Cumulative Prospect Theory

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Jou rna l of Risk and Un cer ta inty , 5 :297-323 (1992)

© 1992 K l uw e r A c a de m i c P ub l i she r s

A dvan ces in P rospect Theory:C um ulative R epresentation of U ncertainty

A M O S T V E R S K Y

Stanford Univers i ty , Depar tm ent o f Psychology , S tan ford , C A 94305-213 0

D A N I E L K A H N E M A N *Univers i ty o f C al i forn ia a t Berke ley , Depa r tment o f Psychology , Berke ley , CA 94720

K e y w or ds : c umul a t i ve p r ospe c t t he o r y

Abstract

W e de ve l op a ne w ve r s i on o f p r ospe c t t he o r y t ha t e mpl oys c umul a t i ve r a t he r t ha n se p a r a b l e de c i s ion w e i gh ts

a nd e x t e nds t he t he o r y i n s e ve r a l r e spe ct s . T h i s ve r si on , c al l e d c umul a t i ve p r ospe c t t he o r y , a pp l i e s t o u nc e r t a i na s w e l l a s t o r isky p r ospe c t s w i t h a ny n um be r o f ou t c ome s , a nd i t a ll ow s d i f f e re n t w e i gh t i ng f unc t i ons f o r ga i ns

an d fo r losses . Two pr inc iples , d iminishing sensi tiv ity a nd loss avers ion, a re invoked to expla in the charac te r i s -

t i c c u r va t u r e o f t he va l ue f unc t i on a nd t he w e i gh t i ng f unc t ions . A r e v i e w o f t he e xpe r i me n t a l e v i de nc e a nd t h e

r e su lt s o f a ne w e xpe r i me n t c onf i r m a d i s t inc t i ve f ou r f o ld pa t t e r n o f r i sk a t t it ude s : r i sk a ve r s ion f o r ga i ns a nd

r i sk seeking for losses o f h igh pro babi l i ty ; r i sk seeking for ga ins an d r i sk avers ion for losses of low probabi l i ty .

Expected ut i l i ty theory re igned for severa l decades as the dominant normat ive and

desc rip tive m o de l o f dec i s ion m aking un de r unce r t a in ty , bu t i t ha s com e und e r se r ious

ques t ion in r ecen t yea r s . The r e i s now gene r a l ag r eem ent t ha t t he theor y does no t

provide an adeq uate descr ipt ion of individual choice : a subs tant ia l body of evidenceshows tha t dec ision mak ers systematically violate its basic tenets . M any al terna t ive mod -

e ls have be en prop osed in respon se to this empir ica l cha l lenge ( for reviews, see C am erer ,

1989; F ishburn, 1988; M achina , 1987) . Som e t ime ago we p resen ted a m od el of choice ,

ca l led p rospec t theory, which expla ined th e ma jor v iola tions of ex pec ted uti li ty theo ry in

cho ice s be tw een risky p r ospec ts w i th a sm all nu m b er o f ou tcom es ( Kahn em an an d Tve r -

sky, 1979; Tversky and K ahn em an, 1986) . Th e key e lemen ts of th is the ory a re 1) a va lue

fun ction tha t is conc ave for gains, convex for losses, an d ste ep er for losses tha n fo r gains,

*A n e a r l i e r ve r s i on o f t h is a r t ic l e w a s e n t i t l e d "Cum ul a t i ve P r ospe c t T he or y : A n A na l ys i s o f D e c i s i on und e rU n c e r t a i n t y . "

T h i s a r t ic l e ha s be ne f i t e d fr om d i sc us s ions w i t h Co l i n Ca me r e r , C he w S oo- H ong , D a v i d F r e e dm a n , a n d D a v i d

H . K r a n t z . We a r e e spe ci al ly g r a t e f u l to P e t e r P . W a kke r f o r h i s inva l ua b l e i npu t a nd c on t r i bu t i on t o t he

a x i oma t ic a nal ys is . W e a r e i nde b t e d t o R i c h a r d G onz a l e z a nd A m y H a ye s f o r r unn i ng t he e xpe r i me n t a nd

ana lyz ing th e d a ta . This work was s upp or ted b y Gra nts 89-0064 and 88-0206 from the A ir Force Of fice of Sc ienti fic

Re se a r c h , by G r a n t S ES -9109535 f r om t he N a t i ona l S c ie nc e F ounda t i on , a n d by t he S l oa n F ounda t i on .



298 A M O S T V E R S K Y / D A N IE L K A H N E M A N

and 2) a nonl inear t ransformat ion of the probabi l i ty sca le , which overweights smal l

probabi l i t ies and underweights modera te and h igh probabi l i t ies . In an impor tant la te rdeve lop me nt , severa l au thors (Quiggin , 1982; Schm eidle r, 1989; Yaar i , 1987; W eym ark,

1981) have advanced a new representa t ion , ca l led the rank -dep end ent or the cumula t ive

functional , that t ransforms cumulative ra ther than individual probabil i t ies . This ar t ic le

p r e se n t s a ne w ve r s ion o f p r ospe c t the o r y tha t i nc o r por a te s the c um ula t ive f unc t iona l

a nd e x tends the the o r y to unc e r t a in a s we l l t o r isky p r ospe c t s w i th a ny nu m b e r o f ou t -

c om e s . The r e su l ting m ode l , c a l le d c um ula t ive p r ospe c t the o r y , c om b ine s som e o f the

a t t rac t ive fea tures of both deve lopm ents ( see a lso Luce an d F ishburn , 1991) . I t g ives r i se

to d i f fe rent eva lua t ions o f ga ins and losses, which a re not d is t inguished in the s tand ard

cumu la t ive mo del , and i t provides a uni f ied t rea tme nt of bo th r isk and uncer ta in ty .To se t the s tage for the present deve lopment , we f i r s t l i s t f ive major phenomena of

choice , which v io la te the s tan dard m ode l and se t a minimal cha l lenge tha t m ust be m et

by any adequ a te descr ip tive theory of choice . Al l these f indings have be en conf i rmed in a

nu m ber o f exper iments , wi th bo th rea l and hypothe t ica l payoffs.

Fram ing effects. Th e ra t iona l theo ry of choice assum es descr ip t ion invariance: equiva-

l e n t f o r m ula t ions o f a c ho ic e p r ob le m shou ld g ive r i s e to the sa m e p r e f e r e nc e o r de r

(Arrow, 1982) . Co ntra ry to th is assumpt ion , th ere i s mu ch ev idence tha t var ia t ions in the

framin g o f option s (e .g., in term s o f gains or losses) yield systematically dif feren t prefer-ences (Tversky and K ahnem an, 1986) .

Non linear preferences. A ccord ing to the expe ctat ion pr inciple, the ut i li ty of a r isky

prosp ec t i s l inear in outc om e probabi l it ies. Al la is 's (1953) famo us exam ple cha l lenged

this pr inc ip le by showing tha t the d i f fe rence be tw een probabi l i t ies of .99 and 1 .00 has

m o r e im pa c t on p r e f e r e nc e s tha n the d i f fe r e nc e be tw e e n 0 .10 a nd 0 .11 . M or e r e c e n t

s tudies observed nonl inear pre fe re nces in choices tha t d o not involve sure th ings (Cam -

erer and Ho, 1991) .

Source dependence. P e op le ' s w i llingness to b e t on a n unc e r t a in e ve n t de pe n ds n o t on ly

on th e de gree o f uncer ta in ty b ut a lso on i ts source . El lsberg (1961) observ ed tha t p eop le

p r e f e r to be t on a n u r n c on ta in ing e qua l num b e r s o f r e d a nd g r e e n bal ls , r a the r tha n on

a n u r n tha t c on ta ins r e d a nd g r e e n ba ll s i n unknow n p r opor t ions . M or e r e c e n t e v ide nc e

ind ic a te s tha t pe op le o f t e n p r e f e r a be t on a n e ve n t in the ir a r e a o f c om pe te nc e ove r a

be t o n a m a tc he d c ha nc e e ve n t, a l though the f o r m e r p r oba b i li t y is va gue a nd the l a t t e r i s

c lea r (He a th and Tversky, 1991).

Risk seeking. Risk avers ion i s genera l ly assumed in economic ana lyses of dec is ion

under uncer ta in ty . However , r i sk-seeking choices a re cons is tent ly observed in two

classes of dec is ion problems. F i r st , peo ple o f ten pre fe r a sm al l probabi l i ty of winning a

large pr ize ov er the exp ected value o f that p rospect . Second, r isk seeking is prevalent w hen

peo ple mu st choo se betw een a su re loss and a substantial probabili ty of a larger loss.Loss' aversion. O ne o f the ba s i c phe n om e na o f cho ic e und e r bo th r isk a nd unc e r t a in ty

is tha t losses loom la rger than g a ins (Kahn em an and Tversky, 1984; Tversky and K ahne-

man, 1991) . Th e o bserv ed asymm etry be tw een ga ins a nd losses i s fa r too ext rem e to be

expla ined by incom e e f fec ts or by decreas ing r isk aversion .



ADVANCES IN PROS PECTTHEORY 299

The present development explains loss aversion, r i sk seeking, and nonl inear prefer-

ences in t e rms o f the va lue and the w eigh ting func t ions . I t i ncorpora tes a f raming p ro -cess , and i t can accommodate source p refe rences . Add i t iona l phenomena tha t l i e be-

yond the scope o f the th eo ry - -an d o f i ts a l t e rna t ives - -a re d i scussed l a te r .

The present ar t ic le i s organized as fo l lows. Sect ion 1 .1 in t roduces the ( two-part ) cu-

mulat ive funct ional ; sect ion 1 .2 d iscusses relat ions to previous work; and sect ion 1 .3

descr ibes the qua l i t a t ive p roper t i es o f the va lue and the weigh t ing func t ions . These

pro per t ies a re tes ted in an extensive study of individual choice, descr ib ed in sect ion 2,

wh ich also add resse s the ques t ion of m one tary incent ives. Imp l icat ions and l imi tations of

the th eory ar e d iscuss ed in sect ion 3. A n axiomatic analysis o f cum ulat ive pro spe ct

theory is p res en ted in the append ix .

1. Theory

Prosp ec t theory d i s t ingu i shes two phases in the cho ice p rocess: f raming a nd va lua t ion . In

the f raming phase , the dec i s ion make r const ruc t s a rep resen ta t ion o f the ac ts , con t ingen-

cies, and o utco m es that are releva nt to the decision. In the valuat ion phase, the de cision

m aker assesses the va lue o f each p rosp ec t and chooses accord ing ly . Al though no fo rmal

theor y of fram ing is available , w e have learne d a fai r am oun t a bo ut the ru les that gove rn

the rep re sen ta t ion o f ac ts , ou tcom es, and con t ingenc ies (Tversky and Ka hnem an , 1986).

The va lua t ion p rocess d i scussed in sub sequ en t sec t ions is app l ied to f ram ed p rospec t s .

1.1. Cumulative prospect theory

In th e classical theory , the u t il ity of an u nce rtain pro spe ct i s the sum of the u t il it ies of the

ou tcomes, each weigh ted by i t s p robab i l i ty . The empi r i ca l ev idence rev iewed above

suggests two m ajor m odif icat ions of this theory: 1) the carr iers o f value are gains and

losses, not f inal assets; and 2) the value of each outcome is mul t ip l ied by a decisionweight , not by an add i t ive probabi l ity . The w eight ing sche m e use d in the or ig inal ve rsion

o f p ro sp ec t t h eo ry an d in o t h e r mo d e l s i s a m o n o t o n ic t r an s fo rma t i o n o f o u t co me p ro b -

abil it ies. This s chem e en cou nters two problem s. Fi rst , i t doe s not a lways sati sfy stochast ic

dom inance , an assum pt ion tha t many theor i s t s a re re luc tan t to g ive up . Second , i t is no t

r eadi ly ex t en d ed t o p ro sp ec t s w i th a l a rg e n u m b er o f o u tco mes . T h ese p ro b l ems can b e

hand led by assuming tha t t ransparen t ly dom ina ted p ros pec t s a re e l imina ted in the ed i t -

ing phase, and by norma l izing th e weigh ts so that the y add to unity. A l ternat ively , both

prob lem s can be so lved by the rank-d epen den t o r cum ula tive func t iona l , fir st p rop osed

by Quiggin (1982) for decision un de r r i sk and by Sch m eidler (1989) for decision unde runcer t a in ty . Ins tead o f t ransfo rming each p robab i li ty separa te ly , th is m odel t ransfo rms

the en t i re cumula t ive d is t r ibu t ion func tion . The p resen t theory app l i es the cumula t ive

funct ional sepa rately to gains and to losses. This deve lopm ent extends pros pec t theo ry to



300 A M O S T V E R S K Y / D A N I E L K A H N E M A N

u n c e r t a i n a s w e ll a s t o r is k y p r o s p e c t s w i t h a n y n u m b e r o f o u t c o m e s w h i le p r e s e r v in g

m o s t o f i ts e s se n t ia l f e a t u r e s . T h e d i f f e re n c e s b e t w e e n t h e c u m u l a t iv e a n d t h e o r ig i n a lv e r s io n s o f t h e t h e o r y a r e d i s c u s s e d i n s e c t i o n 1 .2 .

L e t S b e a f in i t e se t o f s t a t e s o f n a tu r e ; s u b s e t s o f S a r e c a l l e d e v e n t s . I t is a s s u m e d th a t

e x a c t l y o n e s t a t e o b t a in s , w h ic h i s u n k n o w n to t h e d e c i s i o n m a k e r . L e t X b e a s e t o f

c o n s e q u e n c e s , a l s o c a l l e d o u t c o m e s . F o r s im p l i ci ty , w e c o n f in e t h e p r e s e n t d i s c u s s io n t o

m o n e t a r y o u t c o m e s . W e a s s um e t h a t X i n cl u de s a n e u t r a l o u t c o m e , d e n o t e d 0 , a n d w e

in t e r p r e t a l l o th e r e l e m e n t s o f X a s g a in s o r l o ss e s , d e n o t e d b y p o s it i v e o r n e g a t i v e

n u m b e r s , r e s p e c t iv e ly .

A n u n c e r t a in p r o s p e c t f is a f u n c t i o n f r o m S i n to X th a t a s s ig n s t o e a c h s t a t e s e S a

c o n s e q u e n c e f l s ) - - x i n X . T o d e f i n e th e c u m u l a t iv e f u n c t io n a l , w e a r r a n g e t h e o u t c o m e so f e a c h p r o s p e c t in i n cr e a s in g o r d e r . A p r o s p e c t f i s t h e n r e p r e s e n t e d a s a s e q u e n c e o f

p a i r s ( x i , A i ) , w h i c h y i e l d s x i i f A i o c c u r s, w h e r e x i > x j i f f i > j , a n d ( A i ) i s a p a r t i t i o n o f

S . W e u s e p o s i t iv e s u bs c r ip ts t o d e n o t e p o s i t i v e o u t c o m e s , n e g a t i v e s u b s c rip t s t o d e n o t e

n e g a t iv e o u t c o m e s , a n d t h e z e r o s u b s c r i pt t o i n d e x t h e n e u t r a l o u t c o m e . A p r o s p e c t i s

c a l l e d s t r i c tl y p o s i ti v e o r p o s i t iv e , r e s p e c t iv e ly , i f it s o u t c o m e s a r e a l l p o s it i v e o r n o n n e g -

a t iv e . S tr i c tl y n e g a t i v e a n d n e g a t i v e p r o s p e c t s a r e d e f in e d s im i la r ly ; a ll o th e r p r o s p e c t s

a r e c a l l e d m i x e d . T h e p o s it iv e p a r t o f f , d e n o t e d f + , is o b t a i n e d b y l e t t i n g f + ( s) = f ( s ) i f

f ( s ) > 0 , and f + ( s ) = 0 if f ( s ) < O . T h e n e g at iv e p a r t o f f , d e n o t e d f - , is d e fi n e d

similar ly .

A s i n e x p e c t e d u ti li ty th e o r y , w e a s si gn t o e a c h p r o s p e c t f a n u m b e r V ( f ) s u ch t h a t f i s

p r e f e r r e d t o o r i n d i f f e r e n t t o g i ff V ( f ) >_ V ( g ) . T h e f o ll o w i ng r e p r e s e n t a t i o n is d e f i n e d i n

t e rm s o f th e c o n c e p t o f c a p a c i t y ( Ch o q u e t , 1 9 5 5) , a n o n a d d i t i v e s e t f u n c t i o n t h a t g e n e r -

a l izes the s ta nda rd no t ion o f p robab i l ity . A capac i ty Wis a func t ion tha t as s igns to ea ch A C

S a n u m b e r W ( A ) sat is fy ing W((b) = 0 , W ( S ) = 1 , an d W ( A ) >_ W ( B ) w h e n e v e r A D B .

Cu m u la t i v e p r o s p e c t t h e o r y a s s e r t s t h a t t h e r e e x is t a s t r ic t ly i n c r e a s in g v a lu e f u n c t i o n

v : X - - + Re , s a t is f y in g v (x 0) = v ( 0 ) = 0 , a n d c a p a c i t i e s W + a n d W - , s u c h t h a t f o r f = (xi ,

A i ) , - m <- i < n ,

V ( f ) = V ( f + ) + V ( f - ) ,n 0

V ( f + ) = ~'Tr/+v(x,) , V ( f - ) = 2 " rr ,- v( xi ), (1 )i - O i = m

w h e r e t h e d e c i s i o n w e ig h t s "rr + ( f + ) = (n v~ -, . . . , v + ) a n d ~ r - ( f - ) = ('r r_ -m , " " , W o )

a r e d e f in e d b y:

+ = W + = W - ( A - m ) ,

nvi+ = W + ( A i U . . . U A n ) - W + ( A i + I U . . . U A n ) , O < _ i < _ n - 1 ,

"rr - = W - ( A - m U . . . U A i ) - W - ( A - m O . . . U A i - 1 ) , l - m <- i <- O .

L e t t i n g qr = "rr? if/ --> 0 and Tri = q ' r / - i f / < O , e q u a t i o n ( 1 ) r e d u c e s t o

V ( f ) = 2 " rr iP (x i)i = - - m

( 2 )



302 AMOS TVERSKY/DANIELKAHNEMAN

i f fo r a l l s u m s o f m o n e y x , y , U ( x Q y ) = U ( x + y ) . Th i s a s s u m p t i o n a p p e a r s t o u s

i n e s c a p a b l e b e c a u s e t h e j o i n t r e c e i p t o f x a n d y i s t a n t a m o u n t t o r e c e iv i n g t h e i r s u m .Th u s , we e x p e c t t h e d e c i s io n m a k e r t o b e i n d i f f e r e n t b e t w e e n r e c e iv i n g a $ 1 0 b i ll o r

r e c e iv i n g a $ 2 0 b il l a n d r e t u rn i n g $ 10 i n c h a n g e . Th e Lu c e - F i s h b u rn t h e o ry , th e re fo re ,

d i f fe rs f rom ours in two essen t ia l respec t s . F i r s t , it ex tends to comp os i te p rospec t s tha t

a r e n o t t r e a t e d i n t h e p re s e n t t h e o ry . S e c o n d , i t p ra c t i ca l l y fo rc e s u ti l it y t o b e p ro p o r -

t i o n a l t o m o n e y .

Th e p re s e n t r e p re s e n t a t i o n e n c o m p a s s e s s e v e ra l p r e v i o u s t h e o r i e s t h a t e m p l o y t h e

s a m e d e c i s io n we i g h t s fo r a ll o u t c o m e s . S t a rm e r a n d S u g d e n (1 98 9) c o n s i d e re d a m o d e l

i n wh i c h w - (p ) = w + (p ) , a s in t h e o r i g i n a l v e r s i o n o f p ro s p e c t t h e o ry . I n c o n t r a s t, t h e

ra n k -d e p e n d e n t m o d e l s a s s u m e w - (p ) = 1 - w + (1 - p ) o r W - (A) = 1 - W + (S - A) .I f we a p p l y t h e l a t t e r c o n d i t i o n t o c h o i c e b e t we e n u n c e r t a i n a s s et s, we o b t a i n t h e c h o i c e

m o d e l e s t a b l i s h e d b y S c h m e i d l e r (19 89 ), w h i c h is b a s e d o n t h e C h o q u e t i n t e g ra l. 2 O t h e r

a x i o m a t i z a t io n s o f t h i s m o d e l we re d e v e l o p e d b y G i l b o a (19 87 ), N a k a m u ra (19 90 ), a n d

W ak ker (1989a, 1989b). Fo r p robab i li s t ic ( ra the r tha n unce r ta in ) p rospec t s , th i s m ode l

was f i rs t es tab l i shed by Quigg in (1982) and Y aar i (1987), an d was fu r the r an a lyzed by

Ch ew (1989), Sega l (1989) , an d W ak ke r (1990) . A n ea r l i e r ax iom at iza t ion o f th i s m ode l

i n t h e c o n t e x t o f i n c o m e i n e q u a l i t y wa s p re s e n t e d b y W e y m a rk (1 98 1) . N o t e t h a t i n t h e

p re s e n t t h e o ry , t h e o v e ra ll v a l u e V ( f ) o f a m i x e d p ro s p e c t i s n o t a C h o q u e t i n t e g ra l b u t

r a t h e r a s u m V ( f + ) + V ( f - ) of two such in teg ra ls .Th e p re s e n t t r e a t m e n t e x t e n d s t h e o r i g i n al v e r s i o n o f p ro s p e c t t h e o ry i n se v e ra l r e -

s p e ct s. F i r s t, i t a p p l ie s t o a n y fi n i te p ro s p e c t a n d i t c a n b e e x t e n d e d t o c o n t i n u o u s

d is t r ibu t ions . Second , i t app l ies to bo th p robab i l i s t i c and uncer ta in p rospec t s and can ,

t h e r e f o r e , a c c o m m o d a t e s o m e f o r m o f so u r c e d e p e n d e n c e . T h i r d , t h e p r e s e n t t h e o r y

a l lows d i f fe ren t dec i s ion weigh ts fo r ga ins and losses , ther eb y genera l iz ing the o r ig ina l

v e r si o n t h a t a s s u m e s w + = w - . U n d e r t h is a ss u m p t io n , t h e p r e s e n t t h e o r y c o i nc i d es

wi t h t h e o r i gi n a l v e r s i o n fo r a ll two -o u t c o m e p ro s p e c t s a n d fo r a l l m i x e d t h r e e -o u t c o m e

p ro s p e ct s . I t i s n o t e w o r t h y t h a t f o r p ro s p e c t s o f th e fo rm ( x , p ; y , 1 - p ) , w h e r e e it h e r x >

y > 0 o r x < y < 0 , t h e o r ig i n a l t h e o ry is i n f a c t r a n k d e p e n d e n t . A l t h o u g h t h e t wo

m o d e l s y i e ld s im i l a r p r e d i c t io n s i n g e n e ra l , t h e c u m u l a t i v e v e r s i o n - -u n l i k e t h e o r i g in a l

o n e - - s a t i s f i e s s t o c h a st i c d o m i n a n c e . Th u s , i t is n o l o n g e r n e c e s s a ry to a s s u m e t h a t t r a n s -

p a r e n t l y d o m i n a t e d p r o sp e c t s a r e e l i m i n a t e d i n th e e d i t i n g p h a s e - - a n a s s u m p t i o n t h a t

wa s c r i t i c i z e d b y s o m e a u t h o r s . On t h e o t h e r h a n d , t h e p re s e n t v e r s i o n c a n n o l o n g e r

exp la in v io la t ions o f s tochas t ic dom inan ce in no n t ra nsp are n t co n tex t s (e .g ., Tv ersky and

K a h n e m a n , 1 98 6) . A n a x i o m a t i c a n a ly s is o f t h e p re s e n t t h e o ry a n d i ts r e l a ti o n t o c u m u -

l a ti v e u ti li ty th e o ry a n d t o e x p e c t e d u t i li ty t h e o ry a r e d i s c u ss e d i n t h e a p p e n d i x ; a m o re

c o m p re h e n s i v e t r e a t m e n t i s p r e s e n t e d i n W a k k e r a n d Tv e r s k y (19 91 ).

1.3 . Values and weigh ts

In e x p e c t e d u t i l it y t h e o ry , r is k a v e r s io n a n d r is k s e e k i n g a r e d e t e rm i n e d s o le l y b y t h e

u t il i ty fu n c t i o n . I n t h e p re s e n t t h e o ry , a s i n o t h e r c u m u l a t iv e m o d e l s , r is k a v e r s i on a n d

r is k s e e k i n g a r e d e t e r m i n e d j o i n t l y b y t h e v a l u e fu n c t i o n a n d b y t h e c a p a c it ie s , wh i c h i n



A D V A N C E S IN P R O S P E C T T H E O R Y 3 0 3

t h e p r e s e n t c o n t e x t a r e c a l l e d c u m u l a t i v e w e i g h t i n g f u n c ti o n s , o r w e i g h t i n g f u n c t i o n s f o r

s h o r t . A s i n th e o r i g i n a l v e r s i o n o f p r o s p e c t t h e o r y , w e a s s u m e t h a t v is c o n c a v e a b o v e t h er e f e r e n c e p o i n t ( v " ( x ) _< 0 , x _> 0 ) a n d c o n v e x b e l o w t h e r e f e r e n c e p o i n t (v "( x) >_ O, x <_

0 ) . W e a l s o a s s u m e t h a t v i s s t e e p e r f o r lo s s e s t h a n f o r g a in s v ' (x ) < v ' ( - x ) fo rx _> 0 .

T h e f ir s t t w o c o n d i t i o n s r e f l e ct t h e p r i n c i p l e o f d i m i n i sh i n g s e n si ti v it y : t h e i m p a c t o f a

c h a n g e d i m i n i s h e s w i t h t h e d i s t a n c e f r o m t h e r e f e r e n c e p o i n t . T h e l a s t c o n d i t i o n i s

i m p l i e d b y th e p r i n c i p l e o f l o s s a v e r s i o n a c c o r d i n g t o w h i c h l o s s e s l o o m l a r g e r t h a n

c o r r e s p o n d i n g g a in s ( T v e r s ky a n d K a h n e m a n , 1 99 1).

T h e p r i n c i p l e o f d im i n i s h i n g s e n s it iv i ty a p p l i e s t o t h e w e i g h t i n g f u n c t i o n s a s w e l l. I n

t h e e v a l u a t i o n o f o u t c o m e s , t h e r e f e r e n c e p o i n t s e r v e s a s a b o u n d a r y t h a t d i s t i n g u i s h e s

g a i n s f r o m l o s s e s . I n th e e v a l u a t i o n o f u n c e r t a i n t y , t h e r e a r e t w o n a t u r a l b o u n d a r i e s - -

c e r t a in t y a n d i m p o s s i b i l it y - - t h a t c o r r e s p o n d t o th e e n d p o i n t s o f t h e c e r t a i n ty s c a le .

D i m i n i s h i n g s e n s it iv i ty e n t a il s t h a t t h e i m p a c t o f a g i v e n c h a n g e i n p r o b a b i l i t y d i m i n i s h e s

w i t h i t s d i s t a n c e f r o m t h e b o u n d a r y . F o r e x a m p l e , a n i n c r e a s e o f .1 i n t h e p r o b a b i l i t y o f

w i n n i n g a g i v e n p r i z e h a s m o r e i m p a c t w h e n i t c h a n g e s t h e p r o b a b i l i t y o f w i n n i n g f r o m .9

t o 1 .0 o r f r o m 0 t o .1 , t h a n w h e n i t c h a n g e s t h e p r o b a b i l it y o f w i n n i n g f r o m .3 to . 4 o r f r o m

.6 t o .7 . D i m i n i s h i n g s e n s i ti v i ty , t h e r e f o r e , g i v e s r i s e t o a w e i g h t i n g f u n c t i o n t h a t i s c o n -

c a v e n e a r 0 a n d c o n v e x n e a r 1 . F o r u n c e r t a i n p r o s p e c t s , t h i s p ri n c i p l e y i e ld s s u b a d d i t iv i t y

f o r v e r y u n l i k e ly e v e n t s a n d s u p e r a d d i t i v i t y n e a r c e r t a in t y . H o w e v e r , t h e f u n c t i o n i s n o t

w e l l - b e h a v e d n e a r t h e e n d p o i n t s , a n d v e r y s m a ll p r o b a b i l it i e s c a n b e e i t h e r g r e a t l y o v e r -

w e i g h t e d o r n e g l e c t e d a l to g e t h er .

B e f o r e w e t u r n t o t h e m a i n e x p e r i m e n t , w e w i s h t o r e la t e t h e o b s e r v e d n o n l i n e a ri ty o f

p r e f e r e n c e s t o t h e s h a p e o f t h e w e i g h t i n g f u n c t i o n . F o r t h i s p u r p o s e , w e d e v i s e d a n e w

d e m o n s t r a t i o n o f t h e c o m m o n c o n s e q u e n c e e f f ec t i n d e c is i o n s in vo lv in g u n c e r t a in t y r a t h e r

t h a n r is k. T a b l e 1 d is p l a y s a p a i r o f d e c i s i o n p r o b l e m s ( I a n d I I ) p r e s e n t e d i n t h a t o r d e r t o a

g r o u p o f 1 5 6 m o n e y m a n a g e r s d u r i n g a w o r k s h o p . T h e p a r t ic i p a n ts c h o s e b e t w e e n p r o s -

p e c t s w h o s e o u t c o m e s w e r e c o n t i n g e n t o n t h e d i f fe r e n c e d b e t w e e n t h e c lo s in g v a l u e s o f t h e

D o w - J o n e s t o d a y a n d t o m o r r o w . F o r e x a m p l e, f ' p a y s $2 5,0 00 i f d e x c e e d s 3 0 a n d n o t h i n g

o t h e rw i s e . T h e p e r c e n t a g e o f r e s p o n d e n t s w h o c h o s e e a c h p r o s p e c t i s g i ve n in b r ac k e ts . T h e

i n d e p e n d e n c e a x i o m o f e x p e c t e d u ti li ty t h e o r y i m p li es t h a t f i s p r e f e r r e d t o g i f f f ' i s p r e -f e r r e d t o g ' . T a b l e 1 s h o w s th a t t h e m o d a l c h o i c e w a s f in p r o b l e m I a n d g ' i n p r o b l e m I I.

T h i s p a t t e rn , w h i c h v io l a te s i n d e p e n d e n c e , w a s c h o s e n b y 5 3 % o f th e r e s p o n d e n t s .

Table 1 . A t e s t o f i n d e p e n d e n c e ( D o w - J o n e s )

A B C

if d < 30 i f30 _< d <- 35 i f35 < d

Pr ob le m I : f $25,000 $25,000 $25,000 [68]g $25,000 0 $75,000 [32]

Pro ble m l l : f ' 0 $25 ,000 $25 ,000 [23]g ' 0 0 $75,000 [77]

Note: O u t c o m e s a r e c o n t i n g e n t o n t h e d i f f e r e n c e d b e t w e e n t h e c l o si n g v a l u e s o f t h e D o w - J o n e s t o d a y a n d

t o m o r r o w . T h e p e r c e n t a g e o f r e s p o n d e n t s ( N = 1 5 6 ) w h o s e l e c t e d e a c h p r o s p e c t i s g i v e n i n b ra c k e ts .



3 0 4 A M O S T V E R S K Y /D A N I E L K A H N E M A N

E s s e n t ia l ly t h e s a m e p a t t e r n w a s o b s e r v e d i n a s e c o n d s t u d y f ol lo w i n g t h e s a m e d e -

sig n. A g r o u p o f 9 8 S t a n f o rd s t u d e n t s c h o s e b e t w e e n p r o s p e c t s w h o s e o u t c o m e s w e r ec o n t in g e n t o n t h e p o i n t - s p r e a d d i n t h e f o r t h c o m i n g S t a n f o r d - B e r k e l e y f o o t b a ll g a m e .

T a b l e 2 p r e s e n t s t h e p r o s p e c t s i n q u e s t i o n . F o r e x a m p l e , g p a y s $ 1 0 i f S t a n f o r d d o e s n o t

w i n , $ 3 0 i f i t w i n s b y 1 0 p o i n t s o r l e s s, a n d n o t h i n g i f i t w i n s b y m o r e t h a n 1 0 p o i n t s . T e n

p e r c e n t o f t h e p a r t ic i p a n t s , s e l e c t e d a t r a n d o m , w e r e a c t u a l l y p a i d a c c o r d i n g t o o n e o f

t h e i r c h o ic e s . T h e m o d a l c h o i ce , s e l e ct e d b y 4 6 % o f t h e s u b j e ct s , w a s f a n d g ', a g a in i n

d i r e c t v i o l a ti o n o f t h e i n d e p e n d e n c e a x io m .

T o e x p l o r e t h e c o n s t r a i n t s i m p o s e d b y t h is p a t t e r n , l e t u s a p p l y t h e p r e s e n t t h e o r y t o

t h e m o d a l c h o i c e s i n t a b l e 1 , u s i n g $1 , 00 0 as a u n i t . S i n c e f i s p r e f e r r e d t o g i n p r o b l e m I ,

v ( 25 ) > v ( 7 5 ) W + ( C ) + v ( 2 5 ) [ W + ( A U C ) - W + ( C ) ]

o r

v (2 5 ) [1 - W + ( A U C ) + W + ( C ) ] > v ( 7 5 ) W + ( C ) .

T h e p r e f e r e n c e f o r g ' o v e r f ' i n p r o b l e m I I , h o w e v e r , im p l i es

v ( 7 5 ) W + ( C ) > v ( 2 5 ) W + ( C U B ) ;

h e n c e ,

w + ( s ) - w + ( s - B ) > w + ( c u B ) - w + ( O . ( 3 )

T h u s , " s u b t r a c ti n g " B f r o m c e r t a in t y h a s m o r e i m p a c t th a n " s u b t r a c ti n g " B f r o m C U B .

L e t W + ( D ) = 1 - W + ( S - D ) , a n d w + ( p ) = 1 - w + ( 1 - p ) . I t f o l lo w s r e a d il y t h a t

e q u a t i o n ( 3 ) i s e q u i v a l e n t t o t h e s u b a d d i t i v i ty o f W + , t h a t is , W + ( B ) + W + ( D ) >_

W + ( B U D ) . F o r p r o b a b i li s t ic p r o s p e c t s , e q u a t i o n ( 3 ) r e d u c e s t o

1 - w + ( 1 - q ) > w + ( p + q ) - w + ( p ) ,

o r

w + ( q ) + w + ( r ) >_ w + ( q + r ) , q + r < 1.

Table 2 . A t e s t o f i nde pe nde n c e (S t a n fo rd -B e rke l e y foo tba l l ga me )

A B C

i f d < 0 i f 0 < - d < 10 i f l 0 < d

Prob lem I : f $10 $10 $10 [64]g $10 $30 0 [36]

Prob lem II : f ' 0 $10 $10 [34]g' 0 $30 0 [66]

Note: O utc om e s a re c on t i nge n t on t he p o in t - sp re a d d i n a S t a n fo rd -Be rke l e y foo tba l l ga me . The pe rc e n t a ge o f

re spon de n t s (N = 98 ) w ho se l e c te d e a c h p rospe c t i s g ive n i n b ra c ke ts .



ADVANCES IN PROSPECT THEORY 305

Al l a i s 's e x a m p l e c o r r e s p o n d s t o t h e c a s e w h e re p (C ) = . 1 0 ,p (B ) = .8 9, a n d p (A ) = . 01 .

I t is n o t e w o r t h y t h a t t h e v i o l a t io n s o f i n d e p e n d e n c e r e p o r t e d i n ta b l e s 1 a n d 2 a r e a l soi n c o n s i s t e n t w i t h r e g re t t h e o ry , a d v a n c e d b y Lo o m e s a n d S u g d e n (1 98 2, 1 98 7) , a n d w i t h

F i s h b u rn ' s ( 1 98 8) S S A m o d e l . R e g re t t h e o ry e x pl a in s A l l a i s 's e x a m p l e b y a s s u m i n g t h a t

t h e d e c i s i o n m a k e r e v a l u a t e s t h e c o n s e q u e n c e s a s i f t h e t wo p ro s p e c t s i n e a c h c h o i c e a r e

s t a ti s ti c a ll y i n d e p e n d e n t . W h e n t h e p ro s p e c t s i n q u e s t io n a r e d e f i n e d b y t h e s a m e s e t o f

e v e n t s, a s i n ta b l e s 1 a n d 2 , r e g re t t h e o ry ( li k e F i s h b u rn ' s S S A m o d e l ) i m p l ie s in d e p e n -

d e n c e , s in c e i t is a d d it i ve o v e r s t a te s . Th e f i n d in g t h a t t h e c o m m o n c o n s e q u e n c e e f f e c t is

v e r y m u c h i n e v i d en c e i n t h e p r e s e n t p r o b l e m s u n d e r m i n e s t h e i n t e r p r e ta t i o n o f A l l ai s' s

e x a m p l e i n t e rm s o f r e g re t t h e o ry .

T h e c o m m o n c o n s e q u e n c e e f f e c t i m p l ie s th e s u b a d d it iv i ty o f W + a n d o f w + .

O t h e r v i o l a t io n s o f e x p e c t e d u t i li t y t h e o r y i m p l y t h e s u b a d d i t i v it y o f W + a n d o f w +

f o r sm a l l a n d m o d e r a t e p r o b a b i li t ie s . F o r e x a m p l e , P r e l e c ( 1 99 0 ) o b s e r v e d t h a t m o s t

r e s p o n d e n t s p r e f e r 2 % t o w i n $ 2 0 ,0 0 0 o v e r 1 % t o w i n $ 30 ,0 00 ; t h e y a ls o p re f e r 1 % t o

wi n $ 3 0 ,0 0 0 a n d 3 2 % t o w i n $ 2 0 ,0 0 0 o v e r 3 4 % t o w i n $ 2 0,0 00 . In t e rm s o f t h e p re s e n t

theo ry , th ese da ta imp ly tha t w + ( .02 ) - w + ( .01 ) _> w + ( .34 ) - w + ( .33 ) . M or e

g e n e r a l l y , w e h y p o t h e s i z e

w + ( p + q ) - w + ( q ) >_ w + ( p + q + r ) - w + ( q + r ), (4 )

p r ov ide dp + q + r is su f f ic ien t ly smal l . Eq ua t ion (4) s ta tes tha t w + i s concave nea r the

o r ig i n ; a n d t h e c o n j u n c t i o n o f t h e a b o v e i n e q u a l i t ie s i m p l ie s t h a t , i n a c c o rd w i t h d i m i n -

i sh ing sens i tiv i ty , w ÷ has a n in ver te d S-shape : i t is s t eepes t n ea r th e e ndp o in t s a nd

s h a l l o we r i n t h e m i d d l e o f t h e r a n g e . F o r o t h e r t r e a t m e n t s o f d e c i s i o n we i g h t s , s e e

H og ar th an d Ein ho rn (1990), P re lec (1989) , Viscus i (1989) , and W ak ke r (1990). Ex per -

i m e n t a l e v i d e n c e is p r e s e n t e d i n t h e n e x t s e c ti o n .

2 . E x p e r i m e n t

A n e x p e r i m e n t w a s c a r r ie d o u t t o o b t a i n d e t a i l e d i n f o r m a t i o n a b o u t t h e v a l u e a n d

w e i g h t i n g f u n c t i o n s. W e m a d e a s p e c ia l e f f o r t to o b t a i n h i g h - q u a l i ty d a t a . T o t h is

e n d , w e r e c r u i te d 2 5 g r a d u a t e s t u d e n t s f r o m B e r k e l e y a n d S t a n f o r d ( 12 m e n a n d 13

w o m e n ) w i t h n o s p e c i a l t r a i n i n g i n d e c i s i o n t h e o r y . E a c h s u b j e c t p a r t i c i p a t e d i n

t h r e e s e p a r a t e o n e - h o u r s e s s io n s t h a t w e r e s e v e ra l d a y s a p a r t . E a c h s u b je c t w a s p a i d

$ 25 fo r p a r t i c i p a t i o n .

2.1. Procedure

T h e e x p e r i m e n t w a s c o n d u c t e d o n a c o m p u t e r . O n a t y p ic a l t ri al , t h e c o m p u t e r d i s p la y e d

a p ro s p e c t ( e .g ., 2 5 % c h a n c e t o w i n $ 15 0 a n d 7 5 % c h a n c e t o w i n $ 50 ) a n d i ts e x p e c te d

v a l u e . Th e d i s p la y al so i n c l u d e d a d e s c e n d i n g s e r ie s o f s e v e n s u re o u t c o m e s (g a i n s o r

lo ss es ) l o g a r i th m i c a l l y s p a c e d b e t w e e n t h e e x t r e m e o u t c o m e s o f t h e p ro s p e c t . Th e s u b -

j e c t i n d i c a t e d a p r e f e r e n c e b e t we e n e a c h o f t h e s e v e n s u re o u t c o m e s a n d t h e r i s k y

p ro s p e c t . To o b t a i n a m o re r e f i n e d e s t i m a t e o f t h e c e r t a i n t y e q u i v a l e n t , a n e w s e t o f



306 AMOS TVERSKY/DANIEL KAHNEMAN

seven sure outcom es was then shown, l inear ly spaced b e tw een a va lue 25% h igher than

the lowest amount accepted in the f i r s t se t and a va lue 25% lower than the h ighes ta mo un t r e je c t e d . The c e r ta i n ty e qu iva l en t o f a p r ospe c t wa s e s t i ma t e d by t he mi dpo i n t

be t w e e n t he l owe s t a c c e p t e d va lue a nd t he h i ghes t r e j e c te d va l ue in t he se c ond se t o f

choices. W e wish to em phasize tha t a l though the analys is i s bas ed o n cer t a in ty equiva-

l ents , t he d a ta co nsi s t ed of a se ri es of choices be tw een a g iven prosp ec t and severa l sure

outcomes. Thus , the cash equiva lent of a prospec t was der ived f rom observed choices ,

r a t he r t ha n a s se ssed by the sub je c t. The c om put e r m on i t o r e d t he i n t e rna l c ons is te nc y o f

the responses to each prospec t and re j ec ted e r rors , such as the acceptance of a cash

am oun t lower than o ne previous ly re j ec ted . Er rors cau sed the or ig ina l s t a t emen t of the

pr ob l e m t o r e a pp e a r on t he sc re e n . 3

Th e p r e se n t a na lysis f oc use s on a se t o f t wo- ou t c ome p r ospe c t s wi th m one t a r y ou t -

comes and numer ica l probabi l i t i es . Other da ta involving more compl ica ted prospec t s ,

inc luding prosp ec t s def ined by uncer t a in event s , wi ll be r ep or te d e l sewhere . The re w ere

28 pos it ive and 28 nega tive prospec t s . S ix of the prospec t s ( three nonneg a t ive and three

nonposi t ive) w ere re pea ted on d i f fe rent sessions to obta in the es t imate o f the consi s tency

of choice . Ta ble 3 d i splays the p rospec t s an d the m edian cash equiva lent s of the 25

subjects .

A modi f ie d p r oc e du r e wa s use d in e i gh t a dd it iona l p r ob l e ms . I n f ou r o f the se p r ob-

lems, the subjec t s m ade choices regarding the acceptabi li ty of a se t of mixed prospec t s

(e.g. , 50% chance to lose $100 and 50% chanc e to win x) in wh ich x w as sys temat ical lyvar ied . In fou r o ther prob lems, the subjec t s com pared a fixed pros pec t (e.g ., 50% chanc e

to lose $20 and 50% chance to w in $50) to a se t of prosp ec t s (e.g ., 50% chanc e to lose $50

and 50% chanc e to win x) in which x was sys temat ica l ly var ied . (The se p rospec t s a re

pres ented in table 6 .)

2.2. Results

Th e mo st d is tinc tive impl ica t ion o f prospe c t theory i s the four fo ld pa t t e rn of r isk a t t i-tudes . For the no nm ixed prospec t s used in the pre sent s tudy, the shapes o f the va lue and

the w eigh ting fun ction s imply risk-averse an d risk-seeking pre fer en ces , respectively, fo r

ga ins and for losses of modera te or h igh probabi l i ty . Fur thermore , the shape of the

weigh t ing funct ions favors ri sk seek ing for small pro bab i l i ties of gains a nd r isk aversion

for small probab i li t ies of loss, provided the o utcom es a re not ext reme. No te , however ,

tha t pros pec t theory do es not imply per fec t re f lec t ion in the sense tha t the p re fe ren ce

be tw een any two posi tive prospe c t s is reversed w hen ga ins a re replaced by losses. Ta ble

4 present s , for each subjec t, t he percentag e of risk-seeking choices (whe re the ce r t a in ty

equiva lent excee ded exp ec ted va lue) for gains and for losses wi th low (p _< .1) and wi thhigh (p _ .5) probab i l it ies. Ta ble 4 show s that fo rp _> .5, a ll 25 subjects are pre dom i-

nant ly r isk av erse for pos it ive prospec t s an d r isk seeking for nega t ive ones . M oreove r , the

ent i re fo ur fold pa t t e rn i s observ ed for 22 of the 25 subjec ts , wi th so me var iabili ty a t the

level of individual cho ices.A l thou gh the overall p at tern of pre feren ces i s c lear , the individual data , o f cou rse ,

reveal bo th noise an d individual di fferences. Th e correlat ions, acro ss subjects , be tw ee n



A D V A N C E S I N P R O S P E C T T H E O R Y 307

T a b l e 3 . M e d i a n c a s h e q u i v a l e n t s (i n d o ll a r s) f o r a ll n o n m i x e d p r o s p e c t s

P r o b a b i l i t y

O ut co m es .01 .05 .10 .25 .50 .75 .90 .95 .99

(0,50)

(o, -50 )

(o, lOO)

(0, -100)

(0,200)

(0, -200)

(0,400)

(0, -400)

(50, 100)

( - 5 0 , - 1 0 0 )

(50,150)

(-50, -~5o)

( 1 0 0 , 2 0 0 )

( - 1 0 0 , - 2 0 0 )

9 21 37

8 - 2 1 - 3 9

1 4 2 5 3 6 5 2 7 8

- 8 - 2 3 . 5 - 4 2 - 6 3 - 8 4

10 20 76 131 188

- 3 - 2 3 - 8 9 - 1 5 5 - 190

1 2 3 7 7

- 1 4 - 3 8 0

59 71 83

- 5 9 - 7 1 - 8 5

6 4 7 2 5 8 6 1 0 2 1 2 8

- 6 0 - 7 1 - 9 2 - 1 13 - 13 2

118 130 141 162 178

- 1 1 2 - 1 2 1 - 1 4 2 - 1 5 8 - 179

N o t e : T h e t w o o u t c o m e s o f e a c h p r o s p e c t a r e g i v e n in t h e l e f t - h a n d s i de o f e a c h r o w ; t h e p r o b a b i li ty o f t h es e c o n d ( i. e. , m o r e e x t r e m e ) o u t c o m e i s g i v en b y th e c o r r e s p o n d i n g c o l u m n . F o r e x a m p l e , th e v a l u e o f $ 9 i n t h e

u p p e r l e f t c o r n e r i s t h e m e d i a n c a s h e q u i v a l e n t o f t h e p r o s p e c t ( 0 , .9 ; $ 5 0 , .1 ).

t he c a s h e q u i va l e n t s f o r t he s a m e p ros pe c t s o n s uc ce s si ve se s si ons a ve ra ge d .55 ove r six

d i f f e re n t p ro s p e ct s. T a b l e 5 p r e s e n t s m e a n s ( a f t e r t r a n s f o r m a t i o n t o F i s h e r ' s z ) o f t h e

c o r r e l a t i ons be t w e e n t he d i f f e r e n t t ype s o f p ros pe c t s . F o r e xa m pl e , t he r e w e re 19 a nd 17

pros pe c t s , r e s pe c t ive l y , w i t h h i gh p roba b i l it y o f ga i n a nd h i gh p roba b i l i t y o f lo ss . Th e

va l ue o f .06 i n t a b l e 5 is t he m e a n o f the 17 x 19 = 323 c o r r e l a t i ons be t w e e n t he c a s h

e qu i va l e n t s o f t he s e p ros pe c t s .T h e c o r r e la t io n s b e t w e e n r e s p o n s e s w i t h in e a c h o f t h e f o u r t y p es o f p r o s p e c t s a v e r a g e

.41 , s l igh t ly l ow e r t ha n t he c o r r e l a t i ons be t w e e n s e pa ra t e r e s pons e s t o t he s a me p rob -

l e ms . Th e t w o ne ga t i ve va l ue s i n t a b l e 5 i nd i c a t e t ha t t h os e s ub j e c ts w h o w e re m ore r i sk

a v e rs e in o n e d o m a i n t e n d e d t o b e m o r e r i sk s e e k in g in t h e o t h e r . A l t h o u g h t h e i n di v id -

ua l c o r r e l a t ions a r e f a i rl y l ow , t he t r e n d is c ons i st e n t : 78% o f t he 403 c o r r e l a t ions i n

t he s e t w o c e ll s a r e ne ga t i ve . Th e re is a ls o a t e nd e nc y fo r s ub j ec t s w h o a r e m ore r i sk

a ve r s e fo r h i gh -p roba b i l i ty ga i ns t o b e l es s r is k s e e k i ng fo r ga i ns o f l ow p roba b i l it y . Th i s

t r e nd , w h i c h is a bs e n t i n t he ne ga t i ve dom a i n , c ou l d r e f l e c t i nd i v idua l d i f f e r e nc e s e i t he r

i n t h e e l ev a t io n o f t h e w e i g h t in g f u n c t io n o r i n t h e c u r v a t u r e o f t h e v a l u e f u n c t i o n f o rga i ns . The ve ry l ow c o r r e l a t i ons i n t he t w o r e ma i n i ng c e l l s o f t a b l e 5 , a ve ra g i ng . 05 ,

i nd i c a t e t ha t t h e r e i s no ge n e ra l t r a i t o f r is k a ve r s i on o r r i sk s e e k ing . B e c a us e i nd i v i dua l

c ho i c e s a r e qu i t e no i sy , a gg re ga t i on o f p rob l e m s i s ne c e s s a ry fo r t he a na l ysi s o f i nd i v i dua l

d i f f e r e nc e s .

T h e f o u r f o l d p a t t e r n o f r is k a t t it u d e s e m e r g e s a s a m a j o r e m p i r ic a l g e n e r a l i z a ti o n

a bo u t c ho i c e und e r r is k . I t ha s be e n obs e rve d i n s e ve ra l e xpe r i me n t s ( s e e, e .g ., C o he n ,



3 0 8 A M O S T V E R S K Y / D A N I E L K A H N E M A N

Table 4 . Perc enta ge of risk-seeking choices

Gain Loss

Su bjec t p -< .1 p -> .5 p _< .1 p _> .5

1 100 38 30 100

2 85 33 20 75

3 100 10 0 93

4 71 0 30 58

5 83 0 20 100

6 100 5 0 100

7 100 10 30 86

8 87 0 10 100

9 16 0 80 100

10 83 0 0 93

11 100 26 0 100

12 100 16 10 100

13 87 0 10 94

14 100 21 30 100

15 66 0 30 100

16 60 5 10 100

17 100 15 20 100

18 100 22 10 93

19 60 10 60 63

20 100 5 0 81

21 100 0 0 100

22 100 0 0 92

23 100 31 0 100

24 71 0 80 100

25 100 0 10 87

Risk seeking 78 a 10 20 87 a

Risk neutral 12 2 0 7Risk averse 10 88 a 80 a 6

aValues that correspon d to the fourfold pat tern .

Note : The perc enta ge o f risk-seeking choices is given for low (p <_ .1) and high (p -> .5) probabilities of gain

and loss for each subject (risk-neutral choices were excluded). Th e ove rall per cen tage o f risk-seeking, risk-

neutral, and r isk-averse choices for each type o f prospect app ear at the b ot tom of the table .

J a f fr a y , a n d S a i d , 1 9 8 7 ), i n c l u d i n g a s t u d y o f e x p e r i e n c e d o i l e x e c u t i v e s i n v o l v i n g s ig n i fi -

c a n t , a l b e i t h y p o t h e t i c a l , g a i n s a n d l o ss e s ( W e h r u n g , 1 9 8 9) . I t s h o u l d b e n o t e d t h a t

p r o s p e c t t h e o r y i m p l i e s t h e p a t t e r n d e m o n s t r a t e d i n t a b l e 4 w i t h i n t h e d a t a o f in d i v i d u a ls u b j ec t s , b u t i t d o e s n o t i m p l y h i g h c o r r e l a t i o n s a c r o s s s u b j e c ts b e c a u s e t h e v a l u e s o f

g a i ns a n d o f l o ss e s c a n v a r y i n d e p e n d e n t l y . T h e f a i l u re t o a p p r e c i a t e t h i s p o i n t a n d t h e

l i m i t e d r e l i a b i l i t y o f i n d i v i d u a l r e s p o n s e s h a s l e d s o m e p r e v i o u s a u t h o r s ( e . g . , H e r s h e y

an d S ch o em ak e r , 1 9 80 ) t o u n d e re s t i m a t e t h e ro b u s t n es s o f t h e fo u r fo l d p a t t e rn .




Table 5. A v e r a g e c o r r e l a t i o n s b e t w e e n c e r t a i n t y e q u i v a l e n t s in f o u r t y p e s o f p ro s p e c t s

L + H + L - H -

L + . 4 1 . 1 7 - . 2 3 . 0 5

H + . 3 9 . 0 5 - . 1 8

L - .40 .06

H - .44

Note: L o w p r o b a b i l i t y o f g a i n = L + ; h i g h p r o b a b i l i t y o f g a i n = H + ; l o w p r o b a b i l i t y o f l o s s = L ; h i g h

p r o b a b i l i t y o f l o s s = H .

2 . 3 . S c a l i n g

H a v i n g e s t a b l i s h e d t h e f o u r f o l d p a t t e r n i n o r d i n a l a n d c o r r e l a t i o n a l a n a l y s e s , w e n o w

t u r n t o a q u a n t i t a ti v e d e s c r i p t i o n o f th e d a t a . F o r e a c h p r o s p e c t o f t h e f o r m ( x , p ; O , 1 -

p ) , l e t c / x b e t h e r a t i o o f t h e c e r t a in t y e q u i v a l e n t o f t h e p r o s p e c t t o t h e n o n z e r o o u t c o m e

x . F i g u r e s 1 a n d 2 p l o t t h e m e d i a n v a l u e o f c /x a s a f u n c t i o n o f p , f o r p o s i t iv e a n d f o r

n e g a t i v e p r o s p e c t s , r e s p e c t i v e l y . W e d e n o t e c /x b y a c i r c l e i f Ix ] < 2 0 0 , a n d b y a t r i a n g le

i f Ix [ >_ 2 0 0. T h e o n l y e x c e p t i o n s a r e t h e t w o e x t r e m e p r o b a b i l i t i e s ( . 0 1 a n d . 9 9 ) w h e r e a

c i r c l e i s u s e d f o r Ix ] = 2 0 0 . T o i n t e r p r e t f i g u r e s 1 a n d 2 , n o t e t h a t i f s u b j e c t s a r e r is k

n e u t r a l , t h e p o i n t s w i ll li e o n t h e d i a g o n a l ; i f s u b j e c t s a r e r i s k a v e r s e , a ll p o i n t s w i ll l i e

b e l o w t h e d i a g o n a l i n f i g u re 1 a n d a b o v e t h e d i a g o n a l i n f i g u r e 2. F in a l ly , t h e t r i a n g l e s

a n d t h e c i r cl es w il l l ie o n t o p o f e a c h o t h e r i f p r e f e r e n c e s a r e h o m o g e n e o u s , s o t h a t

m u l t ip l y in g t h e o u t c o m e s o f a p r o s p e c t f b y a c o n s t a n t k > 0 m u l ti p li e s it s c a s h e q u iv a -

l e n t c ( k f ) b y t h e s a m e c o n s t a n t , t h a t is , c ( k f ) = k c ( f ) . I n e x p e c t e d u t i li ty t h e o r y , p r e f e r -

e n c e h o m o g e n e i t y g i ve s r i s e t o c o n s t a n t r e l a t iv e r is k a v e r s io n . U n d e r t h e p r e s e n t t h e o r y ,

a s s u m i n g X = R e , p r e f e r e n c e h o m o g e n e i t y is b o t h n e c e s s a r y a n d s u f fi ci en t t o r e p r e s e n t

v a s a t w o - p a r t p o w e r f u n c t io n o f t h e f o r m

v ( x ) = I x '~ ifx _> 0[ - k ( - x ) P i fx < 0. ( 5 )

F i g u r e s 1 a n d 2 e x h i b it t h e c h a r a c t e r i s ti c p a t t e r n o f r is k a v e r s i o n a n d r i sk s e e k i n g

o b s e r v e d i n t a b l e 4 . T h e y a l s o i n d i c a t e t h a t p r e f e r e n c e h o m o g e n e i t y h o l d s a s a g o o d

a p p r o x i m a t i o n . T h e s l ig h t d e p a r t u r e s f r o m h o m o g e n e i t y i n f i gu r e 1 s u g g e s t t h a t t h e c a s h

e q u i v a l e n t s o f p o s it i v e p r o s p e c t s i n c r e a s e m o r e s lo w l y t h a n t h e s t a k e s ( t r i a n g le s t e n d t o

l ie b e l o w t h e c ir c le s ), b u t n o s u c h t e n d e n c y i s e v i d e n t i n f i g u re 2 . O v e r a l l, i t a p p e a r s t h a t

t h e p r e s e n t d a t a c a n b e a p p r o x i m a t e d b y a t w o - p a r t p o w e r fu n c t io n . T h e s m o o t h c u r v e s

i n f i g u re s 1 a n d 2 c a n b e i n t e r p r e t e d a s w e i g h t i n g f u n c t i o n s , a s s u m i n g a l i n e a r v a l u ef u n c t i o n . T h e y w e r e f i t te d u s i n g t h e f o l lo w i n g f u n c t io n a l f o r m :

p V p 6, ~ ' ( P ) = _ p ? ) l / ~ . ( 6 )

w + ( p ) = ( P ~ + (1 - p ) ~ ) l / ~ ( p ~ + ( 1




x

0

O 0

0

¢.D

0

,, ¢

0

0,,I

0

0

0

6

I I I f I

0.0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

Figure I . M e d i a n c /x f o r a l l p o s i t i v e p r o s p e c t s o f t h e f o r m ( x , p ; 0 , 1 - p ) . T r i a n g l e s a n d c i r c l es , r e s p e c t i v e ly ,

c o r r e s p o n d t o v a l u e s o f x t h a t l ie a b o v e o r b e l o w 2 0 0 .

T h i s f o r m h a s s e v e r a l u s e f u l f e a t u r e s : i t h a s o n l y o n e p a r a m e t e r ; i t e n c o m p a s s e s

w e i g h t i n g fu n c t i o n s w i t h b o t h c o n c a v e a n d c o n v e x r e g i o n s; it d o e s n o t r e q u i r e w ( . 5 ) = .5 ;

a n d m o s t i m p o r t a n t , i t p r o v i d e s a r e a so n a b l y g o o d a p p r o x i m a t i o n t o b o t h t h e a g g r e g a t e

a n d t h e i n d iv i d u a l d a ta f o r p r o b a b i l it i e s in t h e r a n g e b e t w e e n .0 5 a n d .9 5.

F u r t h e r i n f o r m a t i o n a b o u t t h e p r o p e r t i e s o f t h e v a l u e f u n c t i o n c a n b e d e r i v e d f r o m

t h e d a t a p r e s e n t e d i n t a b l e 6 . T h e a d j u s t m e n t s o f m i x e d p r o s p e c t s t o a c c e p t a b i l it y ( p r o b -

l e m s 1 - 4 ) i n d i c a t e t h a t, f o r e v e n c h a n c e s t o w i n a n d l o s e , a p r o s p e c t w i l l o n l y b e a c c e p t -

a b l e i f t h e g a i n i s a t le a s t t w i c e a s l a r g e a s t h e l o s s . T h i s o b s e r v a t i o n i s c o m p a t i b l e w i t h a

v a l u e f u n c t i o n t h a t c h a n g e s s l o p e a b r u p t l y a t z e r o , w i t h a l o s s - a v e r s i o n c o e f f i c i e n t o f

a b o u t 2 (T v e r sk y a n d K a h n e m a n , 1 9 91 ). T h e m e d i a n m a t c h e s i n p r o b l e m s 5 a n d 6 a r e

a l s o c o n s i s t e n t w i t h t h i s e s t im a t e : w h e n t h e p o s s i b l e l o s s i s i n c r e a s e d b y k t h e c o m p e n -

s a t in g g a in m u s t b e i n c r e a s e d b y a b o u t 2 k. P r o b l e m s 7 a n d 8 a r e o b t a i n e d f r o m p r o b l e m s

5 a n d 6 , r e s p e c t i v e l y , b y p o s i t i v e t r a n s l a t i o n s t h a t t u r n m i x e d p r o s p e c t s i n t o s t r i c t l y

p o s i t iv e o n e s . I n c o n t r a s t t o t h e l a r g e v a l u e s o f 0 o b s e r v e d i n p r o b l e m s 1 - 6 , t h e r e s p o n s e s

i n p r o b l e m s 7 a n d 8 i n d i c a t e t h a t t h e c u r v a tu r e o f t h e v a l u e f u n c t i o n f o r g a in s i s s li g h t. A



A D V A N C E S IN P R O S P E C T T H E O R Y 311

o .

60 0

O

~O

0

x

C ~

° .( j""

°

I I I I I I

0 .0 0 .2 0 .4 0 .6 0 .8 1 .0

P

F i g u r e 2 . M e d i a n c / x f o r a l l n e g a ti v e p r o s p e c t s o f t h e f o r m ( x , p ; O , 1 - p ) . T r i an g l e s an d c i r c l e s, r e s p ec ti v e l y ,

c o r r e s p o n d t o v a l u e s o f x t h a t l i e b e l o w o r a b o v e - 2 00 .

de c re a s e i n t he s ma l l e s t ga i n o f a s t r ic t ly pos it ive p ro s pe c t is f u ll y c om pe n s a t e d by as li gh tl y l a rge r inc r e a s e i n t he l a rge s t ga i n . Th e s t a nd a rd r a nk -de pe n de n t mod e l , w h i c h

l ac k s th e n o t i o n o f a r e f e r e n c e p o i n t, c a n n o t a c c o u n t f o r t h e d r a m a t i c e f f e ct s o f s m a ll

t r a ns l a t ions o f p ros pe c t s i l l u s t r a te d i n t a b l e 6 .

The e s t i ma t i on o f a c ompl e x c ho i c e mode l , s uc h a s c umul a t i ve p ros pe c t t he o ry , i s

p r o b l e m a t ic . I f t h e f u n c t i o n s a s so c i at e d w i t h t h e t h e o r y a r e n o t c o n s t r a in e d , t h e n u m b e r

o f e s t i ma t e d pa ra m e t e r s fo r e a c h s ub j e c t is t oo l a rge . To r e du c e t h i s num be r , i t is c om-

m on t o a s s um e a pa ra m e t r i c fo rm ( e .g ., a pow e r u t i li t y func t i on ) , bu t t h i s a pp ro a c h

c o n f o u n d s t h e g e n e r a l t e s t o f t h e t h e o r y w i t h t h a t o f t h e s p ec if ic p a r a m e t r i c f o r m . F o r

t h is r ea s o n , w e f o c u s e d h e r e o n t h e q u a l it a ti v e p r o p e r t i e s o f th e d a t a r a t h e r t h a n o np a r a m e t e r e s t im a t e s a n d m e a s u r e s o f f it . H o w e v e r , i n o r d e r t o o b t a in a p a r s i m o n i o u s

d e s c r ip t io n o f t h e p r e s e n t d a t a , w e u s e d a n o n l i n e a r r e g re s s io n p r o c e d u r e t o e s t i m a t e t h e

p a r a m e t e r s o f e q u a t i o n s ( 5) a n d (6 ), s e p a r a t e l y f o r e a c h s u b je c t. T h e m e d i a n e x p o n e n t

o f t he va l ue fu nc t i on w a s 0 .88 fo r bo t h ga i ns a nd l o ss e s, in a c c o rd w i t h d i mi n i s h i ng

s e nsi ti v it y . Th e m e d i a n ?t w a s 2 . 25 , i nd i c a t i ng p ro no un c e d l o ss a ve r si on , a nd t he m e d i a n




T a b l e 6 . A t e s t o f lo s s a v e r s i o n

P r o b l e m a b c x 0

1 0 0 - 2 5 6 1 2 . 4 4

2 0 0 - 5 0 1 01 2 . 0 2

3 0 0 - 100 202 2 .02

4 0 0 - 150 280 1 .87

5 - 2 0 5 0 - 5 0 1 12 2 .0 7

6 - 5 0 150 - 125 301 2 .01

7 50 120 20 149 0 .97

8 100 300 25 401 1 .35

N o t e : I n e a c h p r o b l e m , s u b j e c t s d e t e r m i n e d t h e v a l u e o f x t h a t m a k e s t h e p r o s p e c t ( $ a , ~½; $ b , ~ A) a s a t t r a c t iv e

a s ( $ c , ~ A; $ x, ~ /2). T h e m e d i a n v a l u e s o f x a r e p r e s e n t e d f o r a l l p r o b l e m s a l o n g w i t h t h e f i x e d v a l u e s a , b , c . T h e

s t a t i s t i c 0 = ( x - b ) / ( c - a ) i s t h e r a t i o o f t h e " s l o p e s " a t a h i g h e r a n d a l o w e r r e g i o n o f t h e v a l u e f u n c t i o n .

va l ue s o f ~ / a nd 8 , re s pe c ti ve ly , w e re 0 . 61 a nd 0 .69, in a g re e m e n t w i t h e qua t i ons (3 ) a n d

(4 ) a b o v e . 4 T h e p a r a m e t e r s e s t i m a t e d f r o m t h e m e d i a n d a t a w e r e e s s en t ia l ly t h e s a m e .

F i gu re 3 p lo t s w + a nd w - u s i ng the m e d i a n e s t i ma t e s o f "y a nd 8 .

F i gu re 3 s how s t ha t , f o r bo t h pos i ti ve a nd ne ga t i ve p ros pe c t s , pe op l e ove rw e i gh t low

p r o b a b il it ie s a n d u n d e r w e i g h t m o d e r a t e a n d h i g h p ro b a b il it ie s . A s a c o n s e q u e n c e , p e o -

p l e a r e r e la t ive l y i n s ens i ti ve to p roba b i l i ty d i f f e r e nc e in t h e m i dd l e o f t he r a nge . F i gu re 3

a ls o s how s t ha t t h e w e i gh t i ng func t i ons fo r ga ins a nd fo r l o s se s a r e qu i t e c l o s e, a l t houg h

the fo rm er i s s l ight ly m ore curv ed tha n the l a t t e r ( i .e . , " , /< 8) . Acc ordingly , r i sk avers ion

f o r g a in s is m o r e p r o n o u n c e d t h a n r is k s e ek i n g fo r lo ss es , f o r m o d e r a t e a n d h i g h p r o b a -

b i li ti e s ( s e e ta b l e 3 ). I t is no t e w o r t hy t ha t t he c ond i t ion w + (p ) = w - ( p ) , a s s um e d i n t he

o r ig i n al v e r s io n o f p r o s p e c t t h e o r y , a c c o u n ts f o r t h e p r e s e n t d a t a b e t t e r t h a n t h e a s su m p -

t io n w + ( p ) = 1 - w - ( 1 - p ) , i m p l i ed b y t h e s t a n d a r d r a n k - d e p e n d e n t o r c u m u l a t i v e

func t i ona l . F o r e xa mpl e , ou r e s t i ma t e s o f w + a nd w - s how t ha t a l l 25 sub j e c ts s at is f ie d

t he c on d i t i ons w + ( .5 ) < .5 a nd w - ( .5 ) < .5 , i mp l i e d by t he fo rm e r mod e l , a nd no o nesa t i s fi ed the c on di t ion w + ( .5) < .5 i f fw - ( .5 ) > .5 , imp l ied by the l a t t e r mo del .

M uc h r e s e a rc h on c ho i c e be t w e e n r is ky p ros pe c t s ha s u t i l iz e d t he t r i a ng l e d i a g ra m

(Ma rs c ha k , 1950 ; M a c h i na , 1987) t ha t r e p re s e n t s t he s e t o f a ll p ros pe c t s o f t he f o rm (Xl,

p l ; x 2 , p z ; x 3 , p3 ) , w i t h f ixe d ou t c om e s x l < x2 < x3 . Ea c h po i n t in t he t r i a ng l e r e p re s e n t s

a p r o s p e c t t h a t y i e ld s th e l o w e st o u t c o m e ( X l) w i t h p r o b a b i l i t y p l , t h e h i g h e s t o u t c o m e

(x 3) w i t h p r o b a b i li ty p 3 , a n d t h e i n t e r m e d i a t e o u t c o m e ( x 2 ) w i t h p r o b a b i l it y p z = 1 -

P l - P 3 . A n i n d i f f e r e nc e c u rve is a s e t o f p ros pe c ts ( i. e. , po i n t s) t ha t t he de c i s i on m a k e r

f i nds e qua l l y a tt r a c ti ve . A l t e rna t i ve c ho i c e t he o r i e s a r e c ha ra c t e r i z e d by t he s ha pe s o f

t he i r i nd i f f e r e nc e cu rve s. I n pa r t i c u l a r , t he i nd i f f e r e nc e c u rve s o f e xpe c t e d u t i li t y t he o rya re pa ra l l e l s t ra i gh t li ne s. F i gu re s 4a a nd 4b i l l u s tr a t e t he i nd i f f e r e nc e c u rve s o f c umu l a -

t ive p ros p e c t t h e o r y fo r nonne g a t i ve a nd nonpos i t i ve p ros pe c t s , r e s pe ct ive l y . Th e s ha pe s

o f t h e c u r ve s a r e d e t e r m i n e d b y t h e w e i g h t in g fu n c t i o n s o f f ig u r e 3; t h e v a l u e s o f t h e

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F i g u r e s 4 a a n d 4 b a r e i n g e n e r a l a g r e e m e n t w i t h t h e m a i n e m p i r i c a l g e n e r a l i z a t i o n s

t h a t h a v e e m e r g e d f r o m t h e s t u d i e s o f t h e t r i a n g l e d i a g r a m ; s e e C a m e r e r ( 1 9 9 2 ) , a n d

C a m e r e r a n d H o ( 1 9 9 1 ) f o r re v i ew s . F i rs t, d e p a r t u r e s f r o m l in e a r it y , w h i c h v i o l a t e e x -

p e c t e d u t il it y t h e o r y , a r e m o s t p r o n o u n c e d n e a r t h e e d g e s o f t h e t r ia n g le . S e c o n d , t h e

i n d i f f e r e n c e c u r v e s e x h i b it b o t h f a n n i n g i n a n d f a n n i n g o u t . T h i r d , t h e c u r v e s ar e c o n c a v e

i n t h e u p p e r p a r t o f t h e t r i a n g le a n d c o n v e x i n t h e l o w e r r i g ht . F in a l l y, t h e i n d i f f e r e n c e

c u r v e s f o r n o n p o s i t i v e p r o s p e c t s r e s e m b l e t h e c u r v e s fo r n o n n e g a t i v e p r o s p e c t s r e f l e c t e d

a r o u n d t h e 4 5 ° l i n e , w h i c h r e p r e s e n t s r is k n e u t r a li t y . F o r e x a m p l e , a s u r e g a i n o f $ 1 0 0 i s

e q u a l l y a s a tt r a ct iv e a s a 7 1 % c h a n c e t o w i n $ 2 0 0 o r n o t h i n g ( s e e f ig u r e 4 a ), a n d a s u r el o s s o f $ 1 0 0 i s e q u a l l y a s a v e r si v e a s a 6 4 % c h a n c e t o l o s e $ 2 0 0 o r n o t h i n g ( s e e f ig u r e 4 b ).

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ADVANCE S IN P ROS P E CT T HE OR Y 3 1 5

2.4. Incentives

W e c o n c l u d e t h i s s e c t i o n w i t h a b r i e f d i s c u s s i o n o f t h e r o l e o f m o n e t a r y i n c en t iv e s . I n t h e

p r e s e n t s t u d y w e d i d n o t p a y s u b j e c t s o n t h e b a s i s o f t h e i r c h o i c e s b e c a u s e i n o u r e x p e r i-

e n c e w i t h c h o i c e b e t w e e n p r o s p e c t s o f t h e t y p e u s e d i n t h e p r e s e n t s tu d y , w e d i d n o t f i nd

m u c h d i f f e re n c e b e t w e e n s u b j e c ts w h o w e r e p a i d a f l at f e e a n d s u b j ec t s w h o s e p a y o f f s

w e r e c o n t i n g e n t o n t h e i r d e c is io n s . T h e s a m e c o n c l u si o n w a s o b t a i n e d b y C a m e r e r

( 1 9 89 ) , w h o i n v e s t i g a t e d t h e e f f e c ts o f in c e n t i v e s u s i n g s e v e ra l h u n d r e d s u b j e c t s . H e

f o u n d t h a t s u b j e c t s w h o a c t u a l l y p l a y e d t h e g a m b l e g a v e e s s e n t i a l l y t h e s a m e r e s p o n s e s

a s s u b j e c t s w h o d i d n o t p l a y; h e a l s o f o u n d n o d i f f e r e n c e s i n re l ia b i li ty a n d r o u g h l y t h e

s a m e d e c i si o n ti m e . A l t h o u g h s o m e s tu d i e s f o u n d d i f f e re n c e s b e t w e e n p a i d a n d u n p a i d

s u b j e ct s i n c h o ic e b e t w e e n s i m p l e p r o s p e c ts , t h e s e d i f fe r e n c e s w e r e n o t l a rg e e n o u g h t o

c h a n g e a n y s i g n if ic a n t q u a l i ta t i v e c o n c l u s i o n s . I n d e e d , a ll m a j o r v i o l a ti o n s o f e x p e c t e d

u t i l i t y t h e o r y ( e . g . t h e c o m m o n c o n s e q u e n c e e f f e c t , t h e c o m m o n r a t i o e f f e c t , s o u r c e

d e p e n d e n c e , l os s a v e rs i on , a n d p r e f e r e n c e r e ve r sa ls ) w e r e o b t a i n e d b o t h w i t h a n d w i t h -

o u t m o n e t a r y i n c e n t i v e s .

A s n o t e d b y s e v e r a l a u t h o r s , h o w e v e r , t h e f i n a n c i a l i n c e n t i v e s p r o v i d e d i n c h o i c e

e x p e r i m e n t s a r e g e n e r a l l y s m a l l r e l a t i v e t o p e o p l e ' s i n c o m e s . W h a t h a p p e n s w h e n t h e

s t a k e s c o r r e s p o n d t o t h r e e - o r fo u r - d i g it r a t h e r t h a n o n e - o r t w o - d i g it f i g u r es ? T o a n s w e r

t h is q u e s t i o n , K a c h e l m e i e r a n d S h e h a t a ( 1 9 9 1 ) c o n d u c t e d a s e r i e s o f e x p e r i m e n t s u s i n g

M a s t e r s s t u d e n t s a t B e i ji n g U n i v e r si ty , m o s t o f w h o m h a d t a k e n a t l e as t o n e c o u r s e i n

e c o n o m i c s o r b u s in e s s . D u e t o t h e e c o n o m i c c o n d i t i o n s i n C h i n a, t h e i n v e s t ig a t o r s w e r e

a b l e t o o f f e r s u b j e c t s v e r y la r g e r e w a r d s . I n t h e h i g h p a y o f f c o n d i t io n , s u b j e c t s e a r n e d

a b o u t t h r e e t im e s t h e i r n o r m a l m o n t h l y in c o m e i n t h e c o u r s e o f o n e e x p e r i m e n t a l s e s-

s io n ! O n e a c h t r ia l, s u b j e c t s w e r e p r e s e n t e d w i t h a s im p l e b e t t h a t o f f e r e d a s p e ci f i ed

p r o b a b i l i t y to w i n a g iv e n p ri z e , a n d n o t h i n g o t h e r w i s e . S u b j e c t s w e r e i n s t r u c t e d t o s t a t e

t h e ir c a s h e q u i v al e n t f o r e a c h b e t . A n i n c en t iv e c o m p a t i b l e p r o c e d u r e ( t h e B D M

s c h e m e ) w a s u s e d t o d e t e r m i n e , o n e a c h t r ia l, w h e t h e r t h e s u b j e c t w o u l d p la y th e b e t o r

r e c e i v e t h e " o f f ic i a l" se l li n g p r i c e. I f d e p a r t u r e s f r o m t h e s t a n d a r d t h e o r y a r e d u e t o t h e

m e n t a l c o s t a s s o c i a t e d w i t h d e c is i o n m a k i n g a n d t h e a b s e n c e o f p r o p e r i n c e n ti v e s , ass u g g e s t e d b y S m i t h a n d W a l k e r ( 1 9 92 ) , t h e n t h e h i g hl y p a i d C h i n e s e s u b j e c t s s h o u l d n o t

e x h ib i t t h e c h a r a c t e r i s ti c n o n l i n e a r i ty o b s e r v e d i n h y p o t h e t i c a l c h o i c e s , o r i n c h o i c e s w i t h

s m a l l p a y o f f s .

H o w e v e r , t h e m a i n f i n d i n g o f K a c h e l m e i e r a n d S h e h a t a ( 1 9 9 1 ) i s m a s s i v e r is k s e e k i n g

f o r s m a l l p r o b a b i li t ie s . R i s k s e e k i n g w a s s li g ht ly m o r e p r o n o u n c e d f o r l o w e r p a y o ff s , b u t

e v e n i n t h e h i g h e s t p a y o f f c o n d i t i o n , t h e c a s h e q u i v a l e n t f o r a 5 % b e t ( t h e i r l o w e s t

p r o b a b i l i t y le v e l ) w a s , o n a v e r a g e , t h r e e t i m e s l a r g e r t h a n i ts e x p e c t e d v a l u e . N o t e t h a t i n

t h e p r e s e n t s t u d y th e m e d i a n c a s h e q u iv a l e n t o f a 5 % c h a n c e t o w i n $ 1 00 ( s e e t a b l e 3 )

w a s $1 4 , a l m o s t t h r e e t i m e s t h e e x p e c t e d v a l u e o f t h e b e t . I n g e n e r a l , t h e c a s h e q u i v a l e n t so b t a i n e d b y K a c h e l m e i e r a n d S h e h a t a w e r e h i g h e r t h a n t h o s e o b s e r v e d i n t h e p r e s e n t

s t u d y . T h i s i s c o n s i s t e n t w i t h t h e f i n d i n g t h a t m i n i m a l s e l l i n g p r i c e s a r e g e n e r a l l y h i g h e r

t h a n c e r t a i n t y e q u i v a l e n t s d e r i v e d f r o m c h o i c e ( s e e , e . g . , T v e r s k y , S l o v i c , a n d K a h n e -

m a n , 1 9 9 0 ) . A s a c o n s e q u e n c e , t h e y f o u n d l i t t l e r i s k a v e r s i o n f o r m o d e r a t e a n d h i g h



316 AMOS TVERSKY/DANIEL KAttNEMAN

probab i l ity of winning. Th is w as t rue for the Ch inese subjects , a t both h igh a nd low

payoffs, as wel l as for Canadian subjects , who ei ther p layed for low stakes or d id notreceive any payoff. The m ost st riking resul t in all groups was the m ark ed overweight ing

of smal l probabi l it ies, in acc ord w i th the pres ent analysis .

Evident ly , h igh incent ives do not a lways dominate noneconomic considerat ions, and

the obse rved depar tu re s f rom expec ted u ti li ty the ory canno t b e ra t iona li zed in t e rms o f

the cost of thinking. W e ag ree wi th Smith and W alke r (1992) that m one tary incent ives

could improve per form anc e un de r certain condit ions b y eliminating careless errors. H ow -

ever, w e maintain that m one tary incentives are ne ither necessary n or sufficient to ensure

subjects' cooperativeness, thoughtfulness, or truthfulness. Th e similarity be tw ee n the re-

sults obtained wi th and w i thout mon etary incentives in choice b etw een simple prospe cts

provides no special reaso n for skepticism ab ou t experiments w ithout contingent paym ent.

3 . D i s c u s s i o n

Theo r ies o f cho ice und er uncer t a in ty com mo nly spec i fy 1) the ob jec t s o f cho ice, 2 ) a

valuat ion rule, and 3) the character istics of the funct ions that m ap un certain even ts and

poss ib le ou tcom es in to the i r sub jec tive coun terpar t s . In s t andard app l i ca tions o f ex-

pe cted uti li ty theory , the objects of choice are probab i l i ty d ist ribut ions ove r we al th , the

valuat ion ru le i s expe cted ut il ity, and uti li ty i s a con cave funct ion o f we al th . T he empir i -

ca l ev idence repor ted here and e l sew here requ i res ma jo r rev is ions o f a ll t h ree e l ement s .

W e have p rop osed an a l te rna t ive descr ip t ive theory in which 1 ) the o b jec t s o f cho ice a re

prospec t s f ram ed in t e rms o f ga ins and losses, 2 ) the va lua t ion ru le i s a two-par t cum u-

lat ive functional, and 3) the value funct ion is S-sh aped and the we ight ing funct ions are

inverse S-shaped. The experimental f indings confi rmed the qual i ta t ive propert ies of

these scales, wh ich can be approx im ated by a ( tw o-par t ) pow er va lue func t ion and by

ident ical w eight ing funct ions for gains a nd losses.

Th e curv ature o f the weight ing funct ion explains the ch aracter ist ic ref lect ion pat tern

of at t i tudes to r i sky prospects. Overweight ing of smal l probabi l i t ies contr ibutes to thepopular i ty of bo th lo t ter ies and insurance. U nderw eight ing o f h igh probabi l i t ies contr ib-

u tes bo th to the p reva lence o f ri sk avers ion in cho ices be tw een p ro bab le ga ins and su re

things, and to the prevalen ce of ri sk seeking in choices be tw ee n proba ble and sure losses.

Risk aversion for gains and r isk seeking for losses are fur ther enh anc ed b y the curva ture

of the va lue func t ion in the two domains . The p ronounced asymmet ry o f the va lue

funct ion, which we have labe led loss aversion, explains th e ex t reme reluc tance to acc ept

mixed prospects. The shape of the weight ing funct ion explains the cer tain ty effect and

violations of quasi -convexity. I t a lso explains w hy these p he no m en a are m ost readi ly

observed a t the two ends o f the p robab i l i ty scale, w here the cu rva tu re o f the weigh t ingfunc t ion i s most p rono unce d (Cam erer , 1992).

The new dem onst ra t ions o f the com m on cons eque nce e f fec t, descr ibed in t ab les 1 and

2 , show tha t cho ice un der uncer t a in ty exhib its som e o f the main charac te r is t ics obs erved

in choice unde r r isk . O n the o ther han d, th ere are indicat ions that the decision weights

associated wi th uncertain and w i th ri sky pros pec ts d i ffer in im portan t ways. Fi rst , the re i s

abun dan t ev idence tha t sub jec tive judgm ents o f p robab i l ity do no t conform to the ru les



ADVANCES IN PROSPECT THEO RY 317

o f p ro b ab i l it y t h eo ry (K ah n em an , S lo vic an d Tv e r s k y , 19 82 ). S ec o n d , E l l sb e rg 's ex am p l e

a n d m o r e r e c e n t s tu d i e s o f c h oi c e u n d e r u n c e r t a i n t y in d i c a te t h a t p e o p l e p r e f e r s o m es o u rce s o f u n c e r t a i n t y o v e r o t h e r s . F o r ex am p l e , H ea t h an d T v e r s k y (19 91 ) fo u n d t h a t

i n d iv i d u a ls co n s is t en t ly p re fe r r ed b e t s o n u n ce r t a i n ev en t s i n th e i r a r ea o f ex p e r t is e o v e r

m a t c h e d b e t s o n c h a n c e d e v ic e s, a l th o u g h t h e f o r m e r a r e a m b i g u o u s a n d t h e l a t te r a r e

n o t . Th e p re s en ce o f s y s t em a t i c p re fe ren ces fo r s o m e s o u rce s o f u n ce r t a i n t y ca l ls fo r

d i f f e r en t we i g h t i n g fu n c t i o n s fo r d i f f e r en t d o m a i n s , an d s u g g est s th a t s o m e o f t h e s e

fu n c t i o n s li e en t i r e l y abo v e o t h e r s . T h e i n v es ti g a ti o n o f d ec i s io n we i g h t s fo r u n ce r t a i n

ev en t s em erg es a s a p ro m i s i n g d o m a i n fo r fu t u re r e s ea rch .

T h e p r e s e n t t h e o r y r e t a i n s t h e m a j o r f e a t u r e s o f t h e o r i g i n a l v e r s i o n o f p r o s p e c t

t h eo ry an d i n t ro d u ces a ( t wo -p a r t ) cu m u l a t i v e fu n c t i o n a l , wh i ch p ro v i d es a co n v en i en tm a t h em a t i ca l r e p re s en t a t i o n o f d ec i s i o n we i g h t s . I t a ls o r e l ax es s o m e d es c ri p ti v e ly in ap -

p ro p r i a t e co n s t r a i n t s o f ex p ec t ed u t il i ty t h eo ry . Des p i t e i t s g r ea t e r g en e ra l i t y , t h e cu m u -

l a ti v e fu n c t i o n a l i s u n li k e l y t o b e accu ra t e i n d e t a il . W e s u s p ec t t h a t d ec i s i o n we i g h t s m a y

b e s en s i ti v e t o t h e fo rm u l a t i o n o f t h e p ro s p ec t s, a s we l l a s to t h e n u m b er , t h e s p ac i n g an d

t h e l ev e l o f o u t co m es . In p a r t i cu l a r , t h e re is s o m e ev i d en ce t o s u g g es t t h a t t h e c u rv a t u re

o f t h e w e i g h ti n g f u n c t io n i s m o r e p r o n o u n c e d w h e n t h e o u t c o m e s a r e w i d e l y s p a c e d

(C am ere r , 1 99 2) . Th e p re s en t t h e o ry can b e g en e ra l i z ed t o acco m m o d a t e s u ch e ff ec ts ,

bu t i t is ques t ion ab le w he the r the ga in in descr ip t ive va l id ity , ach ieve d by g iv ing up the

s ep a rab i li t y o f v a l u e s an d we i g h t s , wo u l d ju s t ify t h e l o s s o f p r ed i c t iv e p o w er an d t h e co s t

o f i n c rea s ed co m p l ex it y .

Th eo r i e s o f ch o i ce a r e a t b e s t ap p ro x i m a t e an d i n co m p l e t e . O n e r e a s o n fo r th i s p e s -

s i m is ti c a s s e s s m en t i s t h a t ch o i ce i s a co n s t ru c t i v e an d co n t i n g en t p ro ces s . W h e n f aced

wi t h a co m p l ex p ro b l em , p eo p l e em p l o y a v a r i e t y o f h eu r i s t i c p ro ced u re s i n o rd e r t o

s i m p li fy t h e r ep re s en t a t i o n a n d t h e ev a l u a t i o n o f p ro s p ec ts . Th e s e p ro ced u re s i n c l u d e

co m p u t a t i o n a l s h o r t cu t s an d ed i t i n g o p e ra t i o n s , s u ch a s e l i m i n a t i n g co m m o n co m p o -

nen ts and d i scard ing nonessen t ia l d i f fe rences (Tversky , 1969) . The heur i s t i cs o f cho ice

d o n o t r e ad i l y l en d t h em s e l v es to fo rm a l an a ly s is b ecau s e t h e i r ap p l i ca t i o n d ep en d s o n

t h e fo rm u l a t i o n o f t h e p ro b l em , t h e m e t h o d o f e li c it a t io n , an d t h e co n t ex t o f ch o ice .

P r o s p e c t t h e o r y d e p a r t s f r o m t h e t r a d i t i o n t h a t a s s u m e s t h e r a t io n a l it y o f e c o n o m i cag en t s ; it is p ro p o s ed a s a d e s c r ip t iv e , n o t a n o rm a t i v e , t h eo ry . Th e i d ea l i z ed a s s u m p t i o n

o f r a t i o n a l i ty in eco n o m i c t h eo ry i s co m m o n l y j u s t if i ed o n t w o g ro u n d s : t h e co n v i c ti o n

t h a t o n l y r a ti o n a l b eh av i o r c an s u rv i ve in a co m p e t i t iv e en v i ro n m en t , an d t h e f ea r t h a t

an y t r ea t m en t t h a t ab an d o n s r a t i o n a l i t y w i ll b e ch a o t i c an d i n t r ac tab l e . B o t h a rg u m en t s

a re q u es t i o n ab l e . F i r s t , t h e ev i d en ce i n d i ca t e s t h a t p eo p l e can s p en d a l i f e t i m e i n a

co m p e t i t i v e en v i ro n m en t w i t h o u t a cq u i r i n g a g en e ra l ab i l i ty t o av o i d f r am i n g e f f ec t s o r

t o ap p l y l i n ea r d ec i si o n we i g h t s . S eco n d , a n d p e rh ap s m o re i m p o r t an t , t h e ev i d en ce

i n d i ca t e s t h a t h u m an ch o i ce s a r e o rd e r l y , a l th o u g h n o t a l way s r a t i o n a l in t h e t r ad i t io n a l

s en s e o f t h is wo rd .

App endix: Axiom atic A nalysis

Le t F = { f : S --~ X} b e t h e s e t o f a ll p ro s p ec t s u n d e r s t u d y , an d l e t F + an d F - d e n o t e t h e

p o s it iv e an d t h e n eg a t i v e p ro s p ec ts , r e s p ec t iv e l y. Le t > b e a b i n a ry p re fe re n ce r e l a t i o n



318 AMOS TVERSKY/DANIELKAHNEMAN

on F , a nd l e t ~ a nd > de n o t e i ts s ym me t r i c a nd a s ym me t r i c pa r t s , re s pe c ti ve ly . W e

assum e tha t ~> i s com ple te , t rans i tive , an d s t r ic t ly m on oton ic , tha t is , i f f ~ g and f (s ) ->g ( s ) fo r a ll s ~ S , t h e n f > g .

F o r a n y f , g e F a n d A C S , d e f i n e h = f a g by: h ( s ) = f ( s ) i f s e A , a n d h ( s ) = g ( s ) i f s

S - A . T h u s , f A g c o in c id e s w i t h f o n A a n d w i t h g o n S - A . A p r e f e r e n c e r e la t io n > o n

F satisfies i n d e p e n d e n c e i f for a l l f , g , f ' , g ' e F a n d A C S , f A g > ~ a g ' i f f f ' A g >>. ' A g ' . Thi s

axiom, a l so ca l l ed the sure th ing pr inc ip le (Savage , 1954) , is on e o f the bas ic qua l i t a t ive

p rop e r t i e s und e r l y i ng e xpe c t e d u t i li ty the o ry , a n d i t is v i o l a t e d by A l l a i s 's c om m on c on-

s e q u e n c e e f fe c t. I n d e e d , t h e a t t e m p t t o a c c o m m o d a t e A l l ai s' s e x a m p l e h a s m o t i v a t e d

t h e d e v e l o p m e n t o f n u m e r o u s m o d e l s , in c l u d in g c u m u l a ti v e u t il it y t h e o r y . T h e k e y c o n-

c e p t i n t he a x i oma t i c ana l ys is o f t ha t t he o ry is t he r e l a t i on o f c omo no t on i c i t y , du e t o

Sch m eidle r (1989). A pa i r o f pro spec t s f , g e F a re c o m o n o t o n i c i f the re a re no s , t e S su ch

t h a t f ( s ) > f ( t ) a n d g ( t ) > g ( s ) . N o t e t h a t a c o n s t a n t p r o s p e c t t h a t y ie l d s t h e s a m e

ou t c om e i n e ve ry s ta t e i s c omo no t o n i c w i t h a ll p rospe c t s . O bv i ous ly , c om ono t on i c i t y is

s ym m e t r i c bu t n o t t r a ns i ti ve .

C um ul a t i ve u t il i ty t he o ry do e s n o t s a t i sfy i nd e p e nd e n c e i n ge ne ra l , bu t i t i mp l i e s

i n d e p e n d e n c e w h e n e v e r t h e p r o sp e c ts f A g , f a g ' , f ' A g , a n d f ' A g ' a bove a r e pa i rw i s e

c om ono t o n i c . Th i s p ro pe r t y is c a ll e d c o m o n o t o n i c i n d e p e n d e n c e . 5 I t a l so ho l ds i n c um u-

l a t i ve p ros pe c t t he o ry , a nd i t p l a ys a n i mpor t a n t r o l e i n t he c ha ra c t e r i z a t i on o f t h i s

t he o ry , a s w ill be s how n be l ow . C um ul a t i ve p ros pe c t t he o ry s at is f ie s a n a d d i t i ona l p rop -

er ty , ca l l ed d o u b l e m a t c h i n g : f o r a ll f , g ~ F , i f f + ~ g + a n d f - ~ g - , t h e n f ~ g .

To c ha ra c t e r i z e t he p r e s e n t t he o ry , w e a s s um e t he fo l l ow i ng s t ruc t u ra l c ond i t ions : S is

f i n it e a nd i nc l ude s a t l e a s t t h r e e s t a te s ; X = R e ; a n d t he p r e f e r e n c e o rd e r is c on t i nuou s

i n t he p ro duc t t opo l ogy on R e k , t ha t i s, { f e F : f > g} a nd { f e F : g ~> f} a r e c l o s e d fo r a ny

g e F . T he l a t t e r a s s um pt i ons c a n be r e p l a c e d by r e s t ri c t e d s o lva b i li ty a nd a c om on o t on i c

A r c h i m e d e a n a x i o m (W a k k e r , 1 99 1).

T h e o r e m 1 . S uppo s e (F + , ~> ) a nd ( F - , > ) c a n e a c h b e r e p re s e n t e d b y a c um ul a t i ve

fun c t iona l . T he n (F , ~> ) sa t is f ies cumu la t ive pro spec t th eo ry i f f i t s a ti s fi es do ub le

m a t c h i n g a n d c o m o n o t o n ic i n d e p e n d e n c e .

T h e p r o o f o f t h e t h e o r e m is g i ve n a t th e e n d o f th e a p p e n d ix . I t is b a s e d o n a t h e o r e m

of W a kke r (1992) r e ga rd i ng t he a dd i t i ve r e p re s e n t a t i on o f l ow e r -d i a gona l s t r uc t u re s .

T h e o r e m 1 p r o v id e s a g e n e r i c p r o c e d u r e f o r c h a ra c t e ri z in g c u m u l a ti v e p r o sp e c t t h e o r y .

Ta k e a ny a x i om s ys t e m t ha t i s s u f fi c ie n t t o e s t a b l i sh a n e s s e n t ia l l y un i qu e c um ul a t i ve

( i . e . , r a nk -de pe nde n t ) r e p re s e n t a t i on . A pp l y i t s e pa ra t e l y t o t he p r e f e r e nc e s be t w e e n

pos i ti ve p ros pe c t s a n d t o t he p r e f e r e nc e s be t w e e n ne ga t i ve p ros pe c t s , a nd c o ns t ruc t the

v a l u e fu n c t i o n a n d t h e d e c i si o n w e i g h t s s e p a r a te l y f o r F + a n d f o r F - . T h e o r e m 1 s h o w s

t h a t c o m o n o t o n i c in d e p e n d e n c e a n d d o u b l e m a t c h in g e n s u r e t h a t, u n d e r t h e p r o p e rresca l ing , the su m V ( f + ) + V ( f - ) p r e s er v e s t h e p r e f e r e n c e o r d e r b e t w e e n m i x e d p r o s-

pe c ts . I n o rd e r t o d i s ti ngu i s h m ore s ha rp l y be t w e e n t he c ond i t i ons t ha t g ive r i se t o a

o n e - p a r t o r a tw o - p a r t r e p r e s e n t a t i o n , w e n e e d t o f o cu s o n a p a r t ic u l a r a x i o m a t iz a -

t i o n o f t h e C h o q u e t f u n c t i o n a l . W e c h o s e W a k k e r ' s ( 1 9 8 9 a, 1 98 9b ) b e c a u s e o f i ts

g e n e r a l i ty a n d c o m p a c t n e s s .



ADVANCES IN PROSPECT THEO RY 319

F o r x e X , f e F , and r e S , l e t x{r}fbe th e p rosp ec t tha t y ie lds x in s ta te r and co inc ides

w i t h f i n al l o t h e r s t a te s . F o l lo wi n g W ak k e r (1 98 9a ), we s ay t h a t a p re fe ren c e r e l a t i o nsat isf ies t r a d e o f f c o n s i s t e n c y 6 ( T C ) i f f or a ll x , x ' , y , y ' e X , f , f ' , g , g ' e F , an d s , t e S .

x { s } f <~ y { s } g , x ' { s } f >~ y ' { s } g a n d x { t } f ' > y { t } g ' implyx '{t}f ' ~> y ' { t } g ' .

T o ap p rec i a t e t h e i m p o r t o f t h is co n d i ti o n , s u p p o s e i ts p rem i s e s h o l d b u t t h e co n c l u -

s ion is reversed , tha t i s , y ' { t } g ' > x ' { t } f ' . I t i s easy to ver i fy tha t under expec ted u t i l i ty

theory , the f i r s t two inequal i t i es , invo lv ing {s}, imply u ( y ) - u ( y ' ) >_ u ( x ) - u ( x ' ) ,

wh ereas t h e o t h e r t wo i n eq u a l it i e s, i n v ol v in g {t}, i m p l y t h e o p p o s i t e co n c l u si o n . T r ad e o f f

co n s is t en cy , t h e re fo re , i s n e ed e d t o en s u re t h a t "u t i l it y i n t e rv a l s " can b e co n s i s ten t l y

o rd e re d . E s s en t i a ll y th e s am e co n d i t i o n was u s e d b y T v e r sk y , S a t t a t h , an d S l ov ic (1 98 8)

i n t h e an a l y si s o f p re fe ren ce r ev e rs a l , an d b y T v e r s k y an d K ah n e m an (19 91 ) i n t h e

ch a rac t e r i za t i o n o f co n s t an t l o s s av e rs io n .

A p re fe ren ce r e l a t i o n s a t i s f i e s c o m o n o t o n i c t r a d e o f f c o n s is te n c y ( C T C ) i f T C h o l d s

w h e n e v e r t h e p r o s p e c ts x { s } f , y { s } g , x ' { s } f , an d y '{ s}g a re p a i rw i s e co m o n o t o n i c , a s a r e t h e

p ro s p ec t s x { t } f ' , y { t }g ' , x ' { t } f ' , an d y '{ t}g ' (Wak ker , 1989a). F ina l ly , a p ref ere nce re la t ion

satisfies s i g n - c o m o n o t o n i c t r a d e o f f c o n s i s t e n c y ( S C T C ) i f C T C h o l d s w h e n e v e r t h e c o n s e -

q u en ces x , x ' , y , y ' a r e e i t h e r a ll n o n n eg a t i v e o r a l l n o n p os i ti v e . C l ea r ly , T C is s t ro n g e r

t h an C T C , wh i ch is s t ro n g e r t h an S C T C . In d eed , i t i s n o t d i f f icu l t t o s h o w t h a t i ) ex-

p ec t e d u t i l i ty t h eo ry i m p l i e s T C , 2 ) cu m u l a t i v e u t i li ty t h eo ry i m p l ie s C T C b u t n o t T C ,a n d 3 ) c u m u l a t iv e p r o s p e c t t h e o r y i m p li e s S C T C b u t n o t C T C . T h e f o ll o w in g t h e o r e m

s h o ws th a t , g iv en o u r o t h e r a s s u m p t i o n s , t h e s e p ro p e r t i e s a r e n o t o n l y n eces s a ry b u t a l s o

suff ic ien t to cha rac te r iz e the respe c t ive theor ies .

T h e o rem 2 . As s u m e t h e s t ru c t u ra l co n d i t i o n s d es c r i b ed ab o v e .

a . (W akke r , 1989a) Exp ecte d u t i l i ty theo ry ho lds i f f ~> sa t is f ies TC.

b . (W akke r , 1989b) Cum ula t ive u t il i ty theo ry ho lds i f f > sa t is f ies CT C.

c . C u m u l a t i v e p ro s p ec t t h eo ry h o l d s i f f ~> s a ti sf ie s d o u b l e m a t ch i n g an d S C T C .

A p ro o f o f p a r t c o f t h e t h eo rem is g i v en a t t h e en d o f th i s sec t io n . I t s h o ws t h a t , i n t h e

p re s en c e o f o u r s t ru c t u ra l a s s u m p t i o n s an d d o u b l e m a t ch i n g , t h e r e s t r i c ti o n o f t r ad e o f f

co n s i s ten cy t o s i g n -co m o n o t o n i c p ro s p ec t s y i e ld s a r ep re s e n t a t i o n w i t h a r e f e re n ce -

d e p en d e n t v a l u e fu n c t i o n an d d i f f e ren t d ec is i o n we i g h t s fo r g a i n s an d fo r l os se s .

P r o o f o f th e o r e m 1 . T h e n e c e ss it y o f c o m o n o t o n i c in d e p e n d e n c e a n d d o u b l e m a t c h i n g

is s t ra igh t fo rw ard . To es tab l i sh su f f ic iency , reca l l tha t , by assum pt ion , the re ex is t func-

t i o n s v + , n v - , v + , v , s u c h t h a t V + = ~ ]w + v + a n d V - = ~ v v p re se rv e ~> o n F +

an d o n F - , r e s p ec t iv e l y . F u r t h e r m o re , b y t h e s t ru c t u ra l a ss u m p t i o n s , "rr + an d v - a r eu n i q u e , wh e reas v + an d v - a r e co n t i n u o u s ra t i o sca le s . He n ce , we ca n s e t v + (1) = 1

an d v - ( - 1 ) = 0 < 0 , i n d ep en d en t l y o f each o t h e r .

L e t Q b e t h e s e t o f p ro s p ec t s s u ch t h a t fo r an y q e Q , q ( s ) ~ q ( t ) for any dis t inc t s , t e S.

L e t F g d e n o t e t h e s e t o f al l p r o s p e c ts i n F t h a t a r e c o m o n o t o n i c w i t h G . B y c o m o n o t o n i c

i n d e p e n d e n c e a n d o u r s t ru c t u r a l c o n d it io n s , it f ol lo w s re a d i ly f ro m a t h e o r e m o f W a k k e r




( 1 9 92 ) o n ad d i t iv e r e p r e s e n t a t i o n s f o r l o w er - t r i an g u l a r s u b s e t s o f R e k t h a t , g i v en an y q

Q, t h ere ex i st i n t e rva l s sca l es (Uq i} , w i t h a c o m m o n u n i t, s u c h t h a t U q = ~ i U q i p r e s e r v es_> o n F q . W i t h n o lo ss o f g e n e r a li ty w e c a n s e t Uq i ( O) = 0 f o r a l l / a n d U q ( 1 ) = 1 . S i n ce

V + a n d V - ab o v e a r e ad d i t iv e r ep r e s en t a t i o n s o f ~> o n F q an d F q , r e s p ec ti v e ly , i t

f o ll o w s b y u n i q u en es s t h a t t h e r e ex is t a q , b q > 0 such tha t fo r a ll i, g q i eq u a l s a q ' r r ? v + o n

R e + , a n d U q i eq u a l s b q ~ Z V - o n R e - .

S o f a r th e r e p r e s e n t a t i o n s w e r e r e q u i r e d t o p r e s e r v e t h e o r d e r o n l y w i th i n e a c h F q .

Th u s , w e ca n ch o o s e s ca le s s o t h a t b q = 1 f o r a ll q . T o r e l a t e t h e d i f f e r en t r ep r e s en t a -

t io n s , s e l ec t a p r o s p e c t h ~ q . S i n ce V + s h o u l d p r e s e r v e th e o r d e r o n F + , a n d U q s h o u l d

p r e s e r v e t h e o r d e r w i t h in e a c h F q , w e can mu l t i p ly V + b y a h , a n d r e p l a c e e a c h a q b y

a q / a h . I n o t h e r w o r ds , w e m a y s e t a h = 1 . Fo r any q e Q , se l ec t f e F q , g ~ F h s u ch t h a t

f + ~ g + > 0 , f - ~ g - > 0 , a n d g ~ 0 . B y d o u b l e m a t c h in g , t h e n , f -~ g ~ 0 . T h u s ,

a q V + ( f + ) + V - ( f - ) = 0 , s in c e t hi s f o r m p r e s e r v e s t h e o r d e r o n F q . B u t V + ( f + ) =

V + ( g + ) a n d V - ( f - ) = V - ( g - ) , s o V + ( g + ) + V - ( g - ) = 0 i m p li es V + ( f + ) +

V - ( f - ) = 0. H e n c e , a q = 1 , an d V ( f ) = V + ( f + ) + V - ( f - ) p r e s e r v e s t h e o r d e r

w i t h i n eac h F q .

T o s h o w t h a t V p r e s e r v e s t h e o r d e r o n t h e e n t i re s e t, c o n s i d e r a n y f , g e F a n d s u p p o s e

f > g . By t ransi t iv ity , c ( f ) >_ c ( g ) w h e r e c ( f ) is t h e c e r t a i n t y e q u i v a l e n t o f f . B e c a u s e c ( f )

a n d c ( g) a r e c o m o n o t o n i c , V ( [ ) = V ( c ( f ) ) >_ V ( c ( g ) ) = V ( g ) . A n a l o g o u s l y , f > g im p l ie s

V ( f ) > V ( g ), w h i ch c o m p l e te t h e p r o o f o f t h e o r e m 1.

P r o o f o f th e o r e m 2 ( p a r t c). T o e s t ab l i s h t h e n eces s i t y o f S C T C , ap p l y cu m u l a t i v e

p r o s p ec t t h eo r y t o t h e h y p o t h es e s o f S C TC t o o b t a i n t h e f o l lo w i n g i n eq u a li ti e s :

V (x{ s} f ) = ~rsV(X) + 2 "r r~v (f (r ) )r c S - s

< - W s v ( y ) + 2 W r V( g ( r ) ) = V( y { s }g )r£ S -- s

V (x ' {s } f ) = "rr sV(X') + E w , . v ( f ( r ) )rES - s

>- ~ s V ( y ' ) + ~ v ; v ( g ( r ) ) = V ( y ' { s} g ) .neS - s

T h e d ec i s io n w e i g h t s ab o v e a r e d e r i v ed , a s s u mi n g S C T C , i n acco r d w i t h eq u a t i o n s ( 1 )

an d ( 2 ). W e u s e p r i m es t o d i s ti n g u is h t h e d ec i s i o n w e i g h t s a s s o c i a t ed w i th g f r o m t h o s e

a s s o c ia t e d w i t h f . H o w e v e r , a ll th e a b o v e p r o s p e c t s b e l o n g t o t h e s a m e c o m o n o t o n i c se t.

H e n c e , t w o o u t c o m e s t h a t h a v e t h e s a m e s i g n a n d a r e a s s o c i a t e d w i t h t h e s a m e s t a t e

hav e the sam e dec i s ion weigh t . In par t i cu l a r , t he weigh t s assoc i a t ed w i th x {s}f an d x'{s}f

a r e i d en t i ca l, a s a r e t h e w e i g h ts a s s o c i a t ed w i t h y{s}g a n d w i th y '{s}g . T h e s e a s s u m p t i o n s

a r e i mp l ic i t i n t h e p r e s e n t n o t a t i o n . I t f o l lo w s t h a t

B e c a u s e x , y , x ' , y ' hav e the sam e s ign , a l l t he dec i s ion we igh t s assoc i a t ed wi th s t a t e s

a r e i d en t i ca l , t h a t is , V s = " rr ;. C an ce l l in g t h is co m m o n f ac t o r an d r ea r r an g i n g t e r m s

yields v ( y ) - v ( y ' ) > - v ( x ) - v ( x ' ) .



A D V A N C E S IN P R O S P E C T T H E O R Y 321

Sup pose SC TC i s not va lid , that i s , x{ t} /~> y { t } g ' bu t x'{t}f' < y ' { t } g ' . Apply ing cumu-

lative p r o s p e c t t h e o r y , w e o b t a i n

= + Zr ~ S - t

+ = V ( y { t } g ' )

r e S - t

V ( x ' { t } f ' ) = r r , v ( x ' ) + ~ " " r r r v ( f ' ( r ) )r e S - t

< + : V ( y ' { t } g ' ) .

r e S - t

A d d i n g t h e s e i n e q u a l i t i e s y i e l d s v ( x ) - v ( x ' ) > v ( y ) - v ( y ' ) c o n t r a r y t o t h e p r e v io u s

con clus ion , which es tablishes the necess ity o f SCT C. Th e necessi ty o f dou ble matching is

i m m e d i a t e .

To prove suf fi ciency , note that SC TC impl ie s com ono tonic indepen den ce . Let t ingx =

y , x ' = y ' , a n d f = g in T C y i e l d s x { t ~ ' >~ x { t } g ' impliesx'{t}/" ~> x ' { t } g ' , provided al l the

above prospec ts are pa irwise comonotonic . This condi t ion readi ly enta i l s comonotonic

i n d e p e n d e n c e ( s e e W a k k e r, 1 9 8 9 b) .

T o c o m p l e t e th e p r o o f, n o t e th a t S C T C c o i n ci d e s w i t h C T C o n ( F + , > ) a n d o n ( F - ,

> ) . By part b o f this theo rem , the cumu lative functional holds , separate ly , in the no nn e-

gat ive and in the nonpos i t ive domains . Hence , by double matching and comonotonic

indep end ence , cumulative prosp ec t theory fo l lows from theorem 1 .

N o t e s

1 . I n k e e p i n g w i t h t h e s p i ri t o f p r o s p e c t t h e o r y , w e u s e t h e d e c u m u l a t iv e f o r m f o r g a in s a n d t h e c u m u l a t i v e

f o r m f o r l o s s e s. T h i s n o t a t i o n i s v in d i c a t e d b y t h e e x p e r im e n ta l f i n d in g s d e s c r ib e d i n s e c t i o n 2.

2 . T h i s m o d e l a p p e a r s u n d e r d i f f e r en t n a m e s . W e u s e c u m u l a t i v e u t il i ty t h e o r y t o d e s c r i b e t h e a p p l i ca t io n o f a

Ch o q u e t i n t e g r a l t o a s t a n d a r d u t i l it y f u n c t i o n , a n d c u m u l a t i v e p r o s p e c t t h e o r y t o d e s c r i b e t h e a p p l i c a t io n o ft w o s e p a r a t e C h o q u e t i n t e g ra l s to t h e v a l u e o f g a i n s a n d l o s s e s.

3 . A n I B M d i s k c o n t a i n i n g t h e e x a c t i n s tr u c t io n s , t h e f o r m a t , a n d t h e c o m p l e t e e x p e r i m e n t a l p r o c e d u r e c a n

b e o b t a i n e d f r o m t h e a u t h o r s .

4 . C a m e r e r a n d H o ( 1 99 1 ) a p p l i e d e q u a t i o n ( 6 ) t o s e v e r a l s t u d i e s o f ri sk y c h o i c e a n d e s t im a te d y f r o m

a g g r e g a t e c h o i c e p r o b a b i l i t i e s u s in g a l o g i st i c d i s t r i b u t i o n f u n c t i o n . T h e i r m e a n e s t im a te ( .5 6 ) w a s q u i t e

c lo s e t o o u r s .

5 . W akk er (1989b) ca l le d th is ax iom c o m o n o t o n i c c o o r d i n a t e i n d e p e n d e n c e . S c h m e id l e r (1 9 89 ) u s e d c o m o n o -

t o n i c i n d e p e n d e n c e for the mix ture spa ce ve r s io n of th is ax iom : f >~ g i f f cq" + (1 - o0h > eg + (1 - ~ )h .

6 . W ak ker (1989a, 1989b) ca l led th is p ro pe r ty c a r d i n a l c o o r d i n a t e i n d e p e n d e n c e . H e a l s o i n t r o d u c e d a n

e q u i v a l e n t c o n d i ti o n , c a ll e d t h e a b s e n c e o f c o n t r a d i c t o r y t r a d e o f f s .

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