AParameterFreeElicitationofthe ProbabilityWeightingFunctionin MedicalDecisionAnalysis HanBleichrodt JoseLuisPinto iMTA iBMG Erasmus University P
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AParameterFreeElicitationofthe ProbabilityWeightingFunctionin MedicalDecisionAnalysis HanBleichrodt JoseLuisPinto iMTA iBMG Erasmus University P

O Box 1738 3000 DR Rotterdam The Netherlands Department of Economics Universitat Pompeu Fabra Barcelona Spain bleichrodtbmgeurnl josepintoeconupfes nimportantreasonwhypeopleviolateexpectedutilitytheoryisprobabilityweight ingPreviousstudiesontheprobab

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AParameterFreeElicitationofthe ProbabilityWeightingFunctionin MedicalDecisionAnalysis HanBleichrodt JoseLuisPinto iMTA iBMG Erasmus University P




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AParameter-FreeElicitationofthe ProbabilityWeightingFunctionin MedicalDecisionAnalysis HanBleichrodt JoseLuisPinto iMTA iBMG, Erasmus University, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands Department of Economics, Universitat Pompeu Fabra, Barcelona, Spain bleichrodt@bmg.eur.nl jose.pinto@econ.upf.es nimportantreasonwhypeopleviolateexpectedutilitytheoryisprobabilityweight- ing.Previousstudiesontheprobabilityweightingfunctiontypicallyassumeaspecific parametricform$excludeheterogeneityinindividualpreferences$andexclusivelyconsider

monetarydecisionmaking.Thisstudypresentsamethodtoelicittheprobabilityweighting functioninrank-dependentexpectedutilitytheorythatmakesnopriorassumptionsaboutthe functionalformoftheprobabilityweightingfunction.Weusebothaggregateandindividual subjectdata$therebyallowingforheterogeneityofindividualpreferences$andweexamine probabilityweightinginanewdomain$medicaldecisionmaking.Thereissignificantevi- denceofprobabilityweightingbothattheaggregateandattheindividualsubjectlevel.The modalprobabilityweightingfunctionisinverseS-shaped$displayingbothlowersubadditiv-

ityanduppersubadditivity.Probabilityweightingisinparticularrelevantattheboundaries oftheunitinterval.)omparedtostudiesinvolvingmonetaryoutcomes$wegenerallyfind moreelevationoftheprobabilityweightingfunction.Therobustnessoftheempiricalfindings onprobabilityweightingindicatesitsimportance.Ignoringprobabilityweightinginmodeling decisionunderriskandinutilitymeasurementislikelytoleadtodescriptivelyinvalidtheories anddistortedelicitations. Nonexpected Utility Decision Theory Probability Weighting Utility Assessment Medical Decision Making Itisbynowwidelyacknowledgedthatexpectedutility

theoryisnotvalidasadescriptivetheoryofchoiceun- derrisk.Animportantreasonwhypeopleviolateex- pectedutilitytheoryisthattheirpreferencesbetween risky prospects are not linear in probabilities. In re- sponsetotheobservedviolationsofexpectedutility theory$severalnonexpectedutilitytheorieshavebeen proposed.Themostimportanttheoriesamongthese nonexpected utility theories are rank-dependent ex- pectedutilitytheory+.uiggin/01/$2aari/013-and itsderivativecumulativeprospecttheory+Starmerand Sugden /010$ Luce and Fishburn /00/$ Tversky and 4ahneman/005-.Anessentialcharacteristicofthelat-

tertwotheoriesisthatprobabilitiesdonotenterlin- earlyintheevaluationformula$butaretransformed intodecisionweightsthroughacumulativeprobabil- ityweightingfunction.Thenonlinearityofpreferences inprobabilitymakesitpossibletoexplainchoicepat- ternsthatareatvariancewithexpectedutilitytheory. Adisadvantageofusingtransformedprobabilitiesin theevaluationformulaisthattheelicitationorestima- tionofthemodelbecomesmoreinvolved$becausein 6657-/06086689://8/917;67.66 ANA=EMENT )IEN)E 5666INFO?MS /75:-776/electronicISSN Vol.9:$ No.//$November5666 pp./917--/90:
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BLEICHRODT AND PINTO A

Parameter-Free Elicitation of the Probability Weighting Function additiontotheutilityfunction$theprobabilityweight- ingfunctionhastobeelicited. Previousstudiesthatelicitedtheprobabilityweight- ing function in rank-dependent expected utility the- ory or cumulative prospect theory generally have threecharacteristicsincommon.First$theyestimated boththeutilityfunctionandtheprobabilityweight- ing function by parametric techniques +Tversky and 4ahneman/005$)amererandHo/009$Tverskyand Fox /007$ Wu and =onzalez /00:-. Specific inverse S-shaped functional forms were suggested for the

probabilityweightingfunction+Lattimoreetal./005$ Tversky and 4ahneman /005$ Prelec /001-. A dis- advantage of this approach is that the estimations dependcriticallyontheassumedfunctionalform.If thetruefunctionalformisdifferentfromtheassumed functionalform$thenconclusionsdrawnfromtheesti- mationsneednolongerhold.Severalstudies+)urrim and Sarin /010$ Wu and =onzalez /00:- used non- parametric methods to derive qualitative properties oftheprobabilityweightingfunction.However$these nonparametrictechniqueswerenotusedtoestimate quantitativeprobabilityweights.

Second$thesestudieshavebeenbasedonaggregate dataCTheyeitherusedasingle-agentstochasticchoice modelorfittedtheweightingfunctiontothemedian subject. Thereby$ heterogeneity of individual prefer- encesisruledout. Third$ they focused on one specific outcome do- mainCmoney.Littleisknownaboutthegeneralization of their findings to other decision domains. Previ- ousresearchsuggeststhatprobabilityweightingmay depend on the decision context. )urrim and Sarin +/010-$forinstance$arguethattheoutcomelevelmay affect probability weighting. Several studies report evidencethattheshapeoftheprobabilityweighting

functionindecisionunderuncertaintydependsonthe source of the uncertainty +Heath and Tversky /00/$ TverskyandFox/007$4ilkaandWeber/001-.Sim- ilarly$ the outcome domain can affect probability weighting.WakkerandDeneffe+/00:-foundhigher risk aversion for life duration than for money even though utility curvature was similar for these out- comes.Dnderrank-dependentexpectedutilitytheory$ thiscanonlybeexplainedbyadifferenceinprobabil- ityweighting.?ecently$?ottenstreichandHsee+/000- have presented evidence that probability weighting dependsontheoutcomedomain. This article generalizes the

aforementioned stud- ies on the probability weighting function in rank- dependent expected utility theory in three respects. First$ we elicit the probability weighting function withoutmakinganypriorassumptionsaboutitsfunc- tionalform.Thatis$weprovideaparameter-freeelic- itationoftheprobabilityweightingfunction.Second$ we use both aggregate data and individual subject data. Third$ we examine probability weighting in a newdomainCmedicaldecisionmaking. We apply the trade-off method of Wakker and Deneffe +/00:- to elicit first the utility function and then the probability weighting function$ using

the elicited utilities as inputs. Independently from us$ twootherpapershavealsoprovidedparameter-free assessmentsoftheprobabilityweightingfunctionus- ingbothaggregatedataandindividualsubjectdata. =onzalez and Wu +/000- use an alternating least squaresapproachtoestimatetheprobabilityweight- ingfunctionandtheutilityfunctionsimultaneously. Similar to our study$ Abdellaoui +5666- applies the trade-off method to elicit the utility function first and then the probability weighting function$ using the elicited utilities as inputs. Even though he also usesthetrade-offmethod$Abdellaoui sprocedureis

differentfromours.Wecompareourapproachwith Abdellaoui sapproachin 5.Both=onzalezandWu andAbdellaouiusemonetaryoutcomes. In what follows$ / reviews rank-dependent ex- pectedutilitytheoryandempiricalevidenceonprob- ability weighting. Section 5 describes the trade-off method and our procedure to elicit the probabil- ity weighting function. Section E describes the ex- perimental procedures used to elicit utilities and probability weights. The results are described in 9. Section7concludes. 1. Rank-Dependent Expected Utility Theory Let beasetof outcomes $inourcase$lifeduration.We

studydecisionunderrisk$andthereforeassumeaset ofsimpleprobabilitydistributions definedover /91: M ANA=EMENT )IEN)E Vol.9:$No.//$November5666
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BLEICHRODT AND PINTO A Parameter-Free Elicitation of the Probability Weighting Function Atypicalelementof isthelotteryF ::: G$ whichyieldsoutcome withprobability .Here isapositiveintegerand I/.Twolot- teriesF ::: GandF ::: Gthatin- ducethesamerankingofoutcomesare comonotonic .Let ::: Gbealotteryforwhich where standsfor atleastasgoodas .Therank- dependentexpectedutilityofthislotteryisequalto RDEU ::: GI I/ -$ +/- where I/ I/ -$ in

particular -.The probability weighting function isastrictlyin- creasingfunctionfromF6$/GtoF6$/Gwith +6-I6and +/-I/.If istheidentityfunction$rank-dependent expectedutilitytheoryisidenticaltoexpectedutility theory. Severalstudieshaveprovidedpreferenceconditions allowingdifferentshapesoftheprobabilityweighting function+TverskyandWakker/007$Wuand=onza- lez/00:$Wuand=onzalez/001$Prelec/001-.Tversky andWakkerdefinetwoconditionstocharacterizethe probability weighting functionC lower subadditivity and upper subadditivity. Lower subadditivity means thatalowerintervalF6$ Ghasmoreimpactonade-

cisionmakerthananintermediateintervalF G$ providedthat isboundedawayfromone.Alter- nativelystated$lowersubadditivitysaysthatachange fromimpossibletopossiblehasastrongerimpacton an individualJs decision than an equal change from possibletomorepossible.Thiseffectisreferredtoas the possibility effect. Upper subadditivity saysthatanup- perintervalF/ $/Ghasmoreimpactthananinterme- diateintervalF G$providedthat isboundedaway fromzero.Hence$achangefrompossibletocertain hasmoreimpactthananequalchangefrompossibleto morepossible.Thiseffectisreferredtoasthe certainty effect . The effect of lower

subadditivity and upper subadditivity is to produce an inverse S-shaped probability weighting function$ overweighting small probabilities and underweighting intermediate and high probabilities. In the context of rank-dependent expected utility$ the probability weighting function satisfieslowersubadditivityif -$ provided -isboundedawayfromone$andit satisfiesuppersubadditivityif/ -$ provided -isboundedawayfromzero. Severalparametricspecificationsoftheprobability weightingfunctionhavebeensuggestedinthelitera- ture.Tverskyand4ahneman+/005-proposedthefol- lowingone-parameterspecificationC -I H+/

+5- ThisfunctionismonotonicandhasaninverseS-shape forvaluesof between6.53and/. =onzalezandWu+/000-suggestatwo-parameter specification for the inverse S-shaped probability weightingfunction$adoptedbeforeby=oldsteinand Einhorn +/013-$ Lattimore et al. +/005-$ and Tversky andFox+/007-C -I H+/ +E- InEquation+E-$theparameter primarilycontrolscur- vatureoftheprobabilityweightingfunction$i.e.$the extent to which people are able to discriminate be- tweendifferencesinprobability$andtheparameter primarilycontrolselevation$i.e.$theextenttowhich peoplefindthechancedomainattractive.

Prelec+/001-hasaxiomatizedalternativespecifica- tionsfortheinverseS-shapedprobabilityweighting function.Hisproposedone-parameterspecificationis -Iexp+ ln +9- andhistwo-parameterspecificationis -Iexp+ ln +7- Theinterpretationoftheparameters and issimilar toEquation+E-C primarilycontrolscurvatureand primarilycontrolselevation. Table / summarizes the results of some empirical studies that estimated the parameters in the above specifications.)umulativeprospecttheoryallowsthe parameterstobedifferentforgainsandlosses.Allesti- matesareconsistentwithaninverseS-shapedweight- ingfunction. ANA=EMENT

)IEN)E Vol.9:$No.//$November5666 /913
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BLEICHRODT AND PINTO A Parameter-Free Elicitation of the Probability Weighting Function Table 1 Empirical Studies on the Probability Weighting Function Functional Form Parameter Estimates H+ Tversky and Kahneman (1992): = 0.61 (gains), = 0.69 (losses) Camerer and Ho (1994): = 0.56 (gains) Wu and Gonzalez (1996): = 0.71 (gains) Abdellaoui (2000): = 0.60 (gains), = 0.70 (losses) H+ Wu and Gonzalez (1996): = 0.84, = 0.68 (gains) Gonzalez and Wu (1999): = 0.77, = 0.44 (gains) Tversky and Fox (1995): = 0.77, = 0.69 (gains) Abdellaoui (2000): =

0.65, = 0.60 (gains) Abdellaoui (2000): = 0.84, = 0.65 (losses) =exp ln Wu and Gonzalez (1996): = 0.74 (gains) 2. The Trade-Off Method Theelicitationprocedureconsistedoftwoparts.Inthe firstpart$weelicitedtheutilityfunctionforlifedu- ration.Theelicitedutilitieswerethenusedasinputs in the second part to elicit the probability weights. We used the trade-off method for the elicitation of utilities and probability weights +Wakker and Den- effe/00:-.Thismethodwasselectedbecauseitisnot affected by probability weighting. Alternative tech- niquessuchastheprobabilityequivalencemethod$the

certaintyequivalencemethod$andthelotteryequiva- lencemethod+Mc)ordanddeNeufville/01:-suffer fromthedefectthattheyarevulnerabletoprobabil- ityweightinganddonotprovidevalidutilitiesunder rank-dependentexpectedutility. Part 1: Elicitation of the Utility Function Thetrade-offmethoddeterminesastandardsequence ofoutcomes$whichareequallyspacedintermsofutil- ity.Thefirststepconsistsoftheselectionoftworefer- enceoutcomes and with andastartingout- come .Thenanindividualisaskedtospecify such thatheisindifferentbetweenthelotteriesF ,/ andF ,/ G$with and toensurethat thetwolotteriesarecomonotonic.After

hasbeen elicited$theindividualisaskedtospecifythenumber suchthatheisindifferentbetweenF ,/ andF ,/ G.If andrank-dependentex- pectedutilitytheoryholds$thenthefirstindifference yields ,/ ,/ -HF/ -G -HF/ -G -F -G IF/ -GF -G$ +:a- and$similarly$thesecondindifferenceyields ,/ ,/ -F -GIF/ -GF -G +:b- )ombining+:a-and+:b-gives -I +3- Aslongas $wecanproceedtoaskforindiffer- encebetweenF ,/ GandF ,/ G$in theprocesselicitingastandardsequence+ ::: -for which -I -forall/ i;j =iventheuniquenesspropertiesoftheutilityfunction $thescaleandtheoriginofthefunctioncanbechosen arbitrarily.Weusedthescaling -I6and

-I/$ fromwhichitfollowsthatforall/ -I j=k Part 2: Elicitation of the Probability Weighting Function The probability was held constant throughout the elicitationofthestandardsequence.Thisensuredthat theelicitedutilitieswerenotdistortedbyprobability weighting.Equations+:a-and+:b-showthattheterms -and/ -canceloutif isheldconstant.To elicittheprobabilityweightingfunction$theprobabil- itieshavetobevaried.Probabilityweightswerede- terminedbytwotypesofquestions.Forlowproba- bilities$weaskedforanoutcome suchthatthein- dividualisindifferentbetweenF ,/ Gand ,/ Gwith $and ;x $and ele-

mentsofthestandardsequenceelicitedinthefirstpart. Forhigherprobabilities$weaskedforanoutcome /911 M ANA=EMENT )IEN)E Vol.9:$No.//$November5666
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BLEICHRODT AND PINTO A Parameter-Free Elicitation of the Probability Weighting Function suchthatindifferenceholdsbetweenF ,/ andF ,/ Gwith and $and elementsofthestandardsequence.Weexplainbe- lowwhyweuseddifferentquestionsforlowandhigh probabilities. By rank-dependent expected utility$ the weight of probability isdeterminedfromthefirstindifference as -I -GHF -G +1- andfromthesecondindifferenceas -I -GHF -G +0-

Asmentionedintheintroduction$Abdellaoui+5666- alsousedthetrade-offmethodtoelicitfirsttheutility functionandthentheprobabilityweightingfunction withtheelicitedutilitiesasinputs.Hisprocedurecon- sistsoftheselectionofanelement fromtheelicited standardsequenceandtodeterminetheprobability thatmakestheindividualindifferentbetweenthelot- teriesF ,/ GandF/$ G$whereF/$ Gstands for withcertainty.Dnderrank-dependentexpected utilitytheoryandthechosenscaling$thisindifference determinestheweightofprobability as -I +/6- If isdeterminedbyaprobabilitymatchingques- tion$thentheresponsescalediffersbetweenthetwo

partsoftheelicitationprocedureCInthefirstpartthe outcome dimension is used to elicit indifference$ in the second part the probability dimension. Previous studies have shown that different response scales prime different aspects of the decision problem$ a phenomenonreferredtoas scale compatibility +Tversky etal./011-.Scalecompatibilityhasbeenobservedboth in matching and in sequential choice tasks +DelquiK /00E$/003-.Abdellaoui+5666-makesacarefulattempt to avoid the distorting impact of scale compatibility byelicitingindifferencethroughnonsequentialchoice

questions.Noexperimentalevidenceexistsaboutthe impactofscalecompatibilityinnonsequentialchoice tasks. Toavoidthedistortingimpactofchangingresponse scales$weusedonlytheoutcomedimensiontoelicit indifferences.Ourprocedurehasthreepotentialdis- advantages.First$theoutcomes and inEquations +1-and+0-neednotbelongtothestandardsequence$ inwhichcasetheirutilityhastobeestimatedfromthe utilityofelementsofthestandardsequence.Thisap- proximationmayintroducebias.However$theutility functiondoesnotdeviatestronglyfromlinearityover smallintervals+WakkerandDeneffe/00:-$andalin- ear approximation will be

reasonable as long as the standardsequenceissufficientlyfine. Second$ our procedure imposes bounds on the elicited probability weights. Because can never be less than zero$ Equation +1- forces the probabil- ity weights to lie between zero and +F -GH -G-.Hence$theelicitedprobabilityweights are bounded above and$ therefore$ we only used Equation +1- to elicit the weights of lower probabil- ities. The utility of in Equation +0- only be deter- minedif issmallerthan $thefinalelementinthe standard sequence. Therefore application of Equa- tion+0-leadstoprobabilityweightsthatliebetween -- +F/ -GHF

-G-andone. Thatis$theelicitedprobabilityweightsarebounded below$ and we therefore only used Equation +0- to elicittheweightsofhigherprobabilities.The bound- edness problem can be limited by an appropriate choiceoftheelementsofthestandardsequence.If inEquation+1-isrelativelyfarfrom inthestandard sequenceand and arerelativelyclose$thenthe elicitedprobabilityweightcantakeonvaluescloseto one.Similarly$ifinEquation+0- isrelativelyclose to and isrelativelyfarfrom $thentheelicited probability weight can take on values close to zero. In Eweexplainhowwehavehandledthe bound- ednessproblem

inourexperimentaldesign$andin 9wepresentevidencethatitcausednoproblemsin ourdata. Third$ our method may suffer from error propa- gation. Equations +1- and +0- determine probability weightsbyaratio.Errorpropagationforratioscanbe problematicifthedenominatorisclosetozero$sothat smallerrorsinthenumeratorinducelargeerrorsin thequotient.SuchproblemsdonotoccurinEquations +1- and +0- because the denominator is remote from ANA=EMENT )IEN)E Vol.9:$No.//$November5666 /910
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BLEICHRODT AND PINTO A Parameter-Free Elicitation of the Probability Weighting Function

zero$morethanthenumerator.Additionally$thenu- meratorandthedenominatorarepositivelycorrelated becauseofthecommonterm$whichagainreducesthe overallerrorinthequotient.Theseanalyticalobserva- tionssuggestthaterrorpropagationwillnotbedra- maticinourdesign.Toobtaininsightintotheextent to which our procedure is affected by error propa- gation$ we performed two simulation studies based ontwodifferenterrortheories.Thesestudiesarede- scribedin 9.Theirresultsindicatethaterrorpropa- gationisnotaprobleminourstudy. 3. Experiment 3.1. Subjects Priortothemainexperiment$thequestionnairewas

testedinseveralpilotsessions$usinguniversitystaffas subjects.Fifty-onesubjectsparticipatedinthemainex- periment.Allsubjectswereundergraduateeconomics students from the Dniversity of Pompeu Fabra. The subjects were paid 7$666 Pesetas +approximately E6 D.S.dollars-fortheirparticipation.Becauseweused life duration as the outcome domain$ individual re- sponses to the experimental questions could not be playedoutforreal.Thatis$therewerenorealincen- tives in our study. Several studies have argued and presented empirical evidence that hypothetical and real questions give similar results in decision

under risk+Tverskyand4ahneman/005$BeattieandLoomes /003-. In a recent reviewof the literature on the ef- fect of financial incentives in experiments$ )amerer andHogarth+/000-concludethatincentivesappearto helpmostfrequentlyinjudgmentanddecisiontasks thataredifferentfromthetaskweused.Thetrade- offmethodassessespreferencesbetweenlotteries$and forsuchataskrealincentivesdonotseemtoimprove performance. 3.2. Procedures The experiment was carried out in two personal in- terview sessions separated by two weeks. Personal interviewsessionswereusedinanattempttoobtain high-qualitydata. Both sessions

started with an explanation of the trade-offmethod$bothorallyandinwriting.Subjects weretoldthattheysufferedfromoneoftwodiseases$ butthatitwasrightnowunknownfromwhichdis- ease they suffered. The diseases were anonymously labeled A and B to avoid possible framing effects. Subjectswerefurtherinformedthatitisknownfrom previous medical experience that people with the symptomstheydisplayedhaveDiseaseAhalfofthe timeandDiseaseBhalfofthetime.Thereexisttwo treatmentstobeatthesymptoms$buttheeffectiveness of the treatments depends on the disease. The out- comesofthetreatmentswerenumbersofremaining life

duration. Subjects were told that the remaining lifedurationwasspentingoodhealth.Subjectshadto chooseatreatmentbeforeitwasknownwhichdisease theyactuallyhad. Followingtheexplanationofthedecisionproblem$ thesubjectsweregivenapracticequestionandasked to explain their answer. Their explanation indicated whethertheyunderstoodthequestionsandtheexper- imentaltask.Afterwewereconvincedthatasubject understoodthequestions$wemovedontotheactual experiment. Thefirstexperimentalsessionstartedwiththede- terminationofthestandardsequenceforutility.The referenceoutcomes and weresetat77yearsand 97 years$

respectively. We had learned from the pi- lotsessionsthatthesereferencevaluescreatedastan- dardsequenceinwhichtheelementswerefairlyclose together+/to7years-.Outcome wassetequalto zero$thatis$ correspondedtoimmediatedeath.A standard sequence ::: was elicited by deter- miningthenumberofyears I/$ ::: $:forwhich subjectswereindifferentbetweenF $77, Gand $97, G.Theelicitedstandardsequencetracesthe utilityfunctionforlifeduration.Theutilityfunction wasscaledsuchthat -I j= :. Thepilotsessionshadshownthatpeoplefindthe trade-offmethodeasiertoansweriftheyfirstdeter-

minethelifedurationsforwhichoneofthetwotreat- mentsisclearlysuperiorandthenmovetowardsthe indifferencevalue.Wethereforeaskedsubjectsfirstto comparethetreatmentsF $77, GandF $97, for and for I97. All subjects agreed that the treatment F $ 77, G is better than the treat- ment F $ 97, G$ and all but one subject agreed thatthetreatmentF/$97Gisbetterthanthetreatment /906 M ANA=EMENT )IEN)E Vol.9:$No.//$November5666
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BLEICHRODT AND PINTO A Parameter-Free Elicitation of the Probability Weighting Function $ 77, G. Subjects were then told that these

preferencesimplythatthereshouldbeavalueof between and97forwhichtheirpreferencesbe- tweenthetreatmentsswitch.Theywereaskedtode- terminethis switchingvalue bygraduallyincreas- ing startingfrom andbygraduallydecreasing from97yearsuntiltheyarrivedatarangeofvaluesfor whichtheyfoundithardtochoosebetweenthetreat- ments.Fromthisrangeofvalues$subjectswerethen askedtopickthevalueof forwhichtheyconsidered thetreatmentsmostfinelybalanced.Thisprocedureis similartothatofDubourgetal.+/009-. Overthetwosessions$fivequestionswereaskedto elicittheshapeoftheprobabilityweightingfunction.

Weightswereestablishedforthefollowingfiveproba- bilitiesC6./6$6.57$6.76$6.37$and6.06.Theseprobabili- tieswereselectedtoincludebothprobabilitiesthatare typicallyoverweighted+6./6and6.57-andprobabili- tiesthataretypicallyunderweighted+6.76$6.37$and 6.06-accordingtopreviousresearch. The weights for probabilities 6./6$ 6.57$ and 6.76 wereelicitedbyaskingfortheindifferencevalue in the comparison between F ,/ G and F GandbyapplyingEquation+1-.Thisquestion leadstoanupperboundoftheprobabilityweightof 6.37.Thisupperboundisreachedif I6.Wedenote the responses to the questions for I6 /6$ 6 57$ and 6.76by

$and $respectively. The weights for probabilities 6.37 and 6.06 were elicitedbyaskingfortheindifferencevalue inthe comparisonbetweenF ,/ GandF ,/ and by applying Equation +0-. This question leads to a lower bound of the probability weight of 6.57. Thislowerboundisreachedif .Wedenotethe responsestothequestionsfor I6 37and6.06by and $respectively. Theprocedureusedtoelicit through wassimi- lartotheprocedureusedintheelicitationoftheutility function.Subjectswereencouragedtodeterminefirst thevaluesof I/$ ::: $7$forwhichtheyclearlypre- ferredoneofthetreatments$andfinallythevalueof

ThissubjectpreferredF $77, GtoF/ 97G.Hispreferences violate comonotonicity of the lotteries$ and he was excluded fromtheanalysesforthisreason. forwhichtheyconsideredthetwotreatmentsmost finelybalanced.Theutilitiesof through werede- terminedbothundertheassumptionthatutilityislin- earbetweenpointsofthestandardsequence$thelinear approximation$andundertheassumptionthatutility isapowerfunction$ -I .Apowerfunctionwas selectedbecausethisfunctionisfrequentlyusedinthe literature+e.g.$Tverskyand4ahneman/005-andthere existsempiricalsupportforapowerfunctionforlife

duration+Pliskinetal./016$Stiggelboutetal./009-. Thepowerfunctionwasestimatedusingtheelements ofthestandardsequenceandtheircorrespondingutil- itiesasdatainputs. Inthefirstexperimentalsessionweaskedfor and .Theorderofthequestionswasvariedtoavoid ordereffects.Thesecondsessionservedtoelicit and andtorepeatthreequestionsfromthefirstsessionto testtheconsistencyofsubjects answers.Thequestions thatwererepeatedvariedacrosssubjects.Procedures andmethodsinthesecondsessionwereidenticalto thoseinthefirstsession. 4. Results 4.1. Reliability and Consistency Two subjects were excluded from the analysesC one

wasunabletomakeanytrade-offs,theothersubjectJs responses violated comonotonicity---both and exceeded forthissubject+seealsoFootnote/-. The second session responses slightly exceeded the first session responses. The mean difference be- tweenfirstandsecondsessionresponseswas E53 SE I6 /19-.Thedifferencebetweenfirstandsecond sessionresponsesdidnotreachconventionallevelsof significancebyapaired test.+ /60 331$ I6 631-. 4.2. Utility Curvature Aggregate Data. Figure / displays the elicited utility function for life duration. The difference be- tween successive points of the standard sequence +the

step size-$ increases$ leading to a concave utility function. The step size increases gradu- ally and$ hence$ the linear utility function for life durationisagoodapproximationovershortintervals. ANA=EMENT )IEN)E Vol.9:$No.//$November5666 /90/
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BLEICHRODT AND PINTO A Parameter-Free Elicitation of the Probability Weighting Function Figure 1 The Elicited Utility Function for Life Duration Individual Data. ByL wedenotethedifference -$thatis$thedifferencebetween two successive step sizes of the standard sequence. PositiveL correspondstoaconcaveutilityfunction$ zeroL

toalinearutilityfunction$andnegativeL toaconvexutilityfunction.Foreachsubject$weob- servefivevaluesofL .Thereare50+0-subjectswith at least E +9- positive values of L $ /E +7- subjects withatleastE+9-zerovaluesofL $and/+6-subject withatleastE+9-negativevaluesofL .)learly$the modalshapeoftheutilityfunctionisconcave. 4.3. Probability Weighting Boundedness Problem and Violations of Stochas- tic Dominance. Nosubjectreportedeitheravalueof zerointhequestionsinvolving $and $oravalue of in the questions involving and . We con- cludethattheboundednessoftheprobabilityweights causednoproblemsinourdata.

Someresponsesweresuchthattheoutcomesofone treatmentwerebetterunderbothstatesoftheworld and$hence$thistreatmentstochasticallydominatedthe othertreatment.Becauseitisnotplausiblethatanin- dividualisindifferentbetweentwotreatmentswhere onetreatmentstochasticallydominatestheother$these responseswereinterpretedasreflectingconfusionand wereexcludedfromtheanalyses. Figure 2 The Elicited Probability Weighting Function Note The median probability weights are 0.253, 0.357, 0.526, 0.668, and 0.707 under the linear approximation and 0.224, 0.320, 0.462, 0.630, and 0.677 under the power approximation.

Estimation of the Power Utility Function. Apower functionwasestimatedforeachsubjectbasedonthe criterion minimize the sum of squared residuals This function was used in the power approxima- tion. The mean of the individual optimal estimates of the power coefficient is 6.330 +median I6 3:0, SE I6 6/33-$ which is close to 6.39$ the estimate ob- tainedbyStiggelboutetal.+/009-inagroupofcancer patients. The estimated utility functions fit the data verywellCTheoverallproportionofthetotalvariation explainedbythepowerfunction+ -isequalto6.013. 4.3.1. Aggregate Data. Figure5displaysthemean

probabilityweightsunderthelinearandthepowerap- proximation.Thepatternofprobabilityweightsiscon- sistentwithaninverseS-shapedprobabilityweighting functionCSmallprobabilitiesareoverweightedandin- termediateandlargeprobabilitiesareunderweighted. TheonlydeviationfromtheinverseS-shapeisthatthe slopeoftheweightingfunctionbetween6.76and6.37 exceedstheslopebetween6.37and6.06.Thedegreeof uppersubadditivityexceedsthedegreeoflowersub- additivity.)omparedtoboththeparametricandthe nonparametricstudiesusingmonetaryoutcomes$we findmorelowersubadditivityandsimilaruppersub-

additivity.Theexceptionisthenonparametricstudy /905 M ANA=EMENT )IEN)E Vol.9:$No.//$November5666
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BLEICHRODT AND PINTO A Parameter-Free Elicitation of the Probability Weighting Function by=onzalezandWu+/000-$whofindlowerandup- persubadditivitysimilartoours. Theprobabilityweightsunderthepowerapproxi- mationaresmallerthantheweightsunderthelinear approximation. This happens because most subjects haveaconcaveutilityfunctionforlifeduration.Ifutil- ityisconcave$thenthelinearapproximationwillun- derestimatetheutilityof and .Itcanbeverified

fromEquations+1-and+0-thattheprobabilityweight isnegativelyrelatedtotheutilityof $respectively Hence$ifutilityisconcavethelinearapproximation tendstooverestimatetheprobabilityweight. ThenoteunderFigure5showsthemediandata.The useofmedianinsteadofmeandatadoesnotaffectthe conclusions. 4.3.2. Individual Data. Let denotetheaverage slope of the probability weighting function between probabilities and /C Let denotethechangeintheaverageslopebe- tweensuccessiveprobabilities$i.e.$thedifferencebe- tween and .The canbeusedtoexamine theshapeoftheprobabilityweightingfunction.Forex-

ample$concavityoftheprobabilityweightingfunction correspondstoadecreasingslopeand$hence$toneg- ative .Similarly$linearitycorrespondstozero and convexity to positive . An inverse S-shaped probabilityweightingfunctionisconcaveforsmall andconvexforlarger Foreachsubject$weobservedfivevaluesof .A subject sprobabilityweightingfunctionwasclassified aslowersubadditiveif 57 wasnegative.Theproba- bilityweightingfunctionwasuppersubadditiveif waspositive.Theprobabilityweightingfunctionwas concave if at least three were negative and the subjectdidnotexhibituppersubadditivity$linearifat leastthree

werezeroandthesubjectdidnotdis- playbothupperandlowersubadditivity$andconvex ifatleastthree werepositiveandthesubjectdid notexhibitlowersubadditivity. Table 5 shows the classification of individuals ac- cording to the shape of their probability weighting Table 2 Classification of Subjects According to the Slope of Their Probability Weighting Function Proportion of Proportion of Subjects (Linear Subjects (Power Shape Approximation) Approximation) Concave 10.9% 6.5% Linear 0% 0% Convex 4.4% 8.7% Lower Subadditivity 95.7% 91.3% Upper Subadditivity 86.9% 89.1% Lower and Upper Subadditivity 83.7%

81.4% Table 3 Classification of Subjects Based on 0.10 and 0.90 Proportion of Proportion of Subjects (Linear Subjects (Power Shape Approximation) Approximation) Lower Subadditivity 95.8% 91.7% Upper Subadditivity 91.3% 95.7% Lower and Upper Subadditivity 88.9% 88.9% PE Exceeds CE 47.5% 25.0% PE Equal to CE 10.0% 0% CE Exceeds PE 42.5% 75.0% Note . PE stands for probability effect and CE for certainty effect. function.Thetableshowsstrongevidenceforlower anduppersubadditivity.Asmallminorityofsubjects has a concave or convex weighting function and no subjecthasalinearweightingfunction.

Analternativewaytoexaminelowersubadditivity anduppersubadditivityattheindividualsubjectlevel istolookat +6 /6-and/ +6 06-.Asubjectsatisfies lowersubadditivityif +6 /6- /6$anduppersubad- ditivityif/ +6 06- /6.Thistestiscomparableto themetricusedbyTverskyandFox+/007-.Thismetric alsopermitsatestoftherelativesizesofthepossibility effectandthecertaintyeffectbycomparing +6 /6-to +6 06-.Thepossibilityeffectexceedsthecertainty effectif +6 /6- +6 06-. TableE displaystheanalysisbasedontheabove metric. The table confirms both lower and up- per subadditivity. Among the subjects who sat- isfy both lower

and upper subadditivity$ there are slightlymoresubjectsforwhomthepossibilityeffect exceeds the certainty effect than subjects for whom ANA=EMENT )IEN)E Vol.9:$No.//$November5666 /90E
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BLEICHRODT AND PINTO A Parameter-Free Elicitation of the Probability Weighting Function the certainty effect exceeds the possibility effect underthelinearapproximation.Dnderthepowerap- proximationthisconclusionisreversed.Thisreversal in conclusion occurs because for many subjects the possibilityeffectandthecertaintyeffectareapproxi- matelyequal. Linearity of for Intermediate Probabilities.

Thereexistssomecontroversyintheliteratureabout the question of whether the probability weighting functionisnonlinearthroughouttheunitintervalF6$/G orwhethernonlinearitiesoccuronlyattheboundaries of the unit interval +)amerer /005$ Wu and =onza- lez/00:-.Inthelattercase$theprobabilityweighting functionislinear$andhenceconsistentwithexpected utilitytheory$forintermediateprobabilities.Wuand =onzalez+/00:-foundsupportfornonlinearity$with weights becoming less concave throughout the unit interval$ whereas )amerer s +/005- results support linearityawayfromtheboundaries.Abdellaoui+5666-

obtainsmixedresults. Wetestedforlinearityoftheprobabilityweighting functionbyexaminingthe for I6 57$6 76$6 37$ 06.Iftheprobabilityweightingfunctionislinearin theinterioroftheunitinterval$thenthe should beapproximatelyequal.ThenonparametricFriedman testforrepeatedmeasurementswasusedtotestfor equalityofthe .Neitherunderthelinearnorun- derthepowerapproximationcouldthenullhypoth- esis of equality of the be rejected + +E-I: 536$ I6 600and +E-I5 :77$ I6 991forthelinearand powerapproximation$respectively-.Thissuggestslin- earityoftheprobabilityweightingfunctionandthus$

nosystematicdeviationsfromexpectedutilityforin- termediateprobabilities. Parametric Weighting Functions. We estimated foreachsubjecttheoptimalvaluesoftheparameters in Equations +5---+7-. The estimation criterion was theminimizationofthesumofthesquaredresiduals +SS?-$ I/ $where istheelicitedproba- bilityweightand theestimatedprobabilityweight under the parametric specification. To be efficient$ thisestimationcriterionrequirestheerrortermstobe normallyandindependentlydistributed. Table 4 Mean Estimation Results for the Parametric Specifications of the Probability Weighting Function Functional

Form Linear Power Approximation Approximation -I H+ 0.713 0.025 0.674 0.027 -I H+ 0.573 0.041 0.550 0.036 1.127 0.093 0.816 0.035 -I exp ln 0.589 0.037 0.533 0.031 -I exp ln 0.604 0.053 0.534 0.038 0.938 0.038 1.083 0.036 Note. Standard errors appear in parentheses. Table9 showsthemeansoftheindividualoptimal values.Fortheone-parameterspecificationEquation +5-$ourestimatesarecomparabletothoseobtainedfor monetaryoutcomes.Forthetwo-parameterspecifica- tion$wefindrelativelymoreelevation$inparticular under the linear approximation. An explanation for thisfindingmaybethatsubjectsconsideralldurations

in the experiment as losses$ because they fall below subjects lifeexpectancy.Itiswell-knownthatpeople findthechancedomainmoreattractiveforlossesthan forgains+Tverskyand4ahneman/005$Fennemaand Wakker/003-.)omparedtoTverskyandFox+/007-$ Wuand=onzalez+/00:-$andAbdellaoui+5666-$we findmorecurvature.Anexplanationforthisfinding may be that health is a more affect-rich outcome thanmoney.?ottenstreichandHsee+/000-findmore curvature of the probability weighting function for affect-richoutcomes.However$comparedto=onzalez andWu+/000-$wefindlesscurvature. =oodness of fit of the various specifications was

assessedbytakingthemeanoftheindividualsums ofsquaredresidualsadjustedfordegreesoffreedom. Basedonthiscriterion$thetwo-parameterspecifica- tionsEquations+E-and+7-fitthedatabetterthantheir one-parameter counterparts Equations +5- and +9-. Equation+E-leadstoreductionsinthemeanoftheindividual sumsofsquaredresidualsadjustedfordegreesoffreedomof E9.6Nand59.5Nunderthelinearandthepowerapproximation$ respectively$ compared to Equation +5-. Equation +7- leads to reductionsof5:.7Nand//./Nunderthelinearandthepower approximation$respectively$comparedtoEquation+9-. /909 M ANA=EMENT )IEN)E

Vol.9:$No.//$November5666
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BLEICHRODT AND PINTO A Parameter-Free Elicitation of the Probability Weighting Function Thisbetterfitofthetwo-parameterspecificationsholds inspiteofthefactthatontheaggregatelevelboth and arerelativelyclosetooneandarenotsignifi- cantlydifferentfromoneunderthelinearapproxima- tion.Apparently$thetwo-parameterspecificationfits the data better than the one-parameter specification attheindividualsubjectlevel$butonlyslightlybetter attheaggregatelevel.Thisfindingisconsistentwith =onzalezandWu+/000-. 4.3.3. Propagation of Error. Toobtainmoreinsight

intotheeffectoferrorpropagation$weperformedtwo simulationstudiesbasedontwodifferenterrortheo- ries.Inthefirstsimulation$weassumedthatinevalu- atingthetrade-offquestions$thesubjectmakesaner- rorinhisassessmentofutilitydifferences.Thiserror theoryiscomparabletoHeyandOrme+/009-.Weas- sumedthattheresponseerror isaproportionofthe trueutilitydifference$i.e.$theassessedutilitydiffer- enceisequalto+/H -timesthetrueutilitydifference. Inthesecondsimulation$weassumedthatwhilethe subjectcorrectlyassessesutilitydifferences$hemakes an error in reporting his response. This is$ by acci-

dentthesubjectsometimesreportsthewrongindiffer- encevalue.ThiserrormodeliscomparabletoHarless and)amererJs tremblinghand theory+Harlessand )amerer/009-.Weassumedthattheresponseerror isaproportionofthetrueindifferencelifeduration. Thatis$thereportedindifferencelifedurationisequal to+/H -timesthetrueindifferencelifeduration. Theerrortermswereinbothsimulationsassumed tobenormallydistributed$withmean6.66andstan- dard deviation 6.67. The selected value of the stan- darddeviationisnotimportant.Theaimofthesimula- tionexerciseistoshowthaterrorpropagationisnota

probleminourdata$thatis$thatsmallresponseerrors donottranslateintolargeerrorsintheelicitedprob- abilityweights.Forbotherrormodels$weperformed /$666simulations.Dnderbotherrortheories$thereis noindicationthaterrorpropagationisaproblemfor ourelicitationprocedure.Toillustrate$Table7shows thestandarddeviationsoftheerrorsintheaggregate probabilityweightsand$inparentheses$thestandard deviationsoftheerrorsasaproportionoftheproba- bilityweightsunderthelinearapproximation.Ineach Table 5 Results of the Simulation Studies in Which There Is an Error in the Assessed Utility Difference (Model I) or in the

Reported Indifference Life Duration (Model II) Probability Standard Deviation Error Standard Deviation Error Model I Model II 0.10 0.0019 (0.8%) 0.0117 (4.8%) 0.25 0.0023 (0.6%) 0.0092 (2.6%) 0.50 0.0024 (0.5%) 0.0057 (1.1%) 0.75 0.0022 (0.3%) 0.0098 (1.5%) 0.90 0.0019 (0.3%) 0.0128 (1.8%) Notes. Standard deviations of the errors as a proportion of the probability weights are in parentheses. The table shows the error in the aggregate probability weights under the linear approximation. case$thestandarddeviationislessthan6 67$these- lectedsizeoftheresponseerror. 5. Conclusion The main conclusion

of this article is that probabil- ityweightingisrobust.Wefindsignificantevidence of probability weighting both at the aggregate level andattheindividualsubjectlevel.Thepredominant shapeoftheprobabilityweightingfunctionisinverse S-shaped with the point of inflection lying between 57 and 6 76. Probability weighting is particularly strongattheboundariesoftheunitinterval.Hence$ we observe strong support for lower subadditivity anduppersubadditivity+TverskyandWakker/007-. Thesefindingsareconsistentwithstudiesusingmon- etaryoutcomes.)omparedtothesestudies$wefind

moreelevationoftheprobabilityweightingfunction. Wakker and Stiggelbout +/007- have shown how probability weighting can lead to biases in health utilitymeasurement.Wehaveshownthatprobability weightingaffectsmedicaldecisions.Weurgemedical decisionanalyststoincorporateprobabilityweighting intheiranalyses.Themethodoutlinedinthisarticle canbeusedtoelicittheprobabilityweightingfunc- tion for individual patients. For societal evaluations where elicitation of the probability weighting func- tionisoftennotfeasible$theparametricspecifications Equations+5---+7-withtheparameterselicitedinthis

studycanbeused. Therobustnessoftheempiricalfindingsonprob- ability weighting indicates its importance. Ignoring ANA=EMENT )IEN)E Vol.9:$No.//$November5666 /907
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BLEICHRODT AND PINTO A Parameter-Free Elicitation of the Probability Weighting Function probabilityweightinginmodelingdecisionunderrisk andinutilitymeasurementislikelytoleadtodescrip- tivelyinvalidtheoriesanddistortedelicitations. WearegratefultoPeterWakker$anassociateeditor$andtwo referees for their helpful comments on previous drafts. Han BleichrodtJsresearchwasmadepossiblebyafellowshipofthe

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authors 55 weeks for revisions. /90: M ANA=EMENT )IEN)E Vol.9:$No.//$November5666