cumulative

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2022年11月24日发(作者:对话聊天)

JournalofRiskandUncertainty,5:297-323(1992)

©1992KluwerAcademicPublishers

AdvancesinProspectTheory:

CumulativeReprentationofUncertainty

AMOSTVERSKY

StanfordUniversity,DepartmentofPsychology,Stanford,CA94305-2130

DANIELKAHNEMAN*

UniversityofCaliforniaatBerkeley,DepartmentofPsychology,Berkeley,CA94720

Keywords:cumulativeprospecttheory

Abstract

Wedevelopanewversionofprospecttheorythatemployscumulativeratherthanparabledecisionweights

rsion,calledcumulativeprospecttheory,appliestouncertain

aswellastoriskyprospectswithanynumberofoutcomes,anditallowsdifferentweightingfunctionsforgains

nciples,diminishingnsitivityandlossaversion,areinvokedtoexplainthecharacteris-

woftheexperimentalevidenceandthe

resultsofanewexperimentconfirmadistinctivefourfoldpatternofriskattitudes:riskaversionforgainsand

riskekingforlossofhighprobability;riskekingforgainsandriskaversionforlossoflowprobability.

Expectedutilitytheoryreignedforveraldecadesasthedominantnormativeand

descriptivemodelofdecisionmakingunderuncertainty,butithascomeunderrious

snowgeneralagreementthatthetheorydoesnot

provideanadequatedescriptionofindividualchoice:asubstantialbodyofevidence

ternativemod-

elshavebeenpropodinrespontothimpiricalchallenge(forreviews,eCamerer,

1989;Fishburn,1988;Machina,1987).Sometimeagoweprentedamodelofchoice,

calledprospecttheory,whichexplainedthemajorviolationsofexpectedutilitytheoryin

choicesbetweenriskyprospectswithasmallnumberofoutcomes(KahnemanandTver-

sky,1979;TverskyandKahneman,1986).Thekeyelementsofthistheoryare1)avalue

functionthatisconcaveforgains,convexforloss,andsteeperforlossthanforgains,

*Anearlierversionofthisarticlewantitled"CumulativeProspectTheory:AnAnalysisofDecisionunder

Uncertainty."

ThisarticlehasbenefitedfromdiscussionswithColinCamerer,ChewSoo-Hong,DavidFreedman,andDavid

forhisinvaluableinputandcontributiontothe

ndebtedtoRichardGonzalezandAmyHayesforrunningtheexperimentand

rkwassupportedbyGrants89-0064and88-0206fromtheAirForceOfficeofScientific

Rearch,byGrantSES-9109535fromtheNationalScienceFoundation,andbytheSloanFoundation.

298AMOSTVERSKY/DANIELKAHNEMAN

and2)anonlineartransformationoftheprobabilityscale,whichoverweightssmall

portantlater

development,veralauthors(Quiggin,1982;Schmeidler,1989;Yaari,1987;Weymark,

1981)haveadvancedanewreprentation,calledtherank-dependentorthecumulative

functional,ticle

prentsanewversionofprospecttheorythatincorporatesthecumulativefunctional

andextendsthetheorytouncertainaswelltoriskyprospectswithanynumberofout-

ultingmodel,calledcumulativeprospecttheory,combinessomeofthe

attractivefeaturesofbothdevelopments(ealsoLuceandFishburn,1991).Itgivesri

todifferentevaluationsofgainsandloss,whicharenotdistinguishedinthestandard

cumulativemodel,anditprovidesaunifiedtreatmentofbothriskanduncertainty.

Totthestagefortheprentdevelopment,wefirstlistfivemajorphenomenaof

choice,whichviolatethestandardmodelandtaminimalchallengethatmustbemet

findingshavebeenconfirmedina

numberofexperiments,withbothrealandhypotheticalpayoffs.

ionaltheoryofchoiceassumesdescriptioninvariance:equiva-

lentformulationsofachoiceproblemshouldgiveritothesamepreferenceorder

(Arrow,1982).Contrarytothisassumption,thereismuchevidencethatvariationsinthe

framingofoptions(e.g.,intermsofgainsorloss)yieldsystematicallydifferentprefer-

ences(TverskyandKahneman,1986).

ingtotheexpectationprinciple,theutilityofarisky

's(1953)famouxamplechallenged

thisprinciplebyshowingthatthedifferencebetweenprobabilitiesof.99and1.00has

cent

studiesobrvednonlinearpreferencesinchoicesthatdonotinvolvesurethings(Cam-

ererandHo,1991).

'swillingnesstobetonanuncertaineventdependsnotonly

rg(1961)obrvedthatpeople

prefertobetonanurncontainingequalnumbersofredandgreenballs,ratherthanon

centevidence

indicatesthatpeopleoftenpreferabetonaneventintheirareaofcompetenceovera

betonamatchedchanceevent,althoughtheformerprobabilityisvagueandthelatteris

clear(HeathandTversky,1991).

ersionisgenerallyassumedineconomicanalysofdecision

r,risk-ekingchoicesareconsistentlyobrvedintwo

,peopleoftenpreferasmallprobabilityofwinninga

,riskekingisprevalentwhen

peoplemustchoobetweenasurelossandasubstantialprobabilityofalargerloss.

Loss'hebasicphenomenaofchoiceunderbothriskanduncertainty

isthatlossloomlargerthangains(KahnemanandTversky,1984;TverskyandKahne-

man,1991).Theobrvedasymmetrybetweengainsandlossisfartooextremetobe

explainedbyincomeeffectsorbydecreasingriskaversion.

ADVANCESINPROSPECTTHEORY299

Theprentdevelopmentexplainslossaversion,riskeking,andnonlinearprefer-

rporatesaframingpro-

cess,onalphenomenathatliebe-

yondthescopeofthetheory--andofitsalternatives--arediscusdlater.

n1.1introducesthe(two-part)cu-

mulativefunctional;ction1.2discussrelationstopreviouswork;andction1.3

desc

propertiesaretestedinanextensivestudyofindividualchoice,describedinction2,

ationsandlimitationsof

maticanalysisofcumulativeprospect

theoryisprentedintheappendix.

Prospecttheorydistinguishestwophasinthechoiceprocess:

theframingpha,thedecisionmakerconstructsareprentationoftheacts,contingen-

cies,aluationpha,thedecision

ghnoformal

theoryofframingisavailable,wehavelearnedafairamountabouttherulesthatgovern

thereprentationofacts,outcomes,andcontingencies(TverskyandKahneman,1986).

Thevaluationprocessdiscusdinsubquentctionsisappliedtoframedprospects.

tiveprospecttheory

Intheclassicaltheory,theutilityofanuncertainprospectisthesumoftheutilitiesofthe

outcomes,iricalevidencereviewedabove

suggeststwomajormodificationsofthistheory:1)thecarriersofvaluearegainsand

loss,notfinalasts;and2)thevalueofeachoutcomeismultipliedbyadecision

weight,ghtingschemeudintheoriginalversion

ofprospecttheoryandinothermodelsisamonotonictransformationofoutcomeprob-

,itdoesnotalwayssatisfystochastic

dominance,,itisnot

roblemscanbe

handledbyassumingthattransparentlydominatedprospectsareeliminatedintheedit-

ingpha,atively,both

problemscanbesolvedbytherank-dependentorcumulativefunctional,firstpropod

byQuiggin(1982)fordecisionunderriskandbySchmeidler(1989)fordecisionunder

doftransformingeachprobabilityparately,thismodeltransforms

nttheoryappliesthecumulative

velopmentextendsprospecttheoryto

300AMOSTVERSKY/DANIELKAHNEMAN

uncertainaswellastoriskyprospectswithanynumberofoutcomeswhileprerving

ferencesbetweenthecumulativeandtheoriginal

versionsofthetheoryarediscusdinction1.2.

LetSbeafinitetofstatesofnature;sumedthat

exactlyonestateobtains,atof

conquences,plicity,weconfinetheprentdiscussionto

methatXincludesaneutraloutcome,denoted0,andwe

interpretallotherelementsofXasgainsorloss,denotedbypositiveornegative

numbers,respectively.

AnuncertainprospectfisafunctionfromSintoXthatassignstoeachstateSa

conquencefls)--nethecumulativefunctional,wearrangetheoutcomes

ectfisthenreprentedasaquenceof

pairs(xi,Ai),whichyieldsxiifAioccurs,wherexi>xjiffi>j,and(Ai)isapartitionof

ositivesubscriptstodenotepositiveoutcomes,negativesubscriptstodenote

negativeoutcomes,ectis

calledstrictlypositiveorpositive,respectively,ifitsoutcomesareallpositiveornonneg-

lynegativeandnegativeprospectsaredefinedsimilarly;allotherprospects

itivepartoff,denotedf+,isobtainedbylettingf+(s)=f(s)if

f(s)>0,andf+(s)=0iff(s)

similarly.

Asinexpectedutilitytheory,weassigntoeachprospectfanumberV(f)suchthatfis

preferredtoorindifferenttogiffV(f)>_V(g).Thefollowingreprentationisdefinedin

termsoftheconceptofcapacity(Choquet,1955),anonadditivetfunctionthatgener-

ityWisafunctionthatassignstoeachAC

SanumberW(A)satisfyingW((b)=0,W(S)=1,andW(A)>_W(B)wheneverADB.

Cumulativeprospecttheoryasrtsthatthereexistastrictlyincreasingvaluefunction

v:X--+Re,satisfyingv(x0)=v(0)=0,andcapacitiesW+andW-,suchthatforf=(xi,

Ai),-m<-i

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

n0

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

i-Oi=m

wherethedecisionweights"rr+(f+)=(nv~-,...,v+)and~r-(f-)=('rr_-m,"",Wo)

aredefinedby:

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

nvi+=W+(AiU...UAn)-W+(Ai+IU...UAn),O<_i<_n-1,

"rri-=W-(A-mU...UAi)-W-(A-mO...UAi-1),l-m<-i<-O.

Lettingqri="rr?if/-->0andTri=q'r/-if/

V(f)=2"rriP(xi)•

i=--m

(2)

ADVANCESINPROSPECTTHEORY301

Thedecisionweight7ri+,associatedwithapositiveoutcome,isthedifferencebetween

thecapacitiesoftheevents"theoutcomeisatleastasgoodasxi"and"theoutcomeis

strictlybetterthanxi."Thedecisionweightvi-,associatedwithanegativeoutcome,is

thedifferencebetweenthecapacitiesoftheevents"theoutcomeisatleastasbadasxi"

and:'theoutcomeisstrictlyworthanxi."Thus,thedecisionweightassociatedwithan

outcomecanbeinterpretedasthemarginalcontributionoftherespectiveevent,1de-

finedintermsofthecapacitiesW+andW-.IfeachWisadditive,andhenceaproba-

bilitymeasure,owsreadilyfromthedefini-

tionsof'rrandWthatforbothpositiveandnegativeprospects,thedecisionweightsadd

edprospects,however,thesumcanbeeithersmallerorgreaterthan1,

becauthedecisionweightsforgainsandforlossaredefinedbyparatecapacities.

Iftheprospectf=(xi,Ai)isgivenbyaprobabilitydistributionp(Ai)=Pi,itcanbe

viewedasaprobabilisticorriskyprospect(xi,Pi).Inthisca,decisionweightsare

definedby:

7+=w+(p.),~_-=w-(p_m),

"rr+=w+(pi+.-.+Pn)-w+(Pi+l+...+pn),O

vri=w-(pm+...+Pi)-w-(p-m+...+pi-l),l-m<_i<_O.

wherew+andw-arestrictlyincreasingfunctionsfromtheunitintervalintoitlf

satisfyingw+(0)=w-(0)=0,andw+(1)=w-(1)=1.

Toillustratethemodel,ladieonce

andobrvetheresultx=1,...,ven,youreceiveSx;ifxisodd,youpaySx.

Viewedasaprobabilisticprospectwithequiprobableoutcomes,fyieldsthecon-

quences(-5,-3,-1,2,4,6),eachwithprobability1/,f+=(0,1/2;2,1/6;4,1/6;

6,1/6),andf-=(-5,1/6;-3,1/6;-1,1/6;0,1/2).Byequation(1),therefore,

V(f)=V(f+)+V(f-)

=v(2)[w+(1/2)-w+(1/3)]+v(4)[w+(1/3)-w+(1/6)]

+v(6)[w+(1/6)-w+(0)]

+v(-5)[w(1/6)-w(0)]+v(-3)[w-(1/3)-w-(1/6)]

+v(-1)[w(1/2)-w-(1/3)].

ontopreviouswork

LuceandFishburn(1991)derivedesntiallythesamereprentationfromamore

e,fOgis

thecompositeprospectobtainedbyplayingbothfandg,featureof

theirtheoryisthattheutilityfunctionUisadditivewithrespecttoO,thatis,U(fOg)=

U(f)+U(g)providedoneprospectisacceptable(i.e.,preferredtothestatusquo)and

nditionemstoorestrictivebothnormativelyanddescriptively.

Asnotedbytheauthors,itimpliesthattheutilityofmoneyisalinearfunctionofmoney

302AMOSTVERSKY/DANIELKAHNEMAN

ifforallsumsofmoneyx,y,U(xQy)=U(x+y).Thisassumptionappearstous

inescapablebecauthejointreceiptofxandyistantamounttoreceivingtheirsum.

Thus,weexpectthedecisionmakertobeindifferentbetweenreceivinga$10billor

receivinga$20billandreturning$e-Fishburntheory,therefore,

,itextendstocompositeprospectsthat

,itpracticallyforcesutilitytobepropor-

tionaltomoney.

Theprentreprentationencompassveralprevioustheoriesthatemploythe

randSugden(1989)consideredamodel

inwhichw-(p)=w+(p),rast,the

rank-dependentmodelsassumew-(p)=1-w+(1-p)orW-(A)=1-W+(S-A).

Ifweapplythelatterconditiontochoicebetweenuncertainasts,weobtainthechoice

modelestablishedbySchmeidler(1989),whichisbadontheChoquetintegral.2Other

axiomatizationsofthismodelweredevelopedbyGilboa(1987),Nakamura(1990),and

Wakker(1989a,1989b).Forprobabilistic(ratherthanuncertain)prospects,thismodel

wasfirstestablishedbyQuiggin(1982)andYaari(1987),andwasfurtheranalyzedby

Chew(1989),Segal(1989),andWakker(1990).Anearlieraxiomatizationofthismodel

inthecontextofincomeinequalitywasprentedbyWeymark(1981).Notethatinthe

prenttheory,theoverallvalueV(f)ofamixedprospectisnotaChoquetintegralbut

ratherasumV(f+)+V(f-)oftwosuchintegrals.

Theprenttreatmentextendstheoriginalversionofprospecttheoryinveralre-

,itappliestoanyfiniteprospectanditcanbeextendedtocontinuous

,itappliestobothprobabilisticanduncertainprospectsandcan,

therefore,,theprenttheory

allowsdifferentdecisionweightsforgainsandloss,therebygeneralizingtheoriginal

versionthatassumesw+=w-.Underthisassumption,theprenttheorycoincides

withtheoriginalversionforalltwo-outcomeprospectsandforallmixedthree-outcome

teworthythatforprospectsoftheform(x,p;y,1-p),whereeitherx>

y>0orx

modelsyieldsimilarpredictionsingeneral,thecumulativeversion--unliketheoriginal

,itisnolongernecessarytoassumethattrans-

parentlydominatedprospectsareeliminatedintheeditingpha--anassumptionthat

therhand,theprentversioncannolonger

explainviolationsofstochasticdominanceinnontransparentcontexts(e.g.,Tverskyand

Kahneman,1986).Anaxiomaticanalysisoftheprenttheoryanditsrelationtocumu-

lativeutilitytheoryandtoexpectedutilitytheoryarediscusdintheappendix;amore

comprehensivetreatmentisprentedinWakkerandTversky(1991).

andweights

Inexpectedutilitytheory,riskaversionandriskekingaredeterminedsolelybythe

renttheory,asinothercumulativemodels,riskaversionand

riskekingaredeterminedjointlybythevaluefunctionandbythecapacities,whichin

ADVANCESINPROSPECTTHEORY303

theprentcontextarecalledcumulativeweightingfunctions,orweightingfunctionsfor

eoriginalversionofprospecttheory,weassumethatvisconcaveabovethe

referencepoint(v"(x)_<0,x_>0)andconvexbelowthereferencepoint(v"(x)>_O,x<_

0).Wealsoassumethatvissteeperforlossthanforgainsv'(x)0.

Thefirsttwoconditionsreflecttheprincipleofdiminishingnsitivity:theimpactofa

tconditionis

impliedbytheprincipleoflossaversionaccordingtowhichlossloomlargerthan

correspondinggains(TverskyandKahneman,1991).

Theprinci

theevaluationofoutcomes,thereferencepointrvesasaboundarythatdistinguishes

valuationofuncertainty,therearetwonaturalboundaries--

certaintyandimpossibility--thatcorrespondtotheendpointsofthecertaintyscale.

Diminishingnsitivityentailsthattheimpactofagivenchangeinprobabilitydiminishes

mple,anincreaof.1intheprobabilityof

winningagivenprizehasmoreimpactwhenitchangestheprobabilityofwinningfrom.9

to1.0orfrom0to.1,thanwhenitchangestheprobabilityofwinningfrom.3to.4orfrom

.shingnsitivity,therefore,givesritoaweightingfunctionthatiscon-

ertainprospects,thisprincipleyieldssubadditivity

r,thefunctionisnot

well-behavedneartheendpoints,andverysmallprobabilitiescanbeeithergreatlyover-

weightedorneglectedaltogether.

Beforeweturntothemainexperiment,wewishtorelatetheobrvednonlinearityof

spurpo,wedevidanew

demonstrationofthecommonconquenceeffectindecisionsinvolvinguncertaintyrather

1displaysapairofdecisionproblems(IandII)prentedinthatordertoa

ticipantschobetweenpros-

pectswhooutcomeswerecontingentonthedifferencedbetweentheclosingvaluesofthe

mple,f'pays$25,000ifdexceeds30andnothing

independenceaxiomofexpectedutilitytheoryimpliesthatfispreferredtogifff'ispre-

ferredtog'.Table1showsthatthemodalchoicewasfinproblemIandg'inproblemII.

Thispattern,whichviolatesindependence,waschonby53%oftherespondents.

findependence(Dow-Jones)

ABC

ifd<30if30_

ProblemI:f$25,000$25,000$25,000[68]

g$25,0000$75,000[32]

Problemll:f'0$25,000$25,000[23]

g'00$75,000[77]

Note:OutcomesarecontingentonthedifferencedbetweentheclosingvaluesoftheDow-Jonestodayand

centageofrespondents(N=156)wholectedeachprospectisgiveninbrackets.

304AMOSTVERSKY/DANIELKAHNEMAN

Esntiallythesamepatternwasobrvedinacondstudyfollowingthesamede-

of98Stanfordstudentschobetweenprospectswhooutcomeswere

contingentonthepoint-spreaddintheforthcomingStanford-Berkeleyfootballgame.

mple,gpays$10ifStanforddoesnot

win,$30ifitwinsby10pointsorless,

percentoftheparticipants,lectedatrandom,wereactuallypaidaccordingtooneof

alchoice,lectedby46%ofthesubjects,wasfandg',againin

directviolationoftheindependenceaxiom.

Toexploretheconstraintsimpodbythispattern,letusapplytheprenttheoryto

themodalchoicesintable1,using$1,ispreferredtoginproblemI,

v(25)>v(75)W+(C)+v(25)[W+(AUC)-W+(C)]

or

v(25)[1-W+(AUC)+W+(C)]>v(75)W+(C).

Thepreferenceforg'overf'inproblemII,however,implies

v(75)W+(C)>v(25)W+(CUB);

hence,

w+(s)-w+(s-B)>w+(cuB)-w+(O.(3)

Thus,"subtracting"Bfromcertaintyhasmoreimpactthan"subtracting"BfromCUB.

LetW+(D)=1-W+(S-D),andw+(p)=1-w+(1-p).Itfollowsreadilythat

equation(3)iquivalenttothesubadditivityofW+,thatis,W+(B)+W+(D)>_

W+(BUD).Forprobabilisticprospects,equation(3)reducesto

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

or

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

findependence(Stanford-Berkeleyfootballgame)

ABC

ifd<0if0<-d<10ifl0

ProblemI:f$10$10$10[64]

g$10$300[36]

ProblemII:f'0$10$10[34]

g'0$300[66]

Note:Outcomescentageof

respondents(N=98)wholectedeachprospectisgiveninbrackets.

ADVANCESINPROSPECTTHEORY305

Allais'xamplecorrespondstothecawherep(C)=.10,p(B)=.89,andp(A)=.01.

Itisnoteworthythattheviolationsofindependencereportedintables1and2arealso

inconsistentwithregrettheory,advancedbyLoomesandSugden(1982,1987),andwith

Fishburn's(1988)theoryexplainsAllais'xamplebyassumingthat

thedecisionmakerevaluatestheconquencesasifthetwoprospectsineachchoiceare

eprospectsinquestionaredefinedbythesametof

events,asintables1and2,regrettheory(likeFishburn'sSSAmodel)impliesindepen-

dence,dingthatthecommonconquenceeffectis

verymuchinevidenceintheprentproblemsunderminestheinterpretationofAllais's

exampleintermsofregrettheory.

ThecommonconquenceeffectimpliesthesubadditivityofW+andofw+.

OtherviolationsofexpectedutilitytheoryimplythesubadditivityofW+andofw+

mple,Prelec(1990)obrvedthatmost

respondentsprefer2%towin$20,000over1%towin$30,000;theyalsoprefer1%to

win$30,000and32%towin$20,000over34%towin$20,softheprent

theory,thedataimplythatw+(.02)-w+(.01)_>w+(.34)-w+(.33).More

generally,wehypothesize

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

providedp+q+on(4)statesthatw+isconcavenearthe

origin;andtheconjunctionoftheaboveinequalitiesimpliesthat,inaccordwithdimin-

ishingnsitivity,w÷hasaninvertedS-shape:itissteepestneartheendpointsand

ertreatmentsofdecisionweights,e

HogarthandEinhorn(1990),Prelec(1989),Viscusi(1989),andWakker(1990).Exper-

imentalevidenceisprentedinthenextction.

ment

Anexperimentwascarriedouttoobtaindetailedinformationaboutthevalueand

end,werecruited25graduatestudentsfromBerkeleyandStanford(12menand13

women)bjectparticipatedin

bjectwaspaid

$25forparticipation.

ure

icaltrial,thecomputerdisplayed

aprospect(e.g.,25%chancetowin$150and75%chancetowin$50)anditxpected

playalsoincludedadescendingriesofvensureoutcomes(gainsor

loss)-

jectindicatedapreferencebetweeneachofthevensureoutcomesandtherisky

inamorerefinedestimateofthecertaintyequivalent,anewtof

306AMOSTVERSKY/DANIELKAHNEMAN

vensureoutcomeswasthenshown,linearlyspacedbetweenavalue25%higherthan

thelowestamountacceptedinthefirsttandavalue25%lowerthanthehighest

taintyequivalentofaprospectwastimatedbythemidpoint

betweenthelowestacceptedvalueandthehighestrejectedvalueinthecondtof

toemphasizethatalthoughtheanalysisisbadoncertaintyequiva-

lents,thedataconsistedofariesofchoicesbetweenagivenprospectandveralsure

,thecashequivalentofaprospectwasderivedfromobrvedchoices,

putermonitoredtheinternalconsistencyof

theresponstoeachprospectandrejectederrors,suchastheacceptanceofacash

caudtheoriginalstatementofthe

problemtoreappearonthescreen.3

Theprentanalysisfocusonatoftwo-outcomeprospectswithmonetaryout-

atainvolvingmorecomplicatedprospects,

includingprospectsdefinedbyuncertainevents,ere

heprospects(threenonnegativeandthree

nonpositive)wererepeatedondifferentssionstoobtaintheestimateoftheconsistency

3displaystheprospectsandthemediancashequivalentsofthe25

subjects.

oftheprob-

lems,thesubjectsmadechoicesregardingtheacceptabilityofatofmixedprospects

(e.g.,50%chancetolo$100and50%chancetowinx)inwhichxwassystematically

otherproblems,thesubjectscomparedafixedprospect(e.g.,50%chance

tolo$20and50%chancetowin$50)toatofprospects(e.g.,50%chancetolo$50

and50%chancetowinx)inwhichxwassystematicallyvaried.(Theprospectsare

prentedintable6.)

s

Themostdistinctiveimplicationofprospecttheoryisthefourfoldpatternofriskatti-

nonmixedprospectsudintheprentstudy,theshapesofthevalueand

theweightingfunctionsimplyrisk-averandrisk-ekingpreferences,respectively,for

rmore,theshapeofthe

weightingfunctionsfavorsriskekingforsmallprobabilitiesofgainsandriskaversion

forsmallprobabilitiesofloss,,however,

thatprospecttheorydoesnotimplyperfectreflectioninthenthatthepreference

betwe

4prents,foreachsubject,thepercentageofrisk-ekingchoices(wherethecertainty

equivalentexceededexpectedvalue)forgainsandforlosswithlow(p_<.1)andwith

high(p_.5)4showsthatforp_>.5,all25subjectsarepredomi-

naer,the

entirefourfoldpatternisobrvedfor22ofthe25subjects,withsomevariabilityatthe

levelofindividualchoices.

Althoughtheoverallpatternofpreferencesisclear,theindividualdata,ofcour,

relations,acrosssubjects,between

ADVANCESINPROSPECTTHEORY307

cashequivalents(indollars)forallnonmixedprospects

Probability

Outcomes.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)

(-50,-100)

(50,150)

(-50,-~5o)

(100,200)

(-100,-200)

92137

8-21-39

1425365278

-8-23.5-42-63-84

1

-3-23-89-155-190

12377

-14-380

597183

-59-71-85

6472586102128

-60-71-92-113-132

11813

-112-121-142-158-179

Note:Thetwooutcomesofeachprospectaregivenintheleft-handsideofeachrow;theprobabilityofthe

cond(i.e.,moreextreme)mple,thevalueof$9inthe

upperleftcorneristhemediancashequivalentoftheprospect(0,.9;$50,.1).

thecashequivalentsforthesameprospectsonsuccessivessionsaveraged.55oversix

5prentsmeans(aftertransformationtoFisher'sz)ofthe

mple,therewere19and17

prospects,respectively,

valueof.06intable5isthemeanofthe17x19=323correlationsbetweenthecash

equivalentsoftheprospects.

Thecorrelationsbetweenresponswithineachofthefourtypesofprospectsaverage

.41,slightlylowerthanthecorrelationsbetweenparateresponstothesameprob-

negativevaluesintable5indicatethatthosubjectswhoweremorerisk

ghtheindivid-

ualcorrelationsarefairlylow,thetrendisconsistent:78%ofthe403correlationsin

salsoatendencyforsubjectswhoaremorerisk

averforh

trend,whichisabntinthenegativedomain,couldreflectindividualdifferenceither

intheelevationoftheweightingfunctionorinthecurvatureofthevaluefunctionfor

ylowcorrelationsinthetworemainingcellsoftable5,averaging.05,

eindividual

choicesarequitenoisy,aggregationofproblemsisnecessaryfortheanalysisofindividual

differences.

Thefourfoldpatternofriskattitudemergesasamajorempiricalgeneralization

eenobrvedinveralexperiments(e,e.g.,Cohen,

308AMOSTVERSKY/DANIELKAHNEMAN

tageofrisk-ekingchoices

GainLoss

Subjectp-<.1p->.5p_<.1p_>.5

285332075

310010093

47103058

583020100

610050100

7100103086

887010100

916080100

10830093

0

138701094

0

1566030100

1660510100

0

1960106063

201005081

2110000100

221000092

23100310100

2471080100

2510001087

Riskeking78a102087a

Riskneutral12207

Riskaver1088a80a6

aValuesthatcorrespondtothefourfoldpattern.

Note:Thepercentageofrisk-ekingchoicesisgivenforlow(p<_.1)andhigh(p->.5)probabilitiesofgain

andlossforeachsubject(risk-neutralchoiceswereexcluded).Theoverallpercentageofrisk-eking,risk-

neutral,andrisk-averchoicesforeachtypeofprospectappearatthebottomofthetable.

Jaffray,andSaid,1987),includingastudyofexperiencedoilexecutivesinvolvingsignifi-

cant,albeithypothetical,gainsandloss(Wehrung,1989).Itshouldbenotedthat

prospecttheoryimpliesthepatterndemonstratedintable4withinthedataofindividual

subjects,butitdoesnotimplyhighcorrelationsacrosssubjectsbecauthevaluesof

luretoappreciatethispointandthe

limitedreliabilityofindividualresponshasledsomepreviousauthors(e.g.,Hershey

andSchoemaker,1980)tounderestimatetherobustnessofthefourfoldpattern.

ADVANCESINPROSPECTTHEORY309

ecorrelationsbetweencertaintyequivalentsinfourtypesofprospects

L+H+L-H-

L+.41.17-.23.05

H+.39.05-.18

L-.40.06

H-.44

Note:Lowprobabilityofgain=L+;highprobabilityofgain=H+;lowprobabilityofloss=L;high

probabilityofloss=H.

g

Havingestablishedthefourfoldpatterninordinalandcorrelationalanalys,wenow

hprospectoftheform(x,p;O,1-

p),letc/xbetheratioofthecertaintyequivalentoftheprospecttothenonzerooutcome

s1and2plotthemedianvalueofc/xasafunctionofp,forpositiveandfor

negativeprospects,tec/xbyacircleifIx]<200,andbyatriangle

ifIx[>_yexceptionsarethetwoextremeprobabilities(.01and.99)wherea

circleisudforIx]=rpretfigures1and2,notethatifsubjectsarerisk

neutral,thepointswilllieonthediagonal;ifsubjectsareriskaver,allpointswilllie

y,thetriangles

andthecircleswilllieontopofeachotherifpreferencesarehomogeneous,sothat

multiplyingtheoutcomesofaprospectfbyaconstantk>0multipliesitscashequiva-

lentc(kf)bythesameconstant,thatis,c(kf)=kc(f).Inexpectedutilitytheory,prefer-

heprenttheory,

assumingX=Re,preferencehomogeneityisbothnecessaryandsufficienttoreprent

vasatwo-partpowerfunctionoftheform

v(x)=Ix'~ifx_>0

[-k(-x)Pifx<0.(5)

Figures1and2exhibitthecharacteristicpatternofriskaversionandriskeking

soindicatethatpreferencehomogeneityholdsasagood

ghtdeparturesfromhomogeneityinfigure1suggestthatthecash

equivalentsofpositiveprospectsincreamoreslowlythanthestakes(trianglestendto

liebelowthecircles),l,itappearsthat

othcurves

infigures1and2canbeinterpretedasweightingfunctions,assumingalinearvalue

refittedusingthefollowingfunctionalform:

pVp6

,~'(P)=_p?)l/~.(6)

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

310AMOSTVERSKY/DANIELKAHNEMAN

x

0

O0

0

¢.D

0

,,¢

0

0,,I

0

0

0

6

IIIfI

0.00.20.40.60.81.0

c/xforallpositiveprospectsoftheform(x,p;0,1-p).Trianglesandcircles,respectively,

correspondtovaluesofxthatlieaboveorbelow200.

Thisformhasveralufulfeatures:ithasonlyoneparameter;itencompass

weightingfunctionswithbothconcaveandconvexregions;itdoesnotrequirew(.5)=.5;

andmostimportant,itprovidesareasonablygoodapproximationtoboththeaggregate

andtheindividualdataforprobabilitiesintherangebetween.05and.95.

Furtherinformationaboutthepropertiesofthevaluefunctioncanbederivedfrom

ustmentsofmixedprospectstoacceptability(prob-

lems1-4)indicatethat,forevenchancestowinandlo,aprospectwillonlybeaccept-

rvationiscompatiblewitha

valuefunctionthatchangesslopeabruptlyatzero,withaloss-aversioncoefficientof

about2(TverskyandKahneman,1991).Themedianmatchesinproblems5and6are

alsoconsistentwiththistimate:whenthepossiblelossisincreadbykthecompen-

ms7and8areobtainedfromproblems

5and6,respectively,bypositivetranslationsthatturnmixedprospectsintostrictly

rasttothelargevaluesof0obrvedinproblems1-6,therespons

inproblems7and8indicatethatthecurvatureofthevaluefunctionforgainsisslight.A

ADVANCESINPROSPECTTHEORY311

o.

6

00

O

~O

0

x

C~

°.(j""

°

IIIIII

0.00.20.40.60.81.0

P

c/xforallnegativeprospectsoftheform(x,p;O,1-p).Trianglesandcircles,respectively,

correspondtovaluesofxthatliebeloworabove-200.

decreainthesmallestgainofastrictlypositiveprospectisfullycompensatedbya

ndardrank-dependentmodel,which

lacksthenotionofareferencepoint,cannotaccountforthedramaticeffectsofsmall

translationsofprospectsillustratedintable6.

Theestimationofacomplexchoicemodel,suchascumulativeprospecttheory,is

unctionsassociatedwiththetheoryarenotconstrained,thenumber

cethisnumber,itiscom-

montoassumeaparametricform(e.g.,apowerutilityfunction),butthisapproach

confou

thisreason,wefocudhereonthequalitativepropertiesofthedataratherthanon

r,inordertoobtainaparsimonious

descriptionoftheprentdata,weudanonlinearregressionproceduretoestimatethe

parametersofequations(5)and(6),ianexponent

ofthevaluefunctionwas0.88forbothgainsandloss,inaccordwithdiminishing

ian?twas2.25,indicatingpronouncedlossaversion,andthemedian

312AMOSTVERSKY/DANIELKAHNEMAN

flossaversion

Problemabcx0

100-25612.44

200-501012.02

300-1002022.02

400-1502801.87

5-2050-501122.07

6-5.01

75.97

81.35

Note:Ineachproblem,subjectsdeterminedthevalueofxthatmakestheprospect($a,~½;$b,~A)asattractive

as($c,~A;$x,~/2).Themedianvaluesofxareprentedforallproblemsalongwiththefixedvaluesa,b,

statistic0=(x-b)/(c-a)istheratioofthe"slopes"atahigherandalowerregionofthevaluefunction.

valuesof~/and8,respectively,were0.61and0.69,inagreementwithequations(3)and

(4)above.4Theparameterstimatedfromthemediandatawereesntiallythesame.

Figure3plotsw+andw-usingthemedianestimatesof"yand8.

Figure3showsthat,forbothpositiveandnegativeprospects,peopleoverweightlow

quence,peo-

plearere3

alsoshowsthattheweightingfunctionsforgainsandforlossarequiteclo,although

theformerisslightlymorecurvedthanthelatter(i.e.,",/<8).Accordingly,riskaversion

forgainsismorepronouncedthanriskekingforloss,formoderateandhighproba-

bilities(etable3).Itisnoteworthythattheconditionw+(p)=w-(p),assumedinthe

originalversionofprospecttheory,accountsfortheprentdatabetterthantheassump-

tionw+(p)=1-w-(1-p),impliedbythestandardrank-dependentorcumulative

mple,ourestimatesofw+andw-showthatall25subjectssatisfied

theconditionsw+(.5)<.5andw-(.5)<.5,impliedbytheformermodel,andnoone

satisfiedtheconditionw+(.5)<.5iffw-(.5)>.5,impliedbythelattermodel.

Muchrearchonchoicebetweenriskyprospectshasutilizedthetrianglediagram

(Marschak,1950;Machina,1987)thatreprentsthetofallprospectsoftheform(Xl,

pl;x2,pz;x3,p3),withfixedoutcomesxl

aprospectthatyieldsthelowestoutcome(Xl)withprobabilitypl,thehighestoutcome

(x3)withprobabilityp3,andtheintermediateoutcome(x2)withprobabilitypz=1-

fferencecurveisatofprospects(i.e.,points)thatthedecisionmaker

ativechoicetheoriesarecharacterizedbytheshapesof

icular,theindifferencecurvesofexpectedutilitytheory

s4aand4billustratetheindifferencecurvesofcumula-

tiveprospecttheoryfornonnegativeandnonpositiveprospects,pes

ofthecurvesaredeterminedbytheweightingfunctionsoffigure3;thevaluesofthe

outcomes(Xl,x2,x3)merelycontroltheslope.

ADVANCESINPROSPECTTHEORY313

0.

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P

ingfunctionsforgains(w+)andforloss(w-)badonmedianestimatesofyand8in

equation(12).

Figures4aand4bareingeneralagreementwiththemainempiricalgeneralizations

thathaveemergedfromthestudiesofthetrianglediagram;eCamerer(1992),and

CamererandHo(1991),departuresfromlinearity,whichviolateex-

pectedutilitytheory,,the

,thecurvesareconcave

y,theindifference

curvesfornonpositiveprospectsremblethecurvesfornonnegativeprospectsreflected

aroundthe45°line,mple,asuregainof$100is

equallyasattractiveasa71%chancetowin$200ornothing(efigure4a),andasure

lossof$100iquallyasaversiveasa64%chancetolo$200ornothing(efigure4b).

Theapproximatereflectionofthecurvesisofspecialinterestbecauitdistinguishesthe

prenttheoryfromthestandardrank-dependentmodelinwhichthetwotsofcurves

areesntiallythesame.

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ADVANCESINPROSPECTTHEORY315

ives

Weconcl

prentstudywedidnotpaysubjectsonthebasisoftheirchoicesbecauinourexperi-

encewithchoicebetweenprospectsofthetypeudintheprentstudy,wedidnotfind

muchdifferencebetweensubjectswhowerepaidaflatfeeandsubjectswhopayoffs

econclusionwasobtainedbyCamerer

(1989),w

foundthatsubjectswhoactuallyplayedthegamblegaveesntiallythesamerespons

assubjectswhodidnotplay;healsofoundnodifferencesinreliabilityandroughlythe

ghsomestudiesfounddifferencesbetweenpaidandunpaid

subjectsinchoicebetweensimpleprospects,thedifferenceswerenotlargeenoughto

,allmajorviolationsofexpected

utilitytheory(monconquenceeffect,thecommonratioeffect,source

dependence,lossaversion,andpreferencereversals)wereobtainedbothwithandwith-

outmonetaryincentives.

Asnotedbyveralauthors,however,thefinancialincentivesprovidedinchoice

experimentsaregenerallysmallrelativetopeople'ppenswhenthe

stakescorrespondtothree-orfour-digitratherthanone-ortwo-digitfigures?Toanswer

thisquestion,KachelmeierandShehata(1991)conductedariesofexperimentsusing

MastersstudentsatBeijingUniversity,mostofwhomhadtakenatleastonecourin

heeconomicconditionsinChina,theinvestigatorswere

ighpayoffcondition,subjectarned

aboutthreetimestheirnormalmonthlyincomeinthecourofoneexperimentals-

sion!Oneachtrial,subjectswereprentedwithasimplebetthatofferedaspecified

probabilitytowinagivenprize,tswereinstructedtostate

ntivecompatibleprocedure(theBDM

scheme)wasudtodetermine,oneachtrial,whetherthesubjectwouldplaythebetor

receivethe"official"rturesfromthestandardtheoryareduetothe

mentalcostassociatedwithdecisionmakingandtheabnceofproperincentives,as

suggestedbySmithandWalker(1992),thenthehighlypaidChinesubjectsshouldnot

exhibitthecharacteristicnonlinearityobrvedinhypotheticalchoices,orinchoiceswith

smallpayoffs.

However,themainfindingofKachelmeierandShehata(1991)ismassiveriskeking

ekingwasslightlymorepronouncedforlowerpayoffs,but

eveninthehighestpayoffcondition,thecashequivalentfora5%bet(theirlowest

probabilitylevel)was,onaverage,atin

theprentstudythemediancashequivalentofa5%chancetowin$100(etable3)

was$14,ral,thecashequivalents

obtainedbyKachelmeierandShehatawerehigherthanthoobrvedintheprent

consistentwiththefindingthatminimalllingpricesaregenerallyhigher

thancertaintyequivalentsderivedfromchoice(e,e.g.,Tversky,Slovic,andKahne-

man,1990).Asaconquence,theyfoundlittleriskaversionformoderateandhigh

316AMOSTVERSKY/DANIELKAttNEMAN

struefortheChinesubjects,atbothhighandlow

payoffs,aswellasforCanadiansubjects,whoeitherplayedforlowstakesordidnot

tstrikingresultinallgroupswasthemarkedoverweighting

ofsmallprobabilities,inaccordwiththeprentanalysis.

Evidently,highincentivesdonotalwaysdominatenoneconomicconsiderations,and

theobrveddeparturesfromexpectedutilitytheorycannotberationalizedintermsof

ewithSmithandWalker(1992)thatmonetaryincentives

couldimpr-

ever,wemaintainthatmonetaryincentivesareneithernecessarynorsufficienttoensure

subjects'cooperativeness,thoughtfulness,ilaritybetweenthere-

sultsobtainedwithandwithoutmonetaryincentivesinchoicebetweensimpleprospects

providesnospecialreasonforskepticismaboutexperimentswithoutcontingentpayment.

sion

Theoriesofchoiceunderuncertaintycommonlyspecify1)theobjectsofchoice,2)a

valuationrule,and3)thecharacteristicsofthefunctionsthatmapuncertaineventsand

dardapplicationsofex-

pectedutilitytheory,theobjectsofchoiceareprobabilitydistributionsoverwealth,the

valuationruleixpectedutility,iri-

calevidencereportedhereandelwhererequiresmajorrevisionsofallthreeelements.

Wehavepropodanalternativedescriptivetheoryinwhich1)theobjectsofchoiceare

prospectsframedintermsofgainsandloss,2)thevaluationruleisatwo-partcumu-

lativefunctional,and3)thevaluefunctionisS-shapedandtheweightingfunctionsare

erimentalfindingsconfirmedthequalitativepropertiesof

thescales,whichcanbeapproximatedbya(two-part)powervaluefunctionandby

identicalweightingfunctionsforgainsandloss.

Thecurvatureoftheweightingfunctionexplainsthecharacteristicreflectionpattern

ightingofsmallprobabilitiescontributestothe

eightingofhighprobabilitiescontrib-

utesbothtotheprevalenceofriskaversioninchoicesbetweenprobablegainsandsure

things,andtotheprevalenceofriskekinginchoicesbetweenprobableandsureloss.

Riskaversionforgainsandriskekingforlossarefurtherenhancedbythecurvature

nouncedasymmetryofthevalue

function,whichwehavelabeledlossaversion,explainstheextremereluctancetoaccept

peoftheweightingfunctionexplainsthecertaintyeffectand

explainswhythephenomenaaremostreadily

obrvedatthetwoendsoftheprobabilityscale,wherethecurvatureoftheweighting

functionismostpronounced(Camerer,1992).

Thenewdemonstrationsofthecommonconquenceeffect,describedintables1and

2,showthatchoiceunderuncertaintyexhibitssomeofthemaincharacteristicsobrved

therhand,thereareindicationsthatthedecisionweights

as,thereis

abundantevidencethatsubjectivejudgmentsofprobabilitydonotconformtotherules

ADVANCESINPROSPECTTHEORY317

ofprobabilitytheory(Kahneman,SlovicandTversky,1982).Second,Ellsberg'xample

andmorerecentstudiesofchoiceunderuncertaintyindicatethatpeopleprefersome

mple,HeathandTversky(1991)foundthat

individualsconsistentlypreferredbetsonuncertaineventsintheirareaofexpertiover

matchedbetsonchancedevices,althoughtheformerareambiguousandthelatterare

nceofsystematicpreferencesforsomesourcesofuncertaintycallsfor

differentweightingfunctionsfordifferentdomains,andsuggeststhatsomeofthe

estigationofdecisionweightsforuncertain

eventmergesasapromisingdomainforfuturerearch.

Theprenttheoryretainsthemajorfeaturesoftheoriginalversionofprospect

theoryandintroducesa(two-part)cumulativefunctional,whichprovidesaconvenient

relaxessomedescriptivelyinap-

eitsgreatergenerality,thecumu-

ectthatdecisionweightsmay

bensitivetotheformulationoftheprospects,aswellastothenumber,thespacingand

icular,thereissomeevidencetosuggestthatthecurvature

oftheweightingfunctionismorepronouncedwhentheoutcomesarewidelyspaced

(Camerer,1992).Theprenttheorycanbegeneralizedtoaccommodatesucheffects,

butitisquestionablewhetherthegainindescriptivevalidity,achievedbygivingupthe

parabilityofvaluesandweights,wouldjustifythelossofpredictivepowerandthecost

ofincreadcomplexity.

sonforthispes-

siced

withacomplexproblem,peopleemployavarietyofheuristicproceduresinorderto

roceduresinclude

computationalshortcutsandeditingoperations,suchaliminatingcommoncompo-

nentsanddiscardingnonesntialdifferences(Tversky,1969).Theheuristicsofchoice

donotreadilylendthemlvestoformalanalysisbecautheirapplicationdependson

theformulationoftheproblem,themethodofelicitation,andthecontextofchoice.

Prospecttheorydepartsfromthetraditionthatassumestherationalityofeconomic

agents;itispropodasadescriptive,notanormative,alizedassumption

ofrationalityineconomictheoryiscommonlyjustifiedontwogrounds:theconviction

thatonlyrationalbehaviorcansurviveinacompetitiveenvironment,andthefearthat

guments

,theevidenceindicatesthatpeoplecanspendalifetimeina

competitiveenvironmentwithoutacquiringageneralabilitytoavoidframingeffectsor

,andperhapsmoreimportant,theevidence

indicatesthathumanchoicesareorderly,althoughnotalwaysrationalinthetraditional

nofthisword.

Appendix:AxiomaticAnalysis

LetF={f:S--~X}bethetofallprospectsunderstudy,andletF+andF-denotethe

positiveandthenegativeprospects,>beabinarypreferencerelation

318AMOSTVERSKY/DANIELKAHNEMAN

onF,andlet~and>denoteitssymmetricandasymmetricparts,

assumethat~>iscomplete,transitive,andstrictlymonotonic,thatis,iff~gandf(s)->

g(s)foralls~S,thenf>g.

Foranyf,geFandACS,defineh=fagby:h(s)=f(s)ifA,andh(s)=g(s)ifs

,rencerelation>on

Fsatisfiesindependenceifforallf,g,f',g'eFandACS,fAg>~fag'ifff'Ag>>.f'Ag'.This

axiom,alsocalledthesurethingprinciple(Savage,1954),isoneofthebasicqualitative

propertiesunderlyingexpectedutilitytheory,anditisviolatedbyAllais'scommoncon-

,theattempttoaccommodateAllais'xamplehasmotivated

thedevelopmentofnumerousmodels,con-

ceptintheaxiomaticanalysisofthattheoryistherelationofcomonotonicity,dueto

Schmeidler(1989).Apairofprospectsf,geFarecomonotoniciftherearenos,teSsuch

thatf(s)>f(t)andg(t)>g(s).Notethataconstantprospectthatyieldsthesame

sly,comonotonicityis

symmetricbutnottransitive.

Cumulativeutilitytheorydoesnotsatisfyindependenceingeneral,butitimplies

independencewhenevertheprospectsfAg,fag',f'Ag,andf'Ag'abovearepairwi

opertyiscalledcomonotonicindependence.5Italsoholdsincumu-

lativeprospecttheory,anditplaysanimportantroleinthecharacterizationofthis

theory,tiveprospecttheorysatisfiesanadditionalprop-

erty,calleddoublematching:forallf,g~F,iff+~g+andf-~g-,thenf~g.

Tocharacterizetheprenttheory,weassumethefollowingstructuralconditions:Sis

finiteandincludesatleastthreestates;X=Re;andthepreferenceorderiscontinuous

intheproducttopologyonRek,thatis,{feF:f>g}and{feF:g~>f}areclodforany

terassumptionscanbereplacedbyrestrictedsolvabilityandacomonotonic

Archimedeanaxiom(Wakker,1991).

e(F+,~>)and(F-,>)caneachbereprentedbyacumulative

(F,~>)satisfiescumulativeprospecttheoryiffitsatisfiesdouble

matchingandcomonotonicindependence.

donatheorem

ofWakker(1992)regardingtheadditivereprentationoflower-diagonalstructures.

Theorem1providesagenericprocedureforcharacterizingcumulativeprospecttheory.

Takeanyaxiomsystemthatissufficienttoestablishanesntiallyuniquecumulative

(i.e.,rank-dependent)tparatelytothepreferencesbetween

positiveprospectsandtothepreferencesbetweennegativeprospects,andconstructthe

valuefunctionandthedecisionweightsparatelyforF+andforF-.Theorem1shows

thatcomonotonicindependenceanddoublematchingensurethat,undertheproper

rescaling,thesumV(f+)+V(f-)prervesthepreferenceorderbetweenmixedpros-

rtodistinguishmoresharplybetweentheconditionsthatgiveritoa

one-partoratwo-partreprentation,weneedtofocusonaparticularaxiomatiza-

eWakker's(1989a,1989b)becauofits

generalityandcompactness.

ADVANCESINPROSPECTTHEORY319

ForxeX,feF,andreS,letx{r}fbetheprospectthatyieldsxinstaterandcoincides

ingWakker(1989a),wesaythatapreferencerelation

satisfiestradeoffconsistency6(TC)ifforallx,x',y,y'eX,f,f',g,g'eF,ands,teS.

x{s}f<~y{s}g,x'{s}f>~y'{s}gandx{t}f'>y{t}g'implyx'{t}f'~>y'{t}g'.

Toappreciatetheimportofthiscondition,suppoitspremisholdbuttheconclu-

sionisreverd,thatis,y'{t}g'>x'{t}f'.Itiasytoverifythatunderexpectedutility

theory,thefirsttwoinequalities,involving{s},implyu(y)-u(y')>_u(x)-u(x'),

whereastheothertwoinequalities,involving{t},ff

consistency,therefore,isneededtoensurethat"utilityintervals"canbeconsistently

iallythesameconditionwasudbyTversky,Sattath,andSlovic(1988)

intheanalysisofpreferencereversal,andbyTverskyandKahneman(1991)inthe

characterizationofconstantlossaversion.

Apreferencerelationsatisfiescomonotonictradeoffconsistency(CTC)ifTCholds

whenevertheprospectsx{s}f,y{s}g,x'{s}f,andy'{s}garepairwicomonotonic,asarethe

prospectsx{t}f',y{t}g',x'{t}f',andy'{t}g'(Wakker,1989a).Finally,apreferencerelation

satisfiessign-comonotonictradeoffconsistency(SCTC)ifCTCholdswheneverthecon-

quencesx,x',y,y'y,TCisstronger

thanCTC,,itisnotdifficulttoshowthati)ex-

pectedutilitytheoryimpliesTC,2)cumulativeutilitytheoryimpliesCTCbutnotTC,

and3)lowingtheorem

showsthat,givenourotherassumptions,thepropertiesarenotonlynecessarybutalso

sufficienttocharacterizetherespectivetheories.

thestructuralconditionsdescribedabove.

a.(Wakker,1989a)Expectedutilitytheoryholdsiff~>satisfiesTC.

b.(Wakker,1989b)Cumulativeutilitytheoryholdsiff>satisfiesCTC.

tiveprospecttheoryholdsiff~>satisfiesdoublematchingandSCTC.

sthat,inthe

prenceofourstructuralassumptionsanddoublematching,therestrictionoftradeoff

consistencytosign-comonotonicprospectsyieldsareprentationwithareference-

dependentvaluefunctionanddifferentdecisionweightsforgainsandforloss.

essityofcomonotonicindependenceanddoublematching

blishsufficiency,recallthat,byassumption,thereexistfunc-

tionsv+,nv-,v+,v,suchthatV+=~]w+v+andV-=~vvprerve~>onF+

andonF-,rmore,bythestructuralassumptions,"rr+andv-are

unique,whereasv+,wecantv+(1)=1

andv-(-1)=0<0,independentlyofeachother.

LetQbethetofprospectssuchthatforanyqeQ,q(s)~q(t)foranydistincts,teS.

notonic

independenceandourstructuralconditions,itfollowsreadilyfromatheoremofWakker

ADVANCESINPROSPECTTHEORY320

(1992)onadditivereprentationsforlower-triangularsubtsofRekthat,givenanyq

Q,thereexistintervalsscales(Uqi},withacommonunit,suchthatUq=~iUqiprerves

_>lossofgeneralitywecantUqi(O)=0forall/andUq(1)=

V+andV-aboveareadditivereprentationsof~>onFqandFq,respectively,it

followsbyuniquenessthatthereexistaq,bq>0suchthatforalli,gqiequalsaq'rr?v+on

Re+,andUqiequalsbq~ZV-onRe-.

SofarthereprentationswererequiredtoprervetheorderonlywithineachFq.

Thus,wecanchooscalessothatbq=tethedifferentreprenta-

tions,lectaprospecth~+shouldprervetheorderonF+,andUqshould

prervetheorderwithineachFq,wecanmultiplyV+byah,andreplaceeachaqby

aq/rwords,wemaytah=qeQ,lectfeFq,g~Fhsuchthat

f+~g+>0,f-~g->0,andg~lematching,then,f-~g~,

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

V+(g+)andV-(f-)=V-(g-),soV+(g+)+V-(g-)=0impliesV+(f+)+

V-(f-)=,aq=1,andV(f)=V+(f+)+V-(f-)prervestheorder

withineachFq.

ToshowthatVprervestheorderontheentiret,consideranyf,geFandsuppo

f>sitivity,c(f)>_c(g)wherec(f)ec(f)

andc(g)arecomonotonic,V([)=V(c(f))>_V(c(g))=V(g).Analogously,f>gimplies

V(f)>V(g),whichcompletetheproofoftheorem1.

Proofoftheorem2(partc).ToestablishthenecessityofSCTC,applycumulative

prospecttheorytothehypothesofSCTCtoobtainthefollowinginequalities:

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

rcS-s

<-Wsv(y)+2WrV(g(r))=V(y{s}g)

r£S--s

V(x'{s}f)="rrsV(X')+Ew,.v(f(r))

rES-s

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

neS-s

Thedecisionweightsabovearederived,assumingSCTC,inaccordwithequations(1)

and(2).Weuprimestodistinguishthedecisionweightsassociatedwithgfromtho

r,alltheaboveprospectsbelongtothesamecomonotonict.

Hence,twooutcomesthathavethesamesignandareassociatedwiththesamestate

icular,theweightsassociatedwithx{s}fandx'{s}f

areidentical,asaretheweightsassociatedwithy{s}gandwithy'{s}ssumptions

owsthat

Becaux,y,x',y'havethesamesign,allthedecisionweightsassociatedwithstates

areidentical,thatis,Vs="rr;.Cancellingthiscommonfactorandrearrangingterms

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

ADVANCESINPROSPECTTHEORY321

SuppoSCTCisnotvalid,thatis,x{t}/~>y{t}g'butx'{t}f'

lativeprospecttheory,weobtain

=+Z

r~S-t

+=V(y{t}g')

reS-t

V(x'{t}f')=rr,v(x')+~""rrrv(f'(r))

reS-t

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

reS-t

Addingtheinequalitiesyieldsv(x)-v(x')>v(y)-v(y')contrarytotheprevious

conclusion,essityofdoublematchingis

immediate.

Toprovesufficiency,gx=

y,x'=y',andf=ginTCyieldsx{t~'>~x{t}g'impliesx'{t}/"~>x'{t}g',providedallthe

nditionreadilyentailscomonotonic

independence(eWakker,1989b).

Tocompletetheproof,notethatSCTCcoincideswithCTCon(F+,>)andon(F-,

>).Bypartbofthistheorem,thecumulativefunctionalholds,parately,inthenonne-

,bydoublematchingandcomonotonic

independence,cumulativeprospecttheoryfollowsfromtheorem1.

Notes

ingwiththespiritofprospecttheory,weuthedecumulativeformforgainsandthecumulative

tationisvindicatedbytheexperimentalfindingsdescribedinction2.

umulativeutilitytheorytodescribetheapplicationofa

Choquetintegraltoastandardutilityfunction,andcumulativeprospecttheorytodescribetheapplicationof

twoparateChoquetintegralstothevalueofgainsandloss.

iskcontainingtheexactinstructions,theformat,andthecompleteexperimentalprocedurecan

beobtainedfromtheauthors.

randHo(1991)appliedequation(6)toveralstudiesofriskychoiceandestimatedyfrom

eanestimate(.56)wasquite

clotoours.

(1989b)dler(1989)udcomono-

tonicindependenceforthemixturespaceversionofthisaxiom:f>~giffcq"+(1-o0h>eg+(1-~)h.

(1989a,1989b)introducedan

equivalentcondition,calledtheabnceofcontradictorytradeoffs.

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