Expected utility theory and the real worldWord文档下载推荐.docx

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Expectedwinnings=1/2*£

1+1/4*£

2+1/8*£

4+…=,butpeoplepaymuchless

So,it’stheexpectedutilitythatmatters:

U=u（xi）pi

Expectedutilitytheory（EUT）

EUT（vonNeumannandMorgenstern1947）isbasedupontwomainaxioms

inadditiontothosethatarerequiredbyconsumertheory

Independence-ifX~Y,thenpX+（1-p）ZandpY+（1-p）Zforanyp

Betweenness-ifX~Y,thenX~Y~pX+（1-p）Y,foranyp

Therehavebeenmanyexperimentsineconomicsandpsychologythathavetestedthedescriptivevalidityoftheseassumptions

TheindependenceaxiomandtheAllaisparadox

A:

£

300witha1.0chanceorB:

400witha0.8chance

C:

300witha0.25chanceorD:

400witha0.2chance

AclearmajorityofpeoplechooseAandD

butthisviolatesindependencesinceCandDare'

scaled-down'

versionsofAandB

i.e.theyarebothscaleddownbyafactorof0.25

SomealternativestoEUT

TheAllaisparadoxcanbeexplainedbyaweakenedversionofindependence:

ifX~Y,then,foreveryZandp,thereissomeqforwhichpX+（1-p）Z~qY+（1-q）Z

RegretTheory（LoomesandSugden1982）-

expectedutilitiesaremodifiedbyfeelingsofregretifthingsturnoutworsethantheywouldhavedoneundertheotheroption,orbyrejoicingifthingsturnoutbetter

Theregretfunction,ij,representsthedifferencebetweentheutilityfromxiandtheutilitythatwouldhavebeenderivedfromxj.Thisfunctionhastwokeyproperties:

ij=-ij

x3x1>

x3x2andx3x1>

x2x1wherex3>

x2>

x1

ItcanbeshownthattheAllaisparadoxrequiresthat:

x3x2+x2x1

EUwithrankdependentprobabilityweights（Quiggin1982）-

assumesthatprobabilitiesareweightedaccordingtotherankorderoftheoutcomesEUwillnotholdunlesstheprobabilitytransformationfunctionf（p）isp.

Aconcavef（.）meansthatifX~Y,thenpX+（1-p）Y<

X

i.e.peopledon’tlikeaprobabilitymixtureofXandYwhenX~Y

That'

srisk-whataboutuncertainty?

Knight（1921）madeadistinctionbetween:

Risk–probabilitiescanbeassignedtodifferentoutcomes,and

Uncertainty–noprobabilitiescanbeattachedtopossibleoutcomes

Modelshavebeenproposedwhichimplythatalluncertaintiescanbereducedtorisks

Savage（1954）arguesthatwecaninfersubjectiveprobabilitiesfromactualchoices

e.g.gamblingdecisionsallowustoinfera"

qualitativeprobabilityrelationship"

SubjectiveEUTandSavage'

ssure-thingprinciple

PostulateIIstatesthatthechoicebetweentwoactionsisunaffectedbythepay-offsinaconstantcolumn

ABC

1

a

b

B

2

3

C

4

If1>

2above,then3>

4,whenaandbareunchangedandCtakesanyvalue

TheEllsbergParadox–Experiment2

Anurncontains30redballsand60blackandyellowballsinunknownproportion

Q1.Doyouprefertobetonred（A）orblack（B）?

Q2.Doyouprefertobetonredandyellow（C）orblackandyellow（D）?

MostpeoplechooseAandDbutthisisinconsistentwithSavage'

redblackyellow

A

100

D

Ambiguity

Ellsberg（1961）explainstheseviolationsofthesure-thingprincipleintermsofpeople'

sdislikeforambiguity,orambiguityaversion

Ambiguitycanbedefinedasuncertaintyaboutprobabilitycreatedbymissinginformationthatisrelevantandcouldbeknown-A>

BbecausetheproportionofredballsisknownandD>

Cbecausetheproportionofblackandyellowballsisknown

Ellsbergarguesthatdecisionsaremadeonthebasisof3factors:

1）therelativedesirabilityofthepossiblepay-offs

2）therelativelikelihoodoftheeventsaffectingthem

3）thenatureofone'

sinformationconcerningtherelativelikelihoodofevents

Thejudgementoftheambiguityofone'

sinformationcannotbecharacterisedasriskasrequiredbythepostulatesofSavageandothers-andthereforeliesoutsideofSEUT

Thepreferencereversalphenomenon（PRP）

Conventionaltheorysuggeststhatthevaluationof,andchoiceover,differentalternativesshouldproducethesameordering

e.g.ifaskedtovaluetwolotterytickets,AandB,andthentochoosebetweenthem,thosewhogivethegreatestvaluetoA（B）shouldchooseA（B）

Butexperimentalevidencesuggeststhatthereisasystematictendencyfortherankingrevealedbyvaluationtaskstodifferfromthatrevealedbychoicetasks

PRP-anexample

P-bet:

0.6chanceofwinning£

22$-bet:

0.3chanceofwinning£

40

0.4chanceofwinning£

00.7chanceofwinning£

Q1.1WhatistheminimumamountyouwouldacceptfortheP-bet?

Q1.2Whatistheminimumamountyouwouldacceptforthe$-bet?

Q2Whichgamblewouldyouprefertoplayout?

Overwhelminglypeopleplacethehighestvalueonthe$-betyetchoosetheP-bet

i.e.thereisasystematictendencyforpeopletoplacethehighestvalueonthebetwiththegreaterpossiblegainandtochoosethebetwithgreaterprobabilityofwinning

ExplanationsofPRP

Economists-

Ifrespondentshaveanunderlyingsetofpreferences,thenthereisarealconcernthatthePRPrepresentsaviolationoftransitivity:

if$-bet=r$>

<

C>

rP=P-bet

therefore,ifr$>

rPandP>

$,thenpreferencesareintransitive

RegretTheorycanaccountfortheintransitivepreferencesfoundinexperiments

Psychologists-

Peopleusedifferentcognitiveprocessesinvaluationandchoicetasks

Whenvaluingthegambles,respondentssettheamountiftheywinasananchorandthenadjustdownwardstotakeaccountofthefactthattheymaynotwin

Buttheydonotadjustdownsufficientlyandhenceplaceahighervalueonthe$-betWhenchoosingbetweenthebets,theypaymoreattentiontotherelativechancesofwinningandhencechoosetheP-bet

Conclusions

Thedescriptivevalidityof（S）EUThasbeencalledintoquestiontimeandtimeagainfromexperimentsthatinvolverealmonetarypay-offstorealpeopleintherealworld

Butdoesthismatter?