Neuman and Morgenstern

8/11/2019 Neuman and Morgenstern

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Journal of Risk and Uncertainty, 2: t27-t58 (t989)

1988 Kluwer Academ ic Pub lishers

Retrospective on the Utility Theory of

von Neumann and Morgenstem

PETER C. FISHBURN*

AT&T Bell Laboratories

Key words: von Neumann-Morgenstern utility, ordinal utility, cardinal utility

Abstract

This article offers an exegesis ofthe passages in von N eum ann and Morgenstern (1944,1947,1953) that

discuss their concep tion of utility. It is occasioned by

two

factors. First, as

we

approach the semicenten-

nial of the publication ofTheoryofGames andEconomic

Behavior,

its immense impact on econom

though t in the intervening years encourages serious reflection on its autho rs'

ideas.

Second, misleading

statements about the theory continue to appear. The article will have accomplished its purpose if it

helps others appreciate the genius and spirit ofthe theory of utility fashioned by John von Neumann

and Oskar Morgenstern.

Utilityis one ofthe strangest words in the ann als of economics an d decision theory.

Early app earan ces occu r in the evaluation of risky monetary ventures by G abriel

Cramer (1728) and Daniel Bernoulli (1738), and in Jeremy Bentham's (1823)

qualitative systemization of value in public and private economics:

.. .in practice,

people with common sense evaluate money in proportion to the

utility they can obtain from it (Cramer, 1728; see Bernoulli, 1954, p. 33);

. . . thevalueof an item must not be based onitsprice,but rather on theutility i

yields. (Bernoulli, 1954, p. 24);

By utility is meant that property in any object whereby it tends to produce

benefit, advantage, pleasure, good, or happines . . . o r . . . to prevent the happ en-

ing ofmischief,pain, evil, or unhappiness ... (Bentham, 1823; see Page, 1968,

pp.3-4).

Its subsequent appearance by many names in nineteenth-century consumer

econom ics led Irving Fishe r (1918) to lam ent that the conceptcalled ftnaldegree o

utility

(Jevons),

effective

utility,

specific

utility,and

marginalefficiency

(J. B. Cla


2/32

128 PETER FISHBURN

his books, but was not entirely happy with its ethical connotation, and suggested

wantability

to his colleagues in econom ics for their conside ration. Econom ists

have also had to contend with

measurable utility

as well as

ordinal utility

andca

dinalutility(Hicks and Allen, 1934), and not a little confusion was sown w hen von

Neumann and Morgenstern (1944) endowedutilitywith an entirely new meaning

It is true tha t a similar m eaning was introduced earlier by Ramsey (1931), but wish-

ing to dissociate his theory from the then-popular notion of diminishing marginal

utility, Ram sey spoke ab outvaluesandmeasuring values

num erically.

Moreover,

work went virtually unnoticed until its importance was publicized by Bruno de

Finetti, Kenneth A rrow, and Leonard

J.

Savage aroun d 1950. Mindful of

its

check-

ered history, Fishburn (1964) avoidedutility in favor ofrelativevalue, but has lo

since recanted (Fishburn, 1970).

The hornets' nest stirred up by von Neumann and Morgenstern's new use of

utilityh as, for the most part, been quiescent since the 1950s, than ks to a num ber of

careful expositions of their theory. These include M arschak

(1950),

W eldon (1950)

Arrow (1951), Strotz (1953), Savage (1954), Ellsberg (1954), Luce and Raiffa (1957)

and Baumol (1958): all offer valuable insights and interpretations. My own

favorite, because of its clarity and incisiveness, is Ellsberg (1954).

These and later expositors have told us a great deal about what von Neumann

and Morgenstern meantby

utility,

but apart from a few choice quotes they have

revealed very little about what von Neumann and Morgenstern saidaboututility.

For that it is necessary to return to the source.

I do this for two reasons. First, nearly 50 years have passed since von N eu m an n

and Morgenstern wrote the

TheoryofGamesandEconomic

Behavior: it has been

classic with enormous influence for most of those years, and it seems none too

soon to submit its passages on utility to exegesis. In doing this I draw not only on

the state of utility theory, preference theory, axiom atics, and measurem ent theory

when they wrote their book during the war years of 1940-1942, but also supple-

ment the discussion with developments in these areas since then. This allows a

larger historical frame for their work and reveals its anticipation of later

contributions.

The second reason is the disturbing fact that inaccuracies and misleading

statements about the intentions and theory of von Neumann and Morgenstern

utility con tinue to find their way into print. I hope that this article will help others

appreciate just what it was that von Neumann and Morgenstern said about and

meant by

utility.

The next two sections prepare for the exegesis proper. Section 1 provides a brie

overview of usagesofutilityprior to 1940 and mentions a few things tha t have h ap-

pened since then. Section 2 summarizes salient points about von Neumann-


3/32

UTILITY THEORY OF VON NEUMANN AND MORG ENSTERN 129

behavior that places their treatment of utility within the larger concern of the

theory of games. All quotes from their work are from the 1953 edition.

Sections 4 throu gh 12 con tain extensive quotes from their sections 3.1 through

3.7, interspersed with commentary. All notation and emphases in the quotations

are their own, and for expositional reasons their footnotes are placed in brackets

immediately following the markers. A few footnotes that add little have been

removed.

Section 13 concludes the essay with a postscr ipt.

1.

Overview of utility

Th e history of utility through the early part of this century is covered by the peer-

less review articles by Stigler (1950), the book by Kauder (1965) on the history of

marginal utility, and the collection of readings compiled by Page (1968). A few

highlights from this early period are noted along with more recent developments.

We first defineordinalandcardinalutility, terms introduced by Hicks and A llen

(1934) a few years before von Neumann and Morgenstern began their book, that

provide connecting threads for our discussion. Let > be an is preferred torelation

between elemen ts in a setX We say that a real-valued function

w

o n Z i s anordina

utility function if

for all

X,

y e X,x > y

u{y}. (1)

Necessary and sufficient conditions for the ordinal utility representation (1),

which consist of an ordering axiom for > on X and an order-denseness condition

(Cantor, 1895) for > when X is uncountably infinite, are discussed in Fisburn

(1970) and Krantz et al. (1971). Because v on X satisfies (1) in p lace of u if and

only if

for allX, y e X, v{x) > v{y) u{x) > u{y), (2

an ordinal utility function is said to beunique uptoanorder-preserving

transform

tion

or a (positive, or increasing)

monotone transformation.

If, in addition to(1),

w

is restricted by other properties that imply that

v

also satis

fies (1)andthoseotherpropertiesif, and only if, there are numbersa> 0andbsuc

that

for all X G X, v{x) = auix) + b, (3)


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130 PETER FISHBURN

For Bernoulli (1738), utility connotedanumerical measure of a person's subjec-

tive value of wealth tha t is independen t of an y consideration of prob ability or risk.

He suggested th at utility of wealth ordinar ily increases at a decreasing rate (know n

later

as the

principle

of

diminishing marginal utility

of

wealth)

and,

more

specifically, that the utility adde d by the next increm ent of wealth is inversely pro -

portional to theamount of wealth already on hand. This implies that Mis a

logarithmic function ofwealth,a form that reappears with elaborations in the

theory

of

Allais (1979/1952, 1979).

Bernoulli assumed also that the relative desirabilities of risky ventures o

probability distributions p,q,... onlevelsof wealth were refiected bytheirex-

pected utilitiesSp{x)u(x),2 q(x)u(x),... of probability-weighted riskless utilities

We know

(e.g.,

Fis hb urn, 1988a, p. 5) tha t this o ther p ropertyofinvariance ofthe

orderofexpected utilities impliesinthe presenceof

1)

that is acardinal utility

function. Moreover, this

is

true quite apa rt from any specific form adopted

foru

suchas the logarithmic form, and apart from anypsychological presum ptions

aboutthenatureofutility.

For E urop ean econom ists in the second half of the nineteenth century (Gossen,

Jevons, Menger, Walras, Marshall), utility continued

to be

viewed

as a

riskless

measureof subjective valueofwealth or ofvarious am ounts ofgoods thatone

might purchase. It waspredom inantly thought of as a psychological entity

measurable

in

its own right" (Strotz,

1953,

p .

84),

but the extent of its m easu rability

was actively debated. Onemight ask whether utilityhas a natural zero point,

whetherit is uniquely determined up to thechoiceoforigin and scale unit,or

whetherit ismerely ordinal.

One popu lar form adopted for the utility of a vector(xi,

2

i^Jofquantities

n goods wastheadditive decom position

H ( X I ,

X2, .

.

.,X) =

M , ( X | ) -I- U2iX2)

+ . . . +

M ( X ) , (4)

in which

M,

is

a

utility function for good

/.

This gives ano the r special prope rty foru

which, in conjunction with (1) and some technical conditions (Debreu,1960

Fishburn, 1970; Krantzetal., 1971), implies tha tuis a cardinal utility function.

But, as before withl ,p(x)u(x),the implication of uniqu eness up to a positive line ar

transformation isforcedby(4)andnot by whether utilityism easurableinsome

internal introspective sense.

The pos ition tha t utility is merely ord inal, without special forms tha t constra init

tobecardinal,andwithout assumptions abo ut hu m an psychology that rend erit

measurable regardless of such special forms, gained currency as writers like

Edgew orth (1881), Fisher (1892), Pa reto (1906), Jo hn so n (1913) and Slutsky (1915)


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UTILITY THEORY OF VON NEUMA NN AND MORG ENSTERN 131

x~y

u{x) =u(j>) (5)

so tha t ~ partitions X into indifference classes, with the sam e utility for all vectors

within a given class. Then (1) orders the indifference classes on a greater-than

utility basis. Each indifference class might be referred to as an isoutility locus,

imagined as a smooth (n - l)-dimensional surface in n-dimensional space, and

the ordered collection of all such loci is said to be an indifference map. The in-

dividual isoutility surfaces in the m ap are tightly packed to cover the whole of X,

and no two of them ever touch or cross.

Although Pareto sided with the ordinalists, he also noted that if, in addition to

(1),

it were possible to com pare preferencedifferences,withupreserving the orderof

those com parisons , thenuwould have the characteristics of a card inal utility func-

tion. GrantingX >y and

z

>

w,

let

{x,

y)>

(z,vv)

signify that the difference in

pref-

erence between

x

and

y

exceeds the difference in preference between z and

w.

If we require

{x, y) > (z, w) u{x) - u(y) > u{z) - u{w) (6 )

for positive differences, and assume a few other technical conditions (Fishburn,

1970,

chapter

6),

then

u

is unique up to a positive linear transformation. T his third

way of inducing cardinal utility is more evocative of the classical notion of

measurability tha n the two men tioned earlier. Note also that B ernoulli's analysis

of riskless utility increments that led to his logarithmic form for the utility of

wealth is a difference analysis.

Pareto 's app roach to card ina l utility through com para ble preference differences

was later axiomatized by Frisch (1926), Lange (1934), and Alt (1936). Along with

Ramsey(1931),these articles helped to introduce the axiom atic m ethod into utility

theory. Although their impact was limited by the continuing popularity ofth e or-

dinal position, they appeared to have im planted a strong association between car-

dinal utility and comparable preference differences in the minds of many

econom ists, if in fact this h ad not already been done by the likes of Jevons (1871)

and Marshall (1890) in the preceding century.

W hen von N eu m ann and M orgenstern entered the scene a few years later, they

tried to divorce their approach from this association. Their success in this en-

deavor will be discussed shortly. We shall also comment on a few of their ax-

iomatic pecu liarities before we consider their own words. First, however, we recall

a typical present-day rendition of their theory.

Difficulties and obscurities in von N eum an n and M orgenstern's own presenta-

tion led a number of people, including Marschak (1950), Friedman and Savage


6/32

132

PETER FISHBURN

X,

SO

Xp+ {1- X)qis inPw hen/?,qG P a n d

0

. to

P,

and say that it is aweakorderif it is asymmetric \p > q=>not{q > p)]

both > and ~, defined as

p~q

if neither/? >q norq >/? , are transitive[{p > q, q

r)

=>/?

>

r;

(p~q, q~r) =>/?-/].Jensen's axioms are these: Forall/?,9, r P a n d al

> 1

1 (order): > on P is a weak order;

2 ( independence ) :

p > q ^ Ip + (I - X)r > hj + {I - X)r;

3 (co ntin uity ): /? > ^ > r

=>

a/? + (1 - a )r > ^ > P/? + (1 - ^)r for som e a and

.strictly

between 0 and 1.

These axioms say that preferences are nicely ordered, that similar convex com

binations with a third distribution{rin 2) preserve preference, and that a distribu

tion between two others in preference {q in 3) is also between nontrivial convex

combinations of those two.

The crux ofthe von Neumann-Morgenstern utility theorem is this: Given > on

convex

P,

axiom s 1, 2 an d 3 hold if and only if there is a real-valued function uo

P such that, for all p, q E. P and all 0 q

u{q), (1*

u{-kp+ (1 - X)q) = Xu{p) + (1 - X)u{q), (7

and these conditions on

u

imply tha t it is un ique up to a positive linear transfor

mation. Property (7) is referred to aslinearity, and it is this property that leads

the expected utility form for u(p). In particular, if P contains every finite-suppor

distributioh o n Z a n d we de fin es on Zb y (jc) = (/?) when/?(x) = 1, then a simpl

induction with (7) shows that

"(p) =

2 J ' ( - ^ ) " ( ^ )

for every finite-support p G

P.

Extensions of this form to u(p) = j u{x)dp{x) for other types of probability dis

tributions are discussed by Blackwell and Girshick (1954), Arrow (1958), Jensen

(1967a), and Fis hb urn (1970,1982) am ong others. Such extens ions, which wereno

discussed by von Neumann and Morgenstern, require additional axioms.

One final remark m ay be helpful before we proceed. The preced ing rend ition of

the von Ne um ann -M org ens tern theory, as well as other recent contributions cited

earlier, lies within the broad er subject o ftherepresentational theoryof measureme


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UTILITY THEORY OF VON NEUMANN AND MORGENSTERN 133

the representational theory are given in the books by Krantz et al. (1971) and

Roberts (1979).

The modern representational theory of measurement is very much concerned

with the uniqueness p roperties of representing functions or, as is sometimes said,

withscalesof measurem ent, a topic tha t ha s traces in Felix Klein's (1872) program

for classifying geometric structures by their automorph ism groups. The definitions

of ordinal utility, based on monotone transformations, and cardinal utility, based

on linear transformations, are simple but important examples ofthe uniqueness

theme. Additional information and references to recent developments are avail-

able in the preceding books and the excellent survey article by Luce and

Narens (1987).

2. Preliminaries and interpretations

The later axiom atizations of linear (expected) utility given by Jensen (1967a) and

others tend to look quite different than the von Neumann-Morgenstern original.

Some ofthe differences are cosmetic, but a few are substantive. They will be ad-

dressed in this section after we review other issues that deserve clarification.

2.1. Comparable preference differences

The most enduring interpretational problem ofthe von Neumann-Morgenstern

theory has been the extent to which it em bodies a no tion of com parable preference

differences either between pairs of outcomes or between pairs of probability dis-

tributions on outcomes. For outcomes themselves we have in mind the

risklessin-

tensityview of m easurable utility adop ted by Bernoulli (1738) and many nine-

teenth-century economists that was subsequently axiomatized by Frisch (1926)

and others. The extension of this view to differences between probability dis-

tributions or risky ventures is considered by Hagen (1979) and Allais (1979,

1988).

The formal axiomatic theory of von Neumann and Morgenstern is devoid of

allusion to com parable preference differences. Its axioms are stated solely in terms

ofasimple preference relation on their set f/of things to be compared : there is no

hin t of a relation like in (6). Jensen's axioms 1 through 3 ofthe preced ing sec-

tion illustrate the point. Moreover, their formal theory m akes no reference to out-


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134 PETER FISHBURN

This discussion has not established yet that von Ne um an n and M orgenstern do

not themselves regard their operation, mistakenly, as measuring differences in

satisfaction. The evidence for this is their repeated rejection ofthe notion that an

individual reaches decisions in risk-taking situations by calculating differences

in utilities, their brand or any other... .But much confusion probably stems

from the fact that they are prone to write in large, clear type about comparing

differences in preferences an d to discard such notions in fine print at the bottom

of the page. (p. 551

Ellsberg goes on to illustrate how von Neumann and Morgenstern offer the

comparable-differences bait only to retract it an instant later. We shall note this

during the exegesis.

Others of course support the separation of the von Neumann-Morgenstern

theory from older doctrines. In the words of Baumol (1958),

It isnot the purpose of the Neumann-Morgenstern utility index to set up any

sort of measure of introspective pleasure intensity. (p. 665)

Several years earlier. Arrow (1951) noted that, for von Neumann and Morgen-

stern,

. . . the utilities assigned are not in any sense to be interpreted as some intrinsic

am ount of good in the outcom e (p. 425)

In their account of comm on mistakes that people make about the von N eu m an n-

Morgenstern theory. Luce and Raiffa (1957) offer us

Fallacy 3.Suppose that A > B > C > D and that the utility fuction hastheprop

erty that u(A) - u(B) > u (C) - u(D),then thechangefrom B to A is more pre

ferred than the change from D to

C.

(p. 32

But not everyone agrees that von N eum ann and M orgenstern did not set out to ax-

iomatize an older version of cardinal utility. The most vocal dissenter may be

Allais (1979), who quotes four brief passages on von Ne um an n and M orgenstern

and goes on to say in his own words that

For all these passages... it results that von Neum ann-M orgenstem indeed aim

to determine differences in satisfaction (or utility), that according to them such

differences cannot be determined by direct experiment, and that therefore it is


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UTILITY THEORY OF VON NEUMANN AND MORGEN STERN 135

Allais's severe misinterpretation of von Ne um ann and Morgenstern's intentions

might be exacerbated by his repeated use of

neo-Bemoullian

to charac terize

various renditions of their theory. Bernoulli'sfunctional form 2 p{x)u{x) does

follow from the von Neumann-Morgenstern linear representation whenX is in-

troduced, but his other main proposition, the riskless intensity view of outcome

utility, has no place in their theory. Th is view

is,

however, a cornerstone of Allais's

own theory, which might equally well be referred to asneo-Bemoullian although it

rejects Bernoulli's expectation form. Since the theories of von Neumann-

Morgenstern and Allais each have a different foot in Bernoulli's foundation, it

would seem desirable to avoid the use of

neo-Bemoullian

for either alone.

Others have made misleading statements about von Neumann-Morgenstern

utility or have stumbled over the passages in their book that, when taken out of

context, could suggest their interest in characterizing an interpretation of utility

that they firmly reject. For example, Marschak (1979, p. 165 and footnote 4) gives

the impression that theories like von Neumann and Morgenstern's follow Ber-

noulli in the measurement of outcome utility, but corrects this misrepresentation

on page 166. Friedman and Savage (1948), beginning withA > B > C,add that

If, in addition, the individual should show by his market behavior that he pre-

fers a 50-50 chance of^ or

C

to the certainty

ofB,

it seems natural to rationalize

this behav ior by supposing tha t thedifferencebetween the utilities he attaches to

AandBis greater tha n the

difference

between the u tilities he attaches toBandC,

so that theexpectedutility ofthe preferred com bination is greater than the utility

offi.

(p. 282)

They appear to apologize for this early near-example ofthe Luce-Raiffa

Fallacy 3

in their later paper (Friedm an and Savage,

1952),

and it is not repeated in Savage's

valuable sketch on utility

(1954,

pp . 91-104). Robertson (1954, pa rt II) is hopelessly

confused abou t the relationships am ong von N eum ann-M orgenstern utility, com-

parable differences, and diminishing marginal utility. Jensen (1967a, p. 172)

speaks abo ut "strength o f. .. preference" in a way that m ight suggest the old no -

tion of measurability for his subsequent axioms, and in Jensen (1967b) he claims

that a calculation

clearly shows that von Neu m ann -M orge nste rn's utility measure is a method of

indirect com parison of utility differences. (p. 233)

Although Jensen did not intend to convey an alliance between strength of prefer-


10/32

136 PETER FISHBURN

difference com pariso ns, or notions of "a psychological entity measurable in it

own right."

2.2.

Logical consistency

The von N eum ann -M org ens tern axioms are logically consistent, and their proo

of the linear utility representation in the Appendix added to the 1947 edition i

mathematically faultless. While most later writers do not challenge this, a few

statements deserve comment. Three of these illustrate different sides of the

matter.

In his excellent survey of economic consumption theory, Houthakker (1961

says in regard to theories like that of von Neumann and Morgenstern, which em

phasize choice among "generalized lottery tickets" that

The principal result in this approach, first obtained by Marschak (1950) and

definitively established by Herstein and Milnor

(1953),

is that consistent choic

am ong such tickets implies the existence of a cardinal utility function, (p. 725

This misattribution of priority is perhaps explained by the fact that Houthakke

references only the first edition of von N eum ann and M orgenstern (1944) and , lik

them, makes no mention of Ramsey (1931).

More troublesom e is Bernard 's (1986) claim that there is a flaw in von N eu m an n

and Morgenstern's reasoning (p. 124) or, as he puts it later.

. . . a difficulty in the formal deduction of the function

u

from the ax

oms.

. . . (p 129

This opinion has n othing to do with their mathem atics, but instead reveals a mis

interpretation of their theory.

Pope (1986) writes about

. . . a logical contradiction in von Ne um ann and M orgenstern's postulate

about when events occur. (p_ 215

. . . a timing inconsistency in von N eum ann and M orgenstern's set of ax

ioms, (p 225


11/32


2.3. Timing

Von Neumann and Morgenstern offered their theory in a static or instantaneous

mode. They attempted in their subsection 3.3.3 to avoid complications arising

from "preferences between events in different periods of the future." Nevertheless,

their later comments on their axiom for algebraic simplification of compound

events and on a "utility of gambling" hint at dynamic considerations. Because

phrases such as "reduction of compound lotteries" and comparisons between lot-

teries that are resolved in two or more steps owe more to later writers than to von

Neumann and Morgenstern, I believe it is closer to their own intentions to view

their theory statically. For preference comparisons between probability dis-

tributions on outcomes, this simply m eans that one com pares distributions holis-

tically ra ther than in some multiple-step format. I agree with Allais (1979/1952) on

the holistic perspective, but in any event one can judge for oneself from the von

Neumann-Morgenstern account.

2.4. Uses of utility

In reading that account, it is important to bear in mind that the authors use

utility

in several ways that have qualitative as well as quantitative significance. In speak-

ing of their own theory, von Neumann and Morgenstern most frequently use

utility,

unmodified, in a qualitative sense as an entity in their set f/to which their

preference relation app lies. The represen tational theory of measurem ent sketched

in the preceding section might denote the ir qualitative structure by(U,>). With w,

VG [/, they advise us to readw >vas uis preferable to v"(p.24), and they refer t

w,

V.

... as utilities, (abstract) utilities, events, and combinations (of events).

When von Neumann and Morgenstern talk about whatwerefertotoday as their

linear utility function, they almost always talk aboutnumericalutility,or numerical

valuation (values) of utility. Their utility function is denoted by v, and their expres-

sion of the linearity property (7) is

v(aw + (1 - a)v) = av(w) + (1 - a)v (v) .

Examples of their usage are

Denote the correspondence by

u -* p =

v ( w ) ,


12/32

138 PETER FISHBURN

We will prove

in

this Appendix, that

the

axioms

of

utility enumerated

in3.6.

make utility

a

number

up to a

linear transformation.'

['I.e.

without fixing

a

ze

ora unitof

utility].

(p. 61

Although their languagehasbeen troublesome to some, their theoryis one

the best early examples ofth e representational theoryofm easurementin the s

cial sciences.

2.5. Indifference

and

independence axioms

Readers who are familiar with

a

later rendition ofthe von N eum ann -M orge nster

theory,but notwiththeoriginalor with commentaries that connectthe two, a

likely

to

encounter three mysteries

in the

original:

Mystery

1:

N othing

is

said about

a set of

outcomes;

Mystery

2:

There

is no

indifference relation

(~);

Mystery 3: Thereis no obvious independence axiom.

The following explications are indebted to Malinvaud's (1952) remarkableon

page solution

of all

three mysteries.

1.The vonNeum ann-Morgenstern domainset f/is formulated abstractlyan

endowed with closure and other properties based on an abstract convexitylik

operation writtenasaw+ (1a)v for0< a to

their dom ain

setU.In

other words,

Uis an

abstract

set of in

ference classesandeach utilityuisoneofthe indifference classes. Their Append

notesthepossib ilityofaxiom atizing indifference also,butthey observe that the

approach is "mathematically perfectly sound," which it is.

3.

Using the notation o fthe preceding section,

the

independence axiom adopte

by Jensen (1967a),

i.e.,


13/32

UTILITY THEORY

OF

VON NEUMANN AND MORGENSTERN

139

p>q=^Xp + {\ - X)r>Xq + {\ - X)r (Samuelson, 1952)

p ~ q =>Xp + {\ - X)r ~ Xq + {\ - Xy

(Malinvaud, 1952)

and to the even sparer independence axiom

p ~ q =^ Vip + Vir ~ Viq+Vir

of Herstein and Milnor

(1953).

In the second expression, >denotes the union of >

and ~. The mystery of why nothing like these axioms appears explicitly among

those of von Neumann-Morgenstern is resolved by the answer to mystery2.Sup-

pose/? and^ are in the same member Moft/ andri s in a membervof {/.Then/? ~

q.Moreover, withw = Xu + {\ - X)v,and = viewed astrueidentity, it must be true

that both A/7+ (1-X)randXq + (1 - > )rare invv.Butthen

A/7

+ {}-Xy ~Xq + {\

- Xy, so we obtain Malinvaud's version of Samuelson's strong independence

axiom. Jensen's strict-preference version can be obtained by appealing to axioms

stated explicitly by von Neumann and Morgenstern.

Finally, it must be recalled that these explanations overlaid later developments

on the formal theory of von Neumann and Morgenstern, who were quite content

to bypass indifference for the sake of parsimony in the mathematical presentation

and to leave its discussion to their informal preparation and, in the second edition,

to the Appendix also.

3.

Preface

to the

third edition

The papers by Samuelson (1952) and Malinvaud (1952) were part of a brief sym-

posium on independence in the October 1952 issue

oiEconometrica.

After citing

the symposium papers in their preface to the third edition, von Neumann and

Morgenstern go on to say that

Ln connection with the methodological critique exercised by some of the con-

tributors to the last-mentioned symposium, we would like to mention that we

applied the axiomatic method in the customary way with the customary pre-

cautions. Thus the strict, axiomatic treatment ofthe concept of utility (in Section

3.6. and in the Appendix) is complemented by an heuristic preparation (in Sec-

tions 3.1.-3.5.). The latter's function is to convey to the reader the viewpoints to

evaluate and to circumscribe the validity of the subsequent axiomatic pro-

cedure. In particular our discussion and selection of natural operations in

those sections covers what seems to us the relevant substrate ofthe Samuelson-


14/32

140 PETER FISHBURN

"substrate" constitutes the ideas developed in their "heuristic preparation" tha

support and give substance to the formal theory in section 3.6.

Before they begin their heuristic preparation in section

3.1,

von Neumann an

Morgenstern place their treatment of utility within their larger concern with th

theory of games.

The conceptual and practical difficulties o fthe notion of utility, and particularl

ofthe attempts to describe it as a num ber, are well known and their treatmen t i

not among the primary objectives ofthiswork. We shall nevertheless be force

to discuss them in some instances, in particular in 3.3. and 3.5. Let it be said a

once that the standpoint o fthe present book on this very impo rtant and very in

teresting question will be mainly opportunistic. We wish to concentrate on on

problem which is not that ofthe measurement of utilities and of preferences

and we shall therefore attempt to simplify all other characteristics as far a

reasonably possible. (p. 8

According to Savage (1954),

Von Neum ann and Morgenstern initiated am ong econom ists and , to a lesser ex

tent, also among statisticians an intense revival of interest in the technical utilit

concept by their treatment of utility, which appears as a digression in [the

book].

(p. 97

But what a digression Its impact could scarcely have been imag ined. Shortl

before he died (in 1977; von Neumann passed away in 1957), Morgenstern (1979

wrote that

It is one of the great pleasures of my life that those few passages we wrote o

utility theory have provided so much stimulus for others to concern themselve

deeply and in a fresh m an ne r with the no tion of utility which is forever basic fo

any economic theory. (p. 182

It is to "those few passages" that we now turn.

4.

Initial preparations

The von Neum ann-M orgens tern treatment of utility is mainly found in section 3

titled

TheNotionof Utility,

of their chapter I. The ensuing quotations retain th


15/32

LITY THEORY OF VON NEUMANN AND MORGENSTERN 141

notion of utility. Many economists will feel that we are assuming far too

much ... and that our standpoint is a retrogression from the more cautious

modern technique of "indifference curves."

Here, and later, their references to "indifference curves" signify the indifference-

map approach o fthe ordinalists described in the parag raph surroun ding

5)

in our

section 1.

The next paragraph of3.1.1gives a brief apologia for their "high handed treat-

ment of preferences and utilities." This is followed by

We feel, however, that one part of our assumptions at leastthat of treating

utilities as num erically m easurab le quantities is no t quite as radical as is often

assumed in the literature. We shall attempt to prove this particular point in the

paragraphs which follow. It is hoped that the reader will forgive us for discuss-

ing only incidentally in a condensed form a subject of

so

great a conceptual im-

portance as that of utility. It seems however that even a few remarks may be

helpful, because the question of the measurability of utilities is similar in

character to corresponding questions in the physical sciences.

Their subsequent analogies to measurement in the physical sciences give them

familiar ground for speaking about utility measurement. Their allusions to "radi-

cal"

assumptions "in the literature" concern intensive introspective approaches

and difference comparisons described earlier. This is their first indication that

they are not interested in resurrecting some previous notion of card inal utility. The

narrative continues on a historical note.

3.1.2. Historically, utility was first conceived as quantitatively measurable, i.e.

as a num ber. Valid objections can be and have been made against this view in

its original, naive form. It is clear that every measurementor rather every

claim of measurabilitymust ultimately be based on some immediate sensa-

tion, which possibly cannot and certainly need not be analyzed any futher.'

['Suchas the sensations of light, heat, m uscu lar effort, etc., in the corresponding

branches of physics.] In the case of utility the immediate sensation of prefer-

enceof one object or aggregate of objects as against anotherprovides this

basis.But this permits us only to say when for one person one utility is greater

than another. It is not in itself a basis for numerical comparison of utilities for

one person nor of any comparison between different persons. Since there is no

intuitively significant way to add two utilities for the sam e perso n, the assum p-


16/32


17/32

UTILITY THEORY OF VON NEUMANN AND MORG ENSTERN 143

compare not only events, but even combinations of events with stated prob-

abilities.

By a com bina tion of two events we mean this: Let the two events be denoted

by i5and C and use, for the sake of simplicity, the probability

50%-50%.

Then the

"com bina tion" is the prospect of seeing fi occur vrith a p robability of

50%

and (if

Bdoes not occur) C with the (remaining) probability of

50%.

We stress that the

two alternatives are mutually exclusive, so that no possibility of complemen-

tarity and the like exists. Also, that an absolute certainty of the occurrence of

either

B or C

exists.

Their "combinations of

events"

are essentially the same as what others refer to as

lotteries, gam bles, risky prospects, rand om prospec ts, option s, and probability dis-

tributions. The novelty of their approach lies in their consideration of preference

between such entities. The omission of indifference in the passage refiects its sup-

pression in their later axiomatization.

The end of their next para grap h through footnote 1 "is probably the greatest

single source of m isunders tand ing" (Ellsberg, 1954, p. 552) of their inten tions . The

retraction is in footnote 2.

To restate our position. We expect the individua l u nder cons ideration to possess

a clear intuition whether he prefers the ev en ts to the 50-50 combination of B or

C, or conversely. It is clear that if he p refers^ toBand also toC,then he will pre-

fer it to the above com bination as well; similariy, if

he

prefers fi as well asCtOy4,

then he will prefer the com bina tion too. But if he should

prekrA

to, say

B,

but at

the same time C to A, then any assertion about his preference of A against the

combination contains fundamentally new information. Specifically: If he now

prefers/I to the 50-50 com bina tion of fi and C, this provides a plaus ible base for

the numerical estimate that his preference of AoverB is in excess of his prefer-

ence of C over/l.'^ ['To give a simple example: Assume that an individual pre-

fers the consumption of a glass of tea to that of a cup of coffee, and the cup of

coffee to a glass of milk. Ifwenow want to know whether the last preference

i.e., difference in utilitiesexceeds the former, it suffices to place him in a situa-

tion where he m ust decide this: Does he prefer a cup of coffee to a glass the con-

tent of which will be determined by a 50%-50% chance device as tea or milk.]

^Observe that we have only postulated an individual intuition which permits

decision as to which oftwo"events" is preferable. But we have not directly pos-

tulated any intuitive estimate ofthe relative sizes oftwopreferencesi.e. in the

subsequent terminology, of two differences of

u tilities.

This is impo rtant, since

the former information ought to be obtainable in a reproducible way by

mere "questioning."]


18/32

^^

PETER FISHBUR

I believe that von Neumann and Morgenstern brought differences into the pi

ture for two m ain reasons. The first was to touch base w ith the familiar notion

preference-difference com parisons as a suggestive device, and to also disclaim

relevance to their own theory. To step out of sequence for a moment, here

another example, from their Appendix (pp. 630-631):

... No matter how much the utility

v

exceeds (or is exceeded by) the utility

and no matter how little the utilitywexceeds (or is exceeded by) the utilityu,

is admixed tou with a sufficiently small numerical probability, the differen

that this admix ture makes fromuwill be less than the difference of

w

from

M

[' ... The reader will also note that we are talking of entities like "the excess of

over

M"

or the "excess of u over v" or (to combine the two former) the "d

crepancy of wand v" (u , v, being utilities) merely to facilitate the verb

discussionthey are not part of our rigorous, axiomatic system.]

The second reason for introducing differences was to make the

mathematical

po

that even though their own approach used only simple preference comparisons,

had at least as much power in arriving at card inal utility as the difference metho

of Pareto and others. Since the following passage is based on simple compariso

between co m binations of events and proceeds by analogy to the older me thod,

should not be interpreted as condoning that method. We continue in sequen

from the preceding footnote 2.

If this standpoin t is accepted, then there is a criterion w ith which to com pare t

preference of C overy4 with the preference of ^ overB. It is well known th

thereby utilitiesor rather differences of utilitiesbecome numerically me

surable.

That the possibility of com parison between /I, B,and C only to this extent

already sufficient for a num erical measurem ent of "distances" was first observ

in economics by Pareto. Exactly the same argum ent has been made, however, b

Euclid for the pos ition of poin ts on a line in fact it is the very basis of his class

cal derivation of numerical distances.

The introduction of numerical measures can be achieved even more direct

if use is made of all possible probabilities. Indeed: Co nsider three even ts,C.A.

for which the order ofthe individual's preferences is the one stated. Let a be

real num ber between 0 and 1, such th at /I is exactly equally desirable with th

com bined event consisting of a chance of probability 1 - a for fl and th

remaining chance of probability a for C. Then we suggest the use of a as

num erical estimate for the ratio ofth e preference of^ over

J5

to tha t of

C

over


19/32

LITY THEORY OF VON NEUMANN AND MORGENSTERN 145

6. Time, probability, and indifference curves

Having prepared us for their later discussion of numerical utility, von Neumann

and Morgenstern reflect on other concerns.

3.3.3.

To avoid m isunders tand ings let us state that the "events" which were used

above as the substratum of preferences are conceived as future events so as to

make all logically possible a lternatives equally adm issible. However, it would be

an unnecessary complication, as far as our present objectives are concerned, to

get entangled with the problems ofthe preferences between events in different

periods ofthe future. It seems, however, that such difficulties can be obviated by

locating all "events" in which we are interested at one and the same, standard-

ized moment, preferably in the immediate future.

My comm ents on timing in section 2 refer to this pa rag rap h. See also Pope (1986)

for concerns about certain versus risky options.

Following a brief parag raph on their need for a numerical concept of proba bil-

ity, the authors continue.

Probability has often been visualized as a subjective concep t more or less in the

nature ofan estimation. Since we propose to use it in cons tructing an individual,

numerical estimation of utility, the above view of probability would not serve

our pu rpose . The simplest procedure is, therefore, to insist upon the alternative,

perfectly well founded interpretation of probability as frequency in long runs.

This gives directly the necessary numerical foothold.^ pif one objects to the fre-

quency interpretation of probability then the two concepts (probability and

preference) can be axiomatized together. Th is too leads to a satisfactory num eri-

cal concept of utility which will be discussed on another occasion.]

A particular interpretation of probability for their formal theory is beside the

point, since their dom ain set f/makes no men tion ofprobability pe rse but only of

numbers in the interval (0, 1). However, they do want to avoid a vague concept in

informal interpretation and appeal to the long-run frequency idea developed by

Venn (1962/1886), von Mises (1957/1928), and Reichenbach (1949/1935), among

others. The joint axiom atization ofprob ability and utility mentioned in the foot-

note was not carried out by von Neumann and Morgenstern, but Morgenstern

(1979,p. 176) cites Pfanzagl (1967, 1968) in this regard while overlooking Ram sey

(1931) and Savage (1954).

Further remarks on comparability and indifference come next.


20/32

146 PETER FISHBUR

of individual preferences. It is conceivableand may even in a way be mo

realisticto allow for cases where the individua l is neither ab le to state which

two alternatives he prefers nor that they are equally desirable. In this case th

treatment by indifference curves becomes impracticable too."

[''These

problem

belong systematically in the mathematical theory of ordered sets. The abov

question in particular amounts to asking whether events, with respect to prefe

ence, form a completely or a partially ordered set. Cf. 65.3.]

Their m ain point here is that if two things are incom parab le, then one cann ot eve

ob tain an ord inal utility function as in

(1).

Section

65.3,

mentioned in the footnot

discusses partial orders and acyclic relations. The first axiomatization of a vo

Neumann-Morgenstern-type linear utility function for these cases is due t

Aum an n (1962). Later contribution s are noted in F ishbu rn (1988a, p. 53).

Subsection 3.3.4 concludes by reiterating the poin t m ade a t the open ing of 3.3

that if one goes as far as an ordinal utility representation, then their approach r

quires little more to obtain a cardinal utility representation. And section

3.3

co

cludes with additional comments on the desirability of pursuing the details th

lead to such a representation.

7.

Natural operations

Section

3.4

opens with a repeat of the difference theme and the by-now

familiar retraction.

3.4.1. The reader may feel, on the basis ofthe foregoing, that we obtained

num erical scale of utility on ly by begging the p rinciple, i.e., by really postu latin

the existence of such a scale. We have argued in 3.3.2. that if an individual pre

fers^

to the 50-50 combination offi and C (while preferring Cto y4 andA toB

this provides a plausible basis for the numerical estimate tha t this preference o

A o ver Bexceeds that ofCovcrA. Are we not pos tulating hereor taking it fo

grantedthat one preference may exceed another, i.e. that such statements con

vey a meaning? Such a view would be a complete misunderstanding of ou

procedure.

3.4.2. We are not postulatingor assuminganything of the kind. We hav

assumed only one thingand for this there is good empirical evidence

namely that imagined events can be combined with probab ilities. And therefor

the same must be assumed for the utilities attached to them,whatever the


21/32

LITY THEORY OF VON NEUMANN AND MORGEN STERN 147

3.4.3.

In all these cases where such a "natu ral" operation is given a nam e which

is reminiscent of a mathematical operationlike the instances of "addition"

aboveone must carefully avoid misunderstandings. This nomenclature is not

intended as a claim that the two operation s with the sam e nam e are identical,

this is manifestly not the case; it only expresses the opinion that they possess

similar traits, and the hope that some correspondence between them will ul-

timately be established. Th is of coursewhen feasible at allis done by finding

a mathematical model for the physical domain in question, within which those

quantities are defined by num bers, so that in the model the m athem atical opera-

tion describes the synonymous "natural" operation.

We recognize here an elementary expression of the representational theory of

measurement. A sophisticated treatment of this theory for the physical sciences

appears in chapter 10 of Krantz et al. (1971).

8. Transformations

Section 3.4 continues, after examples of "natural" and ma thematical operations in

physics, with

3.4.4.

Here a further remark must be made. Assume that a satisfactory mathe-

matical model for a physical domain in the above sense has been found, and

that the physical quantities under consideration have been correlated with

numbers. In this case it is not true necessarily that the description (of the

mathematical model) provides for a unique way of correlating the physical

quantities to nu m bers; i.e., it may specify an entire family of such correlations

the mathem atical nam e is mapp ingsany o ne of which can be used for the pur-

poses ofthe theory. Passage from one of these correlations to another amounts

to atransformationo fthe num erical data describing the physical quantities. We

then say that in this theory the physical quantities in question are described by

numbersup tothat system of transformations. T he m athem atical nam e of such

transformation systems isgroups.

A group of transformations for a mathem atical model is tantam ount to the set of

all functions tha t m ap a qualitative structure into a quantitative structure. We have

used theirup todesigna tion to describe the uniqueness cha racters of ordin al utility

and cardinal utility in section 1.

Von Neum ann and Morgenstern go on to describe different types of transform a-


22/32

148 PETER FISHBUR

monotone transformation" but with "the development of th er m om et ry ... th

transformations were restricted to linear ones, i.e. only the absolute zero and ab

solute unit were missing" (p. 23).

For utility the situation seems to be of a similar nature. One may take the a

titude that the only "natural" datum in this domain is the relation "greater," i.

the concept of preference. In this case utilities are numerical up to a monoton

transformation. This is, indeed, the generally accepted standp oint in econom

literature, best expressed in the technique of indifference curves.

To narrow the system of transformations it would be necessary to discove

further "natural" operations or relations in the domain of utility. Thus it wa

pointed out by Pareto'' fK

Pareto,

Manuel d'Economie Politique, Paris,

1907

264.] that an equality relation for utility differences would suffice; in our te

minology it would reduce the transformation system to the linear transfo

mations. However, since it does not seem that this relation is really a "natura

onei.e. one which can be interpreted by reproducible observationsthe sug

gestion does not achieve the purpose.

Thus they reject Pareto's approach to cardinal utility because its difference rel

tion is not "natural." In particular, their approach of choice between com bination

provides direct access to numerical comparisons used in constructing their utilit

funcfion, but there is no thing like this for the preference-difference concep tio

which must rely on introspection.

9. Tbe utility correspondence

The next .paragraph describes their alternative to Pareto 's approach in light o

what they consider "natural."

3.5.1. The failure of one particular device need not exclude the possibility o

achieving the same end by anothe r device. Ou r contention is that the dom ain o

utility contains a "natural" operation which narrows the system of transfo

mations to precisely the same extent as the other device would have do ne. This

the com bina tion of two utilities with two given alternative probab ilities a,1

(0 < a < 1) as described in 3.3.2. The process is so similar to the formation o

centers of gravity mentioned in 3.4.3. that it may be advantageous to use th

same terminology. Thus we have for utilities u,v the "natural"relation u >

(read:uis preferable to

v),

and the "natural"operation a + (1 - a)v, (0 < a <


23/32


The m ain thing they have

not

told us is tha t the u tilities

u

and

v

are being viewed as

indifference classes of combinations or probability distributions. This becomes

clear only when one studies their ensuing axioms and the 1947 Appendix.

Having sketched their qualitative structure, von Neumann and Morgenstern

become notationally more explicit about their quantitative (numerical) structure

and the mapping or "correspondence" that ties the two together.

Denote the correspondence by

u -f p = V ( M ) ,

u

being the utility and

V(M)

the num ber w hich the correspondence attaches to it.

Our requirements are then:

(3 : l : a )

u > v

implies

V(M)

> v(v),

(3 : l :b ) y{au + (1 - a)v) = av(M) + (1 - a)v (v) .'

['Observe that in each case the left-hand side has the "natural" concepts for

utilities, and the right-hand side the conventional ones for numbers.]

If two such correspondences

(3:2:a)

M

^ p =

V(M),

(3:2:b) ^ p ' = V ' (M),

should exist, then they set up a correspondence between numbers

(3:3) p ^ p ' ,

for which we may also write

(3:4) p = 0( p).

Since (3:2:a), (3:2:b) fulfill (3:l:a), (3:l :b) , the correspondence (3:3), i.e. the func-

tion (t)(p) in (3:4) must leave the relation p > o and the operation ap -I- (1 a)o

unaffected. I.e.

(3 :5 :a) p > a implies (t)(p) > (t)(o),

(3:5:b ) (l)(ap + (1 - a)o ) =


24/32

150

PETER FISHBURN

mined up to a linear transformation. I.e. then utility is a number up to a

linear transformation.

The order-preserving part of their representation (3:l:a) does not need the con-

verse implication,v w)> v(v) =*w > v, because it follows from the omission of

ind

ference: see (3:A:a) in the next section. The second paragraph spells out in modest

detail what it means to say that their utility function is unique up to a positive

linear transformation. We add "positive" to be more explicit about COQ > 0. Th

mathematical

termgroup

at the end of the quote from

3.4.4

in section 8 refers her

to the set of transformations of real-valued funcfions established by (3:3) through

(3:5:b) and summarized in (3:6).

10 .

The axioms

This brings us to the axioms for their qualitative structureconsisting ofthe do-

main set t/a n d its "na tural" relation

w

>

v

(preference) and "natu ral" opera tion a

+ (1 - a)v (formation of lotteries)that

... imply the existence of a correspondence (3:2:a) with the properties (3:l:a),

(3:l:b) as described in

3.5.1.

They note that there is some freedom in choosing the particu lar axiom s tha t yield

the desired implication, and emphasize that

... each axiom should have an immediate intuitive meaning by which its ap-

propriateness may be judged directly.

They say that their axiom set "seems to be essentially satisfactory."

3.6.1. Our axioms are these:

We consider a system {/of entitiesu,v,w....ln Uarelationis given,w> v, an

for any n um ber a, (0 < a < 1), anoperation

aw + (1 a)v = w.

These concepts satisfy the following axioms:

(3:A) u > V is acomplete orderingofU.


25/32

UTILITY THEORY

OF

VON NEUMANN

AND

MORGENSTERN

151

W

= V w > V w < V.

(3:A:b)

u > v, v > wimply u > w.

(3:B)

Ordering

and

combining.

(3:B:a) M< v implies that

w

< aw + (1 - a)v.

(3:B:b) w > v implies that w > aw + (1

a)v.

(3:B:c) w < w < v implies the existence ofan a with

aw + (1 a)v w > vimplies the existence ofan a with

aw -I- (1 a)v > w.

(3:C)

Algebra

of

combining.

(3:C:a) aw -I- (1 - a)v = (1 - a)v + aw.

(3:C:b) a(Pw -I- (1 - P)v) + (1 - a)v = yw + (1 - Y)V

where y = ap.

One can show that these axioms imply the existence of a correspondence

(3:2:a) with the properties (3:l:a), (3:l:b) as described in

3.5.1.

Their proof of the final assertion, in the Appendix, runs along conventional lines

and presents no particular difficulties (pp. 26-27). In the Appendix itself they say

that a shorter proof might be found later. This indeed turned out to be true (Her-

stein and Milnor, 1953; Fishburn, 1970), although the substance of these later

proofs is similar to the original.

A few technical remarks about their axioms are in order before we turn to

their interpretations.

The final two axioms (3:C:a) and (3:C:b), explain properties ofthe combining

operation aw + (1 - a)v, for all

w

v C/andall a (0,1). The expression aw + (1 -

a)v = w near the beginning of

3.6.1

says that f/is closed under the operation, and

might also be considered as an axiom. Under closure, (3:C:a) is a symmetry prop-

erty and (3:C:b) is an algebraic reduction property. The authors take care to note

that the coefficients a, p, and y are always strictly between 0 and 1.This was mod-

ified by Herstein and Milnor (1953) who allow0and 1 also and, with the addition

ofthe axiom lw + (1 - l)v = w, refer to their domain set as amixture set

All structure needed for C/ in connection with the combining operation has been

specified. The other axioms, (3:A) and (3:B), bring the preference relation into the


26/32

152 PETER FISHBURN

Axiom (3:B:a) is a monotonicity condition, (3:B:b) is its natural dual, (3:B:c) is

an Archimedean or "continuity" postulate, and (3:B:d) is

its

natura l dua l. The lat

ter pairjs mimicked in Jensen's (1967a) axiom 3 of section 1, but neither Jensen

nor Herstein and Milnor has a monotonicity condition. This is because ex-

pressions like (3:B:a) and (3:B:b) follow from their other axioms, including their

independence conditions . Again, see the end of section2for further discussion, es

pecially in regard to the invisibility of an independence axiom in the von

Neumann-Morgenstern set.

11.

Interpretations

Von Neumann and Morgenstern follow the preceding list of axioms with words

about their intuitive meanings.

3.6.2. The analysis of our postulates follows:

(3:A:a*) Th is is the statem ent of the com pleteness of the system of in

dividual preferences. It is custom ary to assum e this when discuss

ing utilities or preferences, e.g. in the "indifference curve ana lysis

method." These questions were already considered in 3.3.4. and

3.4.6.

(3:A:b*) Th is is the "transitivity" of preference , a plausible an d generally

accepted property.

(3:B:a*) We state here: Ifv is preferable tou,then even a chance 1 -

a

Valternatively to wis preferable. This is legitimate since any

kind of complementarity (or the opposite) has been excluded, cf

the beginning of 3.3.2.

(3:B:b*) Th is is the dual of (3:B:a*), with "less preferable" in place o

"preferable."

(3:B:c*) We state here: Ifw is preferable to w, and an even more prefe

able

V

is also given, then the com bination ofw with a chance

1

-

of

V

will not affectw spreferability to it if this chance is small e

ough. I.e.: However desirable

v

m ay be in

itself,

one can make it

infiuence as weak as desired by giving it a sufficiently smal

chance. This is a plausible "continuity" assumption.

(3:B:d*) Th is is the dual of (3:B:c*), with "less preferab le" in place of

"preferable."

(3:C:a*) This is the statement that it is irrelevant in which order the con-

stituents M,V of a combination are named. It is legitimate, pa


27/32


course the correct arithmetic of accounting for two successive ad-

mixtures ofV withw.]The same things can be said for this as for

(3:C:a*) above. It may be, however, that this postulate has a

deeper significance, to which one allusion is made in

3.7.1.

below.

I have only two comments before we consider the authors' final remarks on the

axioms. First, their further mention of indifference curves in (3:A:a*) and refer-

ence back to

3.3.4

and

3.3.6

were probab ly felt by von Neum ann and Morgenstern

to provide ample indication that they were not going to treat indifference ex-

plicitly, especially in view of

w

=

v

in

(3:A:a).

This is supported by the quotation in

section

3

from their

1953

preface. However, the

true identity

remark in their Appen-

dix and later com ments there on indifference

(p.

631) suggest tha t they were aw are

of difficulties in the m atter prior to Septehiber,

1946,

the date of their preface to the

second edition.

Second, we see in (3:C:b*) the two-step idea for interpreting the algebraic reduc-

tion axiom (3:C:b). This has been widely used by later writers to defend the

reasonableness ofindepen dence axioms like 2 in section

1:

if you prefer/? to

q

a nd

let

X

govern the first step, then you ought to prefer

Ap

+ (1

- Xytolxj + {I - X)r.

Moreover, we know that comparisons between lotteries can be quite sensitive to

whether they are stated hoHstically or in a two-step frame (Allais, 1979/1952;

Ka hne m an and Tversky, 1979; M acCrim m on and Larsson, 1979). Von N eum ann

and Morgenstern's "deeper significance" phrase at the end of (3:C:b*) and their

subsequent discussion ofa"utility of gam bling" (pp. 28 and 629) hint at the latter

possibility. My own views on the matter, with additional references, appear in sec-

tions 2.5 and 8.1 of Fishburn (1988a).

12.

Further remarks

Von N eu m ann and M orgenstern had a good deal more to say about utility, but I

shall be brief

In

3.7.1

they wonder whether there m ay be a positive or negative "utility of the

mere act of'taking a chance,' of gamb ling, which the use ofth e ma thematical ex-

pectation obliterates." They cite (3:C:b) as the axiom tha t comes closest to exclud-

ing a "utility of gambling," say that its adequate consideration would require "a

much more refined system of psychology ... than the one now available for the

purposes of econom ics," and claim that it "cannot be formulated free of contradic-

tion" on the level of their (3:A)-(3:C). Although several attempts to be more sys-


28/32

5 4 PETER FISHBURN

and Ch ipm an (1960). The joint weakening of completeness and continuity, which

is also suggested in

3.7.2,

is treated axiomatically in Fishburn (1979).

It is fitting to give von Neumann and Morgenstem the last word.

3.7.3.

This brief exposition does not claim to exhaust the subject, but

we

hope

to have conveyed the essential points. To avoid misunderstandings, the follow-

ing further remarks may be useful.

(1) We re-emphasize that we are considering only utilities experienced by one

person. These considerations do not imply anything concerning the com-

parisons of the utilities belonging to different individuals.

(2) It cannot be denied that the analysis of the methods which make use of

mathematical expectation ... is far from concluded at present. Our remarks in

3.7.1.

lie in this direction, but much more should be said in this respect. There

are m any interesting questions involved, which however lie beyond the scope of

this work. For our purposes it suffices to observe that the validity ofthe simple

and plausible axioms (3:A)-(3:C) in

3.6.1.

for the relationw > v and the oper

tion

au + (I - a)v

makes the utilities num bers up to a linear transform ation in

the sense discussed in these sections.

13.

Postscript

Because von Neumann and Morgenstern did not have an explicit independence

axiom in their system, they obviously could not talk abou t weaken ing it. However

many others have. Allais (1979/1952, 1979) is famous for his examples that refute

the general reasonableness of independence axioms; his own theory rejects in-

dependence and its associated expected-utility form. Reviews of more recent

theories of preference comparisons under risk that weaken traditional indepen-

dence axioms are given by Machina (1987a, 1987b) and Fishburn (1988a,

1988b).

I would like to add some personal notes. In the abstract I spoke about "the

genius and spirit of the theory of utility fashioned by John von Neumann and

Oska r M orgenstern."

As

an exegete of theirwork.I have come to a new adm iration

for the depth and insight of their "digression" (their own word as well as Savage's)

on utility. It is my hope that the exposition will help others appreciate their

contribution.

I regret that I never met von Neumann, but I was privileged to know Oskar

Morgenstern. H is very personable account of their collaboration on theTheory o

Games andEconomic Behavior

is recorded in Morgenstern (1976).


29/32


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