Ž .GAMES AND ECONOMIC BEHAVIOR 18, 219]245 1997ARTICLE NO. GA970525

Order Independence for Iterated Weak DominanceU

Leslie M. Marx

W. E. Simon Graduate School of Business Administration, Uni ersity of Rochester,Rochester, New York 14627

and

Jeroen M. Swinkels

J. L. Kellogg Graduate School of Management, Northwestern Uni ersity, E¨anston,Illinois 60208

Received April 20, 1993

In general, the result of the elimination of weakly dominated strategies dependson order. We define nice weak dominance. Under nice weak dominance, order doesnot matter. We identify an important class of games under which nice weakdominance and weak dominance are equivalent, and so the order under weakdominance does not matter. For all games, the result of iterative nice weakdominance is an upper bound on the result from any order of weak dominance.The results strengthen the intuitive relationship between backward induction andweak dominance and shed light on some computational problems relating to weakdominance. Journal of Economic Literature Classification Number: C72. Q 1997

Academic Press

I. INTRODUCTION

As is well known, the result of the iterative removal of weakly domi-nated strategies can depend on the order of removal.1 Consider Game G1with x G 1.

* Marx gratefully acknowledges the support of an NSF Graduate Fellowship and a SloanDoctoral Dissertation Fellowship. Swinkels gratefully acknowledges support from the Na-tional Science Foundation. The authors are grateful to two referees, an associate editor, PaulMilgrom, Roger Myerson, and Phil Reny for useful comments and discussions.

1 Let s and r be strategies for player i. Given opponents’ strategy set W , s weaklyi i y i iŽ . Ž . Ž . Ž .dominates r if p s , t G p r , t ; t g W and p s , t ) p r , t for somei i i y i i i y i y i y i i i y i i i y i

t g W , where p is player i’s payoff function.yi y i i

2190899-8256r97 $25.00

Copyright Q 1997 by Academic PressAll rights of reproduction in any form reserved.

MARX AND SWINKELS220

IIL R

T 2, 3 x, 3

I M 1, 0 0, 1

B 0, 1 1, 0

GAME G1

Depending on the order of elimination, the set of strategies that remains� 4after iterative removal of weakly dominated strategies can be T , L ,

� 4 � 4T , R , or T , L, R .In this paper, we define the transference of decisionmaker indifference

Ž .TDI condition. TDI is satisfied by the normal form of any extensive formfor generic assignment of payoffs to terminal nodes and by some importantgames that do not admit generic extensive forms, including discretizedversions of first price auctions. We show that under TDI any two gamesobtained from the original by the iterative elimination of weakly domi-

Ž .nated strategies subject to no more eliminations being possible arestrategically equivalent. That is, the two games differ only by the additionor removal of redundant strategies and a renaming of strategies. In G1,

� 4 � 4TDI is satisfied if and only if x s 2, and if x s 2, then T , L , T , R , and� 4T , L, R are all strategically equivalent ‘‘reductions’’ of G1.

We derive our results in terms of a concept we term nice weak domi-nance. Essentially, a weak dominance is nice if TDI is satisfied for thestrategies involved in that weak dominance, and so in particular, if a gamesatisfies TDI, weak dominance and nice weak dominance are identical.

We show that, regardless of the game, the result of iterative removal bynice weak dominance does not depend on order. So for games satisfyingTDI, the result of weak dominance is also order independent. And ingames not satisfying TDI, we show that the result of weak dominance isalways essentially a subset of the result of nice weak dominance. So, evenfor games where TDI fails, the result allows us to easily identify strategiesthat must be removed regardless of order and to establish an upper boundon the amount by which the results of different orders of elimination candiffer.

We begin by discussing these results for the case of weak dominanceinvolving pure strategies. However, the results translate fairly directly tomixed strategies. We discuss this in Section V.

Backward induction and weak dominance seem intimately related, bothin motivation and in that every sequence of removals of actions bybackward induction is echoed by a sequence of removals of strategies by

ORDER INDEPENDENCE FOR WEAK DOMINANCE 221

weak dominance. An uncomfortable feature of this relationship is thatbackward induction has a deterministic outcome, while weak dominancedoes not. We show that eliminations of actions under backward inductionare in fact eliminations by nice weak dominance. Thus the order indepen-dence of backward induction and the order dependence of iterated weakdominance are related to the fairly simple difference between weakdominance and nice weak dominance, strengthening the intuitive connec-tion between backward induction in the extensive form and weak domi-nance in the normal form.

Finally, we use our result to make a comment on the work of GilboaŽ .et al. 1993 concerning the complexity of iterative weak dominance.

II. THE LITERATURE

A number of previous papers have explored the issue of order indepen-Ž .dence. Gilboa, et al. 1990 give conditions on a dominance operator that

are sufficient for the order of elimination not to matter. These conditionsŽare satisfied by strong dominance, but not by weak dominance or nice

. 2weak dominance .Ž .Rochet 1980 considers the condition

p s s p t « p s s p tŽ . Ž . Ž . Ž .i i j j

for all i , j g N , s, t g Ł S , 1Ž .ig N i

where N is the set of players, S is the set of strategies for player i, and pi iŽ .is player i’s payoff function see Section IV for formal definitions .

Ž . ŽRochet shows that if a game satisfying 1 is dominance solvable when.one eliminates all weakly dominated strategies at every stage , then the

same outcome is obtained regardless of the order of elimination of weaklydominated strategies. Furthermore, Rochet shows that any normal form

2 Gilboa et al. also consider a version of weak dominance which does not require the strictinequality. We shall refer to this as ¨ery weak dominance. They claim, but do not prove, that

Ž .with some additional conditions which are satisfied by both weak and very weak dominance ,arguments similar to theirs can be used to show that order of removal under a relationshipdom will not matter if

w x w xx dom y and y dom x « x and y are payoff equivalent for all players ,

a condition which is similar to our TDI. This claim is false as stated, because for weakŽdominance the antecedent never holds since it can never be the case that x and y each

.weakly dominate the other and so the condition is vacuously satisfied, but order can clearlymatter. The claimed result is correct as stated for very weak dominance, and many of the keyideas in their analysis reappear in the course of our proof. Gilboa et al. note that theircondition is satisfied for zero sum games under very weak dominance.

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game derived from an extensive form of perfect information satisfying theŽ . Žextensive form analogue to 1 i.e., satisfying that one player is indifferent

.over two terminal nodes only if all players are is dominance solvable withthe same outcome as that determined by backward induction on theextensive form. Rochet shows by example that this need not be the casewhen the extensive form does not satisfy this condition.3

Ž .Gretlein 1983 works with games in which each player’s preferencesŽ .over the set of possible outcomes i.e., payoff vectors are strict. In such

games, Gretlein shows that the set of outcomes that results from iteratedŽelimination of weakly dominated strategies subject to no more elimina-

.tions being possible is the same regardless of the order of eliminations.Ž . 4This condition implies 1 .

Ž .Our condition, TDI, is weaker than 1 . In Section III we give examplesŽ .of interesting games satisfying TDI but not 1 . Both Rochet and Gretlein

consider only domination by pure strategies, while we extend our results tomixed strategies. In addition, neither Rochet nor Gretlein has anything to

Ž .say about games for which 1 is not satisfied, but our notion of nice weakdominance allows us to establish a bound on the outcome of iterated weakdominance even for games not satisfying TDI.

III. TRANSFERENCE OF DECISION MAKER INDIFFERENCE

The normal form of a generic extensive form game always satisfiesŽ . Ž . Ž .condition 1 : for p s s p t to hold, it must be that s and t reach thei i

Ž . Ž . Ž .same terminal node. But then p s s p t for all j g N. Moulin 1979j jŽ .identifies a class of voting games satisfying 1 . So, even under condition

Ž .1 , establishing that the result of iterative elimination under weak domi-nance does not depend on order is of considerable use. However, there are

3 Ž .See also Moulin 1986, Chapter 4.2 on Rochet’s robustness result, dominance solvabilityŽ . Ž .using weak dominance , and the relationship between 1 and extensive-form games. MoulinŽ . Ž .1984 gives conditions for a game to be dominance solvable using weak dominance andshows that for a certain class of games, dominance solvability implies Cournot stability.

4 Also, note that Gretlein’s result states only that the set of outcomes is the same, not thatthe games that remain after iterated removal are in any sense equivalent. If one is interested,for example, in Nash equilibria of the game which results from the iterative removal, then itis important to know that different orders cannot yield games which differ as the followingtwo games do:

2 2

1 6, 2 2, 6 1 6, 2 1, 1

1, 1 4, 0 4, 0 2, 6

ORDER INDEPENDENCE FOR WEAK DOMINANCE 223

Ž .important games which do not satisfy 1 to which we would like ourresults to apply.

As an example, consider a first price auction. To make this a finitestrategy game, assume that players receive signals about the value of the

Žobject that are drawn from a finite set V the analysis to follow does not.depend on the manner in which signals are related across players and that

players are restricted to making bids that are integer multiples of a pennyŽ .up to some large maximum. Then, each player’s strategy space is the set

of all maps from signals in V to allowable bids, and so is finite. Let r andis be two strategies for some player i that differ only in that for somei

Ž . Ž .signal v g V, r v - s v ; i.e., let r and s differ only in that, for somei i i isignal, r specifies a smaller bid than s . Consider any strategy profile ti i yifor the other players such that the largest bid possible under t is lessyi

Ž . Ž . Ž .than r v . Then, p r , t s p s , t . To see this, note that when i’si yi i yi yi i yisignal is not v, his behavior is the same under r and s . When i’s signal isi iv, players other than i do not receive the object, and so they receive a

Ž . Ž .payoff of 0 in either case. However, p r , t / p s , t . Thus, thisi i yi i i yiŽ .game does not satisfy 1 .

The condition failed in this example because once a player has lost, he isindifferent about the amount by which he loses. For almost all specifica-tions of how signals map to valuations, this is the only way in which i can

Ž . Ž .be indifferent between pure strategy profiles r , t and s , t }byi yi i yihaving those two strategy profiles differ only in how much i loses by wheni loses. We formalize this in Appendix A. So, while a unilateral change ofpure strategy by player i can change his payoff while leaving his opponentsindifferent, the opposite cannot occur: i cannot be indifferent about thisunilateral change while affecting his opponents’ payoffs. Thus, while the

Ž .game does not satisfy 1 , it does satisfy the weaker condition

p r , s s p t , s « p r , s s p t , sŽ . Ž . Ž . Ž .i i yi i i yi j i yi j i yi

for all i , j g N , r , t g S , s g S . 2Ž .i i i yi yi

Ž .This is weaker than 1 , because the agreement of player i’s payoffsacross two different strategy profiles only implies agreement for the otherplayers if the strategy profiles differ only by the action of player i. So in

Ž . Ž .Game G2, 1 implies b s c s d, while 2 implies b s c. Note in particu-Ž . Ž .lar that while B, L and B, R differ only in the action of one player, it is

Ž .the payoffs of the non-decisionmaker that agree, and so 2 has no power.Ž .We shall refer to 2 as the transference of decisionmaker indifference

Ž .TDI condition: whenever the decisionmaker is indifferent between twoprofiles that differ only in her action, that indifference is transferred to theother players as well.5 We will show for any finite player game satisfying

5 Interestingly, TDI is formally equivalent to the ‘‘nonbossiness’’ condition in social choiceŽ .theory. See for example Satterthwaite and Sonnenschein 1981 .

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T a, b x

I B a, c a, d

GAME G2

TDI that any two games which are achieved by the iterative removal ofŽ .weakly dominated strategies subject to no more removals being possible

are equivalent.Ž . Ž .For what type of game is 2 satisfied, but 1 not? We have already seen

the example of the first price auction. However, the class of games thatŽ . Ž .satisfy 2 but not 1 is much broader. Consider any game in which at

some point some player has the option to quit the game and collect anoutside option. The game then continues with a subgame played betweenthe remaining players. Examples include patent races and oligopoly withan endogeneous number of firms. In such games, the payoff of the exitingplayer is independent of the outcome of the subgame, but the payoffs of

Ž .other players are not. So, 1 is violated. However, because the exitingplayer makes no decisions in the subgame, these payoff ties do not violateŽ .2 . And, if the player making the exit decision is not indifferent over thevarious times at which he might exit, then the game as a whole will satisfyŽ .2 . In the case of the patent race, for example, the exiting playerpresumably incurs costs as long as he is in the race, and so he is notindifferent between different exit times.

As another example, suppose a particular public good will be provided ifat least k players vote that it should be purchased. If purchased, the costof the public good will be shared equally among the people who vote for it.For generic valuations of the public good, if player i is indifferent betweenvoting for the public good and not, it must be that the good is not

Ž .purchased in either case, so everyone is indifferent. However, 1 is clearlynot satisfied.

Consider students competing for grades. The top 30% of the classreceive A’s, the next 60% receive B’s, and the remainder C’s. Students

Ž .simultaneously choose actions effort levels, attendance, study strategiesfrom a finite set. A student’s payoff is a function of her actions and grade.Then, if one student changes her action and moves from one grade level to

Ž .another, the grades of at most two other students will be affected, and soŽ . Ž .1 will fail. However, for generic assignment of payoffs to action, grade

ORDER INDEPENDENCE FOR WEAK DOMINANCE 225

combinations, a student is never indifferent over two strategy profiles thatŽ .differ only in her actions, so 2 will be satisfied. Other examples of this

type of ‘‘placement game’’ might include competition by workers forpromotions or advertising firms for accounts. Note that these games bearsome similarities to first price auctions, and that indeed, first price auctionsare an example of this class.

Next, consider ‘‘group formation’’ games. In these games, there is aninitial stage during which the players are assigned to groups, modeled by aprocess in which each player chooses an action and then, as a function ofthe vector of actions, is assigned to a group. Once the grouping process isfinished, the members of each group play a game among themselves.Payoffs depend only on the members of one’s group and the strategyprofile chosen within that group. If the group formation process is ofinterest to the modeler, then all these ‘‘subgames’’ must be incorporated

Ž .into a single extensive form. This will involve violating 1 because amember of one group will be indifferent over the actions of players inother groups. Thus, there will be many pairs of strategy profiles wheresome but not all players have equal payoffs. However, the game can easily

Ž . 6satisfy 2 .Another interesting example is signaling games in which the payoff of

the receiver is independent of the signal of the sender. Then, for any fixedaction by the receiver and type of the sender, the payoff of the receiver isunchanging as the signal changes, but if signals have different costs, the

Ž . Ž .payoff of the sender is not. Thus 1 is violated, but 2 holds generically.An example in this spirit is the ‘‘burn the dollar’’ game, in which the payoffof the non-burner is dependent only on the outcome in the underlyinggame, while the payoff of the burner depends both on the outcome in the

Ž .underlying game and on whether or not the dollar was burned, so 1 couldŽ .not be satisfied, but for generic payoffs in the underlying game, 2 is.

Ž .There are interesting games for which 2 fails. In a second priceauction, for example, a player is indifferent between non-winning bids, butthe winner may well not be.

6 Ž .Assume for example that a no player is indifferent between any two outcomes whichŽ .occur in two different groupings; b if a player changes his action in the grouping phase and

this affects the group to which anyone belongs, then it also affects the group to which theŽ . Žplayer himself belongs; c players observe the group to which they are assigned and of

. Ž .course recall their own action in the matching stage, but nothing else; and d each subgamesatisfies TDI. Then, consider any pair of strategy profiles which differ only in the actions of

Ž . Ž .some player i. If player i is indifferent between the two outcomes, then by a and b heŽ .cannot have affected the groupings. So, by c any player not in i’s group will get the same

Ž .payoff. Finally, because of d , any change in the outcome in i’s subgame will be consistentwith TDI.

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IV. FORMALITIES AND THE MAIN RESULT

We work with finite strategy, finite player, normal form games. Players� 4i g N ' 1, . . . , n have finite pure strategy spaces S . Payoffs are given byi

p : Ł S ª R n. The payoff function p is extended to mixed strategiesig N iin the standard way. We assume, without loss of generality, that S l S si jB for all i, j g N, i / j. So, without ambiguity, we can drop the playersubscripts on the strategy names. Let S ' D S . For W : S, let theig N istrategies in W that belong to i be denoted by W ' W l S . Say thati iW : S is a restriction of S if ; i, W / B. Note that any restriction W of Sigenerates a unique game given by strategy spaces W and the restriction ofi

Ž .p to Ł W . We will denote this game by W, p . We similarly defineig N iW ' Ł W . A typical element x g W thus specifies a strategyyi j/ i j yi yix g W for each j / i.j j

DEFINITION 1. Let W be a restriction of S, and let r , s g S . Theni i i

Ž . Ž .i r ¨ery weakly dominates s on W, written r VWD s , if p r , xi i i W i i i yiŽ .G p s , x ; x g W , andi i yi yi yi

Ž .ii r weakly dominates s on W, written r WD s , if r VWD si i i W i i W iŽ . Ž .and, in addition, p r , z ) p s , z for some z g W .i i yi i i yi yi yi

This definition is phrased entirely in terms of pure strategies. Of course,when determining if s is weakly dominated on W, it makes no differenceiwhether one considers W or mixtures over W . However, it is possibleyi yithat s is weakly dominated by a mixture over W but not by any elementi iof W . We extend our results to weak dominance by mixtures in Section V.i

DEFINITION 2. A weak dominance or very weak dominance involving riŽ . Ž .and s on W is nice if for all x g W , if p r , x s p s , x theni yi yi i i yi i i yi

Ž . Ž .p r , x s p s , x .i yi i yi

That is, a dominance is nice if TDI is satisfied with respect to thestrategies involved. Write r NWD s if r weakly dominates s on W andi W i i ithe weak dominance is nice. Say that s is nicely weakly dominated on W ifithere is r in W _ s with r NWD s . Define NVWD and nicely ¨eryi i i W i Wweakly dominated on W similarly for very weak dominance.

DEFINITION 3. Let V be a restriction of S, and let W be a restrictionŽ . Ž .of V. Then W is a reduction of V by nice ¨ery weak dominance if

W s V _ X 1, . . . , X m, where ;k, X k ; S, and ; x g X k, 'z g V _1 k Ž . Ž .X , . . . , X such that z nicely very weakly dominates x on V _1 ky1 Ž . Ž .X , . . . , X . W is a full reduction of V by nice ¨ery weak dominance if

Ž . Ž .W is a reduction of V by nice very weak dominance and no strategiesŽ . Ž .in W are nicely very weakly dominated on W.

ORDER INDEPENDENCE FOR WEAK DOMINANCE 227

That is, a reduction is the result of iteratively removing sets of strategiesthat are dominated in the appropriate sense, and a full reduction is one inwhich no dominances of the appropriate sort remain.

If each of the sets X i in the previous definition is a singleton, then W isŽa one-at-a-time reduction of V by whatever dominance relationship is

.under discussion .Clearly NWD is more restrictive than WD and NVWD, which in turn

Ž .are each more restrictive than VWD. So, any one-at-a-time reduction byŽ .NWD is also a one-at-a-time reduction by WD and NVWD, and any

Ž . Ž .one-at-a-time reduction by WD or NVWD is a one-at-a-time reductionby VWD.

Ž .LEMMA A. Let W be a reduction of V by nice ¨ery weak dominance.Ž .Then W is a one-at-a-time reduction of V by nice ¨ery weak dominance.

Ž . Ž .Proof. Clearly nice very weak dominance on W implies nice veryweak dominance on any subset of W. So, the set of strategies removed ateach step can instead be removed one at a time in any order. A simpleinduction then yields the result. B

DEFINITION 4. Let V and W be restrictions of S. V is equi alent to asubset of W if there exist one-to-one maps m : V ª W , i g N, such thati i iŽ . Ž Ž . Ž ..p x s p m x , . . . , m x ; x g V.1 1 n n

Ž Ž ..So, V differs from a subset of W specifically m V only by a renaming.

Obser ation 1. The relation ‘‘equivalent to a subset’’ is transitive.

DEFINITION 5. Let W be a restriction of S, and let r , s g S . Then r isi i i iŽ . Ž .redundant to s on W, written r R s , if p r , x s p s , x ; x gi i W i i yi i yi yi

W . A strategy s is redundant on W if there is r g W _ s with r R s .yi i i i i i W i

Obser ation 2. r NVWD s if and only if either r NWD s ori W i i W ir R s .i W i

Clearly VWD is transitive. In addition,

Obser¨ation 3. NVWD is transitive; i.e., r NVWD s andi W is NVWD t imply r NVWD t .i W i i W i

To see this, suppose r NVWD s and s NVWD t , and consider anyi W i i W iŽ . Ž . Ž . Ž .s g W such that p r , s s p t , s . Then p r , s s p s , syi yi i i yi i i yi i i yi i i yi

Ž . Ž . Ž . Ž .s p t , s , and so p r , s s p s , s s p t , s . Thus,i i y i i y i i y i i y ir NVWD t .i W i

DEFINITION 6. Let W be a restriction of S. A set V is said to be aŽ .reduction of W by nice weak dominancerredundancersubstitution if V can

be obtained from W by letting V 0 s W and performing a finite number of

MARX AND SWINKELS228

jq1 j Ž .iterations of the following process: obtain V from V by either ijq1 j j j Ž . jletting V s V _ s , where s is nicely weakly dominated on V by an

j j Ž . jq1 j j jelement of V _ s ; or ii letting V s V _ s , where s is redundant onj j j Ž . jq1 Ž j j.V to an element of V _ s ; or iii letting V s V _ s j r, where

r g S and r is redundant on V j to s j.

Ž .Note that if W is a restriction of S, then any reduction of W by niceweak dominancerredundancersubstitution is also a restriction of S.

Ž .LEMMA B. Let W be a restriction of S, let T be a reduction of W by nice¨ery weak dominancerredundancersubstitution, and let s g W. Then there

X X Ž .exists s g T such that s nicely ¨ery weakly dominates s on T.

Ž .Proof. Let s g W. Since T is obtained from W by nice very weakdominancerredundancersubstitution, there exist restrictions T 0, T 1, . . . ,T m such that T 0 s W and T m s T and each T jq1 is obtained from T j asin the definition of a reduction by dominancerredundancersubstitution.The lemma is satisfied for T 0 because s g T 0, so s itself serves. Assume

j X j X Ž .that the lemma holds for T . Then there exists s g T such that s nicelyvery weakly dominates s on T j. If sX g T jq1, then sX serves and the lemmaholds for T jq1. Note in particular that if a strategy for another player isremoved in going from T j to T jq1, then since the requisite inequalitiesand equalities hold on T j, they continue to hold on any subset, and inparticular on T jq1, while if a strategy is substituted then the requisiterelations continue to hold by the definition of redundance.

Suppose that sX f T jq1. Then sX is the strategy that was eliminatedjq1 j X Ž .when T was obtained from T . If s was eliminated by nice weak

dominance or redundance by some t g T j _ sX s T jq1, then let sY s t. If sX

was eliminated and replaced by redundant strategy r g W _ sX, then letY Y jq1 Y Ž .s s r. In either case, s g T and s nicely very weakly dominates s

on T jq1. The result follows by induction. B

Lemma C is the key to the results. Roughly, it says that if you can getˆ Ž .from W to W by NVWD, and you can get from W to V by nice very

weak dominance, then you can get from V to what is essentially a subset ofˆ Ž .W by iteratively removing strategies that are either redundant or nicely

weakly dominated.

ˆLEMMA C. Let W be a restriction of S, let W be a reduction of W by niceŽ .¨ery weak dominance, and let V be a reduction of W by nice ¨ery weak

ˆ ˆ ˆdominance. Then there exists V equi alent to a subset of W, where V isŽ .obtainable from V by the iterati e remo¨al of strategies that are either nicely

weakly dominated or redundant.

ORDER INDEPENDENCE FOR WEAK DOMINANCE 229

ˆProof. Since W is a reduction of W by nice very weak dominance, byˆ 1 mLemma A we can write W s W _ x , . . . , x , where for k s 1, . . . , m,

k k Ž . 1 ky1W s W _ x , . . . , x is a one-at-a-time reduction of W _ x , . . . , x by1nice very weak dominance. Let W 0 s W.

� 4We proceed by induction. For some j g 0, . . . , k y 1 assume that wehave V j and m j such that

Ž . j1 V can be obtained from V by iteratively eliminating strategiesŽ .that are either nicely weakly dominated or redundant,

Ž . j j jŽ . j2 m is a one-to-one map such that ; i g N, ;s g V , m s g W ,i i i ij Ž . Ž jŽ . jŽ ..and ; t g Ł V , p t s p m t , . . . , m t , andig N i 1 n

Ž . jŽ j. Ž .3 m V is a reduction of V by nice weak dominancerredun-dancersubstitution.

We will exhibit V jq1 and m jq1 satisfying Assumptions 1]3. Since theassumptions are trivially satisfied for j s 0 by taking V 0 s V and m0 asthe identity map on V 0, this will establish the lemma.

So, given V j and m j, let us construct V jq1 and m jq1.jŽ j. jq1 jq1 j jq1 jIf m V : W , let V s V and m s m .

jŽ j. jq1 jŽ j. j jq1 jŽ j.Suppose m V ­ W . Then since m V : W , x g m V . Leti be the player to whom x jq1 belongs and let r g W jq1 be a strategy thati i

jq1 j jŽ j.nicely very weakly dominates x on W . Since m V is a reduction of VŽ .by nice weak dominancerredundancersubstitution, and since V is a

Ž .reduction of W by nice very weak dominance, Lemma B implies thatjŽ j. Ž . jq1there is y g m V that nicely very weakly dominates r , and thus x ,i

jŽ j. Ž . jq1 jŽ j.on m V . Then, either y nicely weakly dominates x on m V , or yjq1 jŽ j.and x are payoff equivalent for i on m V .

Ž . jq1 jŽ j. Ž j.y1Ž .Assume y nicely weakly dominates x on m V . Then, m yŽ . Ž j.y1Ž jq1. j jq1 jnicely weakly dominates m x on V . So, define V by V _Ž j.y1Ž jq1. jq1 j jq1m x , and let m be the restriction of m to V . ClearlyAssumptions 1]3 are satisfied for V jq1 and m jq1.

jq1 jŽ j.Assume y and x are payoff equivalent on m V . Then, since y veryjŽ j.weakly dominates r on m V , and since r nicely very weakly dominatesi i

jq1 jŽ j. jq1 jŽ j.x on m V , it must be that r and x are redundant on m V .ijŽ j.Now, either r g m V , or not.i

jŽ j. Ž j.y1Ž jq1. Ž j.y1Ž .Assume that r g m V . Then, m x and m r are redun-i ij jq1 j Ž j.y1Ž jq1. jq1dant on V . So, let V s V _ m x , and let m be the restric-

tion of m j to V jq1. Again, Assumptions 1]3 are clearly satisfied for V jq1

and m jq1.jŽ j. jq1 j jq1 jAssume that r f m V . Let V s V , and let m agree with mi

except that the strategy which used to map onto x jq1 is now mapped ontor . Once again, Assumptions 1]3 are clearly satisfied for V jq1 and m jq1,iand we are done. B

MARX AND SWINKELS230

PROPOSITION 1. Let X be a full reduction of S by nice weak dominance,Ž .and let Y be a full reduction of S by nice weak dominance. Then, after the

remo¨al of redundant strategies, Y is equi alent to a subset of X.

Proof. X is a reduction of S by nice very weak dominance and Y is aˆŽ .reduction of S by nice very weak dominance. By Lemma C, a set Y

equivalent to a subset of X is obtainable from Y by the iterative removalŽ .of strategies that are either nicely weakly dominated or redundant.

ˆŽ .Since Y was a full reduction by nice weak dominance, Y differs from Yonly by the removal of redundant strategies. So, after the removal of someredundant strategies, Y is equivalent to a subset of X. B

The obvious application of Proposition 1 is the following:

THEOREM 1. Let X and Y be full reductions of S by nice weak dominance.Then, X and Y are the same up to the addition or remo¨al of redundantstrategies and a renaming of strategies.

Proof. Applying Proposition 1 twice, each of X and Y differs from asubset of the other by the removal of some redundant strategies and arenaming. B

And, since weak dominance and nice weak dominance are equivalent forgames that satisfy TDI, we have the following corollary:

Ž .COROLLARY 1. Let S, p satisfy TDI, and let X and Y be full reductionsof S by weak dominance. Then, X and Y are the same up to the addition orremo¨al of redundant strategies and a renaming of strategies.

However, Proposition 1 has another nice implication. Even in gameswhere TDI fails, Proposition 1 sets an upper bound on the amount bywhich different orders of elimination by weak dominance can matter. Even

Žwhen TDI fails, the outcome of iterated nice weak dominance is essen-.tially unique by Theorem 1. And by Proposition 1, any full reduction of S

by weak dominance is equivalent to a subset of the outcome of iteratednice weak dominance. So, no matter what the order in which weakdominance is applied, the result will be at least as ‘‘tight’’ as the outcomeof iterated nice weak dominance. In applications where weak dominance isimportant, but TDI fails, once should perhaps first apply nice weak

ORDER INDEPENDENCE FOR WEAK DOMINANCE 231

dominance, since the result of this stage is independent of order, and onlywhen all removals by nice weak dominance have been made considerŽ . 7non-nice weak dominance.

V. MIXED STRATEGIES

The results of the previous section dealt solely with eliminations by purestrategies. In this section we extend our analysis to mixed strategies. If W

Ž .is a restriction of S, let s g D W indicate that s is a mixed strategyprofile in which each player i uses only pure strategies in W . In whatifollows, we will often write s where we more properly mean the mixedistrategy that places probability 1 on s .i

If a pure strategy s is weakly dominated on W, then every mixedistrategy placing positive weight on s is also weakly dominated on W. Soiwe initially consider only orders of elimination in which when a purestrategy is removed, so is every mixed strategy using that pure strategy. Ateach stage, the set of remaining strategies is a subset of the pure strategiesplus any mixtures over those pure strategies that have not been eliminated.For sets V of this form, a strategy s is weakly dominated on V if and onlyiif it is weakly dominated versus the pure strategies underlying V.8 Thus,for any sequence of sets of this form, one can consider a sequence in whichat each stage a strategy is removed if and only if a pure strategy in itssupport is removed, except that when no more pure strategies are remov-able, one round of removing weakly dominated mixed strategies occurs.This new sequence will yield the same result. So, for sequences in whichthe removal of a pure strategy implies the removal of every mixed strategy

7 Note that it is not the case that anything achievable by weak dominance can also beŽ .reached by first using nice weak dominance and then not necessarily nice weak dominance:

L R

T 1, 1 2, 0

B 1, 3 1, 1

Only R can be removed by NWD. Once R is gone, there are no further removals by WD.Conversely, under WD, one can remove both B and R simultaneously at the first step.

8 To see this, note that s is weakly dominated on V if and only if there exists s in thei ireconvexification of V that weakly dominates s on V. The mixture s is in the reconvexifi-i ication of V if and only if it is in the reconvexification of the set of pure strategies underlyingV, and s weakly dominates s on V if and only if s weakly dominates s on the purei i i istrategies underlying V.

MARX AND SWINKELS232

using that pure strategy, the ‘‘order does not matter’’ result depends onlyon the order of removal of pure strategies. Thus it is this order to which werestrict attention in the analysis that follows. We later outline how theanalysis can be extended to sequences in which pure strategies may beremoved while leaving behind some mixed strategies using those purestrategies.

We begin by extending the definitions from Section IV to mixed strate-gies and by strengthening the TDI condition. We wish to construct a proofsimilar to that of Theorem 1. Recall that in that proof, we mapped each V j

iinto a subset of W j. Here, we wish to do the same. However, because weiare allowing eliminations of a pure strategy by a mixed strategy, it mayhappen at some stage that the strategy that some s g V j was mapped intoi idisappears because it is, for example, redundant to some mixture over W j.iTo accommodate this, we must expand our notion of equivalent to a subsetto allow members of V j to be mapped onto mixed strategies in W j. Recalli i

jŽ j.also that it was important in Theorem 1 that the images m V could bethought of as coming from V by weak dominance, redundance, or substitu-

jŽ j.tion. Since m V itself may include mixed strategies, our definitions mustthus include the possibility of a mixed strategy eliminating or replacinganother mixed or pure strategy.

DEFINITION 7. For all i g N, let V be a nonempty finite subset ofiŽ . Ž .D S j S , and let V s D V . Let s , t g D S j S . Theni i ig N i i i i i

Ž . U Ui s ¨ery weakly dominates t on V, written s VWD t , if ;g gi i i V i yiŽ . Ž .Ł V ' V , p s , g G p t , g , andj/ i j yi i i yi i i yi

Ž . U U Uii s weakly dominates t on V, written s WD t , if s VWD ti i i V i i V iX Ž X . Ž X .and in addition for some g g V , p s , g ) p t , g .yi yi i i yi i i yi

As in Section IV, t g V is ¨ery weakly dominatedU on V if there isi iŽ . U Us g D V _t where s VWD t , and similarly for weakly dominated .i i i i V i

Recall that V itself may contain mixed strategies. We abuse notationiŽ . Ž .slightly by taking D V to be that subset of D S which can be imple-i i

� 4mented by randomizing over the elements of V . So, if S s L, M, R andi iŽ . Ž . Ž .a is the mixed strategy given by a L s a M s 1r2, then D a , Ri i i i

consists of all mixed strategies that place equal weight on L and M.

DEFINITION 8. A weak dominanceU or very weak dominanceU involvingŽ . Ž .s and t on V is nice if for all g g V , p s , g s p t , g impliesi i yi yi i i yi i i yi

Ž . Ž .p s , g s p t , g .i yi i yi

We define NWDU , nicely weakly dominatedU on V, and NVWDU ,V Wnicely very weakly dominatedU on W, in the obvious way.

ORDER INDEPENDENCE FOR WEAK DOMINANCE 233

Obser ation 4. For all i g N, let W be a nonempty finite subset ofiŽ . Ž .D S j S , and let W s D W . Let s , t g D S j S . For all i g N,i i ig N i i i i i

Ž .let V be a nonempty finite subset of D W j W , and let V s D V .i i i ig N iŽ . U Ž . UThen s N VWD t implies s N VWD t .i W i i V i

To see that niceness is inherited, note that by the definition of niceŽ . U Ž . Ž .very weak dominance on W, p s , y G p t , y for all y gi i yi i i yi yi

Ž . Ž .Ł W . So, if p s , v s p t , v for some v gk g N _ i k i i y i i i y i y iŽ . Ž . Ž .Ł D W , then p s , y s p t , y for all y in the support ofk g N _ i k i i yi i i yi yi

Ž . Ž .v . But, then because of niceness, p s , y s p t , y for all y inyi i yi i yi yiŽ . Ž .the support of v , and so p s , v s p t , v .yi i yi i yi

Ž . Ž .For a restriction W of S, a reduction of W by nice very weakdominanceU is defined as in the previous section to be a set that can be

Ž . Ž .obtained from W by iteratively eliminating nicely very weaklydominatedU pure strategies. Similarly, a one-at-a-time reduction of W byŽ . Ž . U Ž . Ž .nice very weak dominance is a reduction of W by nice very weakdominanceU in which only one pure strategy is eliminated at each round.

DEFINITION 9. For all i g N, let V be a nonempty finite subset ofiŽ . Ž .D S j S , and let V s D V . Let s , t g D S j S . Then s isi i ig N i i i i i i

U U Ž .redundant to t on V, written s R t , if for all g g V , p s , g si i V i yi yi i yiŽ . U Ž .p t , g . A strategy t is redundant on V if there is s g D V _t withi yi i i i i

s RU t .i V i

Ž . UDEFINITION 10. S, p satisfies TDI if for all restrictions W and for alls , if s is very weakly dominatedU on W, then it is either weakly dominatedU

i ion W or redundantU on W.

We were able to state TDI in terms of a simple condition on the payoffsto pure strategies in a game. The same condition on payoffs is not enoughto imply TDIU. Consider Game G3.

Game G3 satisfies TDI, but TDIU fails because R is very weaklyU 1 1 Udominated on S_ B by L q C, but R is neither weakly dominated2 2

nor redundantU on S_ B. Furthermore, a strengthening of TDI is clearlynecessary because, as this game illustrates, even when TDI holds, the

IIL C CR

T 2, 1 2, 3 0, 2

I M 0, 3 3, 1 0, 2XM 1, 4 1, 4 1, 4

B 1, 4 0, 3 0, 2

GAME G3

MARX AND SWINKELS234

order of elimination under WDU can matter: if one first removes B, thenS_ B is a full reduction of S by weak dominanceU , while if one first

X � 4removes R, then B and M can also be removed. But, S_ B and T , M =� 4L, C are clearly not equivalent.

U 1G3 has a highly nongeneric feature: R is weakly dominated by L21q C, but by no other mixture. Consider different assignments of payoffs,2

Žsubject to existing ties in payoffs for each player being maintained eachplayer receives four different payoffs, so these assignments correspond to

8.elements of R . For all such assignments, the equality of payoffs in thethird row will remain, and so, in particular, any mixture of L and C willgive the same payoff as R versus M X. However, for almost all suchassignments, one of two things will occur: either there will be a mixture of

U � 4L and C that strictly dominates R on T , M, B , or there will be noU � 4mixture that even weakly dominates R on T , M, B . In particular, the set

of payoff assignments that yields a mixture of L and C that weaklyU � 4dominates R on T , M, B but does not yield any mixture that strictlyU � 4 8dominates R on T , M, B is a lower dimensional subspace of R . If one

allows perturbations that do not respect some of the existing ties, then thesituation in this game becomes even less likely.

So, even allowing for some ‘‘structural’’ ties in payoffs, for almost allŽ . Ugames, if there is s g D W _ s with s VWD s , then there is anotheri i i i W i

Ž . Umixed strategy t g D W _ s that strictly dominates s except versusi i i ithose opposition strategy profiles s on which all elements in the supportyiof t give the same payoff to i as s does. For such strategy profiles, TDI isi ienough to imply TDIU. So, if a game satisfies TDI, then it will genericallysatisfy TDIU as well. Thus, in particular, TDIU will be generically satisfiedfor the normal form of any given extensive form and for the discrete firstprice auction. See Appendix B for a formal statement and proof of thisresult.

For games satisfying TDIU , weak dominanceU is equivalent to nice weakdominanceU , so the analysis of Section IV goes through fairly directly:

U Ž . ULEMMA A . Let W be a reduction of V by nice ¨ery weak dominance .Ž . UThen W is a one-at-a-time reduction of V by nice ¨ery weak dominance .

Proof. As in the proof of Lemma A. B

Obser ation 5. s NVWDU s if and only if either s NWDU s ori W i i W is RU s .i W i

DEFINITION 11. Let W and V be restrictions of S.9 V is equi alentU toa subset of W if there exists a one-to-one map m such that ; i g N and

9 Recall that by the definition of restriction, W and V contain only pure strategies.

ORDER INDEPENDENCE FOR WEAK DOMINANCE 235

Ž . Ž . Ž . Ž Ž .;s g V , m s g D W j W and ;s g Ł V , p s s p m s , . . . ,i i i i i ig N i 1Ž ..m s .n

DEFINITION 12. Let W be a restriction of S, and ; i g N, let V be aiŽ . 0nonempty subset of D S j S . Let V ' D V . Let V ' W. Then, thei i ig N i

Ž . U Uset V is a reduction of W by nice weak dominance rredundance rsub-stitutionU if V can be obtained from V 0 by performing a finite number ofiterations of the following process: obtain V jq1 from V j by, for some

Ž . jq1 j j j Ž . Ui g N, i letting V s V _t , where t is nicely weakly dominatedi ij Ž j j. Ž . jq1 j j jon V by an element of D V _t ; or ii letting V s V _t , where ti i i i

U j Ž j j. Ž . jq1is redundant on V to an element of D V _t ; or iii letting V si iŽ j j. Ž . j U j jV _t j s , where s g D W _V and s is redundant on V to t .i i i i i i i

Ž .Note that any set V obtained from a restriction W of S by nice weakdominanceUrredundanceUrsubstitutionU is a finite set of pure or mixedstrategies with at least one strategy for each player, and that only thesubstitutionU operation creates new strategies.

LEMMA BU. Let W be a restriction of S, let T be a reduction of W byŽ . U U U Ž .nice weak dominance rredundance rsubstitution , and let g g D W ji

X Ž . X Ž .W for some i. Then there exists g g D T such that g nicely ¨ery weaklyi idominatesU g on T.

Ž . Ž .Proof. Let g g D W j W . Since T is a reduction of W by nice weaki idominanceUrredundanceUrsubstitutionU , there exist sets T 0, T 1, . . . , T m

such that T 0 s W and T m s T and each T jq1 is obtained from T j as inŽ . U Uthe definition of a reduction by nice weak dominance rredundance r

U 0 Ž 0.substitution . The lemma is satisfied for T because g g D T , so g itselfiŽserves if g g S , then the mixed strategy which places weight 1 on gi i

. j X Ž j.serves . Assume that the lemma holds for T . Then there exists g g D TiX Ž . U j X Ž jq1.such that g nicely very weakly dominates g on T . If g g D T ,

then g X serves and the lemma holds for T jq1.X Ž jq1. XSuppose that g f D T . Then g places positive weight on thei

strategy t j that was eliminated when T jq1 was obtained from T j. If t ji i

Ž . U U Žwas eliminated by nice weak dominance or redundance no strategy. Ž j j. Yadded by some z g D T _t , then let g be the strategy that placesi i

XŽ . Ž . XŽ j. j j jweight g s q z s g t on strategies s g T _t . If t was eliminatedi i i iU X Ž j. Yand replaced by redundant strategy z g D W _t , then let g be thei i

XŽ . j jstrategy that places weight g s on strategies s g T _t and weighti iXŽ j. X Y Ž jq1. Y Ž .g t on z . Then g g D T and g nicely very weakly dominates gi i

on T jq1. The result follows by induction. B

MARX AND SWINKELS236

Ž . Ž .Let x g S , and let s , r g D S be such that r x s 0. Then, definei i i i i i i� 4 Ž . Ž . Ž . Ž .s x ª r as the mixed strategy g given by g s s s s q s x r si i i i i i i i i i i i

Ž . � 4for s / x , and g x s 0. That is, s x ª r is obtained from s byi i i i i i i iredistributing the weight on x according to r .i i

U ˆLEMMA C . Let W be a restriction of S, let W be a reduction of W by niceU Ž .¨ery weak dominance , and let V be a reduction of W by nice ¨ery weak

U ˆ U ˆ ˆdominance . Then there exists V equi alent to a subset of W, where V isŽ .obtainable from V by the iterati e remo¨al of strategies that are either nicely

weakly dominatedU or redundantU.

Proof. The proof is similar to that of Lemma C. There are two maindifferences. First, in Lemma C, each V k was mapped onto a subset of W k.i iIn this case, each V k is mapped onto a subset of mixtures over W k.i iSecond, because of the mixed strategies, the step at which we show thatx kq1 can be eliminated is more intricate.

ˆ USince W is a reduction of W by nice very weak dominance , by LemmaU ˆ 1 m kA we can write W s W _ x , . . . , x , where for k s 1, . . . , m, W s

1 k Ž . 1 ky1W _ x , . . . , x is a one-at-a-time reduction of W _ x , . . . , x by nicevery weak dominanceU. Let W 0 s W.

� 4We proceed by induction. For some j g 0, . . . , k y 1 assume that wehave V j and m j such that

Ž . j Ž .1 V : V can be obtained from V by iteratively eliminating pureŽ . U Ustrategies that are either nicely weakly dominated or redundant ,

Ž . j j jŽ .2 m is a one-to-one map such that ; i g N, ;s g V , m s gi i iŽ j. j j Ž . Ž jŽ . jŽ ..D W j W and ; t g Ł V , p t s p m t , . . . , m t , andi i ig N i 1 n

Ž . jŽ j.3 m V can be obtained iteratively from V by eliminating strate-Ž . U Ugies that are either nicely weakly dominated or redundant or by

substitutionU.

So, while V j remains a subset of pure strategies, its image in W j mayinvolve mixtures. Given V j and m j satisfying Assumptions, we will exhibit

jq1 jq1 Ž . Ž .V and m satisfying 1 ] 3 . This will establish the lemma, sinceŽ . Ž . 0 01 ] 3 are satisfied for j s 0 by taking V s V and m as the identitymap.

So, given V j and m j, let us construct V jq1 and m jq1. Let i be thejq1 Ž jq1.player to whom x belongs and let r g D W be a strategy thati i

nicely very weakly dominatesU x jq1 on W j. Then, by Observation 4, riU jq1 jŽ j.nicely very weakly dominates x on m V .

So, by Observation 5, either r nicely weakly dominatesU x jq1 onijŽ j. jq1 U jŽ j. jq1m V or r and x are redundant on m V . Assume r and xi i

U jŽ j. j jŽ .� jq1 4are redundant on m V . Then, for each s g V , m s x ª r isi i i iU jŽ . jŽ j.redundant to m s on m V . Now, it may turn out that there are setsi

ORDER INDEPENDENCE FOR WEAK DOMINANCE 237

of strategies in V j all elements of which are mapped into the same strategyiwhen x jq1 is replaced by r . If so, then these strategies are redundantU oniV j. Construct V jq1 by removing all but one of any such set, and leti

jq1 j jq1 jq1Ž . jŽ .� jq1 4V s V for k / i. For s g V , define m s s m s x ª r ,k k i i i ij jq1Ž . jŽ . Ž . Ž .while for k / i and s g V , let m s s m s . Then, 1 ] 3 arek k k k

jq1Ž jq1. jŽ j.clearly satisfied. In particular, m V is obtained from m V byU Ž jsubstitution and if more than one strategy in V is mapped onto thei

jq1 . Usame strategy when x is replaced by r by redundance .iU jq1 jŽ j.Assume next that r nicely weakly dominates x on m V . Leti

jŽ j. Ž jq1. Ž jŽ j..s g m V have s x ) 0. We claim first that there is g g D m Vi i i i iŽ jq1. Ž .such that g x s 0, and such that g nicely weakly dominates s . Toi i i

U � jq1see this, note first that s is nicely weakly dominated by h ' s x ªi i i4 U Ž jŽ j.. Ž .r . By Lemma B , there is f g D m V which nicely very weaklyi i i

U Ž . Udominates h and therefore nicely weakly dominates s . Let Q be thei iŽ jŽ j.. Ž . Uset of elements of D m V which nicely very weakly dominate h . Q isi i

non-empty since it contains h . It is also clearly compact. So, there isiŽ .g g arg max f a , wherei a g Q ii

f a ' p a , d .Ž . Ž .Ýi i i yij jŽ .d gŁ m Vyi k / i k

Ž . U Ž jŽ j..Clearly, g nicely weakly dominates s , and is in D m V . Assumei i iŽ jq1. U jq1 jŽ j.g x ) 0. Then, since r nicely weakly dominates x on m V ,i i� jq1 4 Ž . U jŽ j.g x ª r nicely weakly dominates g on m V . And, by Lemmai i i

U Ž jŽ j.. Ž . U � jq1B , there is m g D m V which nicely very weakly dominates g xi i i4 Ž . Ž .ª r . But then m g Q, and f m ) f g , a contradiction.i i i i

� j jŽ .Ž jq1. .Let Y ' s g V ¬ m s x ) 0 . By the previous claim, for eachi i i iŽ jŽ j.. Ž . Us g Y , there is g g D m V such that g nicely weakly dominatesi i i i

jŽ . Ž jq1.m s , and such that g x s 0. But, then it must be the case thati iŽ jŽ .. Ž j.y1Ž . Ž . jq1 jg m Y s 0, and so m g g D V _Y . So, define V s V _Y ,i i i i i i i i

V jq1 s V j for k / i, and define m jq1 as the restriction of m j to V jq1. Byk kj Ž .the preceding argument, each s removed from V was nicely weaklyi i

U jq1 jŽ j.dominated by a mixture over V , and each s removed from m V isi i iŽ . U jq1Ž jq1. Ž . Ž .nicely weakly dominated by a mixture over m V , and so 1 ] 3iagain clearly hold. B

PROPOSITION 2. Let X be a full reduction of S by nice weak dominanceU ,Ž . Uand let Y be a full reduction of S by nice weak dominance . Then, after the

remo¨al of redundantU strategies, Y is equi alent to a subset of X.

Proof of Proposition. As in the proof of Proposition 1. B

MARX AND SWINKELS238

As before, we have:

THEOREM 2. Let X and Y be full reductions of S by nice weak domi-nanceU. Then, X and Y are the same up to the addition or remo¨al ofredundantU strategies and a renaming of strategies.

Proof. As for Theorem 1. B

Ž . UCOROLLARY 2. Let S, p satisfy TDI , and let X and Y be full reduc-tions of S by weak dominanceU. Then, X and Y are the same up to theaddition or remo¨al of redundantU strategies and a renaming of strategies.

We now sketch how the analysis of this section can be extended to allownicely weakly dominatedU pure strategies to be eliminated without neces-sarily eliminating all mixtures that use those pure strategies. Let V0, V1,

m 0 Ž .. . . , V , where V s D S , be a finite sequence of rounds of eliminationof nicely weakly dominatedU mixed strategies such that Vm is a fullreduction of S by nice weak dominanceU.10 For each k, let W k be the setof pure strategies s for which there is a mixed strategy in Vk that putsprobability 1 on s. Then, a fairly straightforward induction argumentestablishes that any strategy in Vk that uses a pure strategy not in W k isnicely very weakly dominatedU on Vk by some mixed strategy using only

k � k4elements of W . Consider the sequence C obtained by removing fromeach Vk any mixed strategy using a pure strategy not in W k. By theprevious assertion, this is a valid sequence of removals by nice very weakdominanceU. By Observation 5, any removal that was not by nice weakdominanceU was redundantU , and so the proof of Proposition 2 establishesthat C m, if it contains no nicely weakly dominatedU strategies, depends onthe order or removals only up to redundancyU and renaming. So, we mustshow that C m contains no nicely weakly dominatedU strategies. Considerany strategy s g Vm _C m. By the assertion, s is nicely very weaklydominatedU by some mixture over W m. Since Vm is a full reduction of Sby nice weak dominanceU , s is not nicely weakly dominatedU , and so byObservation 5, s is redundantU to some mixture over W m. But then astrategy in Vm is nicely weakly dominatedU on Vm if and only if it is nicelyweakly dominatedU on C m, which, since Vm is a full reduction of S bynice weak dominanceU , establishes that C m is also a full reduction of S bynice weak dominanceU , establishing the result.

10 Dealing with infinite sequences of removals introduces issues which we would prefer notto deal with here, although the main ideas of this analysis translate directly.

ORDER INDEPENDENCE FOR WEAK DOMINANCE 239

VI. WEAK DOMINANCE AND BACKWARD INDUCTION

In its purest form, backward induction consists of iteratively removingactions that are strictly dominated given the information available whenthat action is taken.

A normal form strategy that is consistent with reaching a particularinformation set and takes a strictly dominated action at that informationset is weakly dominated by one that differs only by taking a dominatingaction at that information set. So, any sequence of removals of actions bybackward induction in the extensive form corresponds to a sequence ofsets of removals of strategies by weak dominance in the normal form.11

This relationship between backward induction in the extensive form andweak dominance in the normal form extends to their respective motiva-tions. Backward induction requires that the action chosen at an informa-tion set must not be strictly dominated when the opponents play in such away that the information set is reached. Weak dominance requires that achosen strategy not be strictly dominated by another strategy when theopponents play in such a way that the choice between these two strategiesaffects the decisionmaker’s payoffs. So, each is about making decisions

Ž .under the possibly counterfactual hypothesis that the opponents have notchosen in such a way that the decision ‘‘does not matter.’’12

Thus, at an intuitive level, there seems to be an intimate relationshipbetween backward induction and weak dominance. However, the order ofremovals under weak dominance can matter, while backward induction isdeterministic.13 This raises the possibility that backward induction andweak dominance differ at some fundamental level that we have failed tounderstand. What is it about these two concepts that leads one to bedeterministic and the other not?

We claim that the distinction between weak dominance and backwardinduction comes down to a fairly minor difference in what the twoconcepts mean by ‘‘does not matter.’’ For backward induction, ‘‘does notmatter’’ means ‘‘makes the information set unreachable,’’ while for weakdominance, ‘‘does not matter’’ means ‘‘makes the choice irrelevant for the

11 Ž .Glazer and Rubinstein 1993 start from this observation in defining ‘‘guides,’’ which areessentially prespecified orders in which to check weak dominance.

12 On the general relation between normal and extensive form motivations and implemen-tations of solution ideas, including a discussion of counterfactual reasoning in normal form

Ž .games, see Mailath et al. 1993 .13 The set of outcomes that survive backward induction remains deterministic even if we

allow iterative elimination in any order of dominated actions at information sets, i.e., even ifthe elimination process does not start at the end of the tree and move up.

MARX AND SWINKELS240

decisionmaker’s payoffs.’’ Of course, if the information set is not reached,the choice is irrelevant for the decisionmaker’s payoffs and every otherplayer’s payoffs. Thus, in backward induction a strategy profile for playersother than i is excluded in player i’s decisionmaking process only if,regardless of the decision i makes, the profile gives the same payoffs to allplayers. So, eliminations in backward induction are eliminations by niceweak dominance and thus are order independent.

Thus, the fact that backward induction is deterministic while weakdominance is not does not reflect some fundamental difference in theirmotivations, but rather the fairly simple difference between nice weakdominance and weak dominance.

The connection between backward induction and nice weak dominanceŽ .allows us to strengthen some of the results of Rochet 1980 . Rochet shows

that for extensive form games with perfect information satisfying anŽ .extensive form version of 1 , the unique backward induction payoff is the

same as the unique payoff from iterated weak dominance in the normalform, which is the same as the unique proper equilibrium payoff. Thisresult extends to games satisfying an extensive form version of TDI. To seethis, note that an extensive form game with perfect information thatsatisfies TDI has a unique backward induction payoff. Since backwardinduction eliminations correspond to nice weak dominance eliminations,by Proposition 1, any full reduction by nice weak dominance or weakdominance yields the unique backward induction payoff. Since any properequilibrium is a backward induction solution, any proper equilibrium alsohas the same payoff.

In addition, Proposition 1 shows that if an extensive form game has aunique backward induction payoff, then any full reduction by nice weakdominance or weak dominance also contains only that payoff, and anyviolations of TDI are non-essential in that they do not affect the outcomeof iterated weak dominance.

VII. WEAK DOMINANCE AND COMPLEXITY

We close with a brief comment on how our results interact with those ofŽ .Gilboa et al. 1993 . They point out that, in general, computational

problems involving weak dominance are hard. In particular, given a fullreduction of a game, the question of whether different choices earlier inthe sequence of weak dominance removals might have led to a strategicallydifferent result remains. To figure out all the strategic implications of weakdominance, one must thus check all possible orders of removal, which

ORDER INDEPENDENCE FOR WEAK DOMINANCE 241

cannot be done in polynomial time. Gilboa et al. interpret this result ascasting additional doubt on the use of weak dominance as a solutionidea.14

Consider, however, nice weak dominance in general games or weakdominance in games that satisfy TDI or some other conditions such thatorder does not matter. Then, once one has arrived at a full reduction, oneknows no other order could have resulted in a strategically different game.Since finding a full reduction is a polynomial problem, nice weak domi-nance avoids some of the computational problems of weak dominance andso for an important class of games weak dominance becomes less suspect.15

APPENDIX A

In this section we formalize the argument that the discrete first priceauction generically satisfies TDI. Think of the auction as being generatedby first fixing a set of players 1, . . . , n, a finite set B of admissible bids, a

� 1 m4 nset of names of signals V s v , . . . , v , a measure r on V , and mapsV : Vny1 = V ª R giving the value of the object to player i when theiother bidders receive signals v g Vny1 and i receives signal v g Vyi iŽnotation aside, having different sets V for different players presents no

. ndifficulties . Since V assigns a value to each of m different signal profiles,ithe function V can be associated in the obvious way with an element ofiR m n

. Each player’s strategy space is the set of all maps from signals in Vto bids in B, and so is finite. Fix a pure strategy profile s for the playersyiother than i. Given this behavior of the other players and any pure

Ž Ž ..strategy s for i, let p v, s v be i’s probability of winning when signalsi iare v and bids are according to s. The expected payoff to i from followings is thusi

p s , s s r v p v , s v V v y s v .Ž . Ž . Ž . Ž . Ž .Ž . Ž .Ýi i yi i i i invgV

ŽThis expression depends on V only directly r and p do not depend oni i.the value assignment , and so it is a linear function of V . Consider twoi

14 Ž .See Samuelson 1992 on another concern with the concept of iterated weak dominance,namely that it cannot be grounded by assuming that it is common knowledge that players donot play weakly dominated strategies.

15 Ž .See Gilboa et al. 1993 for a formal development of this material. In particular, seeSection 4.3 for a proof that a full reduction can be computed in polynomial time. The readermay also wonder whether checking TDI is a polynomial problem. The answer is yes, becausechecking where the antecedent of the TDI condition holds involves checking, for each player

1 < < < < < < Ž .i, S S y 1 S possible equalities this is the number of r , s , s triplets and then, fori i y i i i y i2

each such equality, checking n y 1 further equalities, which is clearly a polynomial problem.

MARX AND SWINKELS242

pure strategies s and t for i and a pure strategy profile s for i’si i yiŽ . Ž .opponents. Suppose that p s , s s p t , s .i i yi i i yi

Ž . Ž . Ž Ž .. Ž Ž .Case i . 'v such that r v ) 0 and p v, s v / p v, t v ,i i i iŽ .. Ž . Ž .s w . Then the equation p s , s y p t , s s 0 has nonzeroyi yi i i yi i i yi

Ž .coefficient on V v and so is satisfied for a set of possible value assign-iments V which is a lower-dimensional subspace of R m n

.i

Ž . Ž . Ž Ž .. Ž Ž . Ž ..Case ii . ;v such that r v ) 0, p v, s v s p v, t v , s v .i i i i yi yiŽ . Ž . Ž . Ž . Ž .If r v ) 0 implies s v s t v , then clearly p s , s s p t , s .i i i i i yi i yi

Ž . Ž . Ž .Suppose r v ) 0 and s v / t v for some v. Without loss of gener-ˆ ˆ ˆ ˆi i i iŽ . Ž .ality, assume s v ) t v . By assumption, i’s probability of winningˆ ˆi i i i

Ž . Ž . Ž .facing s v is the same with s v and t v . But then the highest bidˆ ˆ ˆyi yi i i i tŽ .for the opponents under v must be either less than t v or greaterˆ ˆyi i i

Ž . Ž . Ž .than s v . In either case, player i’s change from s v to t v does notˆ ˆ ˆi i i i i iaffect the payoff of players other than i.

Note that there are a finite number of s , t , s combinations, and soi i yiŽ .the set of V for which the situation of Case i holds is a finite union ofi

m n Ž .lower-dimensional subspaces of R . In the situation of Case ii , TDIholds.

APPENDIX B

In this appendix, we formalize the notion that TDI generically impliesTDIU. We first need to formalize the idea of generic payoffs subject to anygiven set of payoff ties. To do this, retain the previous notation for playersand strategies, but add, for each player i, a finite set O and two functionsif : S ª O and r : O ª R. In games as we have considered them so far,i i i ip is the composition of f with r . Note that any given r can be thoughti i i iof as a member of R <O i <. Genericity statements can then be made at thelevel of r . We interpret O as a set of descriptions of economic outcomesi iplayer i may receive in the game. f describes which outcome is associatediwith each strategy profile in the game, and finally r describes i’s utilityi

Ž . Ž .over outcomes. In this interpretation, if f s s f t , then from player i’si ipoint of view, s and t lead to identical outcomes, and so whatever hisutility function over outcomes, he will be indifferent over s and t. On the

Ž . Ž .other hand, if f s / f t , then the outcomes associated with s and t arei idistinct, and so i is indifferent over s and t only if his utility function

Ž . Ž .happens to be indifferent over f s and f t . For example, in a privatei ivalue auction, the elements of O might describe whether i won the objectior not and how much was paid. r would then capture information aboutii’s attitudes toward money and the object. For any such utility function, i is

ORDER INDEPENDENCE FOR WEAK DOMINANCE 243

indifferent over strategy profiles in which he does not win, but only forvery specific realizations of price will i be indifferent between losing andwinning.

These preliminaries out of the way, we can state and prove our mainresult:

THEOREM. Fix N, and for each i g N, fix S and f . Assume that TDI isi isatisfied at the le¨el of outcomes. That is, assume that for all i, j g N,r , t g S , and s g S ,i i i yi yi

f r , s s f t , s « f r , s s f t , s .Ž . Ž . Ž . Ž .i i yi i i yi j i yi j i yi

U Ž .Let F be the set of r such that TDI fails. Then, Cl F has empty interior,Ž .where Cl ? indicates closure.

Proof. Let W be a restriction of S, and let t g S _W for some i. Leti i iŽ .C ? indicate the carrier of a mixed strategy. We will show that the set of ri

U Ž . Usuch that t is VWD by some s with C s s W , but t is neither Ri W i i i i WU Ž .nor WD by any g with C g s W , has closure with empty interior.W i i i

Ž .Since there are a finite number of W, t pairs, this establishes the result.iŽ . Ž .Assume that 's g W such that ;s g W , f s , s s f t , s .yi yi i i i i yi i i yi

ŽThen, by TDI, f will also be constant, and so regardless of s withj iŽ . . Ž . Ž .C s s W , and regardless of r, p s , s s p t , s will hold. Thus,i i i yi i yi

no failure of TDIU relative to t and W could ever be traced to s . So,i yiwithout loss of generality, assume that W has been chosen in such a way

Ž . Ž .that ~ s g W such that ;s g W , f s , s s f t , s .yi yi i i i i yi i i yiDefine A as the set of all r such that t is not VWDU on W by any sno i i i

Ž .with C s s W . Define A as the set of r such that t is strictlyi i strict i iŽ . 16dominated on W by some s with C s s W . Note that A is open,i i i strict

since if s strictly dominates t on W for r , then it continues to do so fori i iall rX in a neighborhood of r . Let A j A ' A. Then, any failure ofi i no strict

U c Ž c .TDI is contained in the set A . So, it is enough to show that Cl Acontains no open sets.

Let r g Ac. We will examine two cases.i

Ž . Ž . ŽCase i . Assume that for some Y : W , f W = Y : f t =yi yi i i yi i i.Y . Then, letyi

a g arg max r aX ,Ž .iX Ž .a gf t =Yi i y i

16 Ž . Ž .A strategy s g D S strictly dominates s g S on W if, for all s g W , p s , s )i i i i y i y i i i y iŽ .p s , s .i i y i

MARX AND SWINKELS244

Ž . Xand let r g Y be such that f t , r s a. Generate r from r byyi yi i i yi i iŽ .increasing r a by any small positive amount, and otherwise leaving ri i

XŽ Ž .. XŽ Ž ..unchanged. Then, ;s g W , r f s , r F r f t , r with strict in-i i i i i yi i i i yiŽ .equality at least once, since for at least some s g W , f s , r / a. But,i i i i yi

Ž . XŽ . XŽ . Xthen, for any s having C s s W , p s , r - p t , r , where p isi i i i i yi i i yithe payoff function for i generated by f and rX. So, for rX, t is not VWDU

i i i iŽ . Xon W by any s with C s s W , and so r g A . Further, this remainsi i i i no

Y X YŽ X.Xtrue of any perturbation r of r such that a s arg max r a ,i i a g f Ž t =Y . ii i y i

and so for all rY in a neighborhood of rX, rY g A . That is, rX is ini i i no iŽ . Ž .Int A , where Int ? indicates interior. Thus, any neighborhood of r hasno i

Ž . Ž c . Ž .non-empty intersection with Int A . Since Cl A l Int A s B, thisno noŽ c .implies that there is no neighborhood of r contained in Cl A .i

Ž . Ž . ŽCase ii . Assume that for each Y : W , f W = Y ­ f t =yi yi i i yi i i. Ž .Y . Let s be such that C s s W and such that ;s g W ,yi i i i yi yi

Ž . Ž .p s , s G p t , s , where p is the payoff function generated by r .i i yi i i yi i iŽ . Ž . XPick some a g f W = W _f t = W . Generate r from r by in-i i yi i i yi i i

Ž .creasing r a by any small positive amount, and otherwise leaving ri iXŽ . XŽ .unchanged. Then, p s , s ) p t , s for any s g W such thati i yi i i yi yi yi

Ž .a g f W = s , of which there is at least one, while ;s g W ,i i yi yi yiXŽ . XŽ . Xp s , s G p t , s . Let W be the subset of W such that ;s gi i yi i i yi yi yi yiX XŽ . XŽ . XW , p s , s s p t , s . If W is empty, then we are done. Other-yi i i yi i i yi yi

Ž X . Ž X . Y Xwise, choose b g f W = W _f t = W . Generate r from r byi i yi i i yi i iXŽ . Xincreasing r b by a small positive amount, and otherwise leaving ri i

YŽ . YŽ . Xunchanged. Then, p s , s ) p t , s , for any s g W such thati i yi i i yi yi yiŽ .b g f W = s , of which there is at least one, while ;s gi i yi yi

X YŽ . YŽ . YŽ . XŽ .W , p s , s G p t , s . And, by making r b y r b smallyi i i yi i i yi i ienough, the strict inequalities for s in W _W X will be retained.yi yi yiProceeding in this way, we generate r9 . . . 9 arbitrarily close to r underi iwhich s strictly dominates t on W , and so r9 . . . 9 g A . Since Ai i yi i strict strictis open, we have again shown that there is no neighborhood of ri

Ž c .contained in Cl A . B

REFERENCES

Ž .Gilboa, I., Kalai, E., and Zemel, E. 1990 . ‘‘On the Order of Eliminating DominatedStrategies,’’ Oper. Res. Lett. 9, 85]89.

Ž .Gilboa, I., Kalai, E., and Zemel, E. 1993 . ‘‘The Complexity of Eliminating DominatedStrategies,’’ Math. Oper. Res. 553]565.

Ž .Glazer, J., and Rubinstein, A. 1993 . ‘‘Simplicity of Solution Concepts: Subgame PerfectEquilibrium in Extensive Games vs Iteratively Undominated Strategies in Normal Games,’’Working Paper 9]93, Sackler Institute of Economic Studies.

ORDER INDEPENDENCE FOR WEAK DOMINANCE 245

Ž .Gretlein, R. J. 1983 . ‘‘Dominance Elimination Procedures on Finite Alternative Games,’’Int. J. Game Theory 12, 107]113.

Ž .Mailath, G., Samuelson, L., and Swinkels, J. 1993 . ‘‘Extensive Form Reasoning in NormalForm Games,’’ Econometrica 61, 273]302.

Ž .Moulin, H. 1979 . ‘‘Dominance Solvable Voting Schemes,’’ Econometrica 47, 1337]1351.Ž .Moulin, H. 1984 . ‘‘Dominance Solvability and Cournot Stability,’’ Math. Soc. Sci. 7, 83]102.Ž .Moulin, H. 1986 . Game Theory for the Social Sciences, 2nd ed. New York: New York Univ.

Press.Ž .Rochet, J. C. 1980 . ‘‘Selection of an Unique Equilibrium Value for Extensive Games with

Perfect Information,’’ D. P. Ceremade, Universite Paris IX.´Ž .Samuelson, L. 1992 . ‘‘Dominated Strategies and Common Knowledge,’’ Games Econ. Beha¨ .

4, 284]313.Ž .Satterthwaite, M. A., and Sonnenschein, H. 1981 . ‘‘Strategy-Proof Allocation Mechanisms at

Differentiable Points,’’ Re¨ . Econ. Studies 48, 587]597.