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PSYCHOLOGICAL BULLETIN Vol.   51, No. 4, 19 1954 54

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

 



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 -

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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

388 38 8

 

WARD   E D W A RD S

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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389 38 9

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

 

393 39 3

MAKING

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

3944 39

WARD   E D W A R D S

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-

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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

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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

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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

_J

5 2.

 

o

Si

 

Q

0.5

LJ

O

Ld

CO

D  

0

05

I

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

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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

 

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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-

 

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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,

 

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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

 

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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.

 

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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

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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

 

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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,,

 

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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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