3

Owing to (62:8)in 62.3.1.,this meansthat

2 = 0, W V ^ 3 ^ 0.Now (62:6)in 62.3.1.gives

a\\ = w 3,

hencethe above conditioncan bewritten as

(62:18) v g ai ^ w, a* = 0, a8 = w - 01.The secondpart of V (the curve) extendsfrom the smallesta{above

to the absolute minimum of a{ ( V Itsgeometricalshape (cf.

(47:6)in 47.5.5.)can be characterizedby stating that along it 2> <*s areboth monotonic decreasingfunctions of a{. We may again pass from

i 2> to i, 2, as by (62:8)in 62.3.1.Then a\\ varies from its minimumin (62:18)above (v) to its absoluteminimum (u),and 2, s areagain bothmonotone decreasingfunctions of i. Sowe have:

(62:19)u g i g v, c*2, 3 aremonotonic decreasingfunctions of ai.2- 8

Thus the generalsolutionV is the sum of the two setsgiven by (62:18)and1Note that it holds whenever v ^ it>, (62:B)notwithstanding.2They must, of course,fulfill (62:5),(62:6)in 62.3.1.8 As Figure 98shows, the lowest point on the line / coincideswith the highest point on

the curve. I.e.the point a.\\ *= v of (62:18)and of (62:19)is the same.Hencewe couldexcludeai v from either (but not from bothl) of (62:18),(62:19).)))

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(62:19).It will be noted that the functions mentioned in (62:19)arearbitrary (within certainlimits),but that a definite solution(i.e.a definitestandard of behavior)correspondsto a definite choiceof thesefunctions.This situation is entirely similar to those analyzed in (47:A) of 47.8.2.and in 55.124.

62.5.2.(62:18),(62:19)can be used whenever v ^ w (cf.footnote 1onp.570). Forv = w (62:18)simplifiesto

(62:20) a\\ = v, 2 = s = 0.We shall therefore use (62:18),(62:19)only when v < w, and (62:20),(62:19)when v == w.1

If v > w, then we can utilize (62:18),(62:19)by interchanging theplayers2,3 the two buyers and v, w. Then (62:18),(62:19)become(62:21) w ^ ai g v, a2 = v - i, a3 = O.2

(62:23)u g a\\ g w y a*, 3 aremonotonic decreasingfunctions of ai.1Summingup:

(62:C) Assuming (62:16),the generalsolution is given by (62:18),(62:19);(62:20),(62:19);(62:21),(62:23)for v <, = , >wrespectively.

62.6.Discussion

62.6.1.Let us now apply the ordinary, common-senseanalysis to themarket of one sellerand two buyersand one indivisibleunit of a good,inorderto compareits result with the mathematical onestatedin (62:C).

Thelinesof this common-senseprocedureareclearlylaid down* we areactually dealingherewith one of the simplestspecialcasesof the theory of\"marginal pairs.\" Theargument runs as follows:

The selleroffers only one indivisibleunit of the good under consider-ation and therearetwo buyers. Henceone will be includedin the trans-action,and one will be excluded.Clearly the strongerbuyer will be inthe first position exceptwhen the two buyershappento be equally strong,in which caseeitheris eligible.Accordingly the priceat which the trans-actiontakesplacewill liebetweenthe limits of the includedand the excludedbuyer and if they happento beequallystrong,thepricemust bepreciselytheircommon limit. Thelimit of the seller,which must beassumedto be

l The observation of footnote 3 on p.570 concerning (62:18),(62:19)appliesalsoto(62:20),(62:19).Hencewe couldomit (62:20)altogether, but it is more convenient tokeepit, for the sakeof the interpretation in 62.6.

*Note that, owing to the aboveinterchange, (62:4)in 62.1.2.becomes

(62:22) v((l,2,3))- ,and so (62:6)in 62.3.1.becomes

(62:6*) ai + a* -f as = v.

8 Theobservation offootnote 3 on p.570concerning (62:18),(62:19)appliesto (62:21),(62:23)also. Of coursewe must replaceits v by w.)))

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572 GENERAL NON-ZERO-SUMGAMES

lower than that of eitherbuyer in orderto have a genuine three-personmarket,comesin no caseinto play.

In our mathematical formulation the limits of the sellerand of the twobuyerswerew, v, w. Theabove remarkmeans

(62:16) u < v, w.

Thestatementsconcerningthe priceamount to

(62:24) v p w for v < w,

(62:25) p = v for v = w,

(62:26) w g p g v for v > w.

A buyerwho isexcluded,finisheswhere he started i.e.in our normalizationof utility at zero.

Consequentlyour present statements correspondexactly to (62:18),(62:20),(62:21),as providedfor by (62:C).

Sofar the mathematical and the common-senseresultsagree. But thelimit of this agreementis also in evidence:(62:C)providedfor the furtherimputations of (62:19),(62:23),and thereis no traceof thesein the ordinarytreatment,as presentedabove.

What then is the meaning of (62:19),(62:23)?Do they expressaconflict betweenour theory and the common-sensestandpoint?

It is easy to answerthesequestions,and to seethat thereexistsno realconflict, but that (62:19),(62:23)representa perfectly proper extensionof the common-sensestandpoint.

62.6.2.Theamount obtainedby the sellerin a given imputation, i, isclearlythe pricep envisagedwhen that imputation is offered. In (62:19),(62:23),a\\ varies from u to v or w (accordingto which is smaller) i.e.the pricevaries from the seller'slimit to the weakerbuyer's limit. Thereis also a definite (monotonic) functional connectionbetweenthe (variable)amounts obtainedby the two buyers.1

Thesetwo facts strongly suggestgiving (62:19),(62:23)the followingverbal interpretation:The two buyershave formed a coalition,based on adefinite rule of division for any profit obtained,and arebargainingwith theseller. The rule of division is embodiedin the monotonic functions thatoccurin (62:19),(62:23).No bargainingcan depressthe sellerunder hisown limit.2 On the otherhand a priceabove the limit of the weakerbuyerwould excludehim from any possibilityof exertinginfluence.

The specificrules contained in (62:19),(62:23),and the rolesof allparticipants in thesesituationsmay begiven more extendedverbal treat-ment. We shallnot do this here,sincethe above shouldsufficeto establishour main point:On the one hand (62:18),(62:20),(62:21)(i.e.the upperparts of V in Figures98-100)correspondto the competitionof the twobuyers for the transaction in which the strongerplayer, if oneexists,is

1 All these\"amounts\" refer to the utility in which we reckon of the goodsunderconsideration there existsonly one, indivisible, unit.

1Theseller'slimit is his bestalternative use (instead of a sale)for A.)))

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sure to win. On the otherhand, (62:19),(62:23)(i.e.the lower part of Vin Figures98-100,the curves)correspondto a coalition of the two buyers,againstthe seller.

Thus it appears that the classicalargument at leastin the form usedin 62.6.1.gives the first possibilityonly, disregardingcoalitions.Ourtheory, to which the coalitionscontributeddecisivelyfrom the beginning, isnecessarilydifferent in this respect:It embracesboth possibilities,indeedit gives them weldedtogether,as a unit, in the solutionswhich it produces.Theseparation,accordingto schemeswith and without coalitions,appearsonly as a verbal comment on the relatively simple three-persongamethere is no reasonto believe that it can becarriedout for all games,whilethe mathematical theory appliesrigorously in all situations.

63.EconomicInterpretation of the Resultsfor n = 3 :GeneralCase

63.1.Divisible Goods

63.1.1.It remainsfor us to extend the three-personsetup of 62.1.2.inthe sameway that the two-personsetup of 61.2.2.was extendedin 61.5.2.,61.5.3.

Let us accordinglyreturn to the situation describedin 62.1.2.:the seller1and the (prospective)buyers2, 3 in a market. We allow now for trans-actions involving any or all of s (indivisibleand mutually substitutable)units Ai, - - - , A, of a particular good. (Cf. also footnote 1on p.560.)Denotethe value of t (= 0,1, , s) of theseunits by u t for 1,by v t

for 2,and by w t for 3. Thus the quantities

(63:1) t/o = 0, MI, , u,,(63:2) v = 0, vi, , v.,(63:3) MO = 0, wi, - , w,,

describethe variable utilities of theseunits for eachparticipant.As before,we use for eachbuyer his original positionas the zero of his

utility.As in 61.5.2.,61.5.3.and 62.1.2.,we neednot repeatthe considerations

of 61.2.2.,61.3.1.,61.3.2.concerning the rules of the game which modelsthis setup.

It is easy to seewhat its characteristicfunction must be:Sinceeachbuyer canblocksalesto him, and the selleras well as both buyerstogethercan blockall sales(cf.61.5.2.and 62.1.2.),it followsas in 61.3.3.that

(63:4) v((l)) = u., v((2))= v((3))-0,(63:5) v((2,3))= 0.

Denotingthe number of units transferredfrom the seller1to the buyers2,3 by t, r, respectively,it is easy to expresswhat the remaining coalitions(1,2),(1,3),(1,2,3)i.e.the sellerwith eitherone or both buyers can)))

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574 GENERALNON-ZERO-SUMGAMES

achieve. Thefamiliar argumentsgive))

)) = Max,_ ,i......(u._< + v t),) = Maxr_0>1......(_.+ ,),

(63:7) v((l,2,3))= Max,.r_0>l......(u._,_r + v, +))

Thisv(S)isa characteristicfunction. We leave it to the readerto verifythe inequalitieswhich areimpliedthereby.

Thediscussionas to when this game is essentialcan be carriedout as in61.5.2.,61.5.3.and is left to the reader.2 It is also possibleto determinewhen oneof the two buyers2,3becomesa dummy in the senseof our theoryof decomposition.We shall not considerthis either;the result is notdifficult to obtain, and though not surprisingit is not uninteresting.

63*1.2.Restrictingr in the maximum of (63:7)to the value convertsthis into the first maximum of (63:6).Restrictingt thereto the valueconvertsit into u,. By eachone of theseoperationsthe value becomesg ,i.e.we have))

(63:8)If we do the sameto r, t in the reverseorder,we obtain similarly

(63:9) v((l)) v((l,3)) v((l,2,3)).Considerthe first inequality in (63:8).To have equality theremeans

that the first maximum in (63:6)is assumed for t = 0. According tothe usual ideason the subjectthis meansthat the sellerand buyer 2, in theabsenceof buyer 3,would effect no transfers. I.e.,that buyer 2, in theabsenceof buyer 3, is unableto makethe market function.

Consider the secondinequality in (63:8).To have equality theremeansthat the maximum in (63:7)is assumedfor r = 0. Accordingto theusual ideas on the subject,this means that the sellerand buyer 3,in thepresenceof buyer 2, would effect no transfers. I.e.,that buyer 3, inthe presenceof buyer2,is unable to participatein the market.

Summing these up, togetherwith the correspondingstatements for(63:9)which obtain by interchanging buyers2, 3, we have:

(63:A) Equality in any one of the four inequalitiesof (63:8),(63:9)is a signof someweaknessof one of the buyers.

In the first inequality of (63:8),[(63:9)]it means thatbuyer 2 [3], in the absenceof buyer3 [2], is unable to makethemarket function. In the secondinequality of (63:8),[(63:9)]it

means,that buyer 3 [2], in the presenceof buyer 2 [3],is unableto influence the market.

1Theextra condition t -f r ^ under this Max expressesthat the number t -f r ofunits soldcannot exceedthe number of units originally possessedby the seller.

'Thediscussion of the relationship with (62:1)in 62.1.2.or with (62:16)in 62.4.,when 8 -1,canalsobecarriedout easily. Thediscussion of (61:B)at the end of61.5.3.and footnote 3 on p.562should be remembered.)))

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The really interestingcasearises,obviously, when all theseweaknessesareexcluded.It is therefore reasonableto make the hypotheseswhichexpressthis:(63:B:a) We have < in the first inequalitiesof both (63:8)and

(63:9).(63:B:b) We have < in the secondinequalitiesof both (63:8)and

(63:9).63.2.Analysis of the Inequalities

63.2.1.Assume for a moment (63:B:a),but the negation of (63:B:b).Thismeansthat one of the two playersis absolutelystrongerthan the other.Moreprecisely:That he is at least as strongas the other player,even whenhe tries to excludethe other player completelyfrom the market.

Hencewe may expectin this casea result which is similar to thatobtainedin 62.1.2.-62.5.2.,when only one (indivisible)unit A was available.I.e.the divisibility of the supply into units Ai, , A, which we havehere,shouldnow becomeeffective.

Thisis indeedthe case. To prove it, introducethe quantitiesu, v, w by

(63:10) v((l)) = u, v((l,2))= v 9 v((l,3))= w.

Then the secondinequality of (63:8)and of (63:9)and the negation of(63:B:b)give

(63:11) v((l,2,3))= Max (v, w)

while the first inequality of (63:8)and of (63:9)and (63:B:a)give

(63:12) u <v,w.Now we have preciselythe conditionsof 62.1.2.-62.5.2.:(63:12)coincides

with (62:16)in 62.4.,while (63:4),(63:10)give (62:2),(62:3)in 62.1.2.,and(63:11)gives (62:4)in 62.1.2.(when v ^ w) or (62:22)in 62.5.2.(whenv g w).

Consequentlythe resultsof 62.4.and 62.5.2.are valid, with the u, v, w

of (63:10).The generalsolution obtains as described,e.g.in (62:B) in

62.4.,accordingto Figures98-100.63.2.2.From now on we assumethat (63:B:a),(63:B:b)areboth valid.We introducethe quantitiesu, v, w, z by

(63:13) v((l)) = u, v((l,2)) = v, v((l,3))= w,

(63:14) v((l,2,3))= z.

Then (63:8),(63:9)and (63:B:a),(63:B:b)statethat))

(63:15) u<)))

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576 GENERAL NON-ZERO-SUMGAMES

This arrangementdiffers from that of 62.1.2.,but it is neverthelessworth while to comparethem in detail:(63:15)correspondsto (62:1)loc.cit.,and (63:4),(63:13),(63:14)correspondto (62:2)-(62:4)id.

It is now convenient to introduce again the assumptionof decreasingutility, already utilized in 61.5.2.,61.5.3.In fact we need it now at asomewhatearlierstagethan we did then :It is now (at leastin part) usefulin the mathematical part of the theory,1 while we neededit thereonly inthe interpretativepart.

We statethe decreaseof utility for all threeparticipants1,2,3,:(63:16) u\\ Uo > M2 ui > > u9 w,_i,(63:17) vi t; > t> 2 Vi > > v, v._i,(63:18) Wi WQ > wt wi > > w, w,_i.

In the immediateapplicationonly (63:16)will be required. This is it:(63:19) v + w > z + u*

Proof:Owingto (63:6),(63:7)and (63:13),(63:14),the assertion(63:19)canbewritten as follows:Max, , i (w.-<+ vt) + Maxr.0,i (u,-r + w r)

> MaX,, r_ , 1 (Ut-t-r + V t + Wr) + U..t+riConsiderthe t t r for which themaximum on the right-hand sideis assumed.Sincewe have (63:B:b),i.e.< in the secondinequalitiesof (63:8),(63:9),we canconcludefrom the argumentation of 63.1.2.that these/, r are 7* 0.We denotethem by to, r . Henceour assertionis this

Max,., i (u 9-t + vt) + Maxr_ , i , (u,-r + w r)> W.-l -r + Vt 9 + Wr9 + U,.

I.e.,we claim:Thereexisttwo J, r with

W,_l + V t + U9-.r + Wr > W._<o

_r + t\\ + Wr + U,.

Now this is actually the casefor t = / , r = r . Theabove inequalitymay then bewritten as

(63:20) w._r o- w._

<o_ro > ti.- w.-v

It should be conceptuallyclearthat this follows from our assumptionofdecreasingutilities. Formally it obtains from (63:16)in this way: (63:20)statesthat

1But not necessary;the absenceof this property would only complicatethe discussionsomewhat.

*In 62.1.2.this was trivially true. Indeedusing (63:13),(63:14)we obtain in thatcase

u < v ^ w z

and this gives (63:19)immediately.)))

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ECONOMICINTERPRETATION FOR n -3:GENERAL 577

<e

(63:21) (tt.-r.-tfi- W.-r.-4)> 5) (tt.-*fi - U.-i).*-l i-1(63:16)implies

tt,' W,'_i > U9 Ut-\\

whenever s'< s\", hencein particular

tt,_r -i+i ~ M-r-< > Wt_,-+i U._,-,

and from this (63:21)follows.

63.3.Preliminary Discussion63.3.We now apply 60.3.1.,60.3.2.to the present setup. This will

prove to be quitesimilar to the applicationcarriedout in 62.3.for the setupof 62.1.2.Theexpositionwhich followswill therefore be more concise,andis bestread parallelwith the correspondingparts of 62.3.

As to the comparisonof the mathematical result with that of the ordi-nary, common senseapproach, the remarks of 62.2.apply again. Weindicated already there what complicationsthe present setup produces.We shallconsiderthe situation only briefly, although it isa rather importantone. The generalviewpoints were sufficiently illustrated by our earlier,simplerexamples,and the specific,detailed interpretative analysisof thissetup and other,even more generalones will be taken up auo jure in asubsequentpublication.

63.4.TheSolutions

63.4.1.Theimputations in the presentsetup arethe))

with

(63:22) ai ^ u, a* 0, <* 8 ^ 0,(63:28) i + 2 + as = z.

It is again necessaryto introducethe reducedform. Thisamounts to atransformation

(63:24) a'k = a* + ak

as describedin 62.3.Theprocessesdiscussedtheredeterminethe J, aj,aj,so that (63:24)now becomes))

(63:25)i - Bl --, !-,- , i= a.- -

The correspondingchangeson v(/S) areagain given by (59:1)in 59.2.1.;they carry (63:4),(63:13),(63:14)into)))

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578 GENERALNON-ZERO-SUMGAMES

(63:26) v'((l)) = v'((2))= v'((3))= - -jA(63:27) v'((l,2)) = *L^J>, ^^_ 8 - 2.-))

(63:28) v'((l,2,3))= 0.

Thus 7 = -s > and we again refrain from passingto the normalization))

Hencewe must again insert a proportionalityfactor when applying60.3.1.,60.3.2.as describedin 62.3.This proportionalityfactor is nowz u~3~\"

Comparisonwith (60:8)in 60.3.1.showsthat now

2(z - u) 3w - 2z - u 3v - 2z - uai== _, a2 = _ , a8 = ___

Thesixlinesof (60:15)in 60.3.2.which describethe trianglesfrom which wederivedour solutions,becomenow:))

(63:29)))

, z u , z u , z u!= - g-> 2 = ~ -3-, a,= - -3-,' - 2(g - ^) / _ _ 3w - 2z - u

o o, 3v 2 w1 =

Q))

63.4.2.Applying the criterium of 60.3.3.,we find that

fli + 02+ a8 = v + w 2z g 0.Hencewe have again (60:17:b)loc.cit. i.e.theCase(b)id.,and it remainstobe decidedwhich oneof its four subcases,representedby Figures92-95,is present.

Following the same procedureof graphical representationas in 62.3.,we obtain Figure101.The qualitative features of this figure follow fromthe followingconsiderations:

(63:C:a) Theseconda[-linegoesthrough the intersectionof the firsta'2- and aa-lines. Indeed:

2(z u) z u z u _ n3 ~3~ \"3 U'

(63:C:b) Thesecondaj-[a{-]line is to the right [left]of the first one.(63:C:c) Indeed:It has a greateraj-[a,-]value, since)))

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ECONOMICINTERPRETATION FOR n -3:GENERAL 579))

3w 2z u3

3v 2z u))

z - w >0,* - v >0.))

(63:C:d) The first a{-lineliesbelow the intersectionof the secondajand aj-lines.Indeed:

g \"\" u 3w 2g u _3v 22 w_ , __ __ n3 3 3 z+u-v-w

by (63:19)in 63.2.2.Comparisonof this figure with Figures92-95showsthat it is again a

(rotatedand) degenerateform of Figure94 (cf. footnote 1 on p.568),although less degeneratethan the correspondingFigure96 in 62.3.:The))

Figure 101.)) Figure 102.))

V: Theareaand the curve I))

Figure 103.area is again degeneratedto a point (theuppervertex of thefundamentaltriangle A), but the areas , , , arestill undegenerated(the fourareasinto which the fundamental triangle is divided in our figure). Thisdispositionof the five areasof Figure94 is shown in Figure102.Thegeneralsolutionnow obtains as stated at the end of 60,3.3.,by fitting thepictureof Figure86 into the Situation describedby Figure102.Figure103showsthe result (cf.footnote 1 on p.569).)))

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580 GENERAL NON-ZERO-SUMGAMES

Summingup:(63:D) Assuming (63:B:a),(63:B:b)and (63:16),the generalsolution

V is given by Figure103.Comparisonof this figure with thoseof 62.S.-4.showsthat Figure103

is a form intermediatebetweenthoseof Figures98-100,and thosefiguresarein turn degenerateforms of Figure103.

63.6.Algebraic Form of the Result

63.5.The result expressedby Figure103can be stated algebraically,in the sameway as wasdonefor Figure98 in 62.5.1.

In Figure103the solution V consistsof the aream and the curve ~ .Thefirst part of V is characterizedby3w 2z u , z u 3v 2z u , z u

3 ^ a' = 3-' 3 s a'= ~3\"\"'

Owing to (63:25)in 63.4.1.,this meansthat

z w ^ a? ^ 0, z v ^ 3 ^ 0.Now (63:23)in 63.4.1.gives

1 = Z 2 ~ 3,

and so the exactrangeof i isv + w z^ai^z.

(Recallthat v + w z> u by (63:19)in 63.2.2.)We stateall theseconditionstogether,the result beingsomewhatmore complicatedthan itsanalogue(62:18)in 62.5.1.It is this:

v + w z ai z, Q a* z-w, Q a* z - v,. .ai + 2 + <*s = z.

Therangesin the first line of (63:30)arethepreciseonesfor i, 2, s.Thesecondpart of V (thecurve) can bediscussedliterally as in 62.5.1.:

ai varies from its minimum in (63:30)above (v + w z) to its absoluteminimum (u), and a2, as are monotonic decreasingfunctions of ai. Sowe have:(63:31)u g i ^ if+ w z, aa , a8 aremonotonic decreasingfunctions

Ofi.wThus the generalsolutionV is the sum of the two setsgiven by (63:30)

and (63:31).It will benoted that the roleof the functions in (63:31)isthe sameas that discussedat the end of 62.5.1.

1They must, of course,fulfill (63:22),(63:23)in 63.4.1.*As Figure 103shows, the lowest point in the area coincideswith the highest point

on the curve. I.e.the point ai - v -f w - zof (63:30)and of (63:31)is the same.Hencewe could exclude<*\\ v + w z from either one (but not from both!)of

(63:30),(63:31).)))

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ECONOMICINTERPRETATION FOR n -3:GENERAL 581Summing up:

(63:E) Assuming (63:B:a),(63:B:b)and (63:16),the general solu-tion V is given by (63:30),(63:31).

68.6.Discussion63.6.1.Letusnow perform the equivalent of62.6.and applythe ordinary

common-senseanalysisto the market of one sellerand two buyers andindivisibleunits of a particulargood,in orderto compareits resultwith themathematical one stated in (63:E).

Actually the interpretation which ought to be carried out now mustcombinethe ideas of 61.5.2.-61.6.3.with those of 62.6.:the former applybecausewe have divisibility into s units;the latterbecausethe market isoneof threepersons. As indicatedin 63.3.,we do not proposeto go fullyinto all detailson this occasion.

The two parts (63:30),(63:31),of which our presentsolution consistsarecloselysimilarto the two parts (62:18),(62:19)(or (62:20),(62:19),or(62:21),(62:23))obtained in 62.5.(Cf. also (63:E) in 63.5.with (62:C)in 62.5.2.)It appearsmost reasonable,therefore, to interpret them in thesame way as we did in the correspondingsituation in 62.6.2.:(63:30)describesthe situation where the two buyerscompetefor the s units in theseller'spossession,while (63:31)describesthe situation where they haveformed a coalition and face the sellerunited. The readerwill have nodifficulty in amplifying the details,in parallelto 62.6.2.

Thesebeingaccepted,thereis nothing new to besaidabout (63:31),thesituation in which the buyershave combinedand do not compete.(63:30)however, which describestheir competition,still deservessomeattention.

Letusconsiderthe imputations belongingto (63:30),and letus formulatetheir contents in terms of the ordinary conceptof prices.This is theequivalent of what we did at the correspondingpoint in 61.6.1.,61.6.2.

We introduceagain the t, r for which the maximum in

(63:7) v((l,2,3))= Max,,r.,i (u.-*-,+ v t + w r)t+r 9

is assumed:J , r . Sinceour imputations

a ={ {i, 2, s}} with i + 2 + s = v((l,2,3))

actually distribute the amount v((l,2,3)),these< , r Q must representthenumbers of units actually transferred by the sellerto the buyers 2,3,respectively.

The analysis of 61.5.2.,61.5.3.leading up to (61:A), could now berepeatedmutatis mutandis. It would show that numbersof units trans-ferred, f , r , can be describedin accordwith Bohm-Bawerk'scriterion of\" marginal pairs\" just as was doneloc.cit.for the correspondingnumberof transfersto. Sincethis discussionwould bring up nothing new, we shallnot dwell upon the point any further.)))

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582 GENERAL NON-ZERO-SUMGAMES

63.6.2.We now turn to thequestionof prices.Thebuyers2,3received,aswe saw,J , r units, respectively. Theimputation a on the otherhand,ascribesthem the amounts at, s. Thesetwo descriptionscan be harmo-nized only by establishingthe equations

(63:32) vt,- Up - at|

(63:33) u?fg- r*q = 8,

and interpretingp, q as the pricespaid per unit by the buyers2,3respec-tively. (63:32),(63:33)are the equivalentsof (61:24)in 61.6.1.,but itmust be emphasizedthat we obtained two different pricesfor the twobuyers!

Now (63:30)can bestated in terms of p, q l as follows:

(63:34) rK-*+ uOS P r \\,10 J<>

(63:35) 1(w r.- z + v) g q u>v

Theseinequalitiesarethe analoguesof (61:25)in 61.6.1.We could treatthem in a way similar to that there,and comparethem with thoselimitswhich result from the applicationof Bohm-Bawerk'stheory. We shallnotcarry this out in detail for the reasonsstated in 63.3.A few remarksmayneverthelessbe appropriate.

The intervals (63:34)and (63:35)areagain widerthan thoseof Bohm-Bawerk'stheory just as in 61.6.(cf. (61:C)id.).Somenumerical examplesshow, however, that the difference tends to be smaller.It is thereforepossible although nothing has beenproven in this respect that a furtherincreasein the number of buyers may tend to obliteratethis discrepancyin that part of the solution which correspondsto no coalition betweenthebuyers. This surmise must, however, be consideredwith the greatestcaution, sincewe know too well how rapidly the complicationof solutionsincreaseswith the number of participantsand how difficult the interpreta-tion of different parts of the solution may then become.

Itwill beobservedalsothat we had to introducetwo (possibly)differentpricesfor the two buyers,and this in spite of our, still valid, assumptionof completeinformation. This is perfectly in harmony with the interpre-tations of 61.6.3.:We sawtherethat what we calledpriceswerereally onlyaverage pricesof several different transactions,that the sellerand thebuyersmust have beenoperatingwith premiumsand rebates and all thisis necessarilyconducive to a differentiation betweenthe two buyers.

Finally, we may statethe equivalent of the last remark of 61.6.3.All

theseabnormalitiesin the formation of the pricestructurearealsoquite in1I.e.its statements concerning at, ascan be translated by means of (63:32),(63:33)

into statements on p, q.Thestatement of (63:30)concerning a\\ is merely a consequenceof those on at, at

using ai -f at -f- aj z. Henceit neednot be considered.)))

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THEGENERALMARKET 683

agreementwith the fact that the market under considerationis one of amonopoly versusduopoly.

64.TheGeneralMarket

64.1.Formulation of the Problem

64.1.1.The markets which we have consideredso far were veryrestricted:They consistedof two or of threeparticipants. We shall now

go a step further and considera more generalmarket, which consistsofI + m participants:I sellersand m buyers. This is, of course,still not themost generalarrangement:That would have to provide among otherthings for the possibilitythat eachparticipantcan choosewhether he will

buy or sell;or again, that he may be sellerfor one classof goods,and buyerfor another. In this study, however, we shall content ourselveswith theabove case.

Further,we proposeto considerone kind of goodsonly, of which units

A\\ f- - - , A, areavailable.Itis convenient to denotethe sellersby 1, , I, their setby

L = (l,' ,0;the buyersby 1*,,m*, their set by

M= (1*, ,m*);and the setof all participantsby

/ = LuM = (1, - - ,1,1*, , m*).1

Denotethe number of units of the goodsunderconsideration,originallyin the possessionof the selleri by s. Then clearly

i

(64:1) * = .t-i

Denotethe utility of t( = 0,1, , s)units of the goodsto the selleribyu\\ and the utility of t( = 0,1, , s) units of the goodsto the buyerj*byv}*. Thus the quantities

(64:2) it< = 0,u\\, , < (i = 1, , 0,(64:3) 4* = 0,i^*, f t>r (j =!*,,m*),

describethe variable utilities of theseunits to eachparticipant.As before,we usefor eachbuyerhis original positionas the zero of his

utility.As in 61.5.2.,61.5.3.,62.1.2.and 63.2.1.,we need not repeatthe con-

siderationsof 61.2.2.,61.3.1.,61.3.2.concerningthe rulesof the game which

modelsthis setup.1We use this notation instead of the conventional 1, , I, I + 1, , I + m.)))

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584 GENERAL NON-ZERO-SUMGAMES

64.1.2.The determinationof the characteristicfunction v(S) of thisgameis easy:

Clearly S / = L u M. We now consider successivelythree alter-native possibilities.

First:S&L. In this caseS consistsof sellersonly, who can carry outno transactions among themselves.One seesimmediately that v(S)merelystatestheir original position:

(64:4) v(fl) = 2 <tinS

Second: sM. In this caseS consistsof buyersonly, who areequallyunable to carry out any transactionsamong themselves.One seesagainthat v(S)merely statestheir original position:(64:5) v(S) = 0.

Third:Neither ScLnor S cM i.e.S has elementsin common bothwith L and with M. In this caseS containssellersas well as buyers,hencetransactionsbetweenthesearedefinitely possible.On the basis of these,the followingformula obtains:

(64:6) v(5) = Max^-o,i. - , ..(.-inani) ( % u\\ t + t

f-,,-0,1, ,(;*in Sfl M) \\ :in S L'

j* in 5 MS , + S v \" S <

in Sfl L j* in SH If in S fl L

In this expressionSn Z/ is the setof all sellersin S,S n M the setof allbuyers in S,J, the number of units transferredfrom the selleri (in S n L),r,.the numberof units transferredto the buyerj*(in S n Af). 1 Thereaderwill now have no difficulty in verifying the formula (64:6).

64.2.SomeSpecialProperties. Monopoly and Monopsony

64.2.1.We arefar from beingable to discussthe theory of this gamethe market of I sellersand m buyers exhaustively. We have at presentonly somefragmentary information on specialcasesand beyondthis only afew surmisesconcerningwider areas. The problemswhich arisein thisconnectionseemto be of definite mathematical interest,aside from theireconomicimportance.It would seempremature,however, to discussthissubjectbeforethe investigation has penetrateddeeper.

Insteadwe shalldraw someimmediateconclusionsfrom the two simplerof our equations:(64:4), (64:5). Theyareas follows:(64:A) All setsSsL and all setsSsM areflat.

1Thereis no needto state here which selleris transferring eachparticular unit towhich buyer: The resulting utilities which alone enter into v(S) are not affected bythis.

All negotiations between individuals, coalitions, compensation, etc.,must beautomatically taken careof by the application of our theory.)))

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THEGENERAL MARKET 585

Proof:This meansthat

v(5) = v((*)) for ScL and for 8sM.JfcinS

which followsimmediately from (64:4),(64:5).(64:B) Thegame is constantsum if and only if it is inessential.

Proof:Sufficiency:Inessentialityclearly impliesconstant-sum.Necessity:Assume that the gameis constant-sum.As L, M arecomplementarysets

(64:7) v(7) = v(L) + v(M).Now by (64:A) (with S = L,M)

(64:8) v(L) = % v((fc))f v(M) = % v((*)).kinL kinAf

Combining(64:7)and (64:8)we obtain

(64:9) v(7) = v((*)).*in/

Now the modification of (27:B)in 27.4.which appliesaccordingto 59.3.1.in our case,givesjust (64:9)as a criterionof inessentiality.

Itmay beworth noting that the criterionof inessentiality(64:9)becomes,when statedexplicitlyby means of (64:4)-(64:6),this:The maximum in

(64:6)isequalto u\\ .. Now this is the value of theexpressionmaximizedt in L

in (64:6)when t* = s,r, = 0. Sothe statementbecomes,that the maxi-mum in (64:6)is assumedwhen U = r;.= 0,i.e.when no transactionstakeplace.

Hence(64:B)can alsobe formulated as follows:

(64:B*) Thefact that the individual utilitiesof thesellersand buyersaresuch that no transactions takeplaceat all i.e.that themaximum in (64:6) is assumed when J, = s, r, == isequivalent to these:that the gameis constant-sum;or equiva-lently (in this case!)that it is inessential.

Thesalientpoint of this result is that our game,representinga market,can be constant-sum only at the priceof the market beingabsolutelyineffective. Hencethis problembelongs quite intrinsically to games ofnonconstant-sum.

64.2.2.We now continue in a somewhatdifferent direction.(64:C) Considertwo imputations,

a = {{ai, , off, i, ,am'!K)))

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586 GENERAL NON-ZERO-SUMGAMES

Assume that

H 0,S being the set of 30.1.1.for this domination. Then neitherS n L nor Sn M can be empty.1

Proof:Otherwisewe shouldhave S M orS L. HenceS is flat by(64:A)and therefore certainly unnecessary(cf.59.3.2.).

We concludefrom (64:C)that in this case

(64:10) a > ft for at leastone i in L,(64:11) a/. > ft* for at leastone j*in M.

Theseformulae (64:10)and (64:11)have a roleof someinterest,wheneitherL or M is a one-elementset:I = 1or m = 1. This meansthat thereexistspreciselyoneselleror preciselyonebuyer, i.e.that we have monopolyor monopsony.

In thesecasesthe iof (64:10)orthe j*of (64:11)is uniquely determined:i = 1or,;*= 1*. Sewe have:

> >

(64:D) a H implies(64:12) ai > 0i when 1 = 1,(64:13) ai* > 0i* when m = 1.

The remarkablething is that both (64:12)and (64:13)are transitive* *

relations,while domination a H ft is not. Thereis, of course,no con-tradiction in this, (64:12)or (64:13)is merely a necessaryconditionfor

a H . But it is neverthelessthe first time that the domination conceptin an actualgameis so closelylinked to a transitive relation.

This connectionseemsto be a quite essentialfeature of the monopolistic(or monopsonistic)situations.2 It will play a roleof some importancein65.9.1.

1I.e.Smust contain both sellersand buyers.1Theverbal interpretation of (64:12),(64:13)is simple and plausible:No effective

domination is possiblewithout the monopolist (or monopsonist).)))

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CHAPTER XIIEXTENSIONSOF THE CONCEPTSOF

DOMINATIONAND SOLUTION

65.TheExtension. SpecialCases66.1.Formulation of the Problem

65.1.1.Our mathematical considerationsof the n-persongamebeginningwith the definitions of 30.1.1.made use of the conceptsof imputation,domination and solution, which were then unambiguously established.Neverthelessin the subsequentdevelopmentof the theory thereoccurredrepeatedly instanceswhere these conceptsunderwentvariations. Theseinstanceswereof threekinds:

First:It happenedin the courseof our mathematical deductions,basedstrictly on the original definitions, that conceptsroseto importancewhichwereobviously analogousto the original ones (of imputation, domination,solution)but not exactlyidenticalwith them. In this caseit wasconven-ient to designatethem by thosenames,necessarilyrememberingthe differ-ences. Examplesof this areto be found in the investigation of theessentialthree-persongame with excessin 47.S.-47.7.where the discussionof thefundamental triangle is reducedto that of one of the various smallertrianglesin it. Another exampleis offered by the investigation of a specialsimplen-persongamein 55.2.-55.11.,where the discussionof the original domain isreducedto that oneof V in a (cf. the analysisof 55.8.2.,55.8.3.).

Second:In the courseof our considerationson decomposabilityinChapterIX, we explicitlyre-defined (generalized)the conceptsof imputa-tion, domination and solution in 44.4.2.-44.7.4.This correspondedto anextensionof the theory from zero-sumto constant-sumgames. Throughoutwhat followed we emphasizedthat we were investigating a new theory,analogousto,but not identicalwith, the original oneof 30.1.1.

Actually thesetwo types of variations of our conceptsarenot funda-mentally different: Thesecondtype can be subsumedunder the first one.Indeed,the new theory was introducedin orderto handle the problemof

decompositionof the original one more effectively. This motive wasstressedthroughout the heuristicconsiderationswhich ledto thisgeneraliza-tion. In the analysisof imbeddingin 46.10.,particularly in (46:K) and(46:L) there,we establishedrigorously that the new theory can be sub-ordinated to the original one preciselyin this sense.

Third:Theconceptsof imputation, domination and solutionswereagainre-defined(generalized)in Chapter XI, specifically in 56.8.,56.11.,56.12.

587)))

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588 EXTENSIONSOF THECONCEPTS

This correspondedto the final extensionof the theory to generalgames.We again emphasizedthat from thereon we were investigating a new theoryanalogousto,but not identicalwith, the precedingones.

This generalization was,however, fundamentally different from the twoprecedingones:It representeda realconceptualwidening of the theory andnot a meretechnicalconvenience.

66.1.2.Throughout the changesreferred to above it was in evidencethat while the conceptsof imputation, domination and solution varied(particularlyregardingextension),someconnectionamong them remainedinvariant. In orderto acquirea generalinsight into thesechanges andotheranalogousoneswhich may follow it is necessaryto find a preciseformulation of this invariant connection.When this is donewe can permitcompletegeneralityin all respectsand reformulate the theory on that basis.

By recallingthe instancesenumeratedin 65.1.1.,it will appearthat thisinvariant connectionis the processby which the conceptof a solution isderived from thoseof imputation and domination. This is the condition(30:5:c)(or the equivalent ones(30:5:a)and (30:5:b))in 30.1.1.Hencewe reachperfect generality if we releasethe notions of imputation anddomination from all restrictions,but define the solutions in the wayindicated.

In accordancewith this programwe proceedas follows:Insteadof imputations we considerthe elementsof an arbitrary but

fixed domain (set)D.Insteadof domination we consideran arbitrary but fixed relation S

betweenthe elementsx,y of D.lNow a solution (in D for S) is a setV D which fulfills the following

condition:

(65:1) The elementsof V are preciselythoseelementsy of D forwhich xSy holdsfor no elementx of V.2

66.2.GeneralRemarks

66.2.Thesedefinitions providethebasisfor a more generaltheory in thesenseindicated.

It shouldbe noted that our presentconceptof solutionbears the samerelation to that oneof saturationanalyzedin 30.3.and particularlyin 30.3.5.,as the original conceptof 30.1.1.In particular our (65:1)shouldbe com-pared with the fourth examplein 30.3.3.,our presentScorrespondingto thenegationof the (R there. It is especiallysignificant that in the searchforsolutionsall difficultiesconnectedwith the lackof symmetryof the relationconsidered,ariseagain. I.e.,the remarks made in 30.3.6.and 30.3.7.tothis effect apply oncemore.

1x$y expressesthat this relation holds between the specificelements x and y. Thereadershould recallthe discussionsat the beginning of30.3.2.

1Thisis the equivalent of (30:5:c)in 30.1.1.,aspromised.)))

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THEEXTENSION.SPECIAL CASES 689

We shallseesubsequentlyhow thesedifficultiescan beresolvedat leastin somespecificcases.1

In orderto acquirea betterunderstandingof the entiresituation, wemust considersome specializationsof the relation x&y. Indeed,in ourpresentexpositionS is entirely unrestrictedand we cannotexpectto find

any particularly deepresult while 8 remains in this generality. On theotherhand, the original conceptof a solution,as defined in 30.1.1.,remainsthe most important applicationof $ and it seemsvery difficult to discoverany simpledistinguishingpropertiesof this particular relation. Thereforethereis no apparentway to introducespecialization,however desirablethiswould be.

We will neverthelessdiscussthreefrequently usedschemesof specializa-tion for relationsx&y and finally find a fourth onewhich possessesa certainlimited applicabilityto our problemproper. In orderto carry this out,we needa few mathematical preparationswhich follow.

66.3.Orderings,Transitivity, Acyclicity

65.3.1.We first considersuch relationsxSy (with the domain D) whichsharethe essentialfeatures of the concepts\" greater\" and \" smaller.\"Thisorderof ideashas receiveddetailedand careful considerationsin the mathe-matical literatureand thereexiststoday rathergeneralagreementto theeffect that a completelist of thesepropertiesruns as follows:

(65:A:a) Forany two x,y of D oneand only one of the threefollow-ing relationsholds:))

x = t/,

(65:A:b) xSy,ySz togetherimply zSz.2

We call a relation Swith thesepropertiesa completeorderingof D.Examplesof completeorderingsareeasyto give and conform to ordinary

intuition:Theusualconceptof \" greater\" for the setof all realnumbersorfor any part of it.8 Theconceptof \" smaller\" under the sameconditions.Even the points of the plane possesscompleteorderings,e.g.this one:xSymeansthat x must have a greaterordinatethan y orthe sameone,but thenx must have a greaterabscissathan y.4

66.3.2.Theconceptof completeorderingcan beweakenedconsiderablyso that a significant conceptstill remains. This,too,has receivedattentionin mathematical literature5 and is of importancein the theory of utilities.Itobtainsby weakening(65:A:a)above,but retaining (65:A:b)unchanged.

1Cf.the results of 65.4.t 65.5.,and the lesssuperficial onesof65.6.-6S.7.'The reader who substitutes the ordinary \"greater

\" relation x > y for xSy in

(65:A:a),(65:A:b),will verify that theseare indeed the basicpropertiesof \"greater.\"1E.g.the integers, or any interval, etc.4 Without this last proviso, our Swould fall under the next section.Cf.0.Birkhoff: LatticeTheory, loc.cit. Chapt.I. In this book orderings, partial

orderings and similar topicsarediscussedin the spirit of modern mathematics. Exten-sive referencesto literature are given there.)))

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590 EXTENSIONSOF THECONCEPTS

I.e.:(65:B:a) Forany two x, y of D at most oneof the three following

relationsholds:))

x = y, xSy,y$x.

(65:B:b) x&y, y&z togetherimply xBz.

We calla relation S with thesepropertiesa partial orderingof D.1 Twox,y of D for which noneof the threerelationsenumeratedin (65:B:a)holds(sincethe orderingis partial, this is a possibility)arecalledincomparable(with respectto S).

Examplesof partial orderingsareeasy to give:Thepoints of the plane,x&y meaning that the ordinate of x is greaterthan that of y (cf.footnote 4on p.589).We may alsodefine that xSy meansthat the ordinateand theabscissaof xareboth greaterthan their counterpartsfor y.2 Another goodexampleobtains in the domain of positiveintegers,x$y meaning that x isdivisibleby y excludingequality.

65.3.3.The two precedingconceptsof orderingmaintained (65:A:b)inthe sameform, while (65:A:a)wasmodified (weakened)to (65:B:a).Thisemphasizedthe importanceof (65:A:b),the property of transitivity.* Wewill now undertake to weakenthe combination of (65:B:a)and (65:A:b)further, so that (65:A:b)is essentiallyaffected, too.

Notefirst that (65:B:a)isequivalent to thesetwo conditions:(65:C:a) Never xSx.(65:C:b) Never xSy,ySx together.

Indeed (65:B:a)excludesthese three combinations:x = y, xSy;% y> y&x

'ix&y> y&x- Now the first and the secondaremerely two ways

of writing (65:C:a),while the third is precisely(65:C:b).We now prove:

(65:D) Considerthe assertion:(A m) Never xiS^o,x&xi, , xmSxm-i,where x<> = xm and

XQ, xi, , Xm~i belongto D.Then we have:

(65:D:a) (65:B:a)isequivalent to (A i) , (A 2) together.(65:D:b) (65:B:a),(65:Altogetherimply all (A,),(A 2), (A 8),

Proof:Ad (65:D:a):Clearly (Ai) is (65:C:a)and (4,)is (65:C:b).Writing the relationsof (A m) in thereverseorder,and applying(65:A:b)

m 1 times gives xm&xo. As xm = x , this means XO&EO, contradicting(65:B:a).

1Note that the word partial is used in the neutral sense,i.e.,a completeordering is aspecialcaseofthe partial ones,since(65:A:a)implies (65:B:a).

*Note that this is closeto a plausible type of partially orderedutilities in the senseofthe last remark of 3.7.2.Each imagined event may be affected with two numericalcharacteristics,both of which must be increasedin orderto producea clearand repro-duciblepreference.1Someother important relations, not at all in the nature of an ordering, alsopossessthis property: E.g.equality, x -

y.)))

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THEEXTENSION.SPECIALCASES 591This result suggestsconsideringthe total aggregateof all conditions

(Ai), (A 2), (A s), . They are impliedby (65:B:a),(65:A:b),i.e.bypartial ordering,and represent,as will appear,a further weakeningof thisproperty.

We define accordingly:(65:D:c) A relation S is acyclic if it fulfills all conditions(Ai), (Aj),

(A.), -

The readerwill understand why we call this acyclicity: If any (A m)shouldfail, therewould bea chain of relations))

which is a cycle,sinceits last element,xm, coincideswith its first one,x .We have alreadyremarkedthat acyclicityis impliedby partial ordering

(this is,of course,the contentof (65:D:b)); and henceafortiori by completeordering. It remains to show that it is actually a broaderconceptthanpartial ordering,i.e.,that a relation can be acyclicalwithout beinganordering(partial or complete).

Theseareexamplesof the latterphenomenon:LetD be the setof all

positive integers,and x$y the relation of immediate succession,i.e.,x = y + 1. Or,letD bethe setof all realnumbers,and xSt/ the relation olbeinggreaterthan, but not by too much say by no more than 1 i.e.,the relationy + 1^ x > y.

We concludethis sectionby observingthat our examplesof completeancof partial orderingsand of acyclicalrelationscould easily be multipliedSpaceforbidsus to go into this here,but it may besuggestedto the readeias a useful exercise.The referencesto the literaturein footnote 1on pag<62 and footnote 5 on page589 can alsobeconsultedto advantage.

65.4.TheSolutions :For a Symmetric Relation. For a CompleteOrdering

65.4.1.Let us now discusstheschemesof specializationreferredto at th(

end of 65.2.First:S is symmetricin the senseof 30.3.2.In this caseit isexpedien

to go backto the connectionwith saturation,pointedout at the beginningo65.2.Owing to the symmetry of S it will provide all information abousolutionswhich we desire.

Second:Sis a completeordering. In this casewe define as usual:xis i

maximum of D if no y with yBx exists. It is sometimesconvenient to indicate the connectionwith a completeorderingby calling it an absolutmaximum of D. (Cf.this with thecorrespondingplacein the nextremark.ClearlyD has eitherno maximum orpreciselyone.1

Now we have:(65:E) V is a solutionif and only if it is a one-elementset,consistin

of the maximum of D.1Proof:If X, y are both maxima of D, then ySx and x&y being excluded,(65:A a

necessitatesx y.)))

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592 EXTENSIONSOF THECONCEPTS

Proof:Necessity:Let V be a solution. SinceD is not empty, V is notempty either.

Considera y in V. If zSy, then x cannot be in V, hencea u in V with

uBx exists. The transitivity gives u&y which is impossible,sinceu, y areboth in V. Sono x (in D!)existswith xSy,1and y must be a maximum of D.

SoD has a maximum which must be unique (cf.above). HenceV is aone-elementset,consistingof it.

Sufficiency:Let a?o be the maximum of D, V = (XQ). Given a y (of D!),the validity of xy for no x of V amounts simplyto the negation of x&y.Sincey&xo is excluded,this negation is equivalent to y = XQ. Sothesey

form the setV. HenceV is a solution.66.4.2.Thus thereexistsno solutiony if D has no maximum, while a

solutionexistsand is unique if D has a maximum.If D is finite, then the latteris certainly the case. This is intuitively

quite plausibleand also easy to prove. Forthe sake of completenessandalsoto makethe parallelismwith the correspondingparts of the next remarkmore evident,we neverthelessgive the proof in full:

(65:F) If D is finite, then it has a maximum.

Proof:Assume the opposite, i.e.,that D has no maximum. Chooseany Xi in Z), then an #2 with x&x\\, then an x$ with XsSx* etc.,etc. By(65:A:b)xm xn for m > n, henceby (65:A:a)xm T xn. I.e.,the Zi, x2, 8,

areall distinct from eachother,and so D is infinite.Theseresultsshow that both the existenceand uniquenessof V parallel

those of the maximum of D.

66.5.TheSolutions :For a Partial Ordering

66.5.1.Third:Sis a partial ordering. In this casewe take over literallythe definition of a maximum of D from the precedingremark. It is some-times convenient to indicatethe connectionwith a partial orderingbycalling it a relative maximum of D. (Cf.this with the correspondingplaceinthe precedingremark. This contrast is quite useful, footnote 2 belownot-withstanding.) D may have no maximum, it may have one and it may haveseveral.2 Thus relative maxima are not necessarilyunique, while theabsoluteonesare.3

1A similar situation was already discussedin 4.6.2.8 Theargument of footnote 1on p.591fails, sinceit dependson (65:A:a)which is now

weakened to (65:B:a).E.g.,take for D the unit square in the plane and define in it a partial ordering by

either one of the two processesin the two first examplesat the end of 65.3.2.Then themaxima of D form its entire upper edge,or the upper and the right edgestogether,respectively.1Thereaderis warned against mixing up our notion of a relative maximum with thatonewhich occursin the theory of functions: Therea localmaximum is frequently calleda relative one. Sincethe quantities involved there are numerical, hencecompletelyordered,this has nothing to do with our present considerations.)))

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THEEXTENSION. SPECIALCASES 593

Thequestionof existencealsoplaysa different rolefor relative maximathan for absoluteones. Itwill appearthat the decisivepropertynow is this:(65:G) If y in D is not a maximum, then a maximum x with x$y

exists.Forabsolute maxima i.e.,if S is a completeordering (65:G)expressespreciselythe existenceof one.l Forrelative maxima this neednot be thecase,i.e.for a partialorderingthe mere existenceof some(relative) maximaneednot imply (65:G).Examplesof this areeasy to give, but we will notpursuethis matter further. Suffice it to say, that (65:G)will prove to bethe proper extensionof the existenceof an absolute maximum (cf. theprecedingremark)to the caseof relative maxima (cf.below).

Now we have:(65:H) V is a solutionif and only if (65:G)is fulfilled (by D and S!)

and V is the setof all (relative) maxima.Proof:Necessity:Let V be a solution.If y is not in V, then an a; in V with xSy exists,hencey is not a maximum.

Soall maxima belongto V.If y is in Vi then the argument given in the proof of (65:E)in the preced-

ing remarkcan be repeatedliterally, showing that y is a maximum.SoV is preciselythe setof all maxima.If y is not a maximum, i.e.,not in V, then an x in V, i.e.,a maximum,

with xSy exists,so (65:G)is fulfilled (byD and S).Sufficiency:Assume that (65:G)is fulfilled, and let V be the setof all

maxima.Forx,y in V, x&y is impossible,sincey is a maximum. If y is not in V f

i.e.,not a maximum, then by (65:G)an x which is a maximum, i.e.,in V,with x&y exists. SoV is a solution by (65:1).

Thereadershouldverify how this result (65:H)specializesto (65:E)ofthe precedingremarkwhen the orderingis complete.

Our result (65:H)showsthat thereexistsno solutionV if D and Sdonotfulfill the condition (65:G),while a solution existsand is unique if thisconditionis fulfilled.

65.5.2.If D is finite then the latteris certainly the case. We give theproof in full:

(65:1) If D is finite, then it fulfills the condition(65:G).Proof:Assume the opposite,i.e.,that D doesnot fulfill (65:G). Calla y

exceptional,if it is not a maximum and xSy holdsfor no maximum x. Thefailure of (65:G)meansthat exceptionaly exist.

Consideran exceptionaly. Sinceit is not a maximum, an x with xyexists. Sincey is exceptional,this x is not a maximum. If a maximum u

1Proof:SinceDis not empty (65:G)implies the existenceof a maximum.Conversely:Let x<> be the maximum of D. Then for every y not a maximum, i.e.,

y j x , the exclusion of j/8x and the validity of (65:A:a)(completeordering!)give XoSy.)))

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594 EXTENSIONSOF THECONCEPTS

with uSxexisted,this would give by (65:B:b)uSy contradictingthe excep-tional characterof y. Henceno such u exists,i.e.,x too is exceptional.I.e.:(65:J) If y is exceptionalthen thereexistsan exceptionalx with x&y.

Now choosean exceptionalx\\ 9 and exceptionalx* with xaSxi,an excep-tional #3 with x&xi etc.,etc. By (65:B:b)xm xn for m > n, henceby(65:B:a)xm 5^ xn. I.e.,xi,#2, #3, areall distinct from eachotherandso D is infinite.

(Cf.the last part of this argument with the proof of (65:F)in the preced-ing remark. Observe,that we could replaceits (65:A:a)by the weaker(65:B:a).)

Theseresultsshowthat the existenceof a solutionnow doesnot corre-spondto the existenceof a maximum, but to the condition(65:G). This isquite remarkableconsideringthe concludingpart of the precedingremarkin 65.4.2.It corroboratesour earlierobservationthat in the presentcaseof partial ordering (65:G) is the proper substitute for the existenceof amaximum.

Theuniquenessof the solutionis even more remarkable.In the lightof the last part of our precedingremark, it would have seemednatural forthis uniquenessto be connectedwith that one of themaximum. But we seenow that the solution is unique,while the (relative)maximum neednot be,as wasalreadymentioned.1

66.6.Acyclicity and Strict Acyclicity

65.6.1.Fourth:S is acyclic.We know that this casecomprisesthe twoprecedingones,i.e.,that it is more generalthan both.

In thosetwo caseswe determinedthe necessaryand sufficient conditionsfor the existenceof a solutionand we alsofound that when they aresatisfiedthesolutionis unique. (Cf. (65:E)and (65:H).)Furthermore,it wasseenthat when D is finite theseconditionsarecertainlysatisfied. (Cf. (65:F)and (65:1).)

In the acycliccasewe will find conditionswhich aresimilar to theseinmany ways and in somerespectswe will gain deeperinsights than before.Itwill benecessary,however, to vary our standpointsomewhatin the courseof our discussionand our resultswill be subjectto certainlimitations. Thecaseof a finite D will again besettledin an exhaustiveand satisfactoryway.

It is again convenient to introducetheconceptof maxima,2 and not onlyfor D itself but alsofor its subsets. Sowe define:x is a maximum of E(sD)if x belongsto E and if no y in E with y&x exists. We denotethesetof allmaxima of E by Em (z E).

1(65:H)shows that the solution V is not connectedwith any particular (non-unique)maximum, but with the (unique) set of all maxima.

1Sincewe used the qualification \"absolute\" in the second,and \"relative\" in the thirdremark, weshould now employ another still weaker one. It seemsunnecessary, however,to bring in sucha terminological innovation at this occasion.)))

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THEEXTENSION.SPECIALCASES 595

Our discussionswill show that it isof decisiveimportance whether D and$ possessthis property:

(65:K) E j* (for E D) impliesEm 7* Q.I.e.:Every non-empty subsetof D possessesmaxima.1 Primafacie (65:K)does not appear to be related in any way to acyclicity, but thereexistsactually a very closeconnection.Beforewe attack our properobjective,the role of solutionsin the presentcase,we investigate this connection.

65.6.2.Forthis purposewedropall restrictionsconcerningD and S,eventhat of acyclicity.

It is convenient to introducea propertywhich is a variation of the (A m)of (65:D) in 65.3.3.,and which will turn out intrinsically connectedwith

them:(AJ Never xir , x2Sxi,XsSx 2, , wherex , Xi, x2, * * * belongto D.2

We define,for reasonswhich shall appearsoon:A relation Sis strictly acyclicif it fulfills the condition (AJ.We now clarify the relationshipof strict acyclicity i.e.of (A*) both

to (65:K)and to acyclicity,by proving the five lemmas which follow. Theessentialresults are(65:0)and (65:P);(65:L)-(65:N)are preparatoryfor

(65:0).(65:L) Strictacyclicity impliesacyclicity.

Proof:Assumethat Sis not acyclic. Then thereexistx , Xi,' '

>xm_i

and xm = x in Z>, suchthat XiSx , x2Sxi, , xmSxm-i. Now extendthis

sequencex , Xi, , xm_i to an infinite one x , Xi, x2, by putting

X = Xm = X2m = * * * ,Xl = Xm+i = X2m+l = ' ' * ,

Xm_i = X2m __i^ Xsm-1 = * ' * -

Then clearly X&XQ, x&x^rr 8Sx2, etc.,etc.,and sostrict acyelicityfails.

(65:M) Acyclicity without strict acyclicity impliesthis:(J5*) Thereexistsa sequencex , Xi, x2, x, * in D with this

property:1Even if Sis a completeordering, this property (65:K)is of great importance in set

theory. Thosereaderswho are familiar with that theory will observe,that (65:K) ispreciselythe fundamental conceptof well ordering. (In this caseSmust be interpretedas \"before\" instead of \"greater.\") For literature cf.A. Fraenkel, loc.cit.p. 195ff, and

299ff, and F.Hausdorff, loc.cit.p.55ff, both in footnote 1 on p.61; also E.Zermelo loc.cit. in footnote 2 on p. 269. It is remarkable that the same property plays a role in

connection with our conceptof solution for arbitrary relations. Themajor part of theconsiderations which make up the remainder of this chapter dealswith this property andits consequences.

Actually this subject and its ramifications appearto deserveconsiderablefurther

study from the mathematical point of view.1Thesequencea; , *i, 3j, - should beinfinite in the sensethat the indicesmust go

on ad infinitum, but the s themselves neednot all bedifferent from eachother.1Cf.this with footnote 2above,and the last part of this lemma.)))

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596 EXTENSIONSOF THECONCEPTS

For xpSx7 , p = q + 1 is sufficient and p > q is necessary.1

(B*)impliesthat the x , Xi, x2, arepairwisedifferentfrom eachotherand therefore D must be infinite in this case.

Proof:SinceSisnot strictlyacyclic,thereexistx , Xi, x2, in D, suchthat XiSxo, x2Sxi,x3Sx2, ' ' ' Hencep = g + 1is sufficient for xp x q.

Now assume that xpSxg. We wish to prove the necessityof p > q.Assume the opposite:p ^ q. Now xp+iSxp, Xp+jSXp+i, , x

flSxg_i,2

XpSXg and theserelationscontradict(A m) with m = q p + 1:It sufficesto replaceits x , Xi, , xm-iand xm = x by our xp, xp+i, , x

fl

and xp. This conflicts with the acyclicityof S.Thus all parts of (B*)areestablished.Now the consequencesof (J8*):If the x , Xi, #2, ' ' werenot pairwise

distinct, then xp = xq would occurfor somep >q. By (#*) x q+i&x q, henceXg+iSXpj by (5*) this impliesq + 1> p, i.e.g ^ p, but q < p. So theXo, Xi, x2, arepairwisedistinctand therefore D must be infinite.

(65:N) Non-acyclicityimpliesthis:Forsomem(= 1,2, ) we have:

(B*) ThereexistXQ, Xi,- , xm_i and xm = XQ in D with this

property:ForXpSx g, p = q + 1is necessaryand sufficient.8

Proof:SinceS is not acyclic,thereexistx , Xi, , #m_i and xm = xin D such that XI&XQ, x&Xi, , xmSxm_i. Choosesucha systemwith itsm(= 1,2, ) as small as possible.

Clearlyp = q + 1is sufficient for xpSxq. We wish to prove that it isnecessarytoo. Assume therefore xpSxq but p ^ q + 1.

Now a cyclicalrearrangementof the Xo, Xi, , xw_i, xm = x doesnot affect their propertiesand we can apply this so as to make xp the lastelement i.e.,to carry p into m. I.e.,there is no loss of generality in

assumingp = m. Now p 7* q + 1,i.e.,q 7* m 1. We can alsoassumethat q 7* m, sinceq = m could be replacedby q = 0. So q g m 2.After these preparations we can replaceXQ, Xi, , xm_i, xm = x byx , Xi, , xg, xm = XQ

4 without affecting their properties.Thisreplacesm by g + 1,which is < m, and this contradictsthe assumed minimum

propertyof m.Thus all parts of (J5*) areestablished.65.6.3.Summingup:

(65:0)(65:0:a) Acyclicity is equivalent to the negation of all (B*),(fij),

1In connectionwith this result cf.also65.8.3.1Thesearepreciselyq p relations, hencetl^ey do not appearif p q.1Observethat the characterization of the interrelatedness of the XD, x\\, x^ is

completein (BJi), but not in (B*). This will beof importance below.4I.e.,omit)))

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THEEXTENSION.SPECIALCASES 597

(65:O:b) Strictacyclicity is equivalent to the negation of all (fif),(*?), and.of(B*).

(65:O:c) Strictacyclicity impliesacyclicity for all D but it is equiva-lent to it for the finite D.

Proof:Ad (65:O:a):The condition is necessarysince(5*) contradicts(-A),henceacyclicity. Thecondition is sufficientby (65:N).

Ad (65:0:b):Thecondition is necessarysincenon-acyclicitycontradictsstrict acyclicity by (65:L),and (J5*) contradicts (A 00 ) 1 hencestrictacy-clity. The condition is sufficient sincethe negation of strict acyclicitypermitsthe applicationof (65:M)in caseof acyclicity,and the applicationof (65:0:a)above in caseof non-acyclicity.

Ad (65:0:c):The forward implication was stated in (65:L). If D isfinite the reverseimplication and hencethe equivalence resultsfrom thelast remark in (65:M).

Finally we establishthe connectionwith (65:K):(65:P) (65:K)is equivalent to strictacyclicity.

Proof:Necessity:Assume that S is not strictly acyclic. ChooseXQ, Xi, z2, ' ' inDwithziSzo,x&x\\, x8Sx2, . Then#= (so,Zi, z2, )is D and 7* , and it possessesclearlyno maxima.So (65:K)fails.

Sufficiency: Assume that (65:K) fails. Choosea nonempty EzDwithout maxima.l Choosean XQ in E. XQ isnot a maximum in E,sochoosean Xi in E with XI&XQ. Xi is not a maximum in E, sochoosean x* in E with

#2i,etc.,etc. In this way a sequenceZo, Xi, #2, * * in E, hencein D,obtainsand XI&XQ, X&KI, zs&r*, - . This contradictsstrictacyclicity.

Sowesee:Strictacyclicityis the exactequivalent of theproperty(65:K),which we expectto be fundamental. Acyclicity and strictacyclicityarecloselyrelatedto eachother. The particular role of the finite D beginsalready to make itself felt: For finite D the two above conceptsareequivalent.

65.7.TheSolutions :For an Acyclic Relation

65.7.1.We now turn to our main objective:The investigation of thesolutionsin D for S. It is at this point that it will appear,why we attributeto the property (65:K)sucha fundamental importance:(65:K)will turn outto bequite intimately connectedwith the existenceof preciselyone solution.

We beginby showing that thereexistspreciselyonesolution(in D for S)if (65:K) is fulfilled. In proving this we will restrictourselvesto finite

setsD, in which casethe solution can even be obtained by an explicitconstruction.This constructionis effected by finite induction. Thefinite-nessof D is not really necessary,but for an infinite setD the constructionin

questionwould bemore complicated.2

1Thereadershould comparethis proof with that one of (65:F)in 65.4.2.*It would be necessaryto make use of more advanced set-theoreticalconcepts(cf.)))

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598 EXTENSIONSOFTHE CONCEPTS

Sincewe must assume (65:K), this means by (65:P)that D must bestrictlyacyclic.SinceD is finite, this is by (65:O:c)indistinguishablefrom

ordinaryacyclicity. Soit doesnot matterfor themoment, whether westatethat we requireacyclicity or strictacyclicity of D. It is neverthelessappropriate to rememberthat we areusing (65:K),i.e.,strictacyclicity,and that the assumptionof finiteness, which obliteratesthe distinction in

question,couldberemoved.We repeat:Fortheremainderof thisparagraphfinitenessof D isassumed

and the property (65:K) i.e.,acyclicity,i.e.,strictacyclicity.Let us now carry out the inductive constructionreferredto. This will

bedonefirst and theannouncedpropertieswill beestablishedafterwards.We define for every i = 1,2, 3, threesets4,B,C (all D) as

follows:A i = Z>. If for an i (= 1,2,3, - ) A* is already known, thenBi,d and A{+\\ obtain in this way:Bi = A? i.e.,J54 is the setof thosey in A<for which x&y for no x in Ai. C is the setof thosey in A* for which x&y forsomex in B. Finally A+i = A Bi C.

Now we prove:

(65:Q) Bi,d aredisjunct.

Proof:Immediateby theirdefinitions.

(65:R) Ai^O implies A^ cAi. 1

Proof:Ai^QimpliesBi = A? 9* Q by (65:K),2 hence

Ai+i = Ai -Bi-d cA>.

(65:S) Thereexistsan t with Ai = 0.Proof:Otherwiseby (65:R) D = A\\^> A*\"=> A*^> , contradicting

the finiteness of D.(65:T) Let i be the smallesti of (65:S),then

D = AI => At ^ Az ^ => 4,o_i 3A, o= .

Proo/:Restatementof (65:R)and (65:S).(65:U) Bit * ' '

y Bi9-i> C\\, , C,_i, are disjunct sets,with

the sum D.Proof:By the definition of Ai+\\ we have JS u C< = A, A+i. Hence

BIu Ci, , JB _i u Co_i, arepairwisedisjunctand theirsum is

A, - A^ = D - = D.

Combining this with (65:Q)shows that B\\ y C\\ 9 , J3 t-i,Cto_i, i.e.,the referencesof footnote 2 on p.269and footnote 1 on p.595),in particular of transfiniteinduction or someequivalent technique.

Thesematters will be consideredelsewhere.1Thepoint is that we have cand not merely !1Thisis the only but decisive! usewemake of (65:K).)))

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THEEXTENSION. SPECIAL CASES 599

BI, , B,o_i, Ci, , Cf_i, are pairwisedisjunct, and that their

sum is alsoD.65.7.2.We now put

(65:2) Vo = B!U uB,t-i.Then(65:U)gives

(65:3) D-Vo= C!U- - u Cf-i.

Now we prove:

(65:V) If V is a solution (in D for S),then V = Vo.

Proof:We beginby showing that Bi V for all i = 1, , * -i.Assumethe oppositeand considerthe smallesti for which Bi V fails

to be true. Let z be an elementof this B not in V. Then ySz for somey in V. z is a maximum in A t hencey is not in A,. Considerthe smallestk for which y is not in A*. Then fc g i and as y is in D = AI, so k j& 1.Putj = fc 1,then 1 j < i. y is in A; but not in A,-+i = A k, henceit isin B, u C, = Aj A/+I.

2; is in Bi A, cA,-. Soif y were in B,,ySz would imply that z is in C,.This is not so,sincez is in B. Hencey is in C,.

Now necessarilyan x in B,with x&y exists. Sincey is in V, this excludesx from V. Thus Bj V cannot hold. As j < i, this contradictstheassumedminimum propertyof i.

Sowe see:(65:4) B< V for all i = 1, , to -1.

If y is in C,then an x in Bi with xSy exists. Sincethis x is in V by(65:4),y cannot bein V.

Sowe see:(65:5) d s -V for all i = 1, , t - 1.

Comparing(65:4),(65:5)with (65:2),(65:3)aboveshowsthat V mustcoincidewith Vo, as asserted.

(65:W) Vo is a solution(in D for S).

Proof:We prove this in two steps:If x,y belongto Vo then xSj/ is excluded:Assumethe opposite:x, y in))

x, y belongto Vo f say x to Bi and y to Bf. If i j, then y isin B, A, A. x is in B,so xSy impliesthat y is in d. This is not so,sincey is in B,. If t >j, then xis in B< S A cA/, y is a maximum in A,,hencexSyis impossible.

Thus we have a contradictionin any event.)))

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600 EXTENSIONSOF THECONCEPTS

If y isnot in Vo> then xBy for somexin Vo 'y is in Voi hencein somed.Hencex&y for an x in 5<,and this x is in consequencein Vo.

This completesthe proof.Combining(65:V) and (65:W) we can state:

(65:X) Thereexistsone and only one solution (in D for S),the Vo of(65:2)above.

66.8.Uniqueness of the Solutions, Acyclicity and Strict Acyclicity

65.8.1.Let us reconsiderthe last three remarks, still retaining for amoment the assumptionof finiteness,in orderto avoid further complications.It is conspicuousthat they all yielded the sameresult,although undervarying assumptions. In eachcasewe proved the existenceof a uniquesolution,but the hypothesiswas first completeordering,then partial order-ing, and finally (ordinary or strict) acyclicity i.e.,it was weakenedatevery step.

Thisbeingso,it is natural to ask whether we have reachedwith the lastremark the limit of this weakening or whether acyclicitycouldbereplacedby even lesswithout impairing theexistenceof a uniquesolution.

It must beadmitted, that this line of investigation takes us away fromthe theory of games.Indeed,in that theory the existenceof solutionswasof primary importance,but we have learnedthat therecouldbe no questionof uniqueness.

Nevertheless,sincewe now have someresultson existencewith unique-ness,we will continue to study this case. We will seelater,that it has evenindirectlya certainbearingon the theory of games. (Cf.67.)

In the senseoutlined we shouldask therefore this:Which propertiesofthe relation Sarenecessaryand sufficient in orderthat thereexista uniquesolution? It is easy to see,however, that this questionis not likelyto havea simpleand satisfactoryanswer. Indeed,the solution(in D for S)disclosesonly little about the structure of D (togetherwith S). Theacyclicalcaseis lesssuited to judgethis, sinceit is somewhatcomplicated,but the casesofcompleteorpartial orderingmakethe point quiteclear. Therethe solutionis only relatedto the maxima of D and it doesnot expressat all what thepropertiesof the otherelementsof D are.

It is not difficult to eliminate this objection.Considera set E .Dinstead of D. TherelationS in D is also a relationin E and if it was acompleteorderingor apartialorderingor (ordinarily or strictly)acyclicin D,then it will be the samein E.1 Henceour result (65:X) impliesthat in

every ESiDthereexistsa unique solution (for S). Now thesesolutions,when formed for all E Z), tellmuch more about the structureof D. It isbestto restrictourselvesagain to the casesof (completeor partial)ordering.Clearly the knowledgeof the maxima of E for all setsE D gives a verydetailedinformation about thestructureof D (togetherwith S).

1I.e.,at leastthe same it can happen that a partial ordering in Dis completein Eor that an acyclicrelation in Dis an ordering in E.)))

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THEEXTENSION.SPECIAL CASES 60165.8.2.Thus we arrive at the following question:Which propertiesof

the relation Sarenecessaryand sufficient in orderthat thereexistfor eachE D a unique solution (in E for S)? We can show that hereacyclicityand strict acyclicityarethe significant concepts,although the subjectis notcompletelyexhausted. Thetwo lemmas which followcontain what we canasserton this matter.(65:Y) In orderthat thereexistfor eachE&D a unique solution

(in E for S),strictacyclicity is sufficient.Forfinite D this follows from (65:X),and strictacyclicity

may be replacedby acyclicity,owing to (65:0:c).Forinfinite D this is dependentupon the extensionof (65:X)

to infinite sets(cf.the beginningof 65.7.1.)Proof:If D is (ordinarily or strictly) acyclic,then the sameis true of all

E SD (cf. above). Now all assertionsof our lemma becomeobvious.

(65:Z) In orderthat thereexistfor eachE&D a unique solution(in E for S)acyclicityis necessary.

Proof; If D is not acyclic,then (65:O:a)yields the validity of a (B*),m = 1,2, , in (65:N).Form its XQ, z A, , zm_i and xm = XQ andput E =

(XQ, Xi, x2, , xm-i). Then EsiDand (B*)describesS in Ecompletely. Let us considerthe solution V in E (for S).

Considersucha solution V. If Xi is in V, then Xi+i is not, sincea^+iSx,-.If xt is not in V then thereexistsa y in V with ?/,i.e.,y = X, with z,&r.This meansj = i+ I,1so y = Xi+\\, and hencext+\\ is in V- Sowe see:(65:6) Xi is in V if and only if Xi+i is not.

Iterationof (65:6)gives:(65:7) If k iseven, then XQ is in V if and only if Xk is.

If k is odd, then XQ is in V if and only if Xk is not.

As XQ = arm, (65:7)involves a contradictionif m is odd.Hencethereexistsno solutionin E (for S) if m is odd. If m is even, then (65:7)impliesthat V iseitherthe setof all xk with an even fc or the setof all Xk with an oddk. And it is easy to verify that both thesesetsare indeedsolutionsinE (for S).

Sowe have:

(65:8) The number of solutionsin E =(XQ, x\\, , a;m-i)for S

(with the ZQ, Xi, , Zm-i from (B*))is 2 or accordingtowhether m is even or odd.

Consequentlythereis in no casea unique solutionin this E( D).Combining(65:Y)and (65:Z)we see:Theexistenceof a unique solution

(in E for S)for all E D iscompletelycharacterisedfor finite sets:Forthese1If f m, then replaceit by i 0.)))

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602 EXTENSIONSOF THECONCEPTS

it is equivalent to acyclicity,i.e.,to strictacyclicity,which in this caseisthe same thing. For infinite setsD we can only say that acyclicityisnecessaryand strictacyclicityis sufficient.

66.8.3.Thegap which existsin this casecan only bebridgedby a studyof the acyclic,but not strictly acyclic(infinite) setsD and their subsetsE.By comparing (65:0:a),(65:0:b)we see that such a D satisfies (B*).Formits x , XL x* and put D* = (X Q, Xi 9 #2, ). This is alsoacyclicbut not strictlyacyclic,hencewe may study it in placeof D.

Thus the questionhas becomethis:(65:9) Assume that D* = (x , Si,*,)fulfills (B*). Will then

every E D*possessauniquesolution (in E for S)?The answer to (65:9)cannot be given immediately, because(B*)

describesthe relation xSyin D* i.e.xpSxq only incompletely. Thecor-respondingquestionfor (J3*) (m = 1,2, ) wasansweredin the proofof (65:Z)in the negative, but (B*)describedthe relation xSy in its set i.e.XpSx q completely. Thus the answer to (65:9)requires an exhaustiveanalysisof all possibleforms of the relation xp&xq which fulfill (B*). Theproblemappears to be one of considerabledifficulty. 1

66.9.Application to Games:Discretenessand Continuity

65.9.1.Our above resultson acyclicityand on strict acyclicityhave, aspointedout before,no directbearingon the theory of games.

As regardsstrictacyclicity,it suffices to emphasizeits equivalenceto(65:K) (by (65:P)),and to rememberthat in the theory of gameseven Ditself (thesetof all imputations)possessesno maxima (i.e.,undominatedelements).2

Ordinary acyclicitytoo is violated,e.g.,already in the essentialthreepersongame.3

Neverthelessthereweresituations that aroseduring the mathematicaldiscussionof certaingames,where the conceptof acyclicitycouldhave beenapplied.Thesesituationsareto beregardedin thespirit of the first remarkof 65.1.1.and they arespecificallyamong the examplesreferredto there.

Thus in thetrianglesT discussedin 47.5.1.wehave an acyclicalconceptofdomination, as the inspectionof figures 76,77 shows.4 Furtherin the setCt

describedin 55.8.2.thereis an acyclicconceptof domination as the criterion(55:Z)makesapparent.6

Finally, in the market discussedin 64. there is an acyclicconceptofdomination in the caseof monopoly or monoposony,as thediscussionat the

1It lieson the boundary line of combinatorics and set theory, and seemsto deservefurther attention.

*This holds for all essentialgames. Cf. (31:M)in 31.2.3.1Thereaderis invited to verify this, e.g.,on the diagram ofFigure 54. It is easyto

ascertainthat (BJ)holds (and (A m) fails) for all m 3.4 Heredomination implies having a greater ordin ate.1Heredomination implies having a greater n-component, and from this acyclicity

obviously follows.)))

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GENERALIZATIONOF THECONCEPTOF UTILITY 603end of 64.2.2.and in particular (64:12),(64:13)thereshows.1 We mayaccentuatethe concludingremarkmadethereby observingthat an intrinsicconnectionis to besurmisedbetweenthe monopolisticsituationsin theeco-nomic sphereand the mathematical conceptof theacyclicityof domination.

Itis very remarkable,therefore, that in all thesecasesparticularly exten-sive families of solutionswerefound to exist. Indeed,not only numericalparameters,but even highly undeterminedcurves or functions enteredintothosesolutions. Forthis cf. 47.5.5.and Fig.81.in the first instance,and thefifth remark in 55.12.in the secondone. In the third instancewe can onlyrefer to the mathematical discussionof a specialcase:The three-personmarket monopoly versusduopoly which was analyzed in 62.3.,62.4.and63.4.

65.9.2.The greatnumber of solutionsin the acyclicsituationsreferredto above may seemnatural, if the infiniteness of theseD (theset of theimputationsunder consideration)is emphasized.After all, it wasonly forfinite setsD that acyclicityimplied uniquenessof the solution, for theinfinite onesstrictacyclicitybecamethe crucial concept.(Cf.the last partof 65.8.,in particular65.8.2.)And all theseexamplesare,of course,notstrictly acyclic,as can be verified with ease.

The situation is neverthelessparadoxical,for the following reason:Modificationsof the conceptof utility, which will be consideredin 67.1.2.can be applied in such a manner as to make the setsin question finite.Then the acyclicalgamesmentionedwill have unique solutions. Now thesefinite modifications can be madeto resemblearbitrarily closelyto the origi-nal, unmodified games.Hencethe original acyclicgameswith many solu-tions (infinite D!)can be approximatedarbitrarily closelyby the modifiedacyclicalgames with unique solutions (finite D!).How can the uniquesolutionsbe \" arbitrarily close\" approximationsof the non-unique ones?

Thisparadoxicalsituationwill bedescribedin detail in 67. Theanalysiswhich we aregoing to give there will clarify this lackof continuity andpresentan opportunityfor someinterpretationsof a certaininterest.

66.Generalizationof the Conceptof Utility

66.1.TheGeneralization. TheTwoPhasesof the TheoreticalTreatment

66.1.1.In the past sectionswe have generalizedthe conceptof a solu-tion basedon a relation S, which takesthe role of domination in a mostextensiveway. Thesegeneralizationsshould be used in our theory asfollows:Our conceptsof imputation, domination and solutionsrestuponthemore fundamental oneof utility. Now if we desireto vary the formalism

1Herethe domination implies having a greater 1-(or 1*-) component, and from thisacyclicity obviously follows.

If neither monopoly nor monopsony exist,i.e.,if with the notations loc.cit.I,m > 1,then (64:10),(64:11)apply instead of (64:12),(64:13)eod. Itis easy to verify that inthis caseacyclicity doesnot prevail.)))

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604 EXTENSIONSOFTHECONCEPTS

used to describethe latter,we can try to renderthesevariations adequatelyby appropriategeneralizationsof the former concepts.

Of course,we do not wish to carry out generalizationsfor their own sake,but there are certainmodifications which would make our theory morerealistic.Specifically:We have treatedthe conceptof utility in a rathernarrow and dogmaticway. We have not only assumedthat it is numerical

for which a tolerablygoodcasecan bemade (cf.3.3.and 3.5.)but alsothat it is substitutableand unrestrictedlytransferablebetweenthe variousplayers (cf.2.1.1.).We proceededin this way for technicalreasons:Thenumerical utilities wereneededfor the theory of the zero-sumtwo-persongame particularlybecauseof the rolethat expectationvalues had to playin it. Thesubstitutabilityand transferability werenecessaryfor the theoryof the zero-sumn-persongamein orderto produceimputations that arevectors with numerical components and characteristicfunctions with

numerical values. All thesenecessitiespresent themselvesimplicitly in

every subsequentconstructionbuilt upon the precedingones and so in finein our theory of the generaln-persongame.

Thus a modification of our conceptof utility in the nature of a gen-eralization appearsdesirable,but at the sametime it is clearthat definitedifficultiesmust beovercome in orderto carry out this program.

66.1.2.Our theory of gamesdivides clearly into two distinct phases:Thefirst onecomprisingthe treatment of the zero-sumtwo-persongame andleadingto the definition of its value, the secondone dealingwith the zero-sumn-persongame,basedon the characteristicfunction, as defined with thehelp of the values of the two-persongames.We pointed out above, howeachof these two phases makes use of specificproperties of the utility

concept.Therefore,if any of thesepropertiesareto be generalized,modi-fied or abandoned,we must study the effect of sucha changein eachphase.It is therefore indicatedto analyze thesetwo phasesseparately.

66.2.Discussionof the First Phase

66.2.1.The difficulties of generalizing the first phase arevery serious.The theory of the zero-sumtwo-persongame as expoundedin ChapterIIImakesfull useof thenumerical characterof utility.

Specifically:It is difficult to seehow a definite value can be assignedto agame,unlessit is possiblefor eachplayer to decidein all caseswhich of thevarious situationsthat may ariseis preferablefrom hispoint of view. Thismeans that individual preferencemust define a completeorderingof theutilities.

Next the operationof combining utilities with numerical probabilitiescannotbe dispensedwith either. We have seenthat the rulesof the gamemay explicitlyrequiresuch operations,if they provide for chancemoves.Buteven when this isnot thecase,the theory of ChapterIIIleadsin generaltotheuseof mixedstrategieswith thesameeffect. (Cf.17.))))

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GENERALIZATIONOF THECONCEPTOF UTILITY 605

Now it is well known that the completelyorderedcharacterof utilitiesdoesnot imply the numerical one. But we have seenin 3.5.that completeordering in conjunction with the possibilityof combining utilities with

numerical probabilitiesimpliesthe numerical characterof utility.Thuswe have at presentno way to adscribea zero-sumtwo-persongame

a value unlessnumerical utilities areavailable.In the n-persongame the characteristicfunction is defined with the

help of the value in various (auxiliary) zero-sumtwo-persongames.Ourreductionof the generaln-persongamesto the zero-sum ones,in addition,made use of the transferability of utilities from one player to another.

n

Indeed,constructionslike3Cn+ i = ]? 3C*in 56.2.2.can hardly begivenfc-i

any othermeaning. Thus the definition of the characteristicfunction inan n-persongame is technicallytied up with the numerical nature of utilityin a way from which we cannot at presentescape.

The values v(S) of the characteristicfunction of such a game arethevalues of the correspondingsets coalitions of players S. Henceourconclusioncan alsobe stated in this way:Our generalmethod to adscribeavalue to every possiblecoalition of players is essentiallydependentuponthe numerical nature of utility and weareat presentnot ableto remedythis.

We have pointedout before that the hypothesisof the numerical natureof utility is not as specialas it is generally believedto be. (Cf.the discus-sion of 3.) Besides,we can avoid all conceptualdifficulties by referringour considerationsto a strictly monetary economy. Neverthelessit wouldbe more satisfactory,if we couldfree our theory of theselimitations andit must beconcededthat thepossibilityof doingthishasnot beenestablishedthus far.

66.2.2.In spite of this inadequacyin general,thereare many gameswhere the difficulty of defining the characteristicfunction is never serious.Thus the examplesof 26.1.and of 57.3.were such that the characteristicfunction couldbedetermineddirectly, without realneed for the elaborateconsiderationsof the theory of the zero-sumtwo-persongames.It is truethat thesewere examplessynthesizedin orderto obtain a known, pre-assignedcharacteristicfunction hencethe easewith which they can behandled in this respectis scarcelysurprising. However,thereexistotherinstancesof the samephenomenon which areof a certain significance:Thusthe characteristicfunction causesno difficulty whatever throughout thetheory of simplegamesof Chapter X.1

Again the various markets con-sideredin 61.2.-64.2.all had characteristicfunctions which wereeasilyanddirectlyobtained.

In thesecasesit would beeasyto replacethe numerical utilitiesby moregeneralconcepts.We proposeto takethem up on another occasion.

1Thesegames were defined by stating which are the winning coalitions and this

implied an implicit determination of the characteristic function.)))

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606 EXTENSIONSOFTHECONCEPTS

66.3.Discussionof the SecondPhase

66.3.1.If the characteristicfunction is taken for granted,we can pasito the secondphase.

Herethe necessityfor a numerical utility canbeentirelycircumventedWe do not proposeto describethis in completedetail,sincethe entinsubjectdoesnot seemto bemature yet for a final mathematical formalization. Indeed,the first phase is obstructedby unsolved difficulties aidescribedabove. Besides,thereappearsto besomejustification for believing that a moreunified form of the theory, of which we can at presentse<only the outlines,might lead us to the desiredgoal.

We shall therefore give only somegeneralindications relative to th<

treatmentof the secondphase.To beginwith, when we renouncethe transferabilityof utilities,as wel

as when we renouncetheir numerical character,conceptslike zero-sumo]constant-sumgamesarenot immediatelydefined. Henceit is bestto deadirectly with generalgames.

Letus therefore considera generaln-persongame. Sincewe possessth<

theory of ChapterXI, we may forget its origin in the theory of zero-sungamesand try to extendit directlyto the caseof more general(non-numerical,non-transferable)utilities.

Theimputations>

a = {{a!, , an }}will still bevectors,but theircomponentsai, , an may not benumbers. It must be noted, that if we give up the numerical characteroutility, it is bestto concedethat eachparticipant i(=1, , n) has *domain of individual utilities% of his own. I.e.,the Od, , <un will iigeneralbedifferent. In this setupthe componenta must belongto if*.

Itmustbenotedthat even if all utilitiesarenumerical i.e.,if<Vi, , <u,coincidewith eachotherand with the setof all realnumbers we may stilomit the assumption of transferability. Also we may considerthe cas<where transferability exists,but subjectto certainrestrictions.Indeedan exampleof this will be discussedin detailin 67.

66.3.2.Now the restrictionson thesecomponentsa must be consideredThey areof two kinds:Firstthe domainof all imputationswas definedii56.8.2.by(66:1) a, v((t)) for i = 1,- - - , n,

(66:2) a< ^ v((l,- , n)).1*~i

Secondwe defined domination with the help of a conceptof effectivit}basedon

1We prefer to use(56:10)hereinstead of the alternatively possible(56:25)of (56:I:bin 56.12.)))

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GENERALIZATIONOF THECONCEPTOF UTILITY 607

(66:3) ai v(S),tinS

which is (30:3)of 30.1.1.All theseinequalitiesbelongto a common type: A certainsetT is given

(T = (t) in (66:1),T = (1, - - , n) = / in (66:2),T = S in (66:3))and* .the imputation a is requiredto placetheset coalition T into a position

which is at leastas good (in (66:1))or at most as good (in (66:2)and in(66:3))as that one statedby v(!T).

The positionof the coalition T i.e.,the compositepositionof all itsparticipants is expressedin all theseinequalitiesby the sum of their com-ponents: ] ak. Fornon-numerical utilitiesthe domainsa/!, l^may

Jbin Tbe different from eachother,and besides,theremay existno addition in

them thus rendering formations like % ak senseless.But even if thekin T

utilities are numerical, the use of ^ ak in the above contextis clearlykin T

equivalent to assumingunrestrictedtransferability. Indeed,the positionof a coalition can be describedby the sum of the amountsgiven to its mem-bers without any referenceto the individual amounts themselves onlywhen thosemembersareable to distribute that sum among themselvesinany way in which they all agree,i.e.,if thereareno physicalobstructionstotransfers.

In general,therefore, weshall have to forego the useof ]? ak. Instead,fcin T

we must introduce the domain of utilities for the compositeperson,con-sistingof all membersof a given coalition T. Denotethis domain by ^(T).Clearly,^((k)) is the same thing as ^ k. ^(T) must be obtainableby someprocessof synthesisfrom the ^ k of all k in T. It is not at all difficult todevisethe propermathematical procedurerequiredfor this process,but weproposeto discussit on another occasion.

Theaggregateofthe ak) k in T7

, as well as the value v(T)of the charac-teristicfunction must be elementsof this system. Theinequalities(66:1),(66:2),(66:3)refer then to preferencesin that systemof utilities.

66.4.Desirability of Unifying the TwoPhases

66.4.In the hope that the readerwill not find the analysisof 66.3.toosketchy, we now indicatein which way the desiredunification of the twophasesmay be lookedfor. Our theory of the zero-sumtwo-persongamewasreally basedon the samegeneralprinciplesas the subsequentstructureof imputations,domination and solutionsfor zero-sumn-persongamesandeven for generaln-persongames.Specifically,the decisivediscussionofthe inter-relatednessof various strategiesin a zero-sumtwo-persongamecarriedout in 14.5.,17.8.,17.9.i.e.,the analysisof the conceptof a good)))

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608 EXTENSIONSOF THECONCEPTS

strategy is in many ways analogous to our use of dominations ofimputations. *

Now it would seemthat the weaknessof our presenttheory liesin thenecessityto proceedin two stages:To producea solutionof the zero-sumtwo-persongame first and then, by usingthis solution,to define a charac-teristicfunction in orderto be able to producea solutionof the generaln-pergongame,basedon the characteristicfunction. Generalexperiencein mathematicsand in the physicalsciencesindicatesthat sucha two stageprocedurewith an intermediary halt representedin our caseby thecharacteristicfunction has two essentialaspects.In the early stagesofthe investigation it may beadvantageous,sinceit divides the difficulties.In the laterstages,however, when full conceptualgenerality is desired,itcanbea handicap. Therequirementof producinga sharplydefined quan-tity in themiddleof our procedure in our casethe characteristicfunctionmight be an unnecessarytechnicality,saddlingthe main problemwith anextraneousdifficulty.

Toapply this specificallyto our experiencewith games:We had todividethedifficultiesin orderto overcome them and to considersuccessivelyzero-sumtwo-persongames with strictdeterminateness,zero-sum two-persongameswith generalstrict determinateness,zero-sumn-persongames,generaln-persongames.However,all thesesteps but two were finally

mergedinto the generaltheory: Only the zero-sumtwo-persongameandthe generaln-persongame remained. Our insistenceon the characteristicfunction amounts to insistingthat for the zero-sumtwo-persongameanintermediaryresultbeobtainedwhich is much sharper than that onewhichwe acceptedas satisfactoryfor the n-persongame.1 Of course,we wereableto fulfill this requirementin the caseof a numerical,unrestrictedlytransferableutility. However,this may bedifferent when theseassump-tions concerningutility arediscarded.And it seemsratherplausiblethatour difficulty with the n-persongamemay be ascribedto our continuedinsistenceon this specialsetup for the zero-sumtwo-persongame. Ourpresenttechnicalprocedureforces us to insist in this respect,but thisinsistencemay neverthelessbemisplaced.

A unified treatmentfor theentiretheory of then-persongame withoutthe (asit now appears)artificial halt at the zero-sumtwo-persongameandthe characteristicfunction may therefore in fine prove to be the remedyfor thesedifficulties.

67.Discussionof an Example67.1.Descriptionof the Example

67.1.1.We shallnow discussan examplein which the conceptsof utilityand transferability aremodified. Thesemodifications do not representa

1For the zero-sum two-person game we obtained a unique value i.e.,imputation.For the generaln-persongame (aswell as for the zero-sum one)we had only a usuallynot unique solution, and eventhe individual solution is a setof imputations!)))

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DISCUSSIONOF AN EXAMPLE 609

particularly significant broadeningof our standpointwith respectto thoseconcepts.The interestof our exampleis rather that it permits of anapplicationof our results concerningacyclicityand thereby yieldsconclu-sionswhich throw somenew light on the subjectdiscussedat the endof 65.9.Specifically,it is hoped that proceduresof this kind will providea moreadequatemathematical approachto the phenomenon of bargaining.67.1.2.The modification to be consideredis this:We assume thatutility or its monetary equivalent is madeup of indivisibleunits. I.e.we do not questionits numerical characterbut requirethat its value be inappropriateunits an integer. Thustransferstoo arenecessarilyrestrictedto integers,but we do not restrictthem further. We proposeto use thecharacteristicfunction as before,but also with integervalues. The con-ceptsof domination and solution,after this, areunaltered.

If this standpoint is applied to generalone and two-persongames,nosignificant changesoccur;i.e.,everything remainsessentiallyas in our oldtheory. It is therefore unnecessaryto enterupon the detailsof thesecases.Thethree-persongame,on the otherhand, offers somenew features,evenin its old zero-sumform. It gives rise to somequite peculiardifficultieswhich appearto be of considerableinterest,but are not yet sufficientlyanalyzed. We therefore prefer to postpone this discussionfor a lateroccasion.

This excludesan exhaustive discussionof the generalthree-persongamein the new setup. We shall, however, analyze a specialcasewhich bearsdirectly upon the nature of bargaining.This is the three-personmarket,consistingof one sellerand two buyers.

67.1.3.We obtained in our previousanalysisof this casevarious solu-tions, dependingon whether we assumedthat only one(individual) trans-actioncouldtake placeor several,also dependingon the relative strengthof the two buyers. Thesesolutionswere describedin (62:C)of 62.5.2.andin (63:E) of 63.5.In all thesecasesit appearedthat the generalsolutionwasmadeup of two parts:(62:18)(or(62:20),(62:21),(63:30))and (62:19)(or (62:23),(63:31)).Our discussionthereshowedthat the parts of thetype (62:18)correspondto the situationwhere the two buyersarecompetingwith eachother,while the parts of the type (62:19)correspondto the situa-tion wherethey have formed a coalition againsttheseller. Thetype (62:18)part wasuniquely determinedand in essentialagreementwith theordinary,common senseeconomicideas on the subject. The type (62:19)part, onthe otherhand, was defined with the help of somehighly arbitrary func-tional connections.Theseexpressed,as we saw in 62.6.2.,the variouspossibilitiesto setup a rule of division betweenthe alliedbuyers for anyprofit obtained. I.e.they constituted their standard of behavior within

theircoalition.Our presentdiscussionis going to providesomeadditionalinformation concerningthefunctioning of this part of thesocialmechanism.

Inorderto do this effectively, it is reasonableto eliminatefrom ourproblemall thoseelementswhich do not contributetothis aspect. I.e.we)))

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610 EXTENSIONSOF THECONCEPTS

wish to get rid of the type (62:18)part of the solution. We know from62.5.2.,62.6.1.that this part is of the smallestsize and indeedcouldbeomitted altogether(cf. footnote 1on p.571) when v w in thenotationsloc.cit. This meansthat only one (indivisible)transactioncan take placeand that the two buyersareof exactlyequal strength. Thesolution is thengiven by (62:20)and (62:19)of 62.5.,((62:20)beingsuperfluous, cf. above),or equivalently by Figure99.

So we assume v = w in the schemeof 62.1.2.We can simplify thesituation further, without any significant loss,by putting the \"alternativeusefor the seller\"u = 0. In this way the (62:2)-(62:4)of 62.1.2.,definingthe characteristicfunction, simplify to

f v((l)) = v((2))= v((3))= 0,(67:1) v((l,2)) = v((l,3))= w, v((2,3))= 0,

I v((l,2,3))= w.

Theimputations arenow defined by))

a2, aawith

(67:2:a) ttl 0, a,^ 0,a,^ 0,(67:2:b) ttl + 2 + <*3 ^ w .1

67.1.4.We now assume all these quantities to be integers i.e.thegiven w and all permissible i, 2, 3 of (67:2:a),(67:2:b).

We define domination as before, i.e.following 56.11.1.which meansthat we repeatthe definitions of 30.1.1.literally.It is therefore necessary to determine the characterof the setsSzl= (1,2,3)with respectto their rolein defining a domination. It iseasy to show that the sets

S = (1,2),(1,3)are certainly necessary,and all others certainly unnecessary.2 Thus we

1Note that we are using (67:2:b)with and not with -. This is the standpointtaken in the discussion of (66:2)in 66.3.2.In the terminology of (56:I:b)in 56.12.,itamounts to using (56:10)and not (56:25). The reason for this procedureis that theformer condition is the original one (cf.,e.g.,56.8.2.),and the equivalence of the two,madeuseof in 56.12.,fails in the setup to beusednow.

It will beseenin the first remark of 67.2.3.that the ^ and the - in (67:2:b)mustproduce different results, but that this divergence nevertheless fits into the generalpicture. Besides,the use of instead of in (67:2:b)would lead to results whichdiffer only in detailsof secondaryimportance from thosewhich we aregoing to obtain.J Theconditions for certainly necessaryand certainly unnecessary setswerederivedin 31.1.,and reconsideredin 59.3.2.Sinceour standpoint has changed again (cf.above,and particularly footnote 1),it would be necessaryto reconsiderthesethings oncemore.It seemssimpler to take them up denow;

Owing to (67:2:a)above, and the condition (30:3)in 30.1.1.,every Swith v(S) -is certainly unnecessary. This disposesof S- (1),(2), (3), (2,3). Again (67:1),(67:2),(67:2:b)abovegive ai + a2 w -v((l,2)),ai + at S w -v((l,3)),hence8- (1,2),(1,3)arecertainly necessary. And since(31:C)in 31.1.3.is clearlystill valid,this renders8- (1,2,3)certainly unnecessary.)))

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DISCUSSIONOF AN EXAMPLE 611canusethe definition of domination with S = (1,2),(1,3).I.e.:

>

a ^ ft

meansthat

(67:3:a) ai > fa

and

(67:3:b) a2 > fa or a3 > fa-

Thus domination implies(67:3:a),and therefore it is clearly acyclical.(Cf. the correspondingdiscussionof 65.9.)Furthermore,the domain

(67:2:a),(67:2:b)of the a is finite, becausethe componentsa\\, a2, as mustbe integers.1

Now we can apply (65:X) of 65.7.2.:Thereexistsone and only onesolutionVo which ischaracterizedby the formulae (65:2),(65:3),id.

67.2.TheSolution and Its Interpretation

67.2.1.In orderto apply the formulae (65:2),(65:3)of 65.7.2.,wemust determinethe sets5,,C,defined at the beginningof 65.7.1.Let usdo this for J5i, Ci.

BIis the set of those a which cannot bedominated. To dominate awe must increase i and a2 or a8 without violating (67:2:a),(67:2:b)in67.1.3.Theseincreasesareby 1at least,while the otherone of a2, as may

bedecreasedas far as to 0. Hencea can be dominated,if either

(ai+ 1)+ (*s + 1) g w or (ai + 1)+ (a,+ 1) w.

SoBIis defined by

(67:4) (a!+ 1)+ (ai + l)>w, (a!+ 1)+ (a8 + 1) > w.

By (67:2:a),(67:2:b)this impliesas < 2, a2 < 2,i.e.a2, as = 0,1. Now(67:4) gives, in conjunction with (67:2:a),(67:2:b),the followingpossibilities:(67:A) a2 = as = 0, ai = w, w 1;(67:B))) or a2 = , as =(67:C) a2 = as = 1, ai = w -2.

Ci is the setof those a which aredominatedby elementsof B\\ y i.e.by thosein (67:A)-(67:C).It is easy to verify that thesearecharacterized

1This was, ofcourse,not the casein the original continuum setup.)))

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612 EXTENSIONSOF THECONCEPTS

by

(,= <M

(67:D) |or }, a,S w - 2.(3= 0j

67.2.2.Now it is betterto deviate from the schemeof (65:2),(65:3)of65.7.2.; that is, not to continue by determiningJ52, Cz,J5a, Cs, , but tousean inductive processwhich isbettersuited to this particularcase. Thisprocessgoesas follows:

Considerthe a with

(67:E) 2 = or 8 = 0.They makeup exactly(67:A),(67:B),(67:D).We know that among these

Vo containspreciselythe (67:A), (67:B).Theremaining a arethosewith

(67:F) a,,a8 ^ 1;henceundominatedby (67:A), (67:B).So we form Vo by taking (67:A),(67:B)outsideof (67:F),and repeatingthe processof finding a solutionin

(67:F).Compare(67:F) with (67:2:a),(67:2:b)in 67.1.3.Theonly difference

is that 2, s areincreasedby 1. Hencew must betreatedas if it werew 2. Thus Vo now containsfurther

(67:G)

(67:H)

andwe must repeatthe processof finding a solutionin

(67:1) a2, a8 ^ 2.Repetitionof this procedureassigns

(67:J) 2 = a = 2, i = tu 4, w 5;( 2 = 3,aa = 2 \\

(67:K) {or 1, ai = 10- 5;

( a2 = 2, a8 = 3 ]

to Vo> and requiresus to repeatthe processof finding a solutionin

(67:L) 2, 2 ^ 3,etc.,etc.

ThusVoConsistsof(67:A),(67:B),(67:G),(67:H),(67:J),(67:K),Thissetcanbecharacterizedas follows:

(67:M) ai - 0,1, 10;))

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//3T XT\\(67:N)))

DISCUSSIONOFAN EXAMPLE

[ = = -1 if w a\\ is even;))

613))

(67:0)))

W 1 OLl)

u - 1 - ai)

a* 2or)

2or)

2

o + 1- 01)2 '-)

2))

if w ai is odd.))

67.2.3.Theresults (67:M)-(67:O)suggesttheseremarks:First:The values of a\\ + a2 + as in this solution are w and w 1.

Thus we cannot replacethe ^ in (67:2:b)of 67.1.3.by =, the resultstatedin (56:I:b)of 56:12is no longer true. Themaximum socialbenefit is notnecessarilyobtained and this appears as the directconsequenceof theexistenceo an indivisibleunit of utility. 1

Second:This \"discrete\"utility scaleconvergestoward our usual, con-tinuous one,if w oo. (Cf. the correspondingconsiderationsconcerningdiscreteand continuous \"hands\"in Poker,in 19.12.)The difference of

i + 2 + s and w, mentioned above, is at most 1. So it becomesmoreand more insignificant as w - oo , i.e.this aspectof the situation tends towhat it was in the continuous case.

Third: 2, a differ from eachother by at most 1. Sothis differencetootends to insignificanceas w > oo. I.e.,when we approachthe continuouscasethe solutiontends to looklike this:))

(67:P)(67:Q))) 3 =))

^ w,w))

As pointedout in the first part of 67.1.3.,this solutionmust be comparedwith (62:19)in 62.5.1.,usingthe values u = 0,v = w. The two solutionsareindeedsimilar,but our solution coversonly one specialcaseof (62:19):The monotonic decreasingfunctions of a\\ mentionedtherecoincidewith

eachotherand with ^Thosefunctions describe,as discussedin 62.6.2.,the rule of division

upon which the two buyers agreedwhen forming their coalition (which isexpressedby (62:19)).In the continuouscasethis rule was highly arbi-trary. But now, in the discretecase,we find that it is completelydeter-mined the two buyersmust betreatedexactlyalike!

What is the meaning of this symmetry? Are the other distributionrules -i.e.the otherchoicesof the functions in (62:19)really impossiblein the \" discrete\"case?))

1 Cf.this with footnote 3 on p. 513.)))

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614 EXTENSIONSOF THECONCEPTS

67.3.Generalization :Different DiscreteUtility Scales67.3.1.In orderto answerthe above questions,we shall try to destroy

the symmetry (betweenthe two buyers),but conserve,the \"discreteness.\"This will be doneby altering the setup of 67.1.in so far that we assign

the indivisibleunit of utility for the buyer 2 a value different from thatonefor thebuyer3. Specifically:Let us prescribethat the values of i, a2must beintegers,while thoseof as must be even integers. Apart from this,everything in 67.1.remainsunaltered.

We now carry out theequivalent of the considerationsof 67.2.Accord-ingly, we beginby determiningthe setsB\\, Ci of 65.7.

> . >

Biis the setof those a which cannot be dominated. To dominate awe must increasea\\ and a2 or a8 without violating (67:2:a),(67:2:b)in67.1.3.Theseincreasesare1 (for i, a2) or 2 (for a8) at least,while the

>

otheroneof a2, a8 may be decreasedas far as to 0. Hencea can bedominatedif either(on + 1)+ (a2 + 1) w or (ai + 1)+ (a* + 2) g w.SoBIis defined by

(67:5) (ai+ 1)+ (at + 1) > w, (i + 1)+ (as+ 2) > w.

By (67:2:a),(67:2:b)this impliesas < 2, a2 < 3,i.e.a2 = 0,1,2, a8 = 0.Now (67:5)gives, in conjunction with (67:2:a),(67:2:b),the followingpossibilities:(67:R) a2 = 0, as = 0, ai = w, w-1;(67:S) a2 = 1, as = 0, ai = w - 1,w -2;(67:T) a2 = 2, as = 0, ai = w - 2.

Ci is the setof those a , which aredominatedby elementsof BI i.e.bythosein (67:R)-(67:T).It is easy to verify that thesearecharacterizedby

(67:U) a2 = 0, ai g w - 2;(67:V) a2 = 1, ai w -3.

67.3.2.Now we repeatthe variant of 67.2.2.:Insteadof determining5s,C2, B8, Cs, , we usea different inductive process.

Considerthe a with

(67:W) a2 = 0,1.They make up exactly (67:R),(67:8),(67:U),(67:V).4 We know that

among these,Vo containspreciselythe (67:R), (67:S).Theremaining aarethosewith

(67:X) a2 ^ 2;1Note that a* cannot be 1,sinceit must be even.)))

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DISCUSSIONOF AN EXAMPLE 615henceundominated by (67:R), (67:S).Sowe form Vo by taking (67:R),(67:S)outsideof (67:X),and repeatingthe processof finding a solution in

(67:X).Compare(67:X)with (67:2:a),(67:2:b)in 67.1.3.Theonly difference

is that a2 is increasedby 2. Hencew must be treatedas if it werew 2.1Thus Vo now containsfurther

(67:Y) a2 = 2, a8 = 0, ai = w - 2, w -3;(67:Z) 2 = 3, <* 8 = 0, ai = w - 3,w - 4;and we must repeatthe processof finding a solution in

(67:A') 2 ^ 4.

Repetitionof this procedureassigns

(67:B') 2 = 4, a3 = 0, a,= w - 4, w - 5;(67:C') a2 = 5, as = 0, ai = w - 5, w - 6;to Vo and requiresus to repeatthe processof finding a solution in

(67:D') 2 ^ 6,

etc.,etc.Thus Vo consistsof (67:R), (67:8),(67:Y), (67:Z),(67:B'),(67:C'),. This setcan be characterizedas follows:

(67:E') ai = 0,1, w\\

(67:F') 2 = w - ai,w - 1- ai(excludingthe secondonewhen ct\\ = w)\\

(67:G') a8 = 0.67.3.3.Theresults (67:E')-(67:G')suggesttheseremarks:First and second:Concerningthe sum ai + c*2+ a and its relation to w

we may repeatliterally the correspondingparts of 67.2.3.Third:Herethings are altogetherdifferent from 67.2.3.We have

identically a = 0. Approaching continuity, i.e.for w * oo the solutiontends to looklikethis:(67:H') ^ i ^ w,

(67:1') a2 = w - i,(67J;

) as = 0.Repeatingthe comparisonto (62:19)in 62.5.1.,as made in the cor-

respondingpart of 67.2.3.,we seethat the situation is now this:Themono-tonic functions of (62:19),which describethe rule of division betweenthetwo alliedbuyers (cf.loc.cit.)areagain completelydetermined but thistime we find (insteadof the equal treatmentthey receivedin 67.2.3.)theentireadvantage going to buyer2!

1Note the difference between this and the corresponding step in 67.2.2.following(67:F)there.)))

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616 EXTENSIONSOF THECONCEPTS

We must now comparethis result with the correspondingonein 67.2.3.and interprettheentirephenomenon.

67.4.Conclusions Concerning Bargaining

67.4.Theconclusionfrom the results of 67.2.3.,67.3.3.is evident. Inthe former casethe two buyershad exactlyequalpowersof discernmenti.e.equal units of utility and the rule of distributionwas found to treatthem equally. In thelattercasebuyer2 had a betterpowerof discernmentthan buyer3 i.e.2'sunit of utility was half of 3's and in the rule of divi-sion the advantage went in its entirety to buyer2. Clearly,if their abilitieshad beenreversed,the result would have beenalso. We may alsosay: Theadvantage in the rule of division betweenalliedbuyersis equallydividedif

they have equally fine utility scales,and goesentirely to the onewith thefiner utility scaleotherwise.1

This is true in the discretecasewhere eachparticipant has a definiteutility scaleand the rule of division (i.e.the solution)isuniquely determined.In the continuous casethe \" fineness\"of the utility scaleis undefined andthe rule of division can bechosenin many different ways, as we have seen.

So we observefor the first time how the ability of discernmentof aplayer specificallythe fineness of his subjectiveutility scale has a deter-mining influence on his positionin bargaining with an ally.2 It is thereforeto beexpectedthat problemsof this type can only be settledcompletelywhen the psychologicalconditionsreferredto areproperlyand systemati-cally taken into account. The considerationsof the last paragraphmay bea first indicationof the appropriatemathematical approach.

1It is possibleto considermore subtle arrangements: We can assign to 2 and to onranges of varying density. In this casewe have still a unique solution for the samereasonsas before. Thecorrelation of 2, <** when plotted in the 2, s plane will be acombination of the three types describedabove:Symmetric in 2, s, i.e.parallel to thebisectrixof the two coordinateaxes;parallelto the a-axis;parallelto the a2-axis.

It is actually possibleto bring about any desiredcombination of theseelementsbychoosing the ranges of 2 and aa appropriately. Any desiredshapeof the curve can beapproximated arbitrarily well in this manner. In this way the original generality of thecontinuous caseis recovered.

We do not proposeto considerthis matter, and various related ones, in detail here.1This occurs,of course,only when the theory with continuous utilities allows several

different rules of division between allies which is plainly the casewhere bargaining playsa role.)))

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APPENDIX. THE AXIOMATIC TREATMENT OF UTILITY

A.I.Formulation of the Problem

A.I.I.We will prove in this Appendix,that the axiomsof utility enumer-ated in 3.6.1.makeutility a number up to a linear transformation. 1 Moreprecisely:We will prove that those axiomsimply the existenceof at leastone mapping (actually, of course,of infinitely many) of the utilities onnumbersin the senseof 3.5.1.,with the properties(3:1:a),(3:1:b);and wewill also prove that any two such mappingsare linear transforms of eachother,i.e.connectedby a relation (3:6).

Before we undertake this analysisof the axioms(3:A)-(3:C)of 3.6.1.,two further remarksconcerningthem may be useful in dispellingpossiblemisunderstandings.

A.I.2.The first remark is this:Theseaxioms,specifically the group(3:A),characterizethe conceptof completeordering, based on the relations>, <. We do not axiomatize the relation =, but interpret it as trueidentity. The alternative procedure,to axiomatize = also,would bemathe-matically perfectly sound,but sois our proceduretoo. The two proceduresare trivially equivalent and representonly variants in taste. The practiceinthe relevant mathematical and logical literature is not uniform and we havetherefore adheredto the simplerprocedure.

The secondremark is this:As pointed out at the beginningof 3.5.1.,we are usingthe symbol> both for the \"natural\" relation u > v affectingutilities u, v and for the numerical relation p > a affecting numbersp, <r;alsowe are usingthe symbola -f (1 a) both for the \" natural\"operation an + (1 a)v affecting utilities u, v and for the numericaloperation ap + (1 a)cr affecting numbersp, a (a is a number in eithercase). One might objectthat this practicecan lead to misunderstandingsand to confusion; however, it doesnot, providedthat one keepsalways inevidencewhether the quantities involved are utilities (u, v, w) or numbers(a,0,7, , p, <r). This identification of the designationsfor relationsand operationsin the two cases(\"natural\"and numerical) has a certainsimplicityand facilitates keepingtrack of the \"natural\" and numericalpairs of analogs. Forthese reasonsit is fairly generallyacceptedin similarsituationsin the mathematical literature,and we proposeto makeuse of it.

A.1.3.The deductionswhich follow in A.2.are rather lengthy and maybe somewhattiring for the mathematically untrained reader. From thepurelytechnical-mathematicalviewpoint thereis the further objection,thatthey cannot be considereddeep the ideasthat underly the deductionsare

1I.e.without fixing a iero or a unit of utility.617)))

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618 THEAXIOMATICTREATMENT OF UTILITY

quite simple,but unfortunately the technicalexecutionhad to be somewhatvoluminous in orderto be complete.Possiblya shorterexpositionmightbe found later.

At any rate,we are now forced to use the estheticallynot quite satis-factory modeof expositionwhich follows in A.2.

A.2.Derivation from the Axioms

A.2.1.We now proceedto carry out our deductionsfrom the axioms(3:A)-(3:C)of 3.6.1.The deductionwill be brokenup into severalsucces-sive steps and it will be carriedout in this sectionand the four next ones.The final result will be stated in (A:V),(A:W).

(A:A) If u < v, then a < ft implies

(1- a)u + av < (1- P)u + PV.

Proof:Clearly a = 7/3 with < 7 < 1. By (3:B:a)(applied to u, v,1 ft in placeof u y v, a) u < (1 flu + pv, and henceby (3:B:b)(appliedto (1 flu + PV, u, 7 in placeof u, v, a)

-(1- flu + PV > 7(0 - P)u + pv) + (1- y)u.

By (3:C:a)this can be written

(1- p)u + pv > y(pv + (1- flu) + (1- y)u.Now by (3:C:b)(appliedto v, u, 7, 0,a = yp in placeof u, v, a,0,7 = a/3)the right hand side is av + (1 <*)u, hence by (3:C:a)(I a)u + av.Thus (1 a)u + av < (1 flu + pv, as desired.(A:B) Given two fixed UQ, VQ with UQ < VQ, considerthe mapping

a > w = (1 a)uQ + avQ.This is a one-to-oneand monotone mappingof the interval

<a < 1on part of the interval u < w < v .1Proof:The mapping is on part of the interval UQ < w < VQ\\ UQ < w

coincideswith (3:B:a)(appliedto u , v , 1 a in placeof u, v, a), w < v

coincideswith (3:B:b)(appliedto t> , wo, in placeof u, v, a).One-to-onecharacter:Followsfrom the monotony, which we establish

next.Monotonecharacter:Coincideswith (A:A).

(A:C) The mappingof (A:B)actually maps the a of < a < Ion all the w of UQ < w < V Q.

Proof:Assume that this were not so,i.e.that someWQ with UQ < WQ < VQ

were omitted. Then for all a in < a < 1 (1 a)uQ + avQ 7* u> , i.e.(1 a)u + av ^ w . According to whether we have < or >, let a

1Itwill appearin (A:C),that this part is actually the whole interval u < w < t>o-)))

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DERIVATIONFROM THEAXIOMS 619belongto classI or II. Thus the classesI, II,which areclearlymutuallyexclusive,exhausttogetherthe interval < a < 1. Now we observe:

First:ClassI is not empty. This is immediate by (3:B:c)(appliedtoMO, Wo, t>o, 1-~ a i*1placeof u, w, v, a).

Second:ClassII is not empty. This is immediateby (3:B:d)(appliedto Vo, Wo, u , a in placeof u, w, v, a).

Third:If a is in I and ft is II,then a <0. Indeed,sinceI and II aredisjunct, necessarilya ^ ft. Hencethe only alternative would be a > ft.But then the monotony of the mappingof (A:B)would imply, that sincea is in I,ft toomust be in I but ft is in II. Henceonly a < ft is possible.

Consideringthesethreepropertiesof I, II,theremust existan a with

< o < 1which separatesthem, i.e.such that all a of I have a g c* ,and all a of IIhave a ^ ao.1

Now a itself must belongto I or to II. We distinguishaccordingly:First:ainl. Then (1 ao)uo4- a v < w ^ Alsow < VQ. Applying

(3:B:c)(with (1 a )wo + o^o, MO, tfo, y in placeof u, w, v, y) we obtaina ywith < y < 1 and y((l o)^o+ ao^o)+ (1 7)^0 < w , i.e.by (3:C:b)(with t*o, Vo, 7, 1 ao, 1 a = 7(1 a ) in place of u, v, a, 0,7 = aft)(1 a)w + av Q < WQ. Hencea = 1 7(1 a ) belongsto I. Howevera > 1 (1 a ) = a , although we shouldhave a ^ a .

Second:a in II. Then (1 a )wo + <*ot>o > WQ. Also UQ < WQ.

Applying (3:B:d)(with (1 aQ)u Q + aoVo, Wo, MO, 7 in placeof u, w, v, a)we obtaina 7 with < 7 < 1and 7((1 a )w + <* vo) + (1 7)^0 > w ,i.e.by (3:C:a)y(aQv + (1 o)w ) + (1 7)^0>w , henceby (3:C:b)(withVo, Wo, 7, o, = 7o in placeof u, v, a, ft, y = aft) avQ + (1 a)uQ > WQ,

i.e.by (3:C:a)(1 a)uo + avo > WQ. Hencea = ya<> belongs to II.Howevera < ao, although we shouldhave a ^ ao.

Thus we obtain a contradictionin eachcase. Therefore the originalassumptionis impossible,and the desiredpropertyis established.

A.2.2.It is worth while to stop for a moment at this point. (A:B)and(A:C)have effected a one-to-onemappingof the utility interval w < w < VQ

(w , v Q fixed with UQ < v , otherwisearbitrary!) on the numerical interval< a < 1. This is clearlythe first step towardsestablishinga numerical

representationof utilities. However,the result is still significantly incom-pletein severalrespects.Theseseemto be the major limitations:

First:The numerical representationwas obtainedfor a utility intervalWQ < w < v only, not for all utilities w simultaneously. Nor is it clear,how the mappingswhich go with different pairs WQ, #o fit together.

Second:The numerical representationof (A:B),(A:C)has not yet beencorrelatedwith our requirements(3:l:a),(3:l:b).Now (3:l:a)is clearly

1This is intuitively fairly plausible. It is, furthermore, a perfectly rigorous inference.Indeed,it coincideswith one of the classicaltheorems effecting the introduction of irra-tional numbers, the theorem concerning the Dedekind cut. Detailscan be found intexts on real function theory or on the foundations of analysis. Cf.e.g.C.Carathtodoryloc.cit. footnote 1 on p.343. Cf.there p.11,Axiom VII. OurclassIshould besubsti-tuted for the set (a) mentioned there. Theset (A ] mentioned there then contains ourclassII.)))

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620 THEAXIOMATICTREATMENT OF UTILITY

satisfied:It is just another way of expressingthe monotony that is securedby (A:B). Howeverthe validity of (3:l:b)remainsto be established.

We will fulfill all these requirements jointly. The procedurewill

primarily follow a coursesuggestedby the first remark,but in the processthe requirementsof the secondremark and the appropriate uniquenessresultswill alsobe established.

We beginby proving a groupof lemmata which is more in the spirit ofthe secondremark and of the uniquenessinquiry; however it is basic inorderto makeprogresstowardsthe objectivesof the first remark too.(A:D) Let u , V Q be as above:UQ, v fixed, UQ < V Q. Forall w in

the interval UQ < w < t> define the numerical function/(w) = /* ,v,(w)as follows:(i) /(wo) = 0.(ii) /(t>o) = 1.(iii) f(w) for w ?* w , fo, i.e.for UQ < w < y , is the numbera in

< a < 1which correspondsto w in the senseof (A:B),(A:C).(A:E) The mapping

w >/(w)has the following properties:(i') It is monotone.(ii') For < < 1 and w ^ u

/((I- 0)Uo+ 0W) = 0f(w).(iii') For < < 1 and w ^ V Q

/((I- 0> + 0ii>) = 1- + 0/(uO-

(A:F) A mappingof all w with UQ g w g v on any setof num-bers,which possessesthe properties(i), (ii) and either(ii')or(iii'),is identicalwith the mappingof (A:D)

Proof:(A:D)is a definition; we must prove (A:E)and (A:F).Ad (A:E):Ad (i'):ForUQ < w < V Q the mappingis monotone by (A:B).

All w of this interval are mappedon numbers >0,< 1,i.e.on numbers>than the map of UQ and < than the map of v . Hencewe have monotonythroughout u<> g w g v .

Ad (ii'):Forw = t/ :The statementis/((I 0)u + 0t; ) = 0,and thiscoincideswith the definition in (A:B)(with in placeof a).

Forw 7* t> , i.e.u < w < v :Put/(w) = a, i.e.by (A:B)w = (1 a)uQ + at/o.

Thenby (3:C:b)(with , u Qy 0,a. in placeof u, v y a, 0,and using (3:C:a))))

Henceby (A:B)/((I- 0)w + 0^) = 0a = 0/(^), as desired.Ad (iii'):For w = u : The statementis /((I 0)t> + 0^o) = 1 0,

and this coincideswith the definition in (A:B)(with 1 in placeof a andusing(3:C:a)).)))

Page 645: Theory of Games Economic Behavior

DERIVATIONFROM THEAXIOMS 621Forw 7* w , i.e.UQ < w < v Q :Putf(w) = a, i.e.by (A:B)

w = (1 a)w + avg.

Then by (3:C:b)(with w , t> , 0,1- a in placeof w, v, a, ft, and using(3:C:a))(1- ft)v + ftw = (1- ft)v<> + ft((l - o)w + avo) = 0(1- a)w))

. , ,AWhenceby (A:B)/((I- 0)t> + ftw) = 1- 0(1- a) = 1- ft + 0a = 1- + 0/(i,),

as desired.Ad (A:F):Considera mapping

(A:l) w-^Mw)with (i), (ii) and either(ii')or (iii').The mapping(A:2) w->f(w)is a one-to-onemapping of UQ w VQ on a I, henceit can beinverted:(A:3) a->^().Now combine(A:l) with (A:3),i.e.with the inverse of (A:2):(A:4) a^A^(a))= ,().Sinceboth (A:l) and (A:2)fulfill (i), (ii),we obtain for (A:4)(A:5) ^(0)= 0, ^(1) = 1.If (A:l) fulfills (ii') or (iii'),then, as (A:2)fulfills both (ii') and (iii'),weobtain for (A:4)(A:6) v(fta) = M),or(A:7) ^(1- j8 + j8a) = 1 - j8 + j8^(a).Now putting a = 1in (A:6)and using (A:5)gives

(A:8) ^(j3) = j8,

and putting a = in (A:7) and using (A:5) gives <p(l 0) = 1 ft.

Replacingft by 1 ft gives again (A:8).Thus (A:8)is valid at any rate, (ii'), (iii') restrictit to the ft with

< ft < 1. However(A:5)extendsit to ft = 0,1too,i.e.to all ft with

|S 1. Consideringthe definition of v?(ex) by (A:3),(A:4),the generalvalidity of (A:8)expressesthe identity of (A:l) and (A:2),which is pre-ciselywhat we wanted to prove.(A:G) Let w , Vo be as above:u Q, v fixed, u < V Q. Let also two

fixed a , /?o with < fto be given. Forall w in the intervalUQ ^ w g VQ define the numerical function g(w) =

fif\";;S|(^)as follows:

g(w) = (j8o - o)/(w) + o,

) accordingto (A:D)).)))

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622 THEAXIOMATICTREATMENT OF UTILITY

We note:(i)(ii)

(A:H) This mappingw ->g(w)

has the followingproperties:(i') It is monotone.(ii') For < ft < I and w 7* UQ

g((l - ft)u Q + ftw) = (1- 0) + ftg(w).

(iii') For < ft < 1and w ^ VQ

g((l - ft)v<> + ftw) = (1- /3)0o+ 00(w).(A:I) A mappingof all w with w ^ t^ ^ VQ on any setof numbers

which possessesthe properties(i), (ii) and either(ii')or (iii'),is identicalwith the mappingof (A:G).

Proof:Usingthe correspondencebetweenfunctions

gi(w) = (0o- a )/i(t0)+ <*o,orequivalently))

(for fi(w), g\\(w), and also for f(w), g(w)\\ the statementsof (A:G)-(A:I)goover into the statementsof (A:D)-(A:F).Hence(A:G)-(A:I)follow from(A:D)-(A:F).(A:J) Assuming (i) ; (ii) in (A:G),the equation

g((l - ftu + 0v) = (1- P)g(u)+ pg(v)

(u Q g u < v g t; ) with u = UQ, v ^ UQ is equivalent to (ii')in (A:I),and with u 7* V Q, v = v it is equivalent to (iii') in(A:I).

Proof:Ad (ii'):Put u , w, ft in placeof u, v, ft.Ad (iii'):Put w, v , 1 ft in placeof u, v, ft.A.2.3.In (A:G)-(A:J)the mapping of a utility interval UQ ^ w g V Q

on a numerical interval a ^ a ^ /3ohas beengiven its technicallyadequateform, with the necessaryuniquenessproperties.We can now begintofit

the various mappingsw - g(w) = gl'tyw)

together.(A:K) Considergf^;Jj and a WQ with u g W Q v<>. Put))

To = i

Then g^\\(w)coincideswith gj,'ij(w)in the latter'sdomain

UQ ^ w ^ WQ (if 1^0 T* u Q, i.e.UQ < WQ\\ and 0jj(u>)coincides)))

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DERIVATIONFROM THEAXIOMS 623

with 0ij5j(w>) in the latter'sdomain WQ< w < v (if WQ 7* t> ,

i.e.We < t><>).

Proof:Ad 0\"j;i|(w):gl\\(w) possessesthe properties(i), (ii') (of (A:G),(A:H))for a , To,MO, t0 , becausethey coincidewith those for a , Ho, w , v

(sincethey involve only the lower end a , MO). It also possesses(ii) (of(A:G))for a , 70, w, u>o, becauseff'fywo) * 70. Henceit follows from

(A:I) thatg\"j;Jj

fulfills within u% 5w t0 a unique characterization ofd*****w

Adgl**,\\\\ 0C;;J;possessesthe properties(ii),(iii') (of (A:G),(A:H))for

To,Ho, u>o, VQ becausethey coincidewith those for , Ho, ^ , t>o (sincetheyinvolve only the upper end Ho Vg). It also possesses(i) (of (A:G))for

To, , Wo, yo,becauseg*$(wg) TO- Henceit followsfrom (A:I)that g*fyfulfills within WQ & w v* & unique characterizationof *',!*.

(A:L) Considera gCj^J and two u\\ } v\\ with u ^ u\\ < vi ^ v .Put en - ^;;J;(wO,

-Hi -d$(i).Then ^(w) coincides

with g\"lt l(w) in the latter'sdomain ui w v\\.

Proof:Apply first (A:K)to g?***and gjjjj(i.e.with w , o, o, /3o, t>i, /3i in

placeof u , v , a , /8o, ^ , %; note that /8i = jC$(i)) this shows that

^*j;Jj(w;) coincideswith 0Cj;J|(w>)in the latter'sdomain UQ w ^ v\\. Apply

next(A:K)to g\"j;Jjand g^\\ (i.e.with w , i, o, Hi, t^i, i in placeof u , t; ,

a , Ho, wo, TO;note that != gl*(ui)= fif^f;(wO) this showsthat gl*!\\(w),

and hencealso gl**\\(w), coincideswith gl^\\(w) in the latter'sdomainU\\ ^ W g Vi.

(A:L) has to be combinedwith a secondline of reasoning.At thispoint we also assumethat two w*, t>* with u* < v* have been chosen;wewill considerthem as fixed from now on until we get to (A:V) and (A:W).

We now prove:(A:M) If wo ^ u* < v* ^ i> , then thereexistsone and only one

03\",Jj(t0) such that

(i) rf(u) -0,(ii) rf(O * I-

We denotethis g*'(w)by /i^Vf (w).

Proo/:Form the/(w) = /^.(w)of (A:D). As u* < v*, BO/(U*)</(*).Forvariable a , Ho (A:G)gives ^(w) = (Ho - a )/(w)+ o- Hencethe

above (i), (ii) mean that (Ho - o)/(u*)+ o = 0,(Ho - o)/(w*) 4- o = 1,and these two equations determine ao, Ho uniquely.1 Hencethe desired

/(**) 1 /(*)'))a \" ~)))

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624 THEAXIOMATICTREATMENT OF UTILITY

existsand is unique.))

(A:N) If MO u\\ u* < v* g vi g MI, then hVo ,V9 (w) coincides

with AWI(VI (M>) in the latter'sdomain u\\ g w g VL

Proof:Put <*!= A r,t(wi), 0i= ft.,,>i).Then, by (A:L), A ti,t(iiOcoincideswith flC

1;?1^) in the latter'sdomain u\\* w ^ v\\. Applying this

to w = u* and to w = v* gives fl|;J|(M*)= AUO ,VO(M*) = and ^j;Jj(v*)=

*,.,(*>*)= 1- Hence by (A:M), gll*\\(w)= A

Mi ,Vi (w). ConsequentlyA

Wo>t , (w) coincideswith AUi ,Vi (w) in the latter'sdomain HI ^ w ^ v\\.

We can now establish the decisivefact:The functions hU9tVt (w) all fit

togetherto one function. Specifically:

(A:0) Given any w, it is possibleto chooseMO, VG so that M O ^ M*

<v*^ VQ and UQ^w ^ ^o. For all such choicesof MO, *>o,

Ati ,ve(w) has the samevalue. I.e.AWoft ,o (w) dependson w only.

We denote it therefore by h(w).

Proof:Existenceof M O, V Q : M O= Min (M*, M;) and v = Max (v*, w)

obviously possessthe desiredproperties.h

Ui>V9 (w) depends on w only: Choose two such pairs M O , V Q and Mj, vj:MO ^ M* < v* ^ v , UQ ^ w ^ VQ and M ^ M* < v* ^ vj, M ^ w ^ v'.Put MI = Max (MO , M O),Vi = Min (t> , ^o)- Then?/ ^ MI ^ M* <y* ^ t^i ^ t; ,MI ^ M? g vi, and Mj ^ MI ^ M* < v* ^ Vi g v , ?/i ^ w ^ VL Now twoapplicationsof (A:N) (first with M O, VQ, MI, wi, i^, then with wj, vj, MI, Vi, t^)

give AUot1>f (M;) = h U} ,Vl (w} and ft u;f ,;(w) = AU,,,^^). Hence))

as desired.A.2.4.The function A(n;) of (A:O)is defined for all utilities and it has

numerical values. We can now show with little trouble that it possessesallthe propertiesthat we need.

This is mosteasilydone with the help of two auxiliary lemmata.

(A:P) Given any two M, v with M < tf, thereexisttwo M O, v with

MO ^ M* < V* g V , MO ^ M < V g !>o.

Proof:Put MO = Min (M*, M), v = Max (v*, v).

(A:Q) Given any two M, v with M < v, put A(M) = a, A(t>)= /3.

Then a < ft, and A(ti?) coincideswith g\"'%(w) in the latter'sdomain u w v.

Proof:ChooseM O, V Q as indicated in (A:P). By (A:M) hUitV9 (w) is a

QW\\(W) w^ two suitable a , j9o- By (A:0) A(M?) coincideswith A^^w),i.e.with gfy(w), in the latter'sdomain u Q w v Q. Applying this to)))

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DERIVATIONFROM THEAXIOMS 625))

w = u and to w = v gives g^\\(u) = h(u) = a and gl**\\(v)=

A(t>) = 0.Since0j;Jj(w)is monotone, this implies a < /3. Next by (A:L) (withw , t>o, o,0o,w, v, a, in placeof w , v , o, 0o,w,, vi, ai,&i)gl'\\(w)coincideswith g't*(w) in the latter'sdomain u g w ^ v. Consequentlythe sameistrue for h(w).

After thesepreparationswe establish the relevant propertiesof h(w)

(A:R) The mapping

w + h(w)

of all w; on a setof numbershas the following properties:(i) MO-0.(ii) MO= 1.(iii) h(w) is monotone,(iv) For < y < 1and u < v

MU - y)u + yv) = (1- y)h(u) + yh(v)

(A:S) A mappingof all w on any setof numbers,which possessesthe properties(i),(ii) and (iv) is identicalwith the mappingof(A:R).

Proof:Ad (A:R):Ad (i), (ii):Immediateby (A:0)and (A:M).Ad (iii):Containedin (A:Q).Ad (iv) :Chooseu, v accordingto (A:P)and then a, and g*'(w)accord-

ing to (A:Q). Now by (A:H),(ii') (with u, v, v, y in placeof u , v Q , w, y)

0:;5(U- 7)w + yv) = (1- y)g%(u)+ yg*Z(v). Henceby (A:Q)

T)U + yv) = (1- y)h(u) + yh(v)))

as desired.Ad (A:S):Considera mapping

w > hi(w)

of all utilities w on numbers,which fulfills (i), (ii) and (iv). ChoosetwoUo, v with u g u* < v* ^ v , and put = ^i(^*), == ^i(v *)- Then,by (A:I),/ii(tu) coincideswith ^uJ'^C^)in the latter'sdomain w g i^ g VQ.

Putting w = u* and 10= v* we get sC$(u*)= hi(u*) = 0, ^J(w*) =

tn(v*) = 1. Henceby (A:M) gift is fc,^. Thus hi(w) coincideswith

hwiM,i.e.with h(w), in w ^ t^ ^ t>o. By (A:0)this means that hi(w)and h(w) arealtogetheridentical.

A.2.5.(A:R),(A:S)give a mappingof all utilities on numbers,which

possessesplausiblepropertiesand is uniquely characterizedby them, andtherefore we might let the matter restthere. However,we are not yet)))

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626 THEAXIOMATICTREATMENT OF UTILITY

quite satisfied,for the followingreasons:The characterizationin (A:R)doesnot coincidewith that one by (3:l:a),(3:l:b)(A:R) goeslessfar in (iv)(this is assertedin (3:l:b)for all u, v, in (iv) only for those with u < v)',and it introducesan arbitrary normalization in (i), (ii) (by means of thearbitrary u*, v*). In what follows,we will eliminate thesemaladjustments.This will prove fairly easy.

We first extend(iv) in (A:R).

(A:T) Always (1 y)u + yu = u.

Proof:Foru (1 y)u + yu say that y belongsto classI (uppercase)or II (lower case). If y is in classI or IIand if < ft < 1,then

u $ (1- ft)u + ft((l - y)u + yu} $ (1- 7)14+ yu

by (3:B:a)and (3:B:b).(For y in classI or II, respectively:First,u,(1 y)u + yu, 1 ft in placeof u, v, a in (3:B:a)or (3:B:b).Second,(1 y)u + yu, u, ft in placeof u, v, a in (3:B:b)or (3:B:a).)By(3:C:a)and (3:C:b)(with u, u, ft, y in placeof u, v, a, ft)

(1- ft)u + ft((l - y)u + yu) - (1- fty)u + ftyu.

Henceu $ (1- 7)\"+ ftyu (1- ft)u + ftu. Put * * 187. Sinceft isfree in < ft < 1,therefore 6 is free in < 3 < 7. Assuming < 7 < 1,

< 6 < 1,we have therefore:

(A :9) If 7 is in classIor II,then every 5 < 7 is in the sameclassIor II.

(A:10) Underthe conditionsof (A:9)))

(1- t)u + &u $ (1- J)u + yu,

respectively.The expression(1 7)1* + yu is unchangedif we replace7 by 1 7.

As 1 y < I 5 is equivalent to 7 > d, we can put 1 7, 1 6 in placeof 7, a in (A:9). Then(A:9)and (A:10)becomethis:(A:ll) If 7 is in classI or II,then every 6 > 7 is the sameclassI

orII.(A:12) Under the condition of (A:ll)

(1- )u + 81* (1- y)u + yu,

respectively.Now (A:9)and (A:ll)show,that if 7 is classI or II,then every 5(<7

or = 7 or > a) is in the sameclassI or II. I.e.if eitherclassI orII is notempty, then it containsall 6 with < 6 < 1. Assume this to be the case(for classI or II),and considertwo 7, d with 7 < 5. Then by (A:10)(1- t)u + tu $ (I-y)u + yu, and by (A:12)(with d, y in placeof 7, *))))

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DERIVATIONFROM THEAXIOMS 827

(1 5)u + $w ^ (1 y)u + yu. Henceat any rateboth < and > holdin (1 6)w + &u (1 T)W + yu. This is a contradiction.Thereforeboth classesI and IImust be empty.

Consequentlynever u ^ (1 y)u + yu, i.e.always (1 y)u + yu w,as desired.(A:U) Always

*((!- 7)u+ yv) * (1- 7)*(u)+ 7*W

(0 < 7 < 1,any u, ).Proo/:Foru < v this is (A:R),(iv). Foru > v it obtains from the

former by putting v, u, 1- y in placeof u, v, y. Foru = v it followsfrom(A:T).

We can now prove the existenceand uniquenesstheorem in the desiredform, i.e.correspondingto (3:l:a)and (3:l:b).At this point we alsodropthe assumedfixed choiceof w*, v*, which was introducedbefore (A:M).(A:V) Thereexistsa mapping

w > v(w)

of all u? on a setof numberspossessingthe following properties:(i) Monotony.(ii) For < 7 < 1and any u, v

v((l - 7)u + yv) = (1- T)V(U) + yv(v).

(A:W) Forany two mappingsv(w) and v'(w) possessingthe prop-ties(i), (ii),we have

v'(w) = o)ov(iy) + coi,

with two suitablebut fixed w , wi and w >0.Proof:Let u*, v* be two different utilities,1 u* ^ v*.If u* > v*, then interchangeu* and v*. Thus at any rate u* < v*.

Use theseu*, v* for the constructionof h(w), i.e.for (A:L)-(A:U).We now

prove :Ad (A:V)-:The mapping

w > h(w)

fulfills (i)by (A:R),(iii),and (ii) by (A:U).Ad (A:W):Considerv(w) first. By (i) v(u*) < v(v*). Put

v(u;) v(u*)))hi(w) =))

v(*) ~ v(u*)))1Strictly speaking, the axioms permit that there should beno two different utilities.

This possibility is hardly interesting, but it is easily disposedof. If there are no twodifferent utilities, then their number is zeroor one. In the first caseour assertionsarevacuously fulfilled. Assume therefore the secondcase:Thereexists oneand only oneutility u> . A function is just a constant v(u> ) - o- Any such function fulfills (i), (ii)in (A:V). In (A:W), with v(w) - o, v'(u>) - aj, chooseo> - 1 and wi - ai - .)))

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628 THEAXIOMATICTREATMENT OF UTILITY

Then hi(w) fulfills (i),(ii)in (A:R)automatically, and (iii),(iv) in (A:R)by(i), (ii) above. Henceby (A:S)hi(w) = h(w), i.e.(A:13) v(w) = <x*h(w) + lf

where ao, i are fixed numbers: = v(v*) v(w*) >0, i = v(u*).Similarlyfor v'(w):))

where a, a[ are fixed numbers: J = v'(t>*) v'(w*) >0, <x{= v(i/*).

Now (A:13)and (A:14)give together))

(A:15) v'(w)

where wo, <*>i are fixed numbers:wo = ~ > 0,wi = -^-l - This isao <*o

the desiredresult.

A.3.ConcludingRemarksA.3.1.(A:V) and (A:W) are clearly the existenceand uniqueness

theoremscalledfor in 3.5.1.Consequentlythe assertionsof 3.5.-S.6.areestablishedin theirentirety.

At this point the readeris advisedto rereadthe analysisof the conceptof utility and of its numerical interpretation,asgiven in 3.3.and 3.8.Therearetwo points, both of which have beenconsideredor at leastreferredtoloc.cit.,but which seemworth reemphasizingnow.

A.3.2.The first one deals with the relationshipbetweenour procedureand the conceptof complementarity. Simply additive formulae, like(3:l:b),would seemto indicate that we are assumingabsenceof any formof complementaritybetweenthe thingsthe utilitiesof which we are combin-ing. It is important to realize,that we aredoingthis solelyin a situationwheretherecan indeedbe no complementarity. As pointedout in the first

part of 3.3.2.,our u, v are the utilitiesnot of definite and possiblycoexist-ent goods or services,but of imagined events. The w, t; of (3:l:b)inparticularrefer toalternatively conceivedeventsw, y, of which only one canand will becomereal. I.e.(3:1:b) deals with either having u (with theprobabilitya) or v (with the remaining probability1 a) but since thetwo arein no caseconceivedas taking placetogether,they can never com-plementeachother in the ordinary sense.

It shouldbe noted that the theory of gamesdoesoffer an adequatewayof dealing with complementaritywhen this conceptis legitimatelyappli-cable:In calculating the value v(S)of a coalition S (in an n-persongame),as describedin 25.,all possibleforms of complementaritybetweengoodsorbetweenservices,which may intervene, must be taken into account. Fur-thermore,the formula (25:3:c)expressesthat the coalition S u T may beworth more than the sum of the values of its two constituent coalitionsST, and henceit expressesthe possiblecomplementaritybetweenthe services)))

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CONCLUDINGREMARKS 629

of the membersof the coalition S and thoseof the membersof the coalitionT. (Cf.also27.4.3.)

A.3.3.The secondremarkdealswith the question,whether our approachforcesone to value a lossexactly as much as a (monetarily) equal gain,whether it permitsto attach a utility or a disutilityto gambling (even whenthe expectationvalues balance),etc. We have alreadytouchedupon thesequestionsin the last part of 3.7.1.(cf. also the footnotes2 and 3 eod.).However,someadditionaland more specificremarksmay be useful.

Considerthe following example:DanielBernoulli proposed(cf.footnote2 on p.28),that the utility of a monetary gain dx shouldnot only be pro-portional to the gain dx, but also (assumingthe gain to be infinitesimalthat is, asymptoticallyfor very small gainsdx) inversely proportionalto theamount x of the owner's total possessions,expressedin money. Hence

(usinga suitable unit of numerical utility), the utility of this gain isx

The excessutility of owning x\\, over owning x2, is then / = Inxtj X #2

The excessutility of gaining the (finite) amount 77 over losingthe same

amount is In - In _ = In ( 1 - 2 \\ This is <0,i.e.of equalX XT] \\ X /gains and lossesthe latter are more strongly felt than the former. A

50%-50%gamblewith equal risks,is definitely disadvantageous.NeverthelessBernoulli'sutility satisfiesour axiomsand obeysour results:

However,the utility of possessingx units of money is proportionalto In x,and not to x!u

Thus a suitabledefinition of utility (which in such a situation is essen-tially uniquely determinedby our axioms)eliminatesin this casethe specificutility or disutility of gambling, which prima facie appearedto exist.

We have stressedBernoulli'sutility, not becausewe think that it is par-ticularly significant, or much nearerto reality than many other more or lesssimilar constructions.The purposewas solelyto demonstrate,that the useof numerical utilitiesdoesnot necessarilyinvolve assumingthat 50%-50%gambleswith equal monetary risksmust be treatedas indifferent, and thelike.8

It constitutesa much deeperproblemto formulate a system, in which

gambling has under all conditionsa definite utility or disutility, wherenumerical utilitiesfulfilling the calculusof mathematical expectationscannotbe defined by any process,director indirect. In sucha systemsomeof our

1 The50%-50%gamble discussedaboveinvolved equal risks in terms of x, but not interms of In x.

1That the utility of x units of money may bemeasurable, but not proportional to x,was pointed out in footnote 3 on p. 18.

8 As stated in remark (1)in 3.7.3.,we are disregarding transfers of utilities betweenseveral persons. The stricter standpoint used elsewherein this book, as outlined in

2.1.1.,specifically, the free transferability of utilities between persons,doesforceone toassume proportionality between utility and monetary measures. However, this is not

relevant at the present stageof the discussion.)))

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THEAXIOMATICTREATMENT OF UTILITY

axiomsmust be necessarilyinvalid. It is difficult to foreseeat this time,which axiom or groupofaxiomsismostlikely to undergosucha modification.

A.S.4.Thereareneverthelesssome observationswhich suggestthem-selvesin this respect.

First:The axiom (3:A) or, more specifically, (3:A:a) expressesthecompletenessof the orderingof all utilities, i.e.the completenessof theindividual'ssystemof preferences.It is very dubious,whether the idealiza-tion of reality which treatsthis postulateas a valid one,is appropriateoreven convenient.I.e.one might want to allow for two utilities u, v therelationshipof incomparability, denotedby u

\\\\ v, which meansthat neitheru = v nor u > v nor u < v. Itshouldbe noted that the current method ofindifferencecurves doesnot properlycorrespondto this possibility. Indeed,in that casethe conjunction of \" neither u > v nor u < v,\" correspondingto the disjunctionof \" either u = VOTU\\\\V,\" and to bedenotedby u v, canbe treatedas a merebroadeningof the conceptof equality (of utilities,cf.also the remarkconcerningidentity in A.I.2.).

Thus if u ||u',v ||v', then u', v' canreplaceu, v in any relationship,e.g.in this caseu < v impliesu' < v'. Hencein particular u

\\\\u' and v = v'

have this consequence,and u = u' and v\\\\

v' have this consequence.I.e.,writing v, w, u for w, t>, u' and u, v, w for u, v, v', respectively:(A:16) u ||v and v < w imply u < w.(A:17) u < v and v

\\\\

w imply u < w.

However, for the really interesting casesof partially ordered systemsneither (A:16)nor (A:17)is true. (Cf.e.g.the secondexampleat the endof 65.3.2.,which is also dealt with in footnote 2 on page590, where theconnectionwith the conceptof utility is pointedout. This is the orderingof a planeso that u > v meansthat u has a greaterordinate than v as wellas a greaterabscissathan v.)

Second:In the group (3:B)the axioms(3:B:a)and (3:B:b)expressaproperty of monotony which it would be hard to abandon. The axioms(3:B:c)and (3:B:d),on the other hand expresswhat is known in geometricalaxiomaticsas the Archimedeanproperty:No matter how much the utilityv exceeds(or is exceededby) the utility w, and no matter how little theutility w exceeds(or is exceededby) the utility u, if v is admixed to uwith a sufficiently small numerical probability, the difference that thisadmixturemakesfrom u will be lessthan the difference of w from u. It isprobably desirableto requirethis property under all conditions,sinceitsabandonmentwouldbetantamount to introducing infinite utility differences.!

1For a statement of the Archimedean property in an axiomat ixation of geometry,where it originated, cf.e.g.D.Hilbert, loc.cit. footnote 1 on page74. Cf.there AxiomV.I. The Archimedean property has sincebeen widely used in axiomatizations ofnumber systems and of algebras.

Thereis a slight difference between our treatment of the Archimedean property andits treatment in most of the literature weare referring to. We aremaking freeuseof theconceptof the realnumber, while this is usually avoided in the literature in question.Thereforethe conventional approachis to \"majorise\" the \"larger\" quantity by successive)))

Page 655: Theory of Games Economic Behavior

CONCLUDINGREMARKS 681In this connectionit is alsoworth while to make the following observa-

tion:Let any completelyorderedsystemof utilities <u be given, which doesnot allow the combination of events with probabilities,and where theutilities are not numerically interpreted. (E.g.a system based on thefamiliar ordering by indifference curves. Completenessof this orderingobtains,asindicatedin the first remarkabove, by extendingthe conceptofequality i.e.by treating the conceptu v, that we introducedthere,asequality. In this caseu * v means,of course,that u and v lie on the sameindifference curve.) Now introduce events affected with probabilities.Thismeansthat one introducescombinationsof, say,n (= 1,2, ) events

A

with respectiveprobabilitieson, , an (i, , a 0,% a,= 1).-iThis requires the introduction of the corresponding(symbolic)utilitycombinationsot\\ui + + anun (HI, , u n in cu). It is possibletoorder thesectiUi + + anun (any n = 1,2, and any a\\ 9 , aw

and ui, , uw subjectto the above conditions)completely,and withoutmaking them numerical if the orderingis allowed to be non-Archimedean.Indeed,comparing, say, mui + + anun and 0it>i + + ft mvm

we may assumethat n = m and that the ui, , un and the v\\,- , vm

coincide(write aiUi + + <*nW + Ot>i + + Ovm and Ot/i ++ Oun + P\\VI + + /Mm for a^i+ + On un and 0it>i + +$mvm , and then replacen + m; MI, , un , Vi,

- - - , vm \\ ai, , an , 0,, 0;0, , 0,ft, , ft. by n; u,, , u\\ <x h , an ; 0i,, n). Then we comparea^i+ + awwn and fti^ -f +

0i/. Next make, by an appropriate permutation of 1, , n y u\\ >- - > un. After thesepreparationsdefinea\\u\\ + + <xnun > ftiUi +

+ ffnUn as meaning that for the smallesti(= 1, , n) for whicha, 7* fa, say i = to, there is ai > ft

i%.

It is clearthat these utilities are non-numerical.Their non-Archi-medean characterbecomesclearif one visualizes that here an arbitrarysmall excessprobabilityat-t & affecting u

i%will outweigh any potential

oppositeexcessprobabilities of the remainingu t, t* = t'o + 1, * , n,i.e.of utilities < u, f . (This then excludesthe applicationof criterialikethat one in footnote 1 on page 18.)Obviously, they violate our axioms(3:B:c)and (3:B:d).

Sucha non-Archimedean orderingis clearly in conflict with our normalideas concerningthe nature of utility and of preference.If, on the other

addition!of the \"smaller\" one(cf.e.g.Hilbert'a procedureloc.cit.),while we \"minorise\"the \"smaller\" entity (the utility discrepancybetween w and v in our case)by a suitablesmall numerical multiple (the a-fold in our case)of the \"larger\" entity (the utility

discrepancybetween9 and u in our case).Thisdifference in treatment is purely technical and doesnot affect the conceptual

situation. The readerwill alsonote that we are talking of entities like \"the excessofv over u\" or the \"excessof u overv\" or (to combine the two former) the \"discrepancy ofu and v\" (u, , being utilities) merely to facilitate the verbal discussion they are notpart of our rigorous, axiomatic system.)))

Page 656: Theory of Games Economic Behavior

632 THEAXIOMATICTREATMENT OF UTILITY

hand, one desiresto define utilities (and their ordering)for the probability-including system,satisfying our axioms(3:A)-(3:C)and hencepossessingthe Archimedean property then the utilitieswould have to be numerical,sinceour deductionof A.2.applies.

Third:It seemsprobable, that the really criticalgroup of axiomsis(3:C)or, more specifically,the axiom (3:C:b).This axiom expressesthe combination rule for multiple chancealternatives, and it is plausible,that a specificutility or disutility of gambling can only existif this simplecombinationrule is abandoned.

Somechangeof the system (3:A)-(3:C),at any rate involving theabandonmentor at leasta radicalmodification of (3:C:b),may perhapsleadto a mathematically completeand satisfactorycalculusof utilities, whichallowsfor the possibilityof a specific utility or disutility of gambling. Itis hoped that a way will be found to achieve this, but the mathematicaldifficulties seemto beconsiderable.Of course,this makes the fulfillment

of the hope of a successfulapproachby purely verbal means appear evenmore remote.

It will be clearfrom the above remarks,that the current method of usingindifference curves offers no help in the attempt to overcome thesediffi-culties. It merely broadens the conceptof equality (c.f.the first remarkabove),but it gives no useful indications and a fortiori no specificinstruc-tions as to how one should treat situations that involve probabilities,which areinevitably associatedwith expectedutilities.)))

Page 657: Theory of Games Economic Behavior

INDEX OF FIGURES

Figure Page Figure Page Figure Page1 62 35 184 70 4052 63 36 192 71 4083 63 37 192 72 4084 64 38 192 73 4095 64 39 197 74 4096 65 40 203 75 4097 65 41 203 76 4108 65 42 212 77 4109 65 43 212 78 410

10 78 44 212 79 41111 93 45 216 80 41112 94 46 216 81 41213 94 47 217 82 41214 94 48 217 83 41215 99 49 230 84 41416 131 50 252 85 41417 132 51 261 86 41518 133 52 283 87 41619 133 53 283 88 41620 133 54 284 89 47021 133 55 284 90 47022 135 56 286 91 47823 135 57 286 92 55424 136 58 286 93 55425 137 59 286 94 55426 137 60 287 95 55427 169 61 293 93 56828a 175 62 295 97 56828b 176 63 305 Q8 56829 177 64 313 Qg 56930 179 65 331 JJ JJJ

579))31 180 66 33732 181 67 33733 181 68 337 102 57934 182 69 395 103 579))

633)))

Page 658: Theory of Games Economic Behavior

INDEX OF NAMES))

Archimedes,63OBernoulli,D.,28,83, 629Birkhoff, G.,62,63, 64, 66, 340,589B6hm-Bawerk,E.von, 9, 562,564,581,582Bohr,N.,148Bonessen,T.,128Borel,E.,154,186,219Brouwer,L.E.J.,154Burnside,W., 256Caratheodory,C.,343,384,619Chevalley,C.,viD'Abro, A., 148Dedekind,129,619Dirac,P.A. M.,148Doyle,C.,176,178Euclid,23Fenchel,W., 128Frankel,A., 61,595Hausdorff, F.,61,269,595Heisenberg,W., 148Hilbert,D.,74, 76, 63OHurwicz,L.,viKakutani, S.,154Kaplanski,I.,viKepler,4Kdnig,D.,6O))

Kronecker,129Lipschitz,493Loomis,L.H.,viMacLane,S.,340Marschak,J.,viMathewson,L. C.,256Menger,C.,564Menger,K.,28, 176Mohs,22Morgenstern,O.,176,178Morse,M., 95Neumann, J.von, 1,154,186Newton, 4, 5, 6, 33Pareto,V., 18,23, 29Speiser,A., 256Tarski, A., 62Tintner, G.,28Tycho de Brahe,4Vebien, O.,76Ville, J,,154,186,198Wald, A., v, viWeierstrass,129Weyl, H.,76, 128,256Young, J. W., 76Zermelo,E.,269,595))

634)))

Page 659: Theory of Games Economic Behavior

INDEX OF SUBJECTS))

Acyclicity, 589,591,594,595,596,598,600,601,602,603,609;strict, 594,595,597,598,600,601,602,603

Additivity of value, 251,628Adversary

\" found out,\" 105Agreements, 221,224;sanctity of, 223Ally, 221Alternatives, 55;number of, 69Anteriority, 51,52,77, 78, 112,117,124Apportionment, 35,41,504Archimedean property, 630,631Assignment, actual, 75; pattern of, 75Asymmetric, 270,448Austrian School,9Axiomatization, 68,74, 76Axioms, 25, 26, 28, 73; independenceof,

76;logistic discussion of the, 76Backgammon, 52,58,79, 124,125,164Bargaining, 338,501,512,557, 558,572,

616Barter exchange,7Behavior, 34;expected,146;standards of,

seeStandards of behaviorBestway of playing, 100Bid, 557Bidding, alternate, 211Bilateral monopoly, 1,6, 35,504,508,543,

556Bilinear form, 154,156,157,166,233Bluffing, 54, 164,168,186,188,204,205,

206,208,218,541;fine structure of, 209Booleanalgebra, 62Bound, 59,60Bounded, 384Bounds, lower, 100;upper, 100Bridge, 49, 52, 53, 58, 59, 79, 86, 224;

Duplicate, 113;Tournament, 113Buyer, 14, 556,557, 565,569,572, 574,

581,583,585,609,610,613

Calculus,3, 5, 6Calculusof variations, 11,95Calling off, 179,180,541Cartels,15,47Categoricity, 76Centerof gravity, 21,131,303Chance,39,52,87Characteristicfunction, 238 ff., 240,245,

509,510,511,527, 529,530,533,535,557, 574, 584,605,610;extended, 528,529,532,533;game with a given, 243,))

530, 532; interpretation, 538; in thenew theory, 348;normalized form of the,325;reduced,248,325,543,544,545;restricted, 528, 529, 531, 532, 533;strategically equivalent, 536; vectoroperations on, 253;zeroreduced,545

Characteristic set function, 241Chess,49,52,58,59,113,124,125,164

good,125Chief player, seePlayer, chiefChoice,49,51,59,69,222,508;actual, 75;

anterior, 72;pattern of,75;umpire's, 80,81,82,183Choice,axiom of, 269Circularity, 40,42,56Closedset,384Coalition, 15,35,47, 221,222,224,225,

229,234,237, 240,260,276, 289,418,420,507, 509,510,531,533,539,566,572, 573, 605;absolute, 231,238;cer-tainly defeated,440;certainly winning,440;decisive,420;defeated,296;final,315,317;first, 306,307, 315,316,320;interplay of, 291;losing, 420,421,423;minimal winning, 429, 430, 436, 438,445; profitable minimal winning, 442;unprofitable, 437; weighted majority,434;winning, 296,297, 333,420,421,423,436,445,470

Coalitions, competition for, 329;of differ-ent strengths, 227

Closedsystems, 400Column of a matrix, 93,141Combinatorics, 45Commodity, 10,13,560,565Communications, 86;imperfect, 86Commutativity, 91ff.,94. SeealsoSaddle

pointsCompatible, 267 ff.

Compensations, 36,44,47, 225,227, 233,234,235,237, 240,507, 508,510,511,513,533,541,558

Competition, 1,13,15,249,509Complement, 62Complementarity, 18,27, 251,437,628Complementation, 422Completeordering, seeOrdering, completeCompletely defeated,296Composition, 340,359,360,454,548;of

simple games, 455))

635)))

Page 660: Theory of Games Economic Behavior

636)) INDEX OF SUBJECTS))

Conjunction, 66Constituent, 340, 353, 359, 360, 518;

indecomposable,457, 471;inessential,-453,457;simple, 453,455,457

Contribution, 364,366Conventions, 224Convex bodies,128Convexity, 128ff., 275,547Cooperation,221,402,474, 481,508,517;

complete,483Couple,222,226,243,509Crusoe,9, 15,31,87, 555CubeQ,293,295;centerand its environs,313;centerof,316,317,321; corner,303,

304,307,340,429;interior of, 302,303,304;main diagonal of, 302,304,305,312;neighborhood of the center, 321;specialpoints in, 295ff .; three-dimensional partof, 314

Curves,undetermined, 418Cutting the deck,185,186\"Cyclical\" dominations, 39

Decomposability, 342,357, 360;analysisof, 343

Decomposition,242, 292,340,359,360,452,537, 548;elementary propertiesof,381; its relation to the solutions, 384

Decomposition Partition, see Partition,decomposition

Defeated,296;certainly, 440;fully, 436.Seealso Players; Coalitions

Defensive,164,205Determinateness,general strict, 150,155,

158ff.;specialstrict, 150,155;strict,106ff., Illff., 165,178,179

Diagonals,separation of the, 173Differential equations, 6, 45Directsignaling, 54Discrimination, 30,288,289,328,475,476,512.SeealsoSolution, discriminatoryDisjunction, 66Distance,20Distribution, 37, 87, 226,261,263,350,

364,437Domain, 89,90,128,157Domination, 38,264,272, 350,367, 371,

415, 474, 520 ff., 522, 523,524, 587;acyclicalconceptof, 602;asymmetrical,270;extension of the conceptof, 587;intransitive nation of, 37

Double-blind Chess,58,72,79Duality, 104Dummy, 299,301,340,358,397,398,400,

454,455,457, 460,461,508,518,537,538

Duopoly, 1,13,543,603Dynamic equilibria, 45Dynamics, 44,45,189,290))

E(e), Solutions for r in, 393ff.

Ecarte*, 59Economic equilibrium, 4; fluctuations, 5;

models, 12,58; statics, 8Economics, mathematical, 154Economies, internal, 341Economy, planned communistic, 555;

Robinson Crusoe,9; SocialExchange,9 ft.

Effectivity, 272,282,350,367, 524Energy, 21Entrepreneur, 8Equidistributed, 197Equilibrium, 4, 34,45,227,365Equivalence strategic, seeStrategic equiv-

alenceEssentiality, 249,272,351,452Exceptional, 593Excess,364,367,417,418,454,455,548;

distribution of the, 418;limitation ofthe, 365,366;lower limit of, 368;toogreat, 374,380,419;toosmall, 374, 380;upper limit of, 369

Exchange economy, 9, 31Exchange, indeterminateness of, 14Excluded player, seePlayer, excludedExpectation, 12,28,83,539;mathematical,

10,28,29,32,33,83,87, 117,118,126,149,156,157;moral, 28,83;values, 183

Exploitation, 30,329,375Extensive form, 112,119F(<? ), Solutions for T in, 384ff.

Fairness, 166,167,225,255,258,259,470Fictitious player, seePlayer, fictitious\"Finding out\" the other player, 148\"Finds out 11 his adversary, 106,110,148First element, 38,271Fixed payments, 246,281,298,534Fixed Point Theorem, 154Flatness, 276,547Found out, 148Frame of reference,129Fully detached,seeImputation, detached

fullyFunction, 88, 128;arithmetical, 89;char-

acteristic,238 ff.; continuity of, 493;measure, 252;numerical, 89; numericalset, 240,243,530,532; of functions,157;set, 89

Functional, 157Functional Calculus, 88,154Functional operations,88,91Fundamental triangle, 284,405 ff., 552,

553, 569, 570, 587; area, 579, 580;curves in, 412,570, 580;undominatedarea,409

Gain, 33,128,145,539,556,559,629Gambling, 27,28,87, 630,631)))

Page 661: Theory of Games Economic Behavior

INDEXOF SUBJECTS)) 637))

Game and socialorganizations, 43;asym-metric, 334;auxiliary, 101ff.; axiomaticdefinition of a, 73;chancecomponent ofthe, 80; classification of, 46; completeconceptof, 55;completesystem of rulesof, 83; composition, 339 ff.; constant-sum, 346ff., 347,350,351,504,505,535,536,537, 585;decomposable,454,471,518;decomposable,solution of, 358,381;decomposition, 339ff.; direct majority,431,433;elements of the, 49;essential,231, 232, 245, 250, 331, 534, 546;essential three-person, 220 ff., 260 ff.,471, 473; everyday concept of, 32;extensive form of the, 85,105,186,234;extreme, 534,535;fictitious, 240;final

simplification of the description of a,79,81;general,504ff., 505,538;generaldescription of, 57; general formaldescription of, 46-84;general n-person,48,85,112,530,606;general n-pereon,application of theory, 542ff .; imbeddingof a, 398; indecomposable,354; ines-sential, 231, 232, 245, 249,251, 471;\"inflation\" of a, 398; invariant, 257;kernel of, 457, 459; length of the,75; main simple solution of the, 444;majorant, 100, 102, 103, 119,149;majority and the main solutions, 431;minimum length of, 123;minorant, 100,101,119,149; non-isolated characterof a, 366;non-strictly determined, 110;non-zero-sum, 47. SeealsoGame,gen-eral;normal zoneof the, 519;normalizedform, 85, 100,105,119,183,234,239,322,325,452,473;of chance,87, 185;one-person,85, 548; partitions whichdescribe a, 67; perfect information,112ff.; plan of the, 98; plays of, 49;reduced,248,259,473, 543ff.; rules ofthe, 32,49, 59, 80, 113,147,224,226,227, 334, 426, 472; set-theoreticaldescription of a, 60, 67; simple, seeSimple game; simplified conceptof a,48;strategies in the extensive form, 111;strictly determined, 98ff., 106,124,150,165,172,174,516;strictly determined,generally, 150;strictly determined,specially, 150;struggle in, 125;super-position of, 254,255;surprise in, 125;symmetric, 165,167,192,195,334,362;symmetric five-person, 332,334;sym-metry, total of, 259; three-person,35,220ff., 282ff., 403ff., 457, 550;three-person, simple majority of, 222 ff.;totally symmetric, 257;totally unsym-metric, 257;unique, 331;vacuous, 116,546;value of the, 102,103,170,516;weighted majority, homogeneous, 444;zero-sum extension of, 529; zero-sum))

four-person, 291ff.; zero-sum n-person,48, 85, 238 ff.; zero-sum three-person,220ff., 260ff.; zero-sum three-person,solution of essential, 282 ff.; zero-sumtwo-person, 48,85ff., 116,169ff., 176

Geometry, 20, 74, 76; linear, 428;plane,7-point, 469;projective, 469

Goodway (strategy), 103Goodway to play, 108,159Goods,complementary, 437,628;divisible,

560,573Group, 22, 76, 255 ff.; alternating, 258;

invariance, 257;of permutations, 256;set-transitive, 258; symmetric, 256;theory, 256,258,295;totally symmetric,257;totally unsymmetric, 257

Group (of players), privileged, 320

Half-space,130,137,139Hand, 53,186,187,190,197,613Hands, discrete,208Heat,3, 17,21Hereditary, 454Heredity, 396,400Heuristic, 7, 25, 33, 120,238,263,291,

296,298,301,302,307, 316,318,322,333,499,509,511,587

Heuristic argument, 147,181,182,227Higgling, 557,558Homo occonomicus, 228Homogeneity, 433,464Hyperplane, 130, 137, 139; supporting,

134ff.

Identity, 255,617Imbedding, 398,399,400,455,587Imputation, 34,37, 39,240,251,263,264,

350,376, 437, 517,520,527, 566,577,587, 606, 610;composition of, 359;decomposition of, 359;detached,369,370, 375 ff., 413;detachedextended,370,375;detachedextended,fully, 370,372;detached,fully, 369,413;economicconceptof, 435;extended,364ff., 367,368,369,372;single, 34,36,37, 39,40;unique, 608

Imputations, extended,setsof, 368;finite

setof, 465,499;infinite setof, 288,499;isomorphism between, 282;set of, 34,44,608;system of, 36,277, 464

Incomparable, 590,630Indecomposability, 357 [630,631,632]Indifference curves, 9, 16,19,20,27, 29,Indifference of player, 300Indirect proof, 147Individual planning, 15Induction, 112,116;complete, 113,123;

finite, 597; transfinite, 598Inessential games, 44)))

Page 662: Theory of Games Economic Behavior

638)) INDEX OF SUBJECTS))

Inessentiality, 249,272,351,357,454Information, 47, 51,54,55,56,58,67, 71,

109,112,183;absurd, 67;actual, 67,79;chance,182;complete, 30, 541, 582;imperfect, 30, 182;incomplete, 30, 86;pattern of, 67, 69;perfect,51,123,124,164,233;perfect,verbal discussion, 126;player'sactual, 75;player'spattern of,75; sets of, 77; umpire's actual, 75;umpire's pattern of, 75, 77; umpire'sstate of, 115

Initiative, 189,190Inner triangle, 409,413,553Interaction, 341,366,400,483International trade, 7, 341Interval, linear, 131Intransitivity, 39,52Inverted signaling, 54Irrational, 128,523Isomorphic correspondence,281Isomorphism, 149,350,504Isomorphism proof, 281

Just dues,360,361

Kernel, 457Killing the variable, 91\"Kriegsspiel,\" 58

Laisserfaire, 225\"Lausanne\" Theory, 15Linear interpolation, 157Linear transformation, 22,23Linearity, 128ff.

Lipschitz condition, 493Logic,62,66,74, 274Logistic,66Losing,421,426Loss,33,128,145,163,167,168,205,539,

555,559,629

Main condition, 273,274,279Main simple solution, seeSolution, main

simpleMain theorem, 153Majorities, weighted, 432Majority, principle of, 431Majority game, homogeneous, 443;homo-

geneous weighted, 444, 463, 469;weighted, 433,464

Majority principle, homogeneous weighted,46r,469

Majorization of rows or columns, 174,180Mapping, 22,618ff.Marginal pairs, 560,562,563,564,571,581Marginal utility school,7Market, 47, 504,556,557,560,563,581,

605; general, 583; three-person, 564;two-person,555

Mass,20,21))

Matching Pennies,111,143,144,164,166,169,176,178,185;generalized forms,175ff.

Mathematical Method, difficulties of, 2;in Economics,1-8

Mathematical physics, 303Matrix, arbitrary, 153;diagonal of, 173,

174;elements, 93, 138, 141;negativetransposed, 141,142;rectangular, 93,138,140,141;scheme,diagonal of the,173;skew symmetric, 142,143,167

Max operations,89 ff.

Maximum, 88,89,591,593,594;absolute,591, 593; collectivebenefit, 513, 514,541,613;problem, 10,11,13,42,86,87,220,504,517,555;relative, 592;satis-faction, 10

Measurability, 16,343Measure, additive, 343; mathematical

theory of, 252,343Measurement, principles of, 16,20Mechanics,4Method, mathematical, 322Method, saturation, 446Min operations,89 ff.

Min-Max problem, 154Minimal elements of W, 430,448Minimum, 88,89Mistake, 162,164,205Mixing the deck,185Models, axiomatic, 74; economic,12,58;

mathematical, 21ff., 32,43,74Money, 8, 10Monopolist, 474Monopoly, 13,474,543,584,586,602,603Monopsony, 584,586,602,603Monotone transformations, 23Motivation, 43Move, chance,50,69,75, 80,83,112,118,

122,124,126,183,185,190,517,604;removing of, 183;dummy, 127;impos-sible, 72; in a game, 49,55,58,59,72,98, 109,111;of the first kind, 50; ofthe secondkind, 50; personal, 50, 55,70, 75, 112,122,126,183,185,190,223,508,510

Negation, 66Negotiations, 263,534,541New Theory,characteristicfunction in the,

348;decomposability in the, 351;domi-nations in the, 350; essential three-person game in the, 403ff.; essentialityin the, 351;imputations in the, 350;un essentiality in the, 351;solutions inthe, 350

Normal zone,396,399,401,417Numerical utilities axiomatic treatment of,

24. Seealso Utility, numerical)))

Page 663: Theory of Games Economic Behavior

INDEX OF SUBJECTS)) 639))

Numerical utility, 17. See aho Utility,numerical

Numerical weight, 432

Offensive, 164,205Oligopoly, 1,13,47, 504Opposition of interest, 11,220,484Optimality, permanent, 162,205. Seealso

StrategyOptimum, 38Optimum Behavior, 34Orderof society,41,43. SeealsoOrgani-

zation, Standard of behaviorOrdering, 19,37,38,589;complete,19,26,

28,589,591,593,595,600,617;partial,590,591,600;well, 595

Oreographical,95,97Organization, 224, 328, 366, 401, 419;

socialand economic,41,43, 225,318,319,329,358,362,365,402,436,471;social,complexity of, 466

Origin (point in space),129Outsidesource,363,364,366,375,419Overbid, 186.Seealso Poker

Parallelism of interests, 11,220,221Partial ordering, 590. SeealsoOrderingParticipants, 31,33Partition, 60,63,64,66,67,69,84,114ff.;

decomposition, 353,354,356,357, 457,471;logistic interpretation, 66

Pass,95,97Passing,seePokerPatience,86Per absurdum proof, 147,148Permutation, 255, 262, 294, 309, 319;

cyclic,230,470Perturbations, 303,341Physical sciences,21,23,32,43Physics,2, 3, 4, 45,76, 148,401Planning, 86Plateau,rectangular, 97Plausibility considerations,7Play, 49,59,71;actual, 82;courseof the,

68,70,84;individual identity of the, 71;outcomeof the, 82;sequenceof choices,49; value of a, 104,105,124,127,150,163,165,178,238

Player, chief, 473, 474 ff., 481,483,500,502; chief, segregated,500,502;com-posite,231,232,239,516;defeated,418;discriminated, 476,502;excluded,289,301,512;fictitious, 505,506,507, 508,509,511,513,514,516,518,537;found

out, 146;indifferent, 299;isolated,375;privileged, 473; segregated,476, 502,503;self-contained,353, 357; splittingthe, 86; unprivileged, 320; victorious,426))

Players, interchanging the, 104,109,122,165,255;permutation of the, 294,463;privileged group of, 320,464;removablesets of, 533;strategies of the, 49. SeealsoStrategy.

Playing appropriately, 102,103,107,159,167

Plays, set of all, 75Poker, 52, 54, 56, 58, 59, 164,168,186,

187 ff., 557,613;bids, 209;Draw, 187;general forms of, 207; good strategy,196;interpretation of the solutions, 218;mathematical description of all solu-tions, 216 ff.; overbidding, 188, 190;passing, 188,190,191,199;seeing,188,190,191,199,218;solution, 199,202,204;strategies, 191;Stud, 187

Position, 21Positive octant, 133Positive quadrant, 133Postulates,27. SeealsoAxiomsPreferences,15,17, 18,23,522,590,607,

630; completeness of, 29, 630, 631;transitivity of, 27. SeealsoUtility

Preliminarity, 51,52,77, 78, 112,117,124Preliminary condition, 273,471Premiums, 582Price,556,559,562,563,564,571,572,

582,585;average,564,582;unique, 564Privilege, 464Probabilities, choiceof, 145Probability, 11,17,19,39,81,87,128,146,

197;geometrical, 197;numerical, 14,19,27,28,69,75,80,113,145,147,156,183,604;of losing, 144;of winning, 144

Production, 5, 13,504Productivity, 33,34,504,540Profit, 33,47, 572Propersubset, 61Propersuperset, 61Psychological phenomena, mathematical

treatment, 28,77, 169

Quantum mechanics, 3, 33,148,401

Randomness, 146Rational, 9, 517;behavior, 8-15,31,33,

127,150,160,224,225;playing, 54Rationality, 99,128Rebates,582Recontracting, 557,558Reducedform, 248,544Reduction, 322,325Relativity Theory, 23,148Removable set, seeSet,removableRing, 243,530,531Risk, 163Robinson Crusoe,seeCrusoe)))

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640)) INDEX OF SUBJECTS))

Rolling dice,166Roulette, 87Row ofa matrix, 93,141.SeealsoMatrix

Saddle,95, 97; points, 93, 95, 110,153;value, 88, 95, 107

St.Petersburg Problem, 28,83Satisfaction, maximum, 8, 10,15Satisfactoriness,267 ff., 446ff.; maximal,

269Saturation, 266,267 ff., 446ff., 448,591Scalarmultiplication, 129,253,254Seeing,seePokerSegregation, 290Seller,556,557, 565,569,572, 574, 581,

583,585,609,610Separatednumbers, 173Set, 60, 61,114ff.; certainly necessary,

273, 274, 277, 308,309,323,405,430,471, 547; certainly unnecessary, 273,274, 276, 308,309,323,405,430,471,547; completely ordered, 19;convex,131,133;convex spanned, 131;effective,38,264;elements of, 61;empty, 61,241,380;finite, 61;flat, 275, 276, 423,424;minuend, 62;one-element,61;partiallyordered,19;removable, 533,534,535;solo,seeSoloset;splitting, 353,457,518;splitting minimal, 355, 356; splitting,system, of all, 354; subtrahend, 62;theory, 45,60 ff.

Sets,difference of, 62;disjunct, 62;inter-sectionof, 62;logistic interpretation, 66;of imputations, composition of, 359;ofimputations, decomposition of, 359;pairwise disjunct, 62;product of, 62,66;self-contained,354;sum of, 62,66;sys-tems or aggregatesof, 61

Signaling, 51,53;direct, 54; inverted, 54Simple game, 420 ff., 454, 605; adding

dummies to, 462; and decomposition,452;characteristic function, 427;char-acterization of the, 423;complementa-tion in, 422 ff.; enumeration of all,445ff.; for n ^ 4, 461;for n ^ 6, 463;for small n, 457;indecomposable,457;one-elementsetsin the, 425ff .; six maincounter-examples, 464;solution of, 430;strategic equivalence,428;systems W,L, of, 426ff.; with dummies, 461

Simplicity, 433, 452, 454; elementaryproperties of, 428;exact definition of,428

Skew-symmetry, 166Socialexchangeeconomy, 9 ff.

Socialorder, 365Social organizations, see Organizations,

social))

Socialproduct, 46Socialstructure, 484Society,42,320,341,523,540Solitaire,86Soloset,244,530,531Solution, 102,264,350,367,368,417,478,

526,527, 587, 588;areas (two-dimen-sional parts) in the, 418;asymmetric,315,362;composition of, 361;conceptof a, 36;conceptof imputation in form-ing, 435;curves (one-dimensional parts)in the, 417;decomposable,362;decom-position, 361;definition, 39, 264; dis-criminatory, 301, 307, 318, 320, 329,442,511,512;essential zero-sum three-person game, 282; existenceof, 42;extension of the concept of, 587;families of, 329,603; finite, 307, 500;finite sets of imputations, 328; for an

acyclicrelation, 597; for a completeordering, 591;for r in E(eQ), 393 ff.;for T in F(eo), 384 ff.; for a partialordering, 592;for a symmetric relation,591;general games with n 3, of all,548; general three-person game, 551;indecomposable,362; inobjective (dis-criminatory), 290; main simple, 444,464,467, 469;multiplicity of, 266,288;natural, 465; new definition of, 526;non-discriminatory, 290,475,511;objec-tive (non-discriminatory), 290; one-element, 277, 280;set of all, 44;super-numerary, 288;symmetric, 315;unique,594, 600, 601, 603; unsyrametricalcentral, 319

Soundness, 265Space,Euclidean, 21,128,129;half, 137;

linear, 157;n-dimensional linear, 128;positive vector, 254

Specialform of dependence,56Splitting the personality, 53Stability, 36,261,263,266,365;inner, 42,

43,265Standard of Behavior, 31,40,41,42,44,

265,266,271,289,361,365,401,418,472, 478, 501,512,513;discriminatory,

290;multiplicity (of stable,or accepted),42,44,417;non-discriminatory, 290

Statics,44,45,147,189,290Statistics, 10,12,14,144Stone, Paper,Scissors,111,143,144,164,

185Stop rule, 59,60Strategic equivalence, 245,247, 248,272,

281,346,348,373, 426,429,472, 505,535,543;isomorphism of, 504,505

Strategies,44,50,79,80,84,101,117,119;combination of, 159)))

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INDEXOF SUBJECTS)) 641))

Strategy, as move, 84; asymptotic, 210;best, 124,517;choiceof, 82, 145,147;conceptof a, 79; found out, 151,153,158,160,168;good, 108,146,160ff.,161,162,164,170,172,178, 179,183,196,205, 206; higher order of a, 84;mixed, 143ff., 146,148,149,155,157,161,168,174,183,192,232,539,604;optimal, 127,517;permanently optimal,163,164,165;pure, 146,148,155,157,161,168,181,182;statistical, 144,146;strict, 146;structure, continuous, 197;structure, fine, 197.;structure, granular,197

Strict Determinateness, seeDeterminate-ness

Struggle, 249Subpartition, 63,64,69Subset,61Substitution rate, 465Superiority, intransitive notion of, 37Superposition, 64Superset,61Symmetry, 104,109,165,166,190,224,

255,256,258,267,315,446ff., 500,591.Seealso Group

Tautological, 8, 40Temperature, 17,21Theory,extended,structure of the, 368Theory, new, 526, 528. See also New

theoryThermodynamics, 23Thermometry, 22Tie,125,315Topology,154;384Totalvalue, 251Transfer, 30,364,365,401,402Transferability of utility, 8, 608Transfinite induction, 269Transformation, 22,23Transitivity, 38,39,51,589,590Trees,66,67Tribute, 30,402Tug-of-war, 100

Umpire, 69,72, 84Uncertainty, 35))

Understandings, 223,224,237Utilities, comparability of, 29; complete

ordering of the, 19,26,29,604,617ff. ;differences of, 18,631;domain of, 23,,607; nonadditive, 250; non-numerical,16, 606, 607; numerical, 17 ff., 157,605,606,617ff.; numerical, substitut-ability, 604;partially ordered,19,590;system of, 26; transferability, 8, 604,606,608,629;variable, 560

Utility, 8, 15,23,33,47, 83,156,556,563,565,569,572, 573, 583,585,603,608,616,617ff.; axiomatic treatment, 26ff.,617 ff.; decreasing,561,576;discrete,613;expected,30;generalization of the

conceptof, 603ff .; indivisible units, 609,613,614;marginal, 29, 30, 31;scale,fineness of, 616;total, 34,35

Value, economic,252,467, 556,565;of afunction, 88;of a play, seePlay, value of

Variables, 12, 13,88; aggregate of, 239;\"

alien,\" 11;partial setsof, 12ff.

Vector, 129,140;addition, 130,253,254;components, 129,404;coordinate, 129,157;distance, 134; length, 134;oper-ations, 129; quasi-components, 404;spaces,254;zero,129

Victorious, 296. Seealso Player; Coali-tion; Winning; Losing

Virtual existence,36,45,338,484

Wants, 10Wave mechanics, 148Weights, 433,434,463;homogeneous, 435,

444Who finds out whom, 110Winning, 296, 421, 426; certainly, 440;

fully, 436Withdrawal, 364,366

Zero-reducedform of characteristic func-tion, 545

Zero-sum condition, 345Zero-sum extension of r, 505,506,527,

531,538Zero-sum restriction, 84,504)))

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