T H E THEORY O F DECISION M A K I N G 1 W A R D EDWARDS T h e Johns Hopkins University
Many social scientists other than psychologists try to account for the behavior o f individuals. Economists
e c o n o m i s t s call it, the theory of con-
and a few psychologists have produced a large body of theory and a fe w experiments that deal with individual decision making. T h e kind o f decision making with which this body o f theory deals is as f o l l o w s : given two states, A and B into either one of either one w h i c h an individual may put h i m s e l f , the individual chooses A in p r e f e r ence to B (or vice versa). For in a child standing in f r o n t of a stance, stance, candy counter may be considering two states. In state A the child has
literatu rature re is almost v o l u m i n o u s . This lite
$0.25 and no candy. In state B the
review will be divided into five sections: the theory of riskless choices, t h e application of the theory o f riskless choices to welfare economics, the theory o f risky choice choices, s, transitivity transitivity in decision making, and the theory of games and of statistical decision f u nct i o ns. Sinc Since e this liter literature ature is unfamiliar and relatively inaccessible to most psychologists, and since I could not find any thorough bibliography on the theory of choice in the econ o m i c literature, literat ure, this pa paper per incl includes udes
child has $0.15 and a ten-cent candy bar. bar. The economic theory of decision making is a theory about how to predict such decisions. Economic theorists have been concerned with this problem since the of Jeremy Bentham (1748days days 1832). In recent years the development of the economic theory o f consumer's decision making (or, as the 1
This work was supported by Contract
N5ori-166, Task Order I , between t h e Office o f Naval Research and The Johns Hopkins
University. This is Report No. 166-1-182, Project Designation Designation No. NR 145-089, under that contract. I am grateful to the Departm e n t o f Political Political Economy, Econo my, T h e Johns Hopkins University, fo r providing m e w i t h an office adjacent to the Econom Economics ics Librar Lib rar y llais, s, while I was writing this paper. M. A llai M . M. Flood, N. Georgescu -Roegen, K. O. drr e o u , L . J . Savage, and esMay, A . P a p a n d pecially C. H. Coombs have kindly made mu ch u npu blis h ed material available t o m e . A number o f psychologists, eco eco nom i sts, sts, a n d mathematicians have given me excellent, but sometimes unheeded, unheede d, criticism. Especially were helpful C L. . Christ, C . H . Coombs, F . and J. Savage. Mostel l er,
380
sumer's choice) has become exceedingly elaborate, mathematical, and
unknown to psychologists, in spite of sporadic pleas in both psychological (40, 84, 10 103, 3, 10 104) 4) and eco econo nomi mic c (1 (101, 01, 102 102,, 123 123,, 12 128 8, 19 199, 9, 202) lliitera te ra-ture f o r greater communication b e tween the disciplines. The purpose of this paper is to review this theoretical literature, and also t the he rapidly increasing number number of psychologic psycho logical al exp experiment erimentss (perf (performe ormed d by both psychologists and economists) that are relevant to it. The
a rather extens extensive ive bibliograp bibliography hy of the literature since since 19 1930 30..
T H E T H E O R Y O F R I S K L E S S S C H O I C E S Economic man.
The meth method od of those theorists who have been con2
N o complete review o f this literature is available. Kauder (105, 106) ha s reviewed th thee
very early history of utility theory. Stigler (180) an d Viner (194) have reviewed th e literature up to appr approxima oximately tely 193 1930. SamuelSamuel son's book (164) contains an illuminating mathematical exposition o f some of the content this theory. explains con A l l e n o f ( 6 ) t h e re cept o indifference curves. Schul tz (172)
THEORY O F DECISION DECISIO N MAKING
381
c o n t i n u o u s and differentiable. Stone
cerned with t h e theory o f decision making is essentially an armchair method. They make assumption assumptions, s, and f r o m these assumptions they de de-presumably can duc e theorems which presumably be tested, though it sometimes seems u nl i k e l y that the testing will ever occur. The most important set of
(182) has recently shown that they can be abandoned with no serious changes in the theory o f choice. Rationality. The crucial fact about economic man is that he is rational. This means two things: He can weakly order the states into which he
assumptions made in the theory o f riskless choices may be summarized by saying that it is assumed that the person who makes any decision to w hi c h the theory is applied is an economic man. What is an economic man l i ke ? He has three properties, (a) He is completely i n f o r m e d . ( 6 ) H e i s infinitely sensitive, c) He is rational. Complete information. Economic man is assumed to know not only what all the courses of action open to
can get, get, and he makes his choices so as to maximize something. Two things are required in order fo r economic man to be able to put all available states into a weak weak orderin ordering. g. First, given any two states into which he can get, A and B B,, he must always be able to tell either that he prefers A to B, or that he prefers B to A or that he is indi indiffere fferent nt betw between een them. I f preference is operationally defined as choice, then it seems seems unth unthinka inkable ble that this requirement can ever be
him are, but also what the outcome of any action will be. Later on, in the sections on the theory of risky choices and on the theory of games, this assumption will be relaxed somewhat somewhat.. (For the results of attempts to introduce t h e possibility o f learning into this picture, see 51, 77.) Infinite sensitivity. I n most of the work o n choice, it is assumed older older that the alternatives available to an individual are continuous, infinitely divisible f u nct i o ns, that prices are
empirically violated. The second requirement for weak ordering, a more severe one, is that all preferences must b e transitive. I f economic man man prefers A to B and B to C, then h e prefers A t o C Similarly, if he is . indifferent between A and B and between B a n d C , then he is indifferent between A and C. It is not obvious that transitivity w i ll always hold f o r human choices, a n d experiments designed to find out whether or not it does w i l l be described in the
infinitely divisible, and that economic man is infinitely sensitive. sensitive. The only
section on testing transiti transitivity. vity. The second requirement of rationality, and in some ways the more important one, is that economic man must make his choices in such a way as to maximize something. This is the central principle of the theory of choice. In the theory of riskless choices, economic man has usually been assumed to maxi mi z e utility. In the theory of risky choices, he is assumed to maximize max imize expe expecte cted d uti utilit lity. y. In the literature on statistical de-
purpose of these assumptions is to make the f u nct i o ns that they lead to,
views t h e developments up to but not includin g the Hicks-Allen revolution from the point o f view of demand theory. H icks icks''s b o o k (87) siti tioo n o f most is a complete an andd detailed expo si th e mathematical an d economic content of o f o f the th e theory up to 1939. Samuel son (167) h as
reviewed the integrability problem and the revealed preference approach. A n d Wold (204, 205, 206) ha s summed up the mathematical content of the whole field for anyone who is comfortabl y a t h o m e w e w it ithh axiom systems an d differential equations.
cision making and the theory of games, various other fundamental
3822 38
WARD E D W A R D S
principles of decision making a r e considered, but they are all m ax im ization principles o f o n e sort or another. T h e f u n d a m e n t a l c o n t e n t o f th th e n o t i o n o f max imizat ion is that econ o m i c man always chooses the best alternative from a m o n g t h o s e o p e n
gists have tended to reject out of hand the theories that res ul t from these these assum ptio ns. T his isn't fair. fair. S u r e l y t h e assumpt ions cont ained i n H u l l i a n behavior theory (91) or in th e Estes (60) o r B u s h - M o s t e l l e r (36, 37 37)) learning theories are no more realisti realis ticc than these. these. T he m o st useful
tcal o hlanguage, i m , as he sees it. Inthat m o er ec ot encohmniiicthe fact man prefers A t o B implies implies and is implied by the fact that A is higher the w e a k l y o r d e r e d than B in the d s e t m e n t i o n e d a bove. (S o me t heo h eo ri ries es in troduce probabilities into t h e above st at ement , s o that if A is hi higher gher than B in the weak ordering, then econ o m i c man is more likely to choose A than B, but not certain to choose A.) This n o t i o n o f m a x i m i z a t i o n is m a t h e m a t i c a l l y useful, since it m a k e s
thing to ido wit h a tions heory to t s assumpt b uist not criticize rather t o test it s t heorems. If the the t h e o r e m s
fit the data, then the theory has at
least heuristic merit. O f course, o n e trivial theorem deduc ibl e from t h e assumpt ions embodied in the c o n c e p t o f e c o n o m i c man is that in any specific case o f choice these assumpt ions will b e satisfied. F o r instance, i f economic man is a model for real men, then real m e n s h o u l d a l w a y s exhibit transitivity o f real choices.
specify iunique t possible p o i nfor t o ar ta heory u n i q uteo subset o af points a m o n g those available to the decider. It seems to me ps psyy cholo gi gi-cally un objecti on a ble. S o m a ny n y differe n t kinds o f f u n c t i o n s can be maximized that almost a n y point act ually available in an experimental situation can b e regar r egarded ded as a m axim um o f s o m e sort. A ss ssumpt umpt ion s abo ut m ax iimization o n l y b e c o m e specific, a n d therefore possibly wrong, when t hey specify what is being maximized.
m o d e l l fo r real men. Economist s t hemselves hemselves a r e s o m e what dist rust ful o f e c o n o m i c m a n (119, 156), and we will see in subseq u e n t sections the results of a n u m b e r o f attempts t o relax these a s sumptions. utility maximization theory. Early Early T h e school o f phi philo lo sopher-eco sopher- eco no m i st stss started by J e rree m y B e n t h am am and
T here has , incidental been al n o has, of thely , possibility most discussion that th e two parts of the concept o f is c rat ionalit y might conflict. It is c o n ceivable, f o r e x a m p l e , that i t m i g h t ceivable, b e costly in effort (and t heref ore in negative utili utility ty ) to maintain a weakly ordered preference field. U n d e r s u ch ch circumstances, w o u lldd it be rational to have such a field? It is easy fo r a psycho logist logis t t o p o i n t out that an economic man who has t h e properties discussed above is very
il la nand popularized b ygoal J a mof e h s M others held that the um action is t o seek pleasure a n d avoid pain. Every o b j e c t e cont o r act ion ion m a y b e sidered from th e p o i n t o f view o f pleasure- or pain-giving properties. These properties are called the utility o f t h e o b j e c t , a n d pleasure is given b y positive utility a n d pain b y negati tive ve utilit ut ilityy . T he goal g oal o f action, act ion, then , is t o seek t h e m a x i m u m utility. This the future simple hedonism of the future is easii ly translat eas trans lated ed into a theo ry o f
un l ike a real m an. In fac t, it is is so easy to point this out that psycholo-
choice. P eople choo chooopen s e t h et alternative, from among o t he that t hose h e m , that
Transitivity is an assumpt ion, but it are the the o t h e r is directly testable. S o are properties of economic man as a
THEORY OP DECISION M AKIN G
leads to the greatest excess of positive over negative utility. This notion of utility maximization is the essence o f t h e ut ilit y t heory o f choice. I t will reappear in various f orm s throughout this paper. ( B o h n e r t [30] discusses t h e logical structure of the utility concept.)
383
comp l eting goods, like right a n d left shoes, which obviously do not have independent utilities). utilities). E d g e w o r t h (53), w h o w a s concerned with such nonindependent utilities, pointed out that total utility was not necessarily a n additive function of the utilities
attributable attributabl e to separate commodities.
o f chomice a slyses t heory ici ce w em bodied bodie th e fo of alld in This r mal econo analyses ana the early great names in economics. In the hands o f Jevons, Walras, a n d M e n g e r it reached increasingly sophisticated m athem atical express expressiio n and it was embodied in the t h i n k i n g o f Marshall, w h o published the first edition of his great Principles of Economics in 1890, and revised it at intervals for m o r e than 30 years thereafter (137). T he use t o which u t il i l i t y th th e o r y w a s
I n the he introduced no o f process indifference d thus tion curves, a nthe began the gradual destruction of the classsica cla sicall utility theo ry . W e sh shall all return to this point shortly. Although the forces of parsimony have gradually resulted in the elimination of the classical concept o f utility f r om t h e e c o n o m i c t he he o r y o f riskless choices, there have been a
put theorists lish by the these of the was nat ure d e mtoa nestabd f o r
veloped methods of measuring margi u] n a l utility (the change in utility with an infinitesimal change in a m o u n t possessed [Q], i.e., du/dQ) from m a r k e t data, by by m aking ass ass umpt ions about the interpersonal similarity of consumer tastes. Recently M o r g a n (141) (141) h a s used several variants of these techniques, has discusssed m athem atical cu atical and logical logical flaw flaw s in t h e m , and has concluded on th e basis of his empirical results that the techniques require too unrealistic assumptions to be w o r k a b l e . The crux of the problem is that, for these techn i ques to be useful, t h e c o m modities used must b e i n d e p e n d e n t than competing or complet(rather (rather ing), and t he broad br oad c o m m o dity dit y cl clas as-sifications necessary f o r adequate market data are not independent. S a m u e l s o n (164) (164) h a s s h o w n that t h e as s ump tion o f independent utilities, while it it does guatantee interval scale ut ilit ili t y m easures eas ures,, put s unw arrant ably
fe few w attempts to use essentially the
classical theory in an empirical way. Fisher (63) and Frisch (75) have de-
e assumption various goods. goods. O n t h e that the utility of any good is a m o n o t o n i c a l l y increasing negatively accelerated function of the a m o u n t o f is easy to show that the that good, it is easy a m o u n t s o f most goods which a c o n s u m e r will buy are decreasing funct ions o f price, functions which a r e precisely specified o n c e th e shapes o f the utility curves are k n o w n . This is the result the economists needed and is, is , o f co urse, urse, a testa tes table ble theo rem . (For see 87, 159.) more on this, see C omp l exities arise in this theory w h e n the relations between the utilities o f different goods are considered. Jevons, Walras, Menger, and even Marshall had assumed that th e utilities o f different commodities can be combined int o a total utility by simple addition; this a m o u n t s to assuming that t h e utilities o f different aree i n d e p e n d e n t ( in spite o f goods ar the fact that Marshall elsewhere dis- severe restrictions on the nature of cussed th e n o t i o n s o f c o m p e t i n g t h e r e s uull t in functio n. Elsei n g d e m a n d functio goods, like soap soap a n d detergents, a n d wh ere Samuelson (158) presented,
384 38 4
WARD E D W A R D S
primarily as a logical and mathem atical exercise, exercise, a m e t h o d o f m eas uru ri n g marginal util ity by as s uming s ome time-dis count fu ncti o n. Since n o r e a s o n a b l e g r o u n d s can be f o u n d fo r assuming o n e such f u n c t i on rather than another, this procedure holds n o pr omi se of empirical success. Marshall suggested (in his n o t io io n o f consumer's surplus ) ) a m e t h o d d o f utility m e a s u r e m e n t that turns o u t t o b e d e p e n d e n t on the as s ump tion o f c o n stant marginal utility o f m o n e y , a n d which is th erefore quite unwork abl e. Marshall's prestige led to extensive discussion and d e b u n k i n g o f this n o t i o n (e.g., 28), b u t little positive c o m e s o u t o f thi thiss literature. T hurstone (186) i iss c u r r e n t l y a t t e m p t i n g t o determine utility functions f o r c o m m o d i t i e s e x p e r i m e n t a l l y , but has
t h e same a m o u n t o f utility from 10 -apples-and-l-banana as you do f r om 6-apples-and-46-apples-and-4-bananas. bananas. T hen these are two points on an indifference curve, and of course there a r e a n infinite n u m b e r o f other points o n t h e s ame ame curve. curv e. N atural atur al l y , this is no t t h e o n l y indifference curve you may
repIndifference orted n o results a s yet. curves. Edgeworth's introduction of the n o t i o n of indifference curves t o deal with t h e utilities o f n o n i n d e p e n d e n t goods w a s m e n t i o n e d above. A n indifference curve is, in E d g e w o r t h ' s f o r m u l a tion, a constant-utility curve. Suppose that w e consider apples a n d bananas , a n d suppose that y o u g e t
indifference T be h e derived, m ap n o t i o n of can asanEdge as worth derived
_ _J
2
Q
15
1 Ld O
5
1
15
2
25
NUMBER OF BANANAS
F I G . 1. A H Y P O T H E T I C A L I N D I F F E R E N C E MA P
have bananas. a n d you It are inm a y between also b e apples true that different between 13-apples-and-5bananas a n d 5-apples-and-15-banana s . These are two p o i n t s o n a n o t h e r , higher indifference curve. A w h o l e family o f such curves is called an indifference map. Figure 1 presents such a map. O ne p articul articul arl y useful kind o f indifference map has a m o u n t s kind o f a commodity on one axis and amounts of money on the other. M o n e y i s a c o m m o d i t y , too. it , from t h e n o t i o n o f measurable util ity. B ut it does n o t have t o b e . Pareto (146, se see e also 151) wa wass serio u s l y c o n c e r n e d a b o u t t h e assumpti on that util ity w a s measurable u p to a linear transformation. He felt that p eop l e coul d tel l wh eth er they preferred to be in state A or state B B,, b u t c o u l d n o t tel l h o w m u c h they preferred o n e state o ver ve r t h e o th ther. er. I n oth er words , h e h y p o t hhee s iz i z e d a util ity fu o n m utr aubsl e foo nl lloyw ont hae n usual ordin ancti l scale. e aLseur e c o n o m i c l a n g u a g e , a n d call utility measured on an ordinal scale ordinal u t i l i t y , and util ity meas ured on an interval scale, cardinal util ity. It is meaningless t o speak of the s l op e, o r meaningless ma r g i n a l u t i l i t y , of an ordinal util ity f u n c t i o n ; such a function c a n n o t b e differentiated. H o w e v e r , Pareto s a w that t h e s ame ame co ncl us io io ns w h iich ch h a d been d r a w n from marginal utilities c o u l d b e d r a w n f r om indifference curves. A nl y indifference map be drawn s imp by finding al l th can e com-
THEORY OF DECISION M AKIN G
385 38 5
binations of the goods involved among which the person is indifferent. Pareto's formul ation as s umes that h igh er indifference curves have greater utility, but does not n e e d to specify how m u cchh greater that utility
c o n s u m e r d e m a n d w i t h n o reference to the notion of even o rdinal rdinal utility (though o f course th e n o t i o n of an ordinal scale o f preferences w a s still embodi ed in their derivation of indifference curves). This paper was is. fo r economics something like the beIt turns out to be possible to de- haviorist revolution in p sy sy c h o l o g y . from indifference of duce ri rigina ginal l layll dethe theorems that w ere ocurves duced from cardinal utility measures. This banishing o f c ardi ardinal nal util ity w a s furthered cons iderabl y b y splendid mathematical papers by J o h n s o n (97) and Slutsky (177). (In modern econ omi c th eory, it is cus tomary to it y th ink o f an w-dimensional c o m m o d it space, and of indifference h y p e r planes in that space, each such hyper n — 1 d im p l ane ane having, o f cours e, n im e n sions. I n o rder rd er to t o avoid unsatisfacto ry
an ge (116), (116) , s timul t imul at ed by inconH ick s andL ange A llen, po i nted o ut ated another
sistency in Pareto. Pareto had assumed that if a person considered f o u r states, A, B, C, and D, he coul d judge w h e tthh e r t h e difference b e t w e e n the utilities of A and B was greater than, equal t o , o r less than t h e differ ann d D . ence b e tw t w e e n t h e utilities o f C a L a n g e p o i n t e d o u t that if such a comparison was possible for any A, B, C, and D, then utility was car-
dinally m eas urabl ur abl e.
S ince nc e itit seems
preference it is necessary that consumers to assume structures, always
that introspectively such comparisons can obvious be made, this paper have a c o m p l e te t e w eak o rdering rder ing fo r all a ll provoked a flood of protest and comc o m m o d i t y bundl es , o r p o i nts in co c o m - m e n t (7, 22, 117 117, 147 147, 209). N ev erm o di dity ty s p ace ace.. G eo rges rgescucu-Ro Ro egen theless, in spite of all the c o m m e n t , [76], Wold [204, 205, 206, 208], and even in spite of skepticism by a H o u t h a k k e r [90], and S a m u e l s o n distinguished e c o n o m i s t as late as discu scuss ssed ed th is pro ble m .) [167] have di 1953 (153), Lange is s urel y righ t. Pareto was not entirel y cons is tent Psychologists should know this at in h is discussion o f o rdi rdinal nal utilit utilityy . once; such comparisons are the basis is discussion A l t h o u g h h e a b an a n d o n e d t h e a s s u m p - o f th thee p s ych op h ys ical Meth od o f tion that its exact value could be Equal Sens e Dis tances , from which k n o w n , h e c o n t i n u e d d t o t a l k a b o u t a n interval scale scale is derived. ved. (S amuelis deri the sign of the assumed comarginal that utility efficient, which some k n o w l e d g e a b o u t t h e util ity function o t he h e r t h a n p u r e l y o r d in in a l k n o w l e d g e wass available. He also committed wa other inconsistencies. So Hick s and A l l e n (88), in 1934, were led to their classic paper in w hi hich ch t h e y a t t e m p t e d to purge the theory of choice of its last intro intro spective spective elem ents. They a do d o p te t e d t he h e c o n v e n t iioo n a l e c o n o m i c view a b o u t indifference curves as deter mi n ed from a sort o f imaginary questionnaire, and proceeded to derive all of the usual co ncl us u s io io ns abo ut
son [162] has pointed out very teresting aNot qualification. o ninly
m u s t s u ch c h j u d g m e n t s o f difference b e possible, but they must also be transitive in order to define an interval scale.) B ut since since such such judgm en ts o f differences did did no t seem to be necessary for the d e v e l o p m e n t o f c o n su su m e r demand theory, Lange's paper did n o t force the reinstatement of cardinal utility. Indeed, the p e n d u l u m s w u n g further in the behavioristic direction. Samuelson developed a new analytic fo u nd ati o n for the theory o f c o n -
386
WARD E D W A R D S
sumer behavior, the th e essence o f w hi hich ch is that indifference curves and hence t h e entire structure of the theory o f
consumer choice can be derived simply from observation of choices among alternative groups o f p urchases available to a co nsume r (1 (160 60,, 161) 16 1).. T his appro ach h a s b e e n e x -
show that marginal utility could be defined even in an ordinal-utility universe (23, 24, 163; 25, 114). Knight (110), in 1944, argued e x -
tensively for cardinal utility; he based h is a r g u m e n t s in part on introspective considerations and in part
on an examination o f psychophysical
developed by tensively Samuel(50, s on (164, 165, 167, 169) an and d others 90,, 125, 126). Th 90 The e essence of the idea is that each choice defines a p o i n t and a slope in commodity space.
He scaling n u m b e r r procedures. o f replies (29, 4 stimulated 2 ; 111). R eacently Robertson (154) pleaded for the reinstatement of cardinal utility in th e interests o f welfare economics Math ematical ap p roximation meth - (this point will b e discussed again ods mak e it possible to combine a bel ow) . B ut i n general t h e indifferw h o l e family o f such slopes into a n ence curve approach, in its various indifference hyperplane. A family o f forms, has firmly established itself as such h y p e r p l a n e s f orm s a n indiffer- t h e structure of the th eory o f riskless ence map. choice. I n a distinguished b u t inaccessible Experiments on indifference curves. series o f articles, Wold (204, 205, 206; 206; Attempts to measure marginal utility
see mm ary presenta 208 for a s uthe tion)also has presented mathematical content of th e Pareto, Hick s and Al len , a n d revealed preference (Samuelson) approaches, a s well a s Cassel's demand function approach, and has s h own that if the assumption about complete weak ordering o f bundl es o f commodities wh ich ich w a s discussed above is made, then all these approaches a r e math ematical l y equival ent. r cardinal utility. T h e Nostalgia fo
data from mThere a r k e t have were discussed above. been three experimental attempts to meas ure indifference curves. Schultz, who pioneered in de riving riving statistical demand curves, lleague at the Univerinterested his co lleague sity of Chicago, the psychologist Thurstone, in the problem of in-
ity reason cardincrucial a l util was for the abandoning a r g u m e n t of the ordinalists that indifference curve analysis in its analysis its various f o r m s c oul d d o everything that cardinal utility could d o , w i t h fe w e r as s ump tions . S o f a r as the theory o f riskless choice is c o n cerne d, this is so . B ut this is o nl y an an argument for parsimony, and parsim o n y is not a l w a y s w e l c o m e . There was a series o f p e o p l e w h o , for one reason or anoth er, wanted to reinstate cardinal utility, or at least
to a each combination hstandard. e preferred instance, For the s ubject judged wh eth er h e preferred eight hats and eight overcoats to fifteen hats and three overcoats. The same procedure w a s repeated f o r hats a n d shoes, and for shoes a n d overcoats. T h e data w e re r e fitted with indifference curves derived fr o m th e as s ump tions that util ity curves fitted Fechner's L a w a n d that t h e utilities of the various objects were indep endent. T h u r s t o n e says that Fechner's Law
There attempts marginal util ity. invalid were several mathematically to
than t h e o t hb eurt dataio fitted ns b e t ht ee r considered, t h ef unct possible
curves. ves. T hursto ne (18 (185) 5) difference cur per for med a very simple experiment. He gave one subject a series of combinations o f hats a n d overcoats, a n d required t h e s ubject t o j u d g e w h e tthh e r
THEORY THEORY O F DECISI DECISION ON
presents n o evidence f o r this assertion. The crux of the experiment was the attempt to predict the indifference curves between shoes a n d overcoats from t h e other indifference curves. This was done by using the other two indifference curves to infer utility functions fo r shoes and for
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con and eggs. By assuming that all students had the same indifference curves, they were able to to derive a composite indifference ma p for bacon a n d eggs. N o mathematical assumptions were necessary, and the indifference map is not given mathematical f o r m . Some judgments were partly or com or com-
overcoats separately, and then using pletely inconsistent with wit h the final map, these two utility functions to predict but not too many. The only concluthe total utility of variou variouss amounts sion which this experiment justifies is o f shoes a n d overcoats jointly. T h e that it is possible to derive such a prediction worked rather we ll. The composite indifference map. The final attempt to measure an judgme nts o f the the o ne subje subject ct used a r e extraordinarily orderly; there is very indifference curve is a very recent one the psychologists Coombs and by little of the inconsistency and vari- by ability that others w o r k i n g in this M ilh o lland (49). T h e indifference area have f ou n d . Thurstone says, curve involved is one between risk Th The e subject . . . was entirely naive and value of an object, and so w ill be as regards the psychophysical prob- discussed below in the section on the l e m involved and had no knowledge theory o f risky decisions. It is menwhatever curves of the nature of the(18S, that we expected to find p. 154). He adds, I selected as subject a research assistant in my laboratory w h o k n e w nothing about psychophysics. Her work wa wass largely clerical in nature. She had a very even disposition, and I instructed h e r eve ven n motivational attitude attit ude to take an e o n the t he successive occasions . . . I was surprised at the consistency of the j u d g m e n t s that I obtained, but I am pretty sure that they we re the result 3 instruction to assume uniuni oform f careful motivational attitude. aFrom t h e economist's point o f view, t h e main criticism o f this experiment is that it involved imaginary rather than real transactions (200). T h e sec second ond experi experiment mental al measurement o f indifference curves i iss reported by the economists Rousseas and Hart (157). They required large numbers o f students to rank sets o f three comcombinat ions o f different amounts of ba8
Personal communication, Thurstone, L. L. Personal December 7, 1953.
here because the same methtioned ods that the in (which show only difference curve is convex to the origin, and so perhaps should not be called measurement) could equally well be applied to the determination o f indifference curves curves in riskless situations. M e n t i o n should b e made of the extensive extensiv e economic work on statistical demand curves. For some reason the most distinguished statistical demand curve derivers feel it necessary give account consumer's to a n as a preliminary of choice theory to the derivation o f their empirical demand curves. The result is that the two best books in the area (172, 182) ar are e each divided into two parts; the first is a general discussion of the theory o f consumer cons umer's 's choi choice ce and the second a quite unrelated report of statistical economic work. Stigler (179) has given goo good d reasons why the statistical d e m a n d c curves urves are so litt little le related related to t h e demand curves o f economic theory, and Wallis and Friedman (200) argue plausibly that this state
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o f affairs is inevitable. At any rate,
there seems to be little prospect of using large-scale economic data to fill in the empirical content of the theory o f individual decision making. Psychological comments. There are several commonplace observations that are likely to occur to psycholo-
C may be more than one j.n.d. apart, and so one may be preferred to the other. This argument is, of course, wrong. If A has slightly more utility th e individual w i l l than B, then the choose A in preference to B slightly more than SO per cent of the time, even though A and B are less than
as as they try to the gists soon of riskless apply to actual theory choices experimental work. The first is that human beings are neither perfectly consistent nor perfectly sensitive. This means that indifference curves are l i k e l y to be observable as indifference regions, or as probability distributions of choice around a central locus. It would be easy to assume that each indifference curve represents the modal value of a normal sensitivity curve, an and d that choices
one util utility. ity. aThe 50 per apart apart is in in theory precisely cent j.n.d. point defined point, not not a region. It may in fact be difficult to determine because o f inconsistencies in judgments and because of changes in taste with time. The second psychological observation is that it seems impo impossible ssible even to dream of getting experimentally an indifference map in w-dimensional space where n is greater than 3. Even the case of w = 3 presents formidable experimental problems. This is less
should have properties f r o mstatistical as predictable that hypothesis the amounts of the commodities in in (locations (locations product space) are changed. This implies that the definition of indifference between two collections of commodities should be that each collection is preferred over the other 50 per cent of the time. Such a definition has been proposed by an economist (108), and used in experimental work by psychologists p e r cent choice (142). O f course, S O pe
important psychologist to the thetheory who wants to use of choice to rationalize experimental data than to the economist who wants to derive a theory of general static e q u i l ib ib rium. Experiments like Thurstone's (185) involve so many assumptions that it is difficult to k know now what their empirical meaning might be if these assumpnot tions were were not made. Presumably, the best thing thing to do with such experiments is to consider them as tests
been psychological has ao f standard definition i ndi fference since t h e days of Fechner. Incidentally, failure on the part of an an economist to understand that a just noticeable diffe re nce ( j . n . d . ) is a statistical concept has led him to argue that the indifference relation is intransitive, that is, that if A is ind i f f e r e n t to to B and B is i ndi fferent to C , then A need not be i ndi fferent to C (8,, 9, 10). He argues that if A and B (8 ar are e less than one on e j.n.d . apart, then A same of same of will be indifferent to B; the the course is true of B and C; but A and
with the ovalidity. f the assumption was wleast i l l i n gface to Thurstone Thurstone
assume utility maximization maximization and independence of the commodities involved (incidentally, his choice of commodities seems singularly unfortunate for j u s t i f y i n g an assumption of independent utilities), and so used his data to construct a utility f u n c t i o n . O f course, if only ordinal utility is assumed, then experimental indifference curves cannot be used this way. In fact, in an ordinalutility universe neither of the principal assumptions made by made by Thurstone Thurstone
TH EO RY O F D E C I S IIO ON
can be tested b y means o f experim e n t a l indifference curves. S o th thee as s ump tion o f cardi cardinal nal util ity , th ough necessary, seems to lead to connot not siderably m o r e specific uses for exsiderably perimental data. A t a n y rate, f r o m t h e exp erimental p o i n t o f view t h e most interesting is:: W h a t is the observed question is shape o f indifference curves between independent commodities? This question awaits a n exp erimental ans wer. sim ilar T h e notion o f utility is very sim to the L e w i n ia i a n n o t iioo n o f valence (120, 121) 121).. L ew in co nceives nceives o f valence as the attractiveness of an o b j e c t or activity to a person (121). Thus, psychologists might consider the th e ex p e rime ntal s tudy o f util uti l ities ities to tall study o f valences, be the exp erimen ta th erefo re an attempt at q u a n t i f y and therefo in g parts of the L e w i n i a n theoretical
schema.
A P P L I C A T I O N O F T H E T H E O R Y O F RISKLESS C H O I C E S T O W E L 4 FARE EC ONOMI C S FARE
T h e classical utility theorists a s sumed t h e existence o f i n ter per so so n a lly comparable cardinal utility. They were thus able to find a simple answer to the question of how to det e r m i n e t h e best e c o n o m i c p o l i c y : That economic p ol icy is best which results in the m a x i m u m total util ity, s u m m e d o v e r a l l m e m b e r s of the economy. int erp ers o nal T h e a b a n d o n m e n t o f interp comparability makes this answer useless. A sum is meaningl es s if the units being s ummed are of varying sizes an d there i s n o w a y of reducing th em t o s o m e c o m m o n s iz e . T h iiss 4
T he discussion o f w elfar elfaree eco no mics giv given en in this paper is exceedingly sketchy. For a th e complexities o f modern picture o f w h a t the welfare eco no mics are are really like (s (see ee 11, 13, 14, 86 86,, 118, 124, 127, 139, 140, 148, 154, 155, 166, 174).
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p oint has not been universally recognized, and certain economists (e.g., 8 2 , 154) still defend cardinal (but not interpersonally comparable) utility o n g r o u n d s of its necessity fo r w e l fare economics. Pareto's principle. T h e a b a n d o n m e n t o f interpersonal comparability a n d th en o f cardi cardinal nal utility pro duced a search for some other principle to Pareto justify economic policy. (146), who first abandoned cardinal utility, provided a partial solution. He suggested that a change should b e considered desirable if it left everyone a t least a s w e l l off as he w a s before, a n d made a t least o n e person better o f f. f. Compensation principle. Pareto's principle is fine as far as it goes, but it obvious l y does not go very far. The economic decisions which can be m ade on so sim sim ple a principle principle are ar e few an d insignificant. S o welfare economics languished until Kaldor (98) p rop os ed t h e comp ens ation p rinciple. This principle is that if it is possible fo r those w h o gain from a n e c o n o m i c change t o c o m p e n s a t e t h e losers for their losses and still have s ometh ing left o v e r from their gains, then the change is desirable. Of course, if the c o m p e n s at at iioo n is actually paid, then this is simply a case o f Pareto's pri principle nciple . B u t Kaldor asserted that the compensation need not actually be m a d e ; a l l th at w a s necessary w a s that it could be m ade. ade. The fact that i t c o u l d b e made, according t o Kaldor, is evidence that the change g o o d o v e r h a r m , p roduces an excess o f go and so is desirable . S cit citoo vsky (173) observed an inco nsist nsistency ency in Kaldor's position: Some cases could arise arise in which, w h e n a change from A to B which, h a s been made because o f Kaldor's criterion, then a change back f r om B also satisfy satisfy K aldo r' r'ss t o A w o uld also
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criterion. It is customary, t herefore, t o assume that changes which meet t h e original Kaldor criterion a r e o n l y desirable if the reverse change does n o t also meet t h e Kaldor criterion. It has g r a d u a l l y b e c o m e o b v i o u s that the Kaldor-Scitovsky criterion does not solve the problem of welfare 18,, 99). It aseconomics (see e.g., 18 sumes that the unpai unpaidd com pensati pensatioo n does as much good to the person who gains it as it w o u l d if it were paid to the people who lost by the change. For instance, suppose that an industrialist can earn $10,000 a year more from his plant by using a new ma chi n e, b u t that t he he i n t r o d u c t i o n o f the machine throws two people irretrievably o u t o f w o r k. k . If the salary o f each worker prior to the change w a s $4,000 a year, t hen the industrialist could compensate compensate t h e w o rkers rkers and stil stilll m ake a pro fi fit. t. B ut i f h e does n o t compensate t h e w o r k ers, then t h e added satisfaction h e gets from h is extra $10,000 m a y b e m u c h less than than t h e misery misery h e p r o two workers. This e x duces in his two duces ample only illust rat es t h e principle; it does not make much sense in these days o f pro pro gressi gressive ve inco m e taxes, taxes, u n e m p l o y m e n t compensat ion, high e m p l o y m e n t , a n d strong unions. Social welfare functions. From Fr om here her e on the subject o f welfare economics gets too complicated and too r e m o t e from psychology to merit extensive exploration in this paper. T h e line that it has taken is the assumption o f a social welfare function (21), a function w hi hich ch co m bines individ individual ual utilities in a way whic h satisfies Pareto's principle but is otherwise undefined. I n spite of its lack o f definition, it is possible to draw froo m such a funccertain conclusions fr tio tio n (see (see e.g., e.g., 164 164). H o w ever, A rrow ( 1 4 ) has r e c e n t l y s h o w n that a social welfare function that meets certain
EDWARDS
very reasonable requirement s about being sensitive in some way to the wishes of all the p e o p l e affected, etc., cannot in general be f oun d in t h e absence o f i n t e r p e r s o n a l l y c o m parable utilities (see also 89). Psychological comment. S o m e e c o n omists are willing to accept the fact that t hey a r e i n e x o r a b l y c o m mitted to making moral j udgment s w h e n t he h e y r e co c o m m e n d e c o n o m ic policies (e.g., 152, 153). Others still he imperson al amo rali l ong fo ong fo r tthe ralitt y o f a utility m easure easure (e.g., 154) 154).. H o w ever desirable int erpersonally comparable cardinal util utility ity may be, it seems U t opia n to h o p e that any experimental procedure will ever give infor ma ti on abo ut individual utilities utilit ies that could be of any practical use in guiding large-scale economic policy. T H E T H E O R Y O F R I S K Y C H O IC E S
Risk a n d uncertainty. Economists a n d statisticians distinguish between 6
Strotz (183) (183) an d A lc hian ( 1 ) present present no n-
technical and sparkling expo siti sitioo ns o f the the von N e u m a n n and Morgenstern utility measurem en t propo sals. sals. G eo rges rgescucu-Ro Ro egen (78) critically discusses vari varioo us axiom systems so as as to som om e of the assumption assu mption s underl ying this bring s clearr focus. Allais kind o f cardinal utility into clea kind (3) reviews some o f these ideas in the course o f criticizing t h e m , A r ro ro w (12, 14) reviews parts o f the th e field. There is a large psychological literature o n o n e kind o f risky decision making, th thee kind which results w h e n psychologists use partial reinforcement. T his his literature h as been re viewed by Jenk ins and S tanley (96 (96)). Recen tly a number of experimenters, including Jarrett (69, 70), 70), B ilodeau (27), and m y(95), Floo d (69, self (56) have been performing experiments on h u m a n subjects who are required to choose repetitively between t w o o r more alternatives, each o f which each which has a probability o f reward greater than zero a lesss than o ne. T h e prob ann d les lems raised by these experiments are too complicated and too far plicated far removed from conventional utility theory to be dealt with in this paper. This line line o f experime ntat ntation ion may eventually provide the link which ties together utility theory theory a n d re inforc e m e nt t h e o r y .
THEORY O F DECISION M A K I N G
risk and uncerta uncertainty. inty. There There does not seem to be any general agreement about which concept should be associated with which word, but the following definitions definitions make t h e most important distinctions. everyone w o u ld A l m o s t everyone w l d agree that w h e n I toss a coin the probability that I will get a head is .5 .5.. A proposihicc h a n umtion about t h e future t o w hi ber can be attached, a number that represents the likelihood that the proposit ion is true, m a y b e called a aree fo r first-order risk. What t h e rules ar attaching such numbers i s a much debated question, which will b e avoided in this paper. Some propositions may depend on m o r e than o n e probability distribution. tion. For instance, I may decide that i f I get a tail, I will put the coin back whereas if I ge t a head, in m y p o c k e t , whereas again. Now, Now , t h e probI will toss it again. ability of the proposition I will g e t a head on my second toss is a function o f t w o probability distributions, the distribution corresponding to the first toss and that corresponding to toss. This might be called the second toss. a second-order risk. Similarly, risks o f any order may be constructed. It is a mathematical characteristic of all higher-order risks that they may be compounded into first-order risks by means o f th thee usual theorems fo r comproba bilities. es. (Some (Some econ econoopounding probabiliti mists have argued against this pro[83],, essenti ess entially ally o n the cedure [83] the grounds a y have more inf o rmat io n that y o u m ay by the time t h e second risk comes around. Such problems can best be dealt with b y means o f vo v o n Neumann an d Morgenster Morge nstern n s [19 [197] concept o f strategy, wh ich is discussed below. They become in general problems o f uncertainty, rather than risk.) S o m e propositions about abo ut t h e future exist to which no generally accepted probabilities can be attached. What
391
is the probability that the f ol l owin g proposition is true: Immediately after finishing this paper, y o u will drink a glass of beer? Surely it is neit her impossible n o r certain, so it between o u g h t to have a probability between zero and one, but it is impossible fo r what that proby o u or me to find find out what ability might be, or even to set up generally acceptable rules about h o w out. t. Such propositions are to find ou considered cases o f uncertainty rather than o f risk. This section section deals only with the subject of first-order risks. T h e subject o f uncertainty will arise again in connection with t h e theory o f games. Expected utility maximization. T h e traditional mathematical notion f o r dealing with games o f chance (and so with risky decisions) is the notion that choices should b e made so as to maximize expected value. T he expected value of a bet is f o u n d b y mul tip l ying t h e value o f each possible o u t c o m e by its probability o f o c currence and summing these products across all possible outcomes. In symbols: where p stands for probability, $ stands for the value of an outcome, • +£n = l . a n d pi+p*+ • • +£ The assumption that people actually behave the way this mathematical notion says they should is contradicted by observable behavior many ny risky situations. situation s. People People a r e in ma insuranc rance, e, even even though willing t o buy insu the person who sells the insurance profit. it. Peopl Peoplee are willing to m a k e s a prof buy lottery tickets, even though the lottery makes a pro fi fit. t. Consideration o f th e problem o f insurance and o f the St. Petersburg paradox led Daniel B e r n o u l l i , a n eighteenth century mathematician, to propose that they could b e resolved b y assuming that
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people act so as to max imize expected utility, rather than expected value (26). ( H e also assumed that utility f o llo we d a function that m o r e than a c e n t u r y later w a s proposed b y Fechn er for subjective magnitudes in general and is now called Fechner's L a w . ) This was the first use of the notion of expected utility. T h e literature o n risky decision m a k i n g prior t o 1944 consists primaril y of t he S t . Petersburg paradox and ot her gambling a n d probabilit y literature in mathematics, some literary discussion in economics (e.g., 109, 187), one economic paper on lotteries (189), and the early literature of the 32, 33, 34, 195), theory of games (31, 32, w hi ch did not use the n o t i o n o f ut ilit y. y. T h e modern period period in the study o f risky decision making began w i t h th e publicat ion in 1944 of von N e u m a n n a n d Morgenstern's m o n u m e n t a l b o o k Theory of Games a nd Economic Behavior (196, se e also 197), which w e w i ll dis discuss cuss m o re fully l at a t er e r . V o n N e u m a n n a n d M o r ggee n stern pointed out that the usual ass u m p t i o n that e c o n o m i c man can a l w a y s s a y w h e t h e r h e prefers o n e state t o anot her or is indifferent b e t ween t hem needs only to be slightly modified in order t o imply cardinal util ity. T h e modification consists o f adding that e c o n o m i c c m a n c a n also completely order probability combinations of states. T hus, suppose suppose that an economic man is indifferent between the certainty of $7.00 and a 50-50 chance o f gaining $10.00 o r n o t h i n g . W e can assume that h is indifference between these two prospects m e a n s that they have the same utility f o r h i m . W e m a y define t h e utility of $0.00 as zero utiles (the he u n i t o f ut ilit y , jjust usual n am a m e fo r tthe ust a s sone is the name for the unit o f
arbitrary definitions correspond t o defining t he t wo undefined constants
ory loudness), and the utility as 10 utiles?, tw o oaudit f $10.00 These
m ea n in gamong f ul . Finally ally choices , it means risky alternatives
w hi ch a r e permissible since cardinal to a linear utility i s m e a su su re r e d o n l y up to t r a n s f o r m a t i o n . T h e n w e m a y calculate t h e u t i l i t y o f $7.00 b y using t h e concept o f expected utility a s f o l lows:
17(17.00) = .5 +. . 5 5.E7($0.00) . 55 £7($10.00) ( 1 0 ) + . 5 ( 0 + ) = = . T h u s w e have det ermined t h e cardin a l utility o f $7.00 a n d f o und that it is 5 ut iles. By varying t he probabilby using ities and by ities using t h e already f o u n d utilities it is possible t o discover t h e uti li ty o f any o t h e r a m o u n t o f m o n e y , using only the two permissible arbitrary definitions. It is even more convenient if instead o f +$10.00, — $10.00 o r some other loss is used as o n e o f th thee arbitrary utilities. A variety of implications is embodied i n t his apparent ly simple n o t ion. In the attempt t o e x a m i n e a n d exhibit clearly what these implications are, a n u m b e r o f axio axio m s y stems, stems, differing fr o m v o n N e u m a n n a n d differing Morgenst ern's b u t leading to the same result, have been developed (73, 74, 85, 135, 136, 171). This paper will n o t attempt t o g o into th e co m ple x discussi discussioo ns ( e.g ., 130, 131, 168, 207) of these various alternative axiom systems. Ohnaes recent discussion discussion o f them (78) concluded, o n reasonable grounds, that t h e o r ig ig in in a l v o n N e u m a n n a n d M o r of axioo ms is still t h e best. genstern set of axi genstern ms is still It is profit able, however, to exa m i n e what the meaning of this notion is f r om t h e empirical point o f view if it is right. First, it means that risky propositions can be ordered in desirability, just as riskless ones can. Second, it means that t h e c o n cept of expected utility is behavior-
T HEOR Y O F DECISION
are made in such a way that they
maximize expected utility. I f this model is to be used t o predict actual choices, what could go wrong with it? It might be that the probabilities by which the utilities are multiplied should not be the objective pr o ba bi li ti t i es; in in other words, a decider s estimate of the subjective importance of a probability may not be the same as the numerical value of that probability. It might be that t h e method o f combination o f probabilities and values should not be simple multiplication. It might be that t h e method o f combination o f the probability-value probabili ty-value products should not be simple addition. It might be that the process of gambling has some positive o r negative utility o f its own. It might be that the whole is wrong, people just that were approach trying d o n o t behave as if they to maximize expected utility. We shall examine some o f these possibilities in greater detail below. Economic implications o f maximizEconomic T h e utilityin g expected utility. measurement notions of von Neumann and Morgenstern were enthusiastically welcomed by many economists (e.g., 73, 193), though a fe few w (e.g., 19 were at least temporarily (20) unconvinced. The most
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clearly willing to accept fair insurance (i.e., insurance with zero expected pecte d money value) val ue) because the serious loss against which he is insurin g w o u l d have a lower expected utility than the certain loss of the insurance premium. (Negatively accelerated total utility curves, like that f r om the origin to /, are what y o u g e t when marginal utility d e creases; thus, decreasing marginal
OLL RS FIG. 2. H Y P O T H E T I C A L U TI LI TY C U R V E FO R M O N E Y , PROPOSED B Y FRIEDM FRIEDMA A N A N D SAVAGE
utility is consistent with the avoidance o f risks.) T h e person would also
interesting economic use u se oand f them w as proposed Friedman Savage by (73), w h o were concerned with t h e question o f w h y t h e same person w h o buys insurance (with a negative expected money v a l u e ) , a n d t heref ore is willing to pay in order not to take risks, will also buy lottery tickets (also with a negative expected money value) in which he pays in order to take risks. They suggested that these facts could be reconciled by a doubly inflected utility curve f o r money, like
be lottery since illing illing to buy the w expected utility oftickets, the lottery ticket is greater than the certain loss
in ent 2. income, If / represents that person pers on s Fig. curr current then hethe is
person on sasutilit utihis lity curve f o r origin a pers money of be taken y customary
o f t h e cost of the ticket, because o f t h e rapid increase in the height of the utility f un c t ion . Other consideraconsiderations make it necessary that the uti li ty curv curvee turn tu rn down again. Note that this discussion assumes that gambling has no inherent utility. M a r k o w i t z (132) suggested an important modification in this hypothesis. He suggested that the
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financial status, and that on b o t h
which differ in some w a y from t h e sides of the origin the curve be as- objective probabilities, as w e l l as on sumed first concave and then convex. utilities different f r om t h e objective I f t h e p erson's cust omary state o f values of the objects involved. weal th changes, then the shape of his Mosteller and Nogee (142) carried utility curve w i ll t hus remain gen- o u t the first ex periment to apply the erally t h e same with respect t o where v o n N e u m a n n - M o r g e n s t e r n m o d e l . h e now is, and so his risk-taking b e - They presented presented H arvard arvard unde rgraduhavior will remain pretty m u c h h t h e ates a n d N a t i o n a l G u a r d s m e n w i t h same instead o f changing with every bets stated in terms of rolls at poker change o f w e a l t h as in the Friedman- dice, w hich each each subjec t co co uld accept o r refuse. E a c h b e t gave a hand Savage f o r m u l a t i o n . Criticism of the expected-utility a t poker dice. If the subject could maximization theory. It is fairly easy beat the hand, he won an amount t o const ruct ex amples o f behavior stated in the bet. I f n o t , h e lost a that violate t h e v o n N e u m a n n - nickel. S ubjects play ed w ith $1.00, $1.00, M o r g e n s t e r n a x i o m s ( f o r a part ic- which they were given at the beularly i ngenious exam exam ple, see 183). It ginning of each experimental session. is especially easy to do so when the T h e y w e r e r u n t oget her in g r o u p s o f a m o u n t s o f m o n e y i n v o l v e d a r e very fi five; ve; b ut each decided a n d rolled t h e large, or when t he probabilit ies or poker dice f o r himself. S ubjects ubject s w ere pro bability diff difference erence s involved a r e provided with a table in which the e x t r e m e l y s m a l l . A l l a is is (5) has c o n - m a t h e m a t i c a l l y fair bet s w ere er e show n, structed a quest ionnaire f u l l o f it ems s o that a s u b j e c t c o u l d i m m e d i a t e l y o f this type. Fo r an e c o n o m i s t in - tell b y referring to the table whet her terested in using these axioms as a a given bet was fair, or better or basis for a completely general theory worse than fair. I n the data analysis, the first step o f risky choice, these examples m a y b e significant. B u t psychological i n - was the d e t e r m i n a t i o n o f indifferFor each probability terest in this model is more modest. ence offers. T he ps psyy cholo gically gically imp o rt ant quesques- used and for each player, t h e a m o u n t tion is: Can such a m o d e l b e used t o o f m o n e y w a s f o u n d fo r which that a c c o u n t f o r simple ex periment al e x - pla playy er w o uld accep ac ceptt th thee bet S O p er amples o f risky decisions? cent of the time. Thus equality w a s O f course a utility f u n c t i on derived defined a s S O p e r cent choice, as it b y v o n N e u m a n n -M - M o r g e n st s t e rn r n m e a n s is likely to be in all psychological e x is n o t necessarily t h e same a s a classi- periment s s o f this sort. Then t h e c a l utility function (74, 203; se e also utility o f $0.00 w as defined as 0 utiles, and the utility of losing a 82). Experiment on the von Neumann- nickel w as defined as — 1 utile. With Morgenstern model. A n u m b e r of ex- these definitions and the probabilities periments on risky deci decisi sioo n m aking involved, it was easy to calculate the h av a v e b e e n p e r f o r m e d . O n l y the first uti li ty corresponding to the a m o u n t o f t h e m , b y M o s t e l l e r a n d N o g e e o f m o n e y i n v o l v e d in the indifference (142), h a s been in the simple fr ame - offer. I t t u r n e d o u t that, i n gene ra) ra),, w o r k of the model described above. the Harvard undergraduates had A l l t h e rest have in some way or diminishing m arginal utilities, utiliti es, w hile
on thef o rconcept of another centered probabilities effective behavior
t h e N a t imarginal o n a l G uutilities. a r d s m e n n had increasing
THEORY THEORY O F D E C I S IO IO N
thus thus calculated were The utilities utilities
used in predicting t h e results o f m o r e comp lex lex bets. It is hard to evaluate success o f these predictions. A t t h e success any rate, an auxiliary pairedcomparisons exp eriment showed that the hypothesis that subjects maximized expected utility predicted choices better than the hypothesis that subjects maximized expected money value. The utility curve that Mosteller a n d Nogee derive is different from the one Friedman and Savage (73) talking lking abo ut. S upp up p ose that a were ta subject's utility curve were of the 1 Friedman-Savage type, as in Fig . 2 , a n d that he had had e n o u g h m o n e y to put h im a t p o i n t P . I f h e n o w wins o r loses a bet, then he is m o v e d to a different location on t he indifference
395
periment. Consequently, their conclusion that t h e a m o u n t o f m o n e y possessed by the subjects was not
seriously imp ortant c a n o n l y b e true i f their utility curves a r e utilityfor-w-more dollars curves and if the shapes of such curves are not affected changes in the r o f dollars b y changes the n u m b e r discus scussi sioo n ex hibits a o n han d. T his di type o f p r o b l e m which m u s t always arise in utility measurement a n d which is new in psychological scaling. which T h e effects o f p revious judgments o n p resent judgments are a familiar story in psychophysics, but they are usually assumed to be contaminating influences that can be minimized o r eliminated by proper experimental design. desi gn. In utility scaling, scaling, the fundamental idea of a utility scale is such that the w h o l e structure o f a subject's
the a m ounts e y i nQv .o(lNv eo d dt e that smaller ocurve, f m o n say are ar e much than in the original Friedman-Savage curve.) H o w ever, the the conu se o f this curve.)
as a choices should be altered result cho ic icee (if the ochoices f each previous
curve assumes that the individual is always at the same point on h is utility uti lity curve, nam el y th t h e o rigin. rigin. This means that t h e curve is really of the Markowitz (132) typ e discussed
one available at present, and that
struction o f a M o s t e l l e rr- N o g e e u t ili l i ty ty
abo ve,
o f the Friedm anSavage type. The curve is not really e y in general, a curve o f utility o f m o n ey
M AKING
instead instead
a r e real ones involving money gaina or losses). The Markowitz solution to this p roblem is the m o s t practical
solution is not entirely satisfactory since all it does is to assume that people's utilities f o r m o n e y operate in such a way that the problem does not really exist. This assumption is plausible for money, but it geta rapi ra pidly dly les less plausible plausible w hen o ther
b u t rather itdollars. is a curve of so, the itutilityfor-w-more dollars . E ven must b e assumed f urt h e r that as the total a m o u n t o f m o n e y possessed by the subject changes during t h e experim e n t , t h e util ity-for-«-more dollars curve does does n o t change. Mosteller a n d N o g e e a rrgg u e , on the basis o f detailed examination o f s o m e o f their data, that the amount of money p ossessed b y t h e subjects did not seriously influence their choices. T h e utility curves cur ves they repo rted rt ed s how ed chang-
a less commodities continuous are with considered character instead. Probability preferences. In a series o f recent experiments (55, 57, 58, 59), the w riter riter has s shh o w n that subjects, w h e n they bet, prefer some probabilities to others (57), and that these accounted f o r preferences cannot b e accounted
within tehxeai nmg o u nmarginal t s o f m o n utility e y usdd within in their
bets w e r e o f three parisons. T h e expected kinds: positive value, nega nega-
b y utility considerations (59). A l l th e experiments were basically o f the same design. Subjects were required to choose between pairs of bets according to the m e t h o d o f paired com-
3966 39
WARD E D W A R D S
tive expected value, value, a n d zero zero e x p ected ected value. v alue. T he tw tw o m em bers o f each pair o f bets had the same e x pected value, s o that there w a s never ( in t h e main experiment [57, 59]) 59]) a n y objective reason to expect that ch choo o s in g o n e b e t w o u l d b e more desirable than choosing th t h e other. Subjects made their choices under three thr ee co nditi nditioo ns: just im agining they th ey were betting; betting f o r worthless chips; a n d betting f o r r e a l m o n e y . They p aid a n y losses fr o m their o w n funds, b u t t h e y w e r e run in extra sessions after the main experiment to bring their w i n n i n g s up to $1.00 p e r hour. The results showed that two factors were most imp ortant in determining choices: gene ral ra l preferences o r dislikes f o r risk-taking, a n d specific preferences amo ng pr proo babi bab ili liti ties. es. A n ex ample of the first kind o f factor is that subjects strongly p referred l o w probabilities o f losing l a r g e a m o u n t s o f m o n e y to high probabilities o f losing s sm m a l l a m o u n t s o f m o n e y — t h e y just didn't like t o lose. I t also turned o u t that o n positive expected value weree mo re w i lling lli ng to accept bets, they wer l o n g shots w h e n p l a y i n g f o r real m o n e y t h a n w h e n j u s t i m a g i n i n g o r p laying f o r worthless chips. A n e xxamp le of the second kind o f factor is that they consistently preferred bets involving a 4/8 probability o f thers, ers, a n d consistently w i n n i n g t o all o th avoided bets involving a 6/8 p robability o f winning. These p references w e r e reversed fo r negative expected value bets. These results were independent o f t h e a m o u n t s o f m o n e y i n v o l v e d in t h e bets, so long as the c o n d i t i o n o f constant expected value was mainhi ch tained (59). When pairs of bets w hich differed from o n e a n o t he h e r in expected
p e c t e d a m o u n t o f m o n e y a n d betting at the preferred probabilities (58). A n attemp t was made to construct individual utility curves adequate to account for the results o f s several everal sub jects. F o r this purpose, t h e utility o jects. utility o f $0.30 was defined as 30 utiles, and it was assumed that subjects cannot discriminate utility differences smallthese a s e r than half a utile. U nder these curves sumpt io ns, no individual utility curves consistent w i t h t h e data could be r i o u s m i n o r e x p e r im im e n t s d r a w n . V a ri s h owed that these results were reliato various possible b l e and not due to artifacts (59). N o a tt tt e m p t w a s m a d e t o generate generat e a m athem at hem atica at icall m o del o f p ro bability babili ty p references. ref erences. T h e existence o f pro bability bability preferences means that the simple von N e u m a n n - M o r g e n s t e r n m e t h o d o f utility measurement cannot succeed. Choices between bets will be deterts o f mined n o t o n l y b y t h e a m o u n ts m o n e y i n v o l v e d , , b u t also also b y t h e preferences the subjects have among the pro babiliti babilities es i nvolved. O nly an ex periment al p rocedure which holds o n e o f these variables constant, o r otherwise allows for it, can hope to measure t h e o t h e r . T h us u s m y experim e n t s c a n n o t be regarded as a way o f measuring probability preferences; they show only that such preferences preferences exist. It may nevertheless be possible to get an interval scale of the utility o f m o n e y fr froo m gambling experiments b y designing a n exp e rim ri m ent which measmeasures utility a n d p robability preferencess si ence sim m ultaneo usly . S uch exp eri eri ments are likely to be complicated a n d difficult t o r u n , b u t they can be designed. Subjective probability. First, a clarification o f terms is necessary. The phrase subjective probability h a s
value choices were c o m p r owere m i s e used, b e t w etehne m a x i m i zwere i n g e xa-
been in otwo fo r a used school f t hwo ua gy sh: t aasb oa u tn a tmh e
THEORY O F D E C I S IIO ON
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397
logical basis o f m a t h e m a t i c a l p r o b ability (51, 52, 80) and as a n a m e fo r a t ransformat ion on the scale of mathematical probabilities which is somehow related to behavior. O n l y the latter usage is intende d here. The clearest distinction between these t w o not ions arises from considera-
ect i ve pro babilit babili t y) a att a b o u t equals ob j ecti equals ob 0.2. Griffith (81) f o u n d s o m e w h a t similar results in an analysis o f p arimutuel bet t ing at race tracks, as d id Attneave (17) in a guessing game, an d Sprowls (178) in an analysis o f various lotteries. The M o s t e l l e r and N o g e e data (142) can, o f course, b e
ject ion probability of what happens when an o b(e.g., tive can be denned
subjective f o r subjective analyzed probabilities instead of Mostel values. l e r a n d N o g e e per for med such an analysis and said that their results were in general agreement wit h P r e s t o n and Baratta's. H o w e v e r , M o s t e l l e r a n d N o g e e f o u n d no indifference p o i n t f o r their H arvard students, whereas the N a t i o n a l Guardsmen had an indifference point at abo ut 0 .5 .5.. T hey are not able to reconcile these differences in results. T h e n o t i o n o f subjective probabil-
in a gam e o f craps). craps). If the subjective probability is ass assum um ed to be different from the o bjectiv bject ivee pro bability, bability, then th e concept is being used in its seco n d , or psycholo gical, gical, s ense. ense. O ther t erms wit h t h e same meaning have also been used: personal probability,
psychological probability, expecta-
tion (a poor term because of the danger o f confusion with expected value). (For a m o r e e l a b o r a t e m e n t t o f concept s s in in this area, st reeea t192.) I n 1948, prior to the M o s t e l l e r and
difficulties. ity ity has so so m eofserious log ical probability The scale objective is
sidered t o represent awinning value bidder o f play money such that the is indifferent bet ween it and the bet he is bidding for, and if it is further assumed that utilities are identical with t h e m o n e y v a l u e with e of the the play m o n e y and that all players have the same subjective probabilities, then these data can be used to construct a subjective subjecti ve pro bab ility sc scale. ale. P resto resto n and Baratta constructed such a scale. The subjects, according to the scale, sca le, o veresti verestim m at atee lo w prob abiliti abilities es
in a moment have occasionally led people to think of a subjective probabilit y scale bounded at 0 but not at 1. This is surely arbitrary. certainty is T h e c o n c e p t t o f absolute certainty neith er more nor less indeterminate
o f event A is P, and that of A not occurring is Q Q, t, hen P+Q=1. S h o u l d
aindifference n d underestimate ones, with a n point high (where subjective
this rul e Intuitively h o l d f o r subjective bilities? it seems probaneces-
b o u n d e d by 0 and 1. S h o u l d a subN o g e e e x p e r i m e n t , P r e s t o n a n d jective probability scale b e similarly Baratta (149) used essentially similar bounded, bounded, o r n o t ? ? I f n o t , t h e n m a n y logic and a somewhat similar experi- different subjective probabilities will m e n t to measure subjective prob- correspond to the objective probaabilities instead o f subjective values. bilities 0 and 1 (unless some transThey required subjects to bid c o m - format ion is used so that 0 and 1 obpetitively for the privilege of taking jective probabilities correspond to a bet. A l l bids were in p l a y m o n e y , infinite subjec tive pro babilities, babilities, w hich hi ch and the data consisted of the w i n n i n g seems un li kely). Considerat ions o f bids. I f each w i n n iinn g bid can be c o n - t h e addit ion t heorem t o b e discussed
than is the concept of absolute im-
possibility. E ven m o re dr dras astt ic ic logical logical pro blem s arise in connection with the addition t h e o r e m . If the objective probability
398
WARD E D W A R D S
sary that if we know t h e subjective probability of A we ought to be able to figure out the subjective probability o f not-^4, and the only reasonable rule for figuring it out is subtraction of the subjective probability o f A f r o m that o f complete certainty. But the acceptance o f this addition
culties is to to stop thinking about a scale o f subjective probabilities and, instead, to think of a weighting function applied to the scale o f objective probabilities which whi ch weights these these objective probabilities according to their ability to control behavior. PrePres u m a b l y , I w a s studying this ability
theorem f o r subjective plus the idea o f boundedprobabilities subjective probabilities means that the subjective probability scale scale must mus t be identical with the objective probability scale. Only for a subjective probability scale identical with the objective probability scale scale will the subjective probabilities o f a collection of events, one of which must h a p p e n , add up to 1. In the special case where only two events, A and not-A, are considered, a subjective
my experiments on probability in eren pref c es (55, 57 57,, 58, 58, 59) 59).. There is n o reason why such weighted probabilities should add up to 1 or should obey any other simple combinatory principle. V i e w s s a n d experiments which combine utility a n d subjective probability. The philosopher Ramsey published in 1926 (reprinted in 150) an essay on the subjective f o u n d a t i o n s of the theory o f probability; this contained an axiom system in which both utility
S I o r S 2 in probability Fig. 3 wouldscale meet like the requirements o f additivity, and this fact has led to some speculation about such scales, particularly about 51. But such scales do not not meet the additivity requirements when more than two events are considered. O n e w a y o f avoiding these diffi-
and subjective probability probability appeared appeared. . H e used 0.5 subjective as probability a reference point from from which to det e r m i n e utilities, and then used these utilities to determine other subjective probabilities. Apparently, economists did not discover Ramsey's essay until after von Neumann and Morgenstern's book aroused interest in the t he subject. The only other fo r mal axiom system in which both utility and subjective probability play a part is one proposed by Savage
H
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5 2.
o
Si
Q
0.5
LJ
O
Ld
CO
D
0
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O B J E C T I V E E P R O B A B I L I T Y FIG. 3. H Y P O T H E T I C A L S U B J E C T IV E P R O B -
A B I L I T Y CURVES
(171), which is concerned with u n certainty, rather than risk, and uses t h e concept o f subjective probability in its theory-of-probability sense. The most extensive and important experimental work in the whole fi fiel el d o f decision making under risk a n d uncertainty is now being carried out by Coombs and his associates at the University o f Michigan. Coomb Coombss s thinking about utility and subjective probability is an outgrowth of his t h i n k i n g about psychological scaling general. l. (For (For a discussion of in genera of h is views, see 43, 44, 45, 46, 47.) The
THEORY O F D E C I S IO IO N
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399
objective probability. probability. T h e subject measure both utility and subjective from w h o s e j u d g m e n t s t h e ordered probability on an ordered metric metric utility measurement was conscale. A n ordered metric scale has all structed was then presented w i th t h e properties of an ordinal scale, imaginary bets involving these ob and it t u r n e d and, i n addition, t h e distances and, distances b e - jects and probabilities, and chose the t w e e n some or all of the stimuli can. out that she almo st alw ays chose b e r an a n k o r de de r e d. d . C o o m b s has de- o n e with the higher expected utility. essence o f his work is the attempt to
veloped fvarious experimental cedures o r obtaining such i n forproma t i o n a b o u t t h e spacings o f stimuli. I n the most important article on utility and subjective probability to c o m e e out of the C o o m b s a p p r o a c h , C o o m b s and Beardslee (48) present a n analysis o f g am bling deci dec is io ns in volving three independent variables: utility f o r prize, utility f o r stake, a n d subjective probability. A l l three a r e assumed measurable only up to an ordered metric, although it is as-
This
x p e r i m e n t isof significant only as an e illustration the application o f t h e m e t h o d ; t h e conclusion that subjects attempt to maximize e x pected utility cannot very c o m f o r t ably be generalized to other subjects and to real choices without better evidence. C o o m b s and Milholland (49) did a m u c h m o r e e l a b o r a t e e x p e r i m e n t in which they establi established shed ordered m etric scales, b o t h for the utilities o f a collection o f o b j e c ttss and for the subjec-
of a collection of ng the tive probabilities sumed hpsy ps cholo giocal (e.g., Robin Roberts will ability othat f lo si sing t th e ystake isgical ne mpro i n ub-s statements t h e psychological probability o f w in 20 games next year). Statements winning the prize, an assumption that and objects were combined into limits the perm is issi sible ble unde rly ing bets, and the two subjects subjects f o r psychological probability functions w h o m the ordered metric scales had to shapes shapes like tho tho se in in Fi Fig. 3. A n been established were asked to m a k e elaborate graphic analysis of the in- judgments about which bet they difference surfaces in this three- w o u l d most, a n d which they w o u l d dimensional space is given, contain- least, prefer from among various se j u d g m e n t s w e r e in g far too many interesting relation- triads o f bets. T h e se ships to summarize h e re r e . A n e x - examined to discover whether or not periment based on this model was de- they demonstrated th e existence o f
signed. si gned. rel uctan t to use us o f Cmooonmbs o besums e y asis the valuable jects in his experiments because o f the danger that subjects will respond to the num erical t he a m o u nt eric al value o f the nt o f dollars rather than to the p s y c h o value. lue. T herefore h e used logical va various desirable objects (e.g., a radioo ) as stim ul i, and m easured their radi utility by the techniques he has developed to obtain ordered metric scales. He used simple numerical statements of probability as the
at least one indifferen eren ce prob convex curve between utility and indiff subjective
ability (the requirements f o r d em em o n strating the convexity of an indifference curve b y m e a n s o f ordered metric judgments are fairly easy to state). A n u m b e r o f cases consistent with a convex indiff indiffere ere nce curve were f o und, but a retest of the ordered metric data revealed changes which eliminated all of the cases consistent with a convex indifference curve f o r o n e subject, and all but one case f o r
the other. It is not possible to m a k e probability probability stimuli, and assum ed that subjective ass wasum equal to a statistical test of w h e t h e r or not
4
W A R D EDWARDS
that one case might have come about by chance. No evidence was f o u n d fo r t h e existence o f concave indifference curves, which are certainly inconsistent with the theory of risky decisions. This experiment is a fine example of the strength and weak of the the Coombs approach. It ness ness
the idea that perhaps only two subjective probability f u n c t i o n s are necessary. Santa Monica Seminar. In the s u m m e r of 1952 at Santa Monica, C a l i f o r n i a , a group of scientists conferred o n problems o f decision making. They met in a two-month semi-
makes almost assumptions, takes very little fornogranted, and avoids t h e concept o f error o f judgment; a s a result, m u c h of the potential inf o r m a t i o n in the data is unused and rarely can any strong conclusion co nclusionss be drawn. A most disturbing possibility is raised rai sed by experi experiment mentss by Mar Marks ks (13 133) 3) and Irwin (94) which suggest that the shape of the subjective probability f u n c t i o n i s influenced by t h e utilities involved in the bets. I f utilities a n d
nar of sponsored University Michigan and by the the Office o f Naval Research. The dittoed reports of these meetings are a gold mine of ideas for the student of this problem. Some of the work done at this seminar is now being prepared for a book on Decision P rocesses edited by R. M. rocesses edited Thrall, C. H. Coombs, and R. L. Davis, of the University of Michigan. Several minor exploratory experiments were done at this seminar. V a il (190) did an experiment in which
subjective probabilities are not inde- h choice f o u r o children e gave t h e bets o f f various they h side pendent, then there is no hope of pre- whic dicting risky decisions unless their wanted to be on. On the assumption l a w o f combination is known, and it o f linear utilities, he was able to comprobabilities ties for these seems very difficult to design an ex- pute subjective probabili children, ren, howperiment to discover that law of com- ch ildre n. The same child bination. However, the main dif- ever, were used as subjects for a ferences that Marks and Irwin f o u n d number o f other experiments; s o , w e r e between probabilities attached w h e n Vail later tried them out on to desirable and undesirable alterna- some other bets, he f o u n d that they tives. tives. It is perf ec t l y possible that consistently chose the bet with the probability of winning, rehighest there is one subjective probability highest f u n c t i on f o r bets with positive e x - gardless of the amounts o f money i n -
pected values and a different one for bets with negative expected values, just as the negative branch of the Markowitz utility f u n c t i o n i s likely to be different f r o m the positive branch. The results of my probability pref eren c e experiments showed very great differences between the probability probabili ty preference patte patterns rns f o r positive and for negative expected value bets (57), but little difference between probability preferences at different expected-value levels so
volved. Whe When n 50-50 bets were involved, one subje volved, subject ct consistently ch chos ose e the bet with the lowest expected value. No generalizable conclusions can be drawn f ro r o m these experiments. Kaplan and Radner (100) tried out a questionnaire somewhat like Coombs's method o f measuring subjective probability. Subjects were asked to assign numbers to various The The statements. statements. numbers could be anything f r o m 0 to 100 and were to the e likelihood that th the e represent th
long as zero expected value was not crossed (59). This evidence supports
statement was true. The hypotheses t o be tested were: ( a ) fo r sets of ex-
THEORY OF DEC ISIO ISIO N
M AKIN G
401
and mutually exclusive haustive and haustive statements in which the numbers assigned (estimates o f degree o f belief) w e r e nearly equal, the sums of these numbers over a set would increase with t h e number o f alternatives (because low probabilities would be overestimated) ; (b (b) ) fo r sets with t h e same numbers of alternatives, those those with one high num n umbe berr as assig signed ned w o u l d have a lower set sum than those with no high numbers. The first prediction was verified; the the second was not. A n y judgments of this sort are so m u c h more likely to be made on the basis o f number preferences a n d similar variables than on subjective probabilities that they offer very little hope as a method of measuring subjective probabilities. Variance preferences. Allais ( 2 , 3, 4 ) and Georgescu-Roegen (78) have argued that it is not enough to apply a transform on on objective objective value and value and on objective probability in order to predict risky decisions f r o m expected utility (see also 188); it is also necessary to take into account at least the variance, and possibly the higher moments, of the utility distribution. There are instances in which this argume arg ument nt see seems ms convincing. You would probably prefer the certainty o f a million dollars to a 50-50 chance
ples of thi thiss type. However, from a s i m p l e - m i n d e d psychological point o f vie w, these examples are irrelevant. It is enough if the theory o f choice can predict cho choice icess involving famil familiar iar amounts o f money a n d familiar probability diff di fferences— erences— choices such as those w h iicc h people are accustomed to making. It may be necessary for e c o n o m i c theory that the theory of choice be universal and exceptionless, but experimental psychologists need not be so so ambitious. This is fortunate, because the introduction of the variance and higher moments of the utility distribution makes the probl e m o f applying t h e theory experimentally seem totally insoluble. insoluble. It is difficult enough to derive reasonable methods o f measuring utility alone f r o m risky choices; when it also becomes necessary to measure subjective probability and to take the higher moments of the utility distribution into account, the problem seems see ms hope hopeless less.. Allais ap appa paren rently tly hopes to defeat this problem b by y using psychophysical methods to measure utility (and presumably subjective probability also). This is essentially what Coombs has done, but Coombs has recognized that such procedures are unlikely to yield satisfactory interval sca scales les.. The doll dollar ar sca scale le of
nothoing. f getting f o u r that mi lli on o rprefer I do either not think this ence is due to the fact that the ex-
pected utility utility of the the 50-50 bet is less than the utility of one million dollars to to you, although this is possible. A more likely explanation is simply that the variances of the two propositions are different. Evidence in favor o f this is the fact that if y o u k ne n e w y o u would b e offered this choice 2 0 times in succession, you would probably take the 50-50 bet each
the of that is so thoroughly value money taught to us it seems almost impossible to devise a psychophysical situation in which subjects would j u d g e the utility, rather than th the e dollar value, of dollars. They might j u d g e the utility of other valuable objects, but since dollars are the usual measure o f value, such judgments would be less useful, and even be likely these the se jjud udgme gments nts w would ould be likely to be contaminated contamina ted by the dolla dollarr values of the objects. I would get more utility
time. Allais Allais (5) has has constructed a number o f more sophisticated exam-
shaver than I from w o u l d af rnew o m a electric ne w washing machine,
4022 40
WARD EDWARDS
b u t because o f m y k n o w l e d g e o f the relative money values values o f these these o b jects, I would certainly choose t h e washing machine if given a choice between them. Somewhat similar arguments can be applied against using psychophysical m e t h o d s t o m easure easure subjective subjecti ve pro bability . A final point is that, since these subjective scales are to be used to predict l d b e best if they coul d choices, it w o u ld b e derived from similar choices. (175) Other approaches. S hackle (175) h a s proposed a t h e o r y o f decision making under risk a n d u n c e r t a i n t y . This theory is u n i q u e in that it does n o t assume a n y kind o f maximizing behavior. For every possible outc o m e o f a decisi decisioo n made in a risky o r uncertain situation, Shackle assumes that there is a degree o f p otential surprise that this, rather than some o t h e r , o u t c o m e w o u l d o ccur ccur.. E very very outcome-potential surprise pair is ranked in accordance with it s ability stim m ulate t h e mind (stimulation in t o sti i n creases with increasing outcome a n d decreases with increasing potential surprise). The highest-ranking positive outcome-potential surprise pair a n d the highest-ranking negative pair a r e f o u n d , a n d these t w o possibilities alone determine what the individual will d o . Semi-mathematical methods are used to predict the o u t c o m e of consideration o f possible lines of action. A l t h o u g h attempts have been made to relate it to W a l d 's 's m i n i m a x principle f o r statistical decision functions (see b e l o w ) , t h e fact remains that most critics of the Shackle point view ew have judged it to be either t o o o f vi vague to be useful, or, if specified in detail, too conducive to patently absurd predictiona (e.g., 201). Shackle's point of view was developed primarily to t o deal w ith it h uniq ue
criticized c o n v e n t i o n a l utility thecriticized o r y ' s attack on this p r o b l e m . S i nc nc e t h e usual f re que ncy theory o f p robability conceives of the probability as limit of the s o f a large th e limit the o u t c o m e s n u m b e r r o f similar trials, it is questionable that notions which u s e probability in the ordinary sense (like t h e n o t i o n o f maximizing expected utility) a r e applicable t o un i que choices. H o w e v e r , , this seems to be an experir o b l e m . I f n o t io io n s w h iicc h u s e m e n t a l p ro o r d i n a r y p r o b a b i l i t y a r e incapable o f predicting actual unique choices, then it will be ne cessary cessary to seek other theoretical tools. But so l o n g as a generally acceptable probability c a n b e defined (e.g., as in the unique toss o f a coin), it is not necessary to assume a p r i o r i that theories based on conventional p robabilities will be inadequate. When n o g e n e r a l l y a c ceptable p robability can be defined, then the p roblem becomes very different. C a r t w r i g h t and Festinger (38, 41) have proposed a t h e o r y a b o u t the time it takes to make decisions which is in some ways similar to those discussed in cussed section. T h e m ain ai n difin this section. ference is that they add the concep t o f restraining forces, a n d that they conceive of all subjective magnitudes as fluctuating r a n d o m l y a r o u n d a this they deduce dedu ce m e a n value. From this various propositions about decision times and the degree o f certainty which subjects wil l feel about their decisions, and apparently these propositions w o r k o u t e x p e r i m e n t a l l y pretty well (38, 3Q , 6 1 , 62). T h e orientation L e w i n i a n n theoretical seems to lead to this kind of m o d e l ; L e w i n , Dembo, Festinger, a n d Sears (122) present a f o r m a l l y similar theo theo ry abo ut level level of aspi aspira rati tioo n. O f course, the notion of utility is very
can similarly be made choices—choices o nly once. A llai lla iswhich (3) has
to the Lewinian notion of similar valence.
T H E O R Y O F D E
Landahl (115) has presented a mathematical model fo r risk-taking behavior based on the conceptual n e u r o l o g y of the mathematical biophysics school. Psychological comments. T h e area o f risky decision making is full o f fascinating experimental problems. O f these, the development of a satisfactory scale o f utility o f m o n e y an d o f subjective probability must come first, since th e theory o f risky d e cision making is based o n these n o tions. T h e criterion f o r satisfactoriness of these scales m ust ust be that they successfully predict choices other than those from which they were d e rived. To be really sati satisf sfact actoo ry , it is desirable that they should predict choices in a wide variety o f differing situ situat atii o ns. U nl ike ike t h e subjective
I S IO IO N
MAKING
4 3
small pilot experiments o n their sons, laboratory assistants, or secretaries. Such experiments are too seldom adequately controlled, and are almost never used a s a basis fo r largerscale, well-designed experiments. Whether a n ill-designed Whether ill-designed a n d haphazardly executed little experiment experiment is better than n o experiment at all is questionable. T h e results o f such pilot experiments to t o o o ft ften en a r e picked u p a n d written into t h e literature w i t h o u t adequate warning about the conditions under which they were performed and the co nsequent lim it itaations on the the significance o f the the results. T H E T R A N S I T I V I T Y O F C H O I C E S
In the section o n riskless choices this paper presented presented a definition o f e c o n o m i c m a n . T h e most important
scales that found these usually in psychophysics, it is likely scales will differ widely from person t o person,
part de finiti fithat nitio n can be summed o f this up by saying oeconomic man is
subjective am Does bling itselfprobabilities? and ghow have utility, m u c h ? To w hat hat extent can these subjective jecti ve scales cales be changed by le arning? To what degree do people differ, a n d c a n these differences b e correlated with environmental, historical, or personality differences? Finally , psypsychologists might b e able t o shed light on the complex economic problem o f interacting utilities o f diff differe ere nt goods. T h e area o f risky decision making, like th e area of the theory o f games,
Two economists have designed to periments specifically intended exintended test the transitivity of choices. P apandreo andr eo u perfo rm ed an ela elabo bo rate rate and splendidly controlled experiment (145) designed to discover whether o r n o t intransitivities occurred in im agined-choice situations. H e prepared triplets o f hypothetical bundles of admissions to plays, athletic contests, concerts, etc., a n d required h is subjects to choose between pairs o f bundles. Each bundle consisted o f
tends int o it encourage those inter ino f carry the custom o ut ested carr y ing
a total e.g., o f f our admissions two events, 3 plays and 1 totennis
rational. T h e concept o f rationality involves t w o parts: that o f a w e a k so a new determination o f each scale involves m u s t b e made fo r each n e w subject. ordering o f preferences, a n d that o f axim iz izee som eIt can o n l y b e hoped that t h e scales choosing so as to m axim do not change in time to any serious thing. O f these concepts, the one thee o ne degree; if they d o , then they a r e which seems m o s t dub io us is th o f a weakly ordered preference field. useless. is dubious because because itit implies O n ce scales o f utility a n d subjec- This is choices a r e transitive; that is, if tive probability are available, then that choices many interesting questions arise. A is preferred t o B, a ann d B is preferred , A is preferred to C . What about about t h e addition theorem theorem f o r to C then
404
WARD EDWARDS
the main experit o u r n a m e n t . . In the ment, each bundle is com pared pa red w ith ith t w o others involving t h e same kinds o f events, but in the better designed auxiliary experiment, experiment, a total of six different events a r e used, so that each th b u n d l e has no events i n c o m m o n w i th t h e o t h e r t w o b u n d l e s in its triplet. Since there a r e three bundles in each triplet, there are three choices between pairs pairs f o r each triplet, a n d these choices may, o r m a y n o t , b e transitive. The subjects were permitted to say that they were indifferent betw een tw o b undle s; consecons eq u e n t l y t h e r e w e r e 2 7 possible configurations o f choices, o f w h i c h o n l y 1 3 satisfied th e transitivity axiom. per cent I n t h e main exp eriment, 5 per o f t h e triplets o f judgments were the auxiliary experiintransitive; in the
f ewer intransitivities if he had permitted t h e indifference j u d g m e n t . I f subjects are really indifferent a m o n g a l l three of the elements o f a triad o f objects, but are required to choose between them in pairs and do so by chance, then they will choose in transitively one-fourth of the time. Papandreou's stochastic model gives o n e t he h e o r y a b o u t w h a t h a pp pp e n s w h e n preferences diverge just slightly from indifference, b u t p r e s u m a b l y a more detailed model can be worked o ut. P ap andreou's a ndreou's mo del p e rm its only three states: prefer A to B, prefer B to A, and indifferent. It o u g h t to be possible to base a m o d e l fo r such situations on t he cumulative and nd thus to p e r m i t any normal curve, a degree o f p reference. F o r every combination o f degrees o f p reference, such a model w o u l d predict the fre 4 per m e n t , o n l y a st cent. cent apandr develops stoo chasti chas ticc. Pmapandreo o del eo fo ur quency o f intransitive choices. choices under such conditions; the In the paired comparisons a m o n g results a r e certaihly consistent with bets (57) described in the section on elaborate i n the am o unt o f intransi intr ansitiv tivit ityy per- risky choices, quite elaborate mitted by his mo del del.. P ap andr andreo eo u transitivities could and did occur. concludes that a t least fo r his his specific H o w e v e r , it is easy to show that any experimental conditions, transitivity intransitivity involving f our o r m o r e does exist. objects in a paired comparisons M a y (138), using different kinds o f j u d g m e n t situation will necessarily st imuli in a less elaborate experiment, produce at least one intransitivity involving ng three thr ee o bjects bjec ts.. C o nsequently, c o m e s u p with results less consistent volvi wit h transitivity. M a y required a t h e intransitive triplet o r circular classroom to marriage groupthree make pairwise choices between part-
is the bestin of analysis for triad unit intransitivities these more com-
ners w h o were identified o n l y b y saying h o w intelligent, good looking, a n d rich they w er ere. e. Judgm ents o f indifference were n o t permitted. T h e results were that 27 per cent of the subjects gave intransitive triads o f choices. M a y suggests, very plausib l y , that intransitive choices m a y b e expected t o occur whenever more than one dimension exists in the stimuli along which subjects may
plicated judgment situations. I counted t h e fr equen cy o f occurrence o f circular triads a n d f o u n d that they regularly occurred about 20 per cent o f t h e total n u m b e r o f times they could occur. ( O f course, n o indiffere n c e j u d g m e n t s c o u l d d b e permitted.) T h e exp eriment f u l f i l l s May's criterion for the occurrence o f intransi-
oMrder rdaer thoeir their H o wgotten ever, y w u l d preferences. p r o b a b l y have
each could account a ntake d subjects be expectedbet, to both into
tivities, since both probability and
a m o u n t o f money were p resent in
THEORY THEORY OF DECISION MAKING
405
tio n co uld certai cert ainly nly b e tested w h e n making choices. I t m i g h t b e This no tio supposed that t h e difference b e t w e e n a n d made more specific b y approprit h e imaginary choices o f t h e Pap an- ate experiments. A final contribution in a related, dreou and May experiments and the ail's sto chastic chastic real choices in my e x p e r iim m e n t w o u l d b u t different, area is V ail's lead to differences i inn the frequency o f utility model (191). V ail ail as assu sum m es occurrence o f intransitivities, b u t that choices a r e dep endent o n utilithere were n o substantial differences ties that oscillate in a random manin m y e x op fe roccurrence i m e n t b e t w eine n the t h e justfrequencies imagining sessions and in the real gambling sessions, a n d what differ theree w ere, w ere in the di directio rectio n ences ther o f greater transitivity when really gambl ing. T hese hese facts facts sho uld faciliexperim ents on this probtate furth e r experim lem.
I n one sense, transitivity can never b e violated. A m i n i m u m o f three required to choices is is required to d e m o n s t r a t e intran sitivity sitivity . S ince these choices will e m a d that e in sequence, i t c a nnecessarily always b e bargued t h e person may have changed h is tastes b e t w e e n the first choice and the third. H o w e v e r , unless t h e assump tion o f constancy o f tastes o v eerr the period o f th e period o
ner around aplus m ean value. From Fr om th this is reasonassumption a few other able ones, he deduces that if the over-all preference is 1>2>3, and if
1 is preferred to 2 more than 2 is preferred to 3, then th e frequencies o f then the occurrence of the six possible transitive orderings should be ordered as
fo l l o w s : 123>132>213>312>231 hiss result is certainly easy easy >321. T hi t o test e x p e r i m e n t a l l y , a n d sounds
plausible.
THE THEORY OF GA MES AN D OF
DECISION F U N C T I O N S '
This section will not go into t h e
theory o f ga gam m es es o r into into t h e intimately related subject o f statistical decision functions at all t h o r o u g h l y . These are ar e m ath athem em atical atical subjects o f a highl y
exp erimentation is made, n o experiments o n choice c a n ever b e m e a n 6 h e o r y o f choice ingful, and the w h o l e t he M ars arschak chak (134 (134), ), H urw icz (92), N eis eisser ser becomes emp ty (see 184 for a similar (143), Stone (181), and Kaysen (107) pubbecomes andd situation). S o this quibble can be re- lished reviews o f T he Theory of Games an Economic Behavior which present t h e fundajected at once. mental ideas in much simpler language than U t ilil i ty ty m a x im im i za z a ti t i o n will n o t w o r k th e o rigina riginall sourc source. e. M arschak w o rks out in except preference field. with C o n s e qa u etransitive n t l y , if the models
discussed in this paper are to predict experimental data, it is necessary that intransitivities in these data be
inf re que nt enough to be considered as errors. H o w e v e r , from a slightly different p o i n t o f view (54) th t h e occurrence o r n o n o c c u r r e n c e o f transitive choice patterns is an experimental p h e n o m e n o n , a n d presumably a lawf u l o n e . M ay has ha s suggested what that law is: Intransitivities occur
w h e n there along a r e conflicting judge. dimensions which tostimulus
detail the possible of a and complicated solutions three-person bargaining game, thereby
illustrates the general na ture o f a soluti solution on . The tw o v olu mes o f Contributions to the Theory of Games (112, 113), plus McKinsey's b o o k o n
provide an excellent excellent bibliogthe subject (129), provide raphy o f the mathematical literature. M c K i n sey's b o o k is an exposition o f the fu nd amental concepts, intended as a textbook, which is
simpler than v o n N e um u m a n n a n d Morgenstern and pursues certain topics further. Wald's book (198) is, of course, the classical work on statistical decision functions. Dross's b o o k (35) presents the fundamental ideas about decisi sion on function s mo re si sim m ply , and statistical deci with a somewhat different emphasis. Girshick and and Blackwell's book (79) is expected to be a
very useful presentation of the field.
4066 40
WARD EDWARDS
technical sort, with few statements which lend th them em selves selves to exp e ri rim m ent a l test. R ather, the p urp ur p o se o f this thi s to show section is to section show h o w these these subjects relate to w h at a t has g o n e b e fo r e , to give a brief s u m m a r y of the c o n t e n t s o f Theory of Games a nd Economic Be havior b y v o n N e u m a n n a n d M o r g e nn-
among them. In the case of tic-tactoe, these rules are trivial, since either player player c a n force a draw. B u t in more comp licated games o f strategy, these rules m a y b e useful. I n particular, t h e theory o f games m a y b e helpful in analyzing p rop er strategy in games having random ele-
stern to describe a to describe cards, bo er p e r i m e(197), n t s in and the area o f g am a m a e pfew l a y iexn g tments, h e t h r olike w i ntgh eo f shuffling dice. I t o f should — e x ppee r im im e n t s wh ich a r e s t i m u l a t e d n o t e d that t h e c o n c e p t o f a g am a m e is an by th e t h e o r y o f g a m e s a l t h o u g h n o t exceedingly gen eral co ncep t. A sci scien en directly relevant to it. y m a y b e tist in his his l a b o r a t o r y e c o n T h e theory of games. T he theory o f sidered to be p l a y i n g a game against games p robably originated in the N a tu t u re r e . ( N o t e , h o w e ve v e r, r , that w e w o r k o f B o r e l (31, 32, 33, 34; see also c a n n o t e x p e cctt N a t u re r e to try to defeat von the scientis scientist.) t.) N eg o tiators tiators in a l a b o r 71 , 72) in the 1920's. In 1928, von (19 5),, w o rking indep end- dispute are p l a y i n g a game against N e u m a n n (195) e n t l y o f B o r e l , p u b l i s h e d the first o n e another. A n y situation in which proof of the fundamental theorem in m o n e y (or (or some valuable equivalent) the t h e o r y , a a t h e o r e m that B o r e l had may be gained as the result of a not believed to be generally true. H o w e v e r , the subject did not b e c o m e
proper choice o f strategy can be c o n sidered as a game. imp ortant until 1944, when v o n To talk about game theory, a few N e u m a n n and Morgenstern p ub- technical term s are necessary. necessary. A lished their epoch-making book (196). strategy is a set of personal rules fo r ( A second edi ti on , w i t h a n appendix p l a y i n g the g a m e . Fo Forr each possible o n cardinal utility measurement, first m o v e o n y o u r part, y o u r r o p came out in 1947 [1 [197 97]. ].)) Th Thei eirr pur- ponent will have a possible set of repose was to analyze mathematically a sponses. For each possible response very general class o f p r o b l e m s , w hi hich ch b y y o u r o p p o n e n t , y o u w i ll have a set might b e called problems o f strategy. o f responses, and so on through th e C o nsi nsi der a gam e o f tic-t tic-tacac-toe. toe. Y o u game. A strategy is a l ist w hich hi ch specia t y o u r m o v e w i ll be for every k n o w at any m o m e n t in the g a m e fies w h at the to opw p ohnaetn t a rmoves e , b u tavailable y o u d o n oyt o uk rn o w which o n e h e will choose. T h e o n l y information you have is that h is choice will n o t , in general, b e c o m pletely r an a n d o m ; h e will m a k e a m o v e which is designed in s o m e way to increase h is chance o f w i n n i n g and diurs. T hus the situati situatioo n is minish y o urs. o n e o f uncertainty rather than risk. Y o u r g o a l s are similar to y o u r opp o n e n t 's 's . Y o u r p r o b l e m i s: s: w h a t strategy should you a d o p t ? The
set are of mplaying. conceivable previous o v e s o f game you the particular Needless to say, only for the simplest games (e.g., matching pennies) does this concept concept o f strategy have have a n y em p iri i rica call m eaning. A ss ssoo ci ciat ated ed w ith strat strategies egies are imn is a set of putations. A n i m p u t a t i o n p a y m e n t s m a d e a s a result o f a game, o ne to each each playe r. In general, different imputations will be associated w i th different sets o f strategies, b u t fo r a n y given set of strategies there
theory o f games offers n o practical h el p i n developing strategies, but it
may be m o r e than one imputation alitioo ns). ns) . ( in games invo lving co aliti
does offer r u l e s a b o u t h o w t o choose
I m p u t a t i o n X is said t o dominate
ON THEORY O F D E C I S IIO
i m p u t a t i o n n F if e of the if o ne or or m o r e players has separately greater gains ( o r smaller losses) in X than in F and can, by acting together (in the case of m o r e than*one player), enforce t h e o ccurr ccu rrence ence o f X , o r o f s o m e o t he h e r im putation at least as good. The relationship o f d o m i n a t i o n is not transitive. A solution is a set of imput at ions, n o n e o f which dominates another, such that every imput at ion outside the solut ion is d o m i n a te t e d by at least
o n e i m p u t a t i o n w i t h i n t h e solut ion. V o n N e u m a n n a nd n d M o r g e n st s t e rn r n a sssert that t h e task of the t h e o r y o f games is to find solutions. For any game, there may be one or more than o n e . O n e b ad ad feat ure o f the t heory o f games is that it frequently gives a large, o r even infinite, n u m b e r o f solutions for a game. The abo ve de fi finition nition s m ake clear that t he only det erminer of behavior is in games, according to this theory, is t h e a m o u n t s o f m o n e y w h iicc h m a y b e or the expected amounts w o n or lost, or in games ga mes w ith it h rando rand o m element s. s. T he fun o f playing, if a n y , is irrelevant. T h e minimax loss principle. T h e n o t i o n s o f d o m i n a ti t i o n and of solut io io n imp l y a n e w f u n d a m e n t a l r u l e f o r decision m a k in in g— g — a rule sharply different from t h e rule o f maximizing
M AKING
4 7
w h e n you consider that the o t h e r player player is out to get y o u , and so will d o his best to m a k e the worst possible outcome f or yo u occur. I f this rule is expressed geometrically, it asserts that the p o i n t you should seek is a saddle-point, like t h e highest point point in a m o unt a in pas pass (t he best r u l e f o r crossing mountains is to minimize th e maximum height, so explorers seek out such saddle-points). B ef ore w e g o a n y further, w e need a few few m o re definit defi nitions. ions. G ames m ay be am am o ng any a ny num be r o f play pla y ers, ers, but the simplest game is a two-person game, and it is this kind o f game which h a s been most extensively a n d most successfully analyzed. Fundam e n t a l l y , t w o kinds o f payoff a r are possible. The simrangements rangements plest and most common is the one in which o n e play pla y er w ins w hat t h e o t h e r player loses, or, more generally, the o n e f o r whic h the sum of all the pay result of the game game ments made as aa result is is called zero-sum game. zero. This is called a zero-sum I n nonzero-sum games, analytical co m plexities plexities ari arise. se. T hese hese can be diminished by assuming the existence o f a fictitious fictitious ex t ra player, w h o w i n s o r loses enough t o bring the sum sum o f payments back to zer zeroo . S uch uch a fictit ious player cannot be assumed to have a strategy and cannot, of co urse, urse,
utility or expected utility with which this paper has been concerned up to this section. This r u l e is the rule o f minimizing t h e m a x i m u m lo l o ss s s, o r , m o r e briefly, minimax loss. In o t her h er words, t h e rule is to consider, fo r each possible strategy that you could adopt , what the worst possible outcome is, and then to select that strate g y which would have have t h e least illeffects if the worst possible outcome h a p p e n e d . A n o t h e r w a y o f p u t t i n g the same idea is to call it the principle
interact with any of the other players. In zero-sum t wo-person games, w h a t will happ en ? E ach ac h player, according to the theory, should pick h is minimax strategy. B u t will this r e sult in a stable s o l u t io i o n ? N o t a l w a y s . S omet imes the surface representing the possible outcomes of the game
o f maximizing th e minimum gain, o r maximin gain. This rule makes considerable sense in two -pers -person on games games
his opponent's strategy, he can gain m o r e by s o m e o t h e r strategy. T h u s the game has no solution.
does not have a saddle-point. In this
case, if p l a y e r A choo ses h is m i n i m a x strategy, t h e n p l a y e r B will have an i ncenti ncent i ve no t t o use us e hhii s o w n m inimax strategy, because having f oun d o u t
40 8
W A R D EDWARDS
Various resolutions of this problem are possible. Von Neumann and Morgenstern chose to introduce the notion of a mixed strategy, which is a probability distribution o f t w o o r pure ure strategies. The f u n d a m e n more p more t a l theorem of the theory o f games i s that if both players in a zero-sum
agree with this), nor is it likely to be o f a n y practical use in telling y o u h o w to play a complicated game; the crux of the theory o f games is the p rincip le o f choosing tfre strategy w h i c h m i n i m i z e s t h e maximum e x pected financial loss; and the theory defines a solution of a game as a set
two-person game adopt mixed strategies which mi ni mi z e the the maximum expected loss, then the game will always have a saddle-poin saddle-point. t. Thu Thuss each person w i ll get, in the long r u n , h i s expected loss, and w i l l have no incentive to change his behavior even i f h e should discover what his opponent's mixed strategy is. Since A is already getting the minimum possible under the strategy he chose, any change i n strategy b y B w i l l only i n crease A's p a y o f ff,, and therefore cause and B to gain less or lose more than he w o u l d by his own minimax strategy. same is true of B. The The Games involving more than two p e o p l e introduce a new element—the possibility that two or more players w i l l cooperate to beat the rest. Such a cooperative agreement is called a coalition, and it f r e q u e n t l y involves side-payments among members o f the coalition. T h e method o f analysis f o r three-or-more-person games is to consider all possible coalitions and to solve the game for each coalition on the principles of a two-person game. This works fai r l y w e l l f o r three-percomplicated cated s o n games, but gets more compli a n d less satisfactory f o r still more
o f imputations which satisfies this principle for all players.
people. This is the end of this exposition o f
the content of von Neumann and the Morgenstern's Morgenste rn's book. It is of course impossible to condense a tremendous and difficult book into one page page.. The m a j o r points to be emphasized are these: the theory of games is not a m o d e l o f h o w people actually play games (some game theorists will dis-
In their book von N e u m a n n and Morgenstern say We have . . . assumed assumed that [utility] is n u m e r i c a l . . . substitutable and unrestrictedly restrictedl y tra transfe nsferab rable le between the various players. (197, p. 604.) Game theorists disagree about what this and other similar sentences mean. One l i k e l y interpretation is that they assume utility to be linear with the physical value o f money involved in a game and to be interpersonally comparable compa rable.. The linear util ut ilit ity y curves seem to be necessary for solving twoperson games; the interpersonal comparability is used for the extension to n per person sons. s. Att Attemp empts ts are being made t o develop solutions free o f these a s sumptions (176). Assumptions,
Statistical decision functions.
V on N e u m a n n (195) first used the mi ni m a x principle in his first publication 1928 28.. Ne Neym yman an o n game theory in 19
and Pearson mentioned its applicability to statistical decision prob1933 33 (144). Wal Wald d (198), who l e m s in 19 p r i o r to his recent death was the central figure in the statistical decis i o n - f u n c t i o n literature, first first seriously ap p lie d the m i n i m a x principle to statistical problems problems in 19 1939 39.. Ap Appa parre n t l y , all these uses of the principle w e r e completely independent o f o n e another. A f t e r Theory of Games a nd Economic Behavior appeared in 1944, Wald (19 198 8) rref efor ormu mula late ted d the proble problem m o f statistical decision making as one o f playing a game again against st Natu Na ture re,,
THEORY OF DECISION
M AKIN G
409
each o f which has a possible gain o r loss. I n s o m e cases, all of these gains and losses and the cost of observing can be exactly calculated, as in indust rial qualit y cont rol. I n o t h e r
confined their discussions to cases in which th e concepts o f minimax loss and minimax regret amount to the sam sam e thing . O ther suggest suggested ed principles are: maximizing the m a x i m u m expected gain, and maximizing some weighted average of the m a x i m u m and m i n i m u m expected gains (93).
a b o u t t h e cost o f b e i n g w r o n g and the gain of being right. At any rate, when they are put in this f o r m , it is o bvious that t h e ingredients of the p r o b l e m o f statistical decision making have a gamelike sound. Wald applied the m i n i m a x principle t o t hem in a way essentially identical with game theory. A very fr e qu e nt criticism of the m i n i m a x approach to games against N a t u r e is that N a t u r e is not hostile, as is the the o p p o n e n t t in a two-person g a m e . N a t u r e will n o t , i n general, use a minimax strategy. For this reason, other principles o f decision ma ki n g hav havee been suggest suggested. ed. T he simple principle o f maximizing e x pected utility (which is the essence o f the Bayes's theorem [15, 198] solution o f th e p r o b l e m ) is no t always applicable because, even though Nature is n o t hostile, s h e does n o t offer a n y w ay o f assi assigning gning a pro bability bability to each each possible outcome. In other words, statistical decision making is a p r o b l e m o f uncertainty, rather than o f risk. Savage has suggested the principle o f minimaxing regret, w h e r e r e gret is defined as the difference b e t ween the m a x i m u m w h i c h can be gained u n d e r any strategy given a certain state of the w o r l d and the amount gained under the strategy adopted. S avage avage be lieve s (170, also also personal c o m m u n i c at a t io io n ) that neither v o n N e u m a n n a nd n d M o r g e n st st e rn rn n o r Wald actually intended to propose th e principle o f minimaxing loss; loss; they
None commands can be general acceptance; each made to show peculiar consequences under some conditions (see 170). Experimental games. The concept s o f th thee theory o f games suggest a new field o f ex periment at ion: H o w d o people behave in game situations? S uch ex periment at ion w o u l d cent er o n t h e d e v e lo l o p m e n t o f strategies, strategies, pa rticularly mixed strategies, and, in th ree-or-more-p ers on games, on th e d e v e l o p m e n t o f coalit ions and o n the bargaining process. You should* rem e m b e r that t h e t heory o f g a m e s does n o t offer a m a t h e m a t i c a l m o d e l predicting the outcomes of such games (except in a fe w special cases); all it does is offer useful concepts a n d language for talking about t hem, and predict that cert ain out comes will n o t occur. A f e w minor ex periment s o f this kind have been conducted by Flood, a mat hemat ician, while he was at Rand C o rporat rpora t io io n. H e usually us ually used used colleagues, many o f w h o m w e r e e x perts in game theory, and secretaries as subjects. T h e general design o f his experiments was that a group of subjects were s h o w n a g r o u p of desirable objects on a table, and told that they, as as a group, could have the first object t hey removed from t h e table, and that they should decide a m o n g themselves which object to choose and how to allocate it. In the first experiment (64) t h e allocation p r o b l e m did no t arise arise because en o ugh duplicate objects were provided so that each subject could have o n e o f
The statistician must decide, on the basis o f observations which cost something to make, between policies,
it is cases, as in theoretical research, necessary to make some assumpt ion
of these principles
4100 41
WA RD E D W A R D S
be tter. P hysical hysical iso iso latio latio n o f o n e subject from a n o t h e r w o u l d m a k e it posgraduates, and the final selection was sible to match each subject against a made b y negotiation a n d voting. I n standard bargainer, t h e ex periment er the second experiment (65), in which o r a stooge, who bargains by a fixed th e subjects were colleagues a n d sec- set of rules that a r e u n k n o w n to the subject. ect. Flood (perso (perso nal co m m unicaretaries, a long negotiation process subj eliminated some of the objects, but a tion) is conducting experiments o f t h e kind o f obj ect t h e group selected. T h e subjects were Harvard under-
tfrom ime alimit a select ion negotiby lot t h e rest. m o n g t gforced Further ations to solve the allocation problem were t erminat ed b y a secretary, w h o snatched t h e o b j e c t , a n n o u n c e d that it w a s hers, a n d then tried t o sell i t . N o o n e w a s willing t o b u y , so the experiment terminated. O ther experiexperiments (66, 67) showed that coalitions s o m e t i m e s f o r m , that a sophisticated subject could blackmail t h e g r o u p f o r a n extra side-payment b y t hreat ening
sort.
Fo Forr
this s, A sch's t hree-or-more-person game sch's (16) (16 ) tec hn ique o f using a group consisting o f o n l y o n e real subject and all the rest stooges might w e l l b e used. I t w o u l d d b e interesting, fo r insta instance, nce, to see how the pro bability of a coalition between two players changes as the n u m b e r and power of players united against them increase. T h e t h e o r y o f games is the area a m o n g those described in this paper in which t h e u n c o n t r o l l e d a n d casuto change his vote, and that the ally planned pilot experiment is having restrained succeeded olanrccen e , ous had secretary, to be physically in subsequent sessions sessions t o prevent m o r e larc larceny eny . T h e general co nclusio nclusio n
most toleast o ccur. S uch experiexperi are at as dangerous m e n t s likely herea s they are in the area o f risky d e cisi cisioo n m aking . Floo d' d'ss results sugsug-
suggested by all these experiments is that even experts on game game t heo h eo ry are
gest that it is especially important to
free an d ex periment s left bargainers used physical objects, whose utilities p r o b a b l y vary w i d e l y fr froo m subject to subject, as stim ul i to bargain o ver. This is naturalistic, but produces data t o o c o m p le il e x x and t o o n o n n u m e r ic al fo r easy analysis. A simp ler si situatuation in which t h e p o ss s s ib l e c o m m u n i cations f r om one bargainer t o another are limited (perhaps by means o f an artificial v o c a b u l a r y ) , in w h ic ic h t he h e subj ect s do no t see o ne ano t her, and in which t h e object bargained
matical theories about about h o w paltereople make choices among desirable natives. nativ es. T hese hese theo ries cente r on the n o t i o n of the subjective value, o r utility, of the alternatives among which t h e decider must choose. They assume that people behave rationally, that is, that they have transitive preferences a n d that they choose in such a way as to maximize utility or expected utility. T he traditi traditioo nal theo ry o f risk riskless less choices, a straightforward theory theory o f
over sum of mpreferably m e r e l y is a simple, o n e y , w o u being l d b e
w a s th ut ilit challenge chall a th t h edby they dmax the e menge e m oimizat n s tr t r at a tion, io io n that
u s e naive subjects and to use t hem o nly o nce, unless unles s th t h e effects o f expertness a n d experience are the m a j o r concern of the ex periment .
less rational rational a n d m o r e c o n v e n t i o n a l than game theory might lead experim e n t e r s t o expect. Psychological comments. T h e m o s t SUMMARY nut rit ive research problems in this For a long t ime, economist s and area see seem m to be the socia sociall pro blem s o f ho w bargaining takes place. Flood's others have been developing mathe-
THEORY O F DECISION DECISION
matical tool o f indifference curves made it possible to account for riskless choices w i t h o u t assuming that utility could be measured on an in-
terval scale. T h e theory o f riskless choices predicted from indifference curves h a s been worked o ut in detail. Experimental determination o f indif-
M AKING
411
fo r utility a n d subjective probability measured on an ordered metric scale, and did some experiments to test implications of the model. E c o n o m i s t s have become worried the assumption that choices about about Experiments have are transitive. shown that intransitive patterns of
so stochastic curves is occur, ference and has choice been possible, But utility measmodels do been and which attempted. have developed ured on an interval scale is necessary p e r m i t occasional intransitivities. (though n o t sufficient) f o r welfare The theory of games presents an elaborate mathematical analysis o f economics. Attention was turned to risky t h e problem o f choosing choosing from from among choices b y v o n Neumann a n d Mor- alternative strategies in games o f genstern s demonstration that com- strategy. This paper summarizes the plete weak ordering o f risky choices main concepts o f this analysis. T h e implies the existence o f utility meas- theory o f games h a s stimulated in a n interval scale. Hosteller terest in experimental games, and a urable o n an Nogee experimentally experimen tally deter- fe w bargaining experiments which a n d Nogee mined utility curves f o r money from can be thought of in game-theoretical
decisions, them t o gambl a n d used predictingother gambling decisions. Edwards demonstrated t h e existence o f preferences among probabilit probabilities ies in gambl ing situations, which complicates the experimental measurement o f utility. Coombs developed a model
terms performed. these been topics represent a new A l l have
an d rich field for psychologists, in which a theoretical structure has already been elaborately worked o u t ma ny experiments need need and in which many t o be p e r f o r m e d .
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