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)
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.
O0
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¢D
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s.
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s"o'~
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,,e3f'"
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,°s
.'J
IIIIII
0.00.20.40.60.81.0
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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