Tversky - Ambiguity and Competence

Journalof RiskandUncertainty,4:5-28(1991) 1991 KluwerAcademicPublishers

PreferenceandBelief:AmbiguityandCompetencein ChoiceunderUncertainty

CHIP HEATHStanfordUniversity

AMOS TVERSKYDepartmentofPsychology,StanfordUniversity, JordanHall, Bldg. 420, Stanfor4CA 94305-2130

Key words: ambiguity,uncertainty,preferences

Abstracf

We investigatethe relation betweenjudgmentsof probability and preferencesbetweenbets.A seriesofexperimentsprovidessupportfor thecompetencehypothesisthatpeoplepreferbettingon theirownjudgmentover anequiprobablechanceeventwhentheyconsiderthemselvesknowledgeable,but not otherwise.Theyevenpay a signilicant premiumto beton theirjudgments.Thesedatacannotbe explainedby aversiontoambiguity,becausejudgmentalprobabilitiesaremore ambiguousthanchanceevents.We interprettheresultsintermsoftheattributionofcreditandblame.Thepossibilityof inferringbeliefs frompreferencesis questioned.

Theuncertaintyweencounterin theworld is notreadilyquantified.Wemayfeelthatour favorite football teamhasa goodchanceto win thechampionshipmatch,that thepriceof goldwill probablygo up, andthat the incumbentmayoris unlikely to be re-elected,butweare normallyreluctantto assignnumericalprobabilitiesto theseevents.However,to facilitatecommunicationandenhancethe analysisof choice,it is oftendesirableto quantify uncertainty.Themostcommonprocedurefor quantifyinguncer-tainty involvesexpressingbeliefin the languageof chance.Whenwesaythatthe proba-bility of anuncertaineventis 30%, forexample,weexpressthebeliefthatthis eventisasprobableasthe drawingof a redball from abox thatcontains30 redand70 greenballs.An altemativeprocedureformeasuringsubjectiveprobabilityseeksto inferthe degreeof belief frompreferenceviaexpectedutility theory. Thisapproach,pioneeredby Ram-sey (1931) and further developedby Savage(1954) and by AnscombeandAumann(1963),derivessubjectiveprobability from preferencesbetweenbets.Specifically, thesubjectiveprobability of an uncertaineventF is said to bep if the decisionmakeris

This work wassupportedby Grant89-0064from TheAir ForceOffice of Scientific Researchto StanfordUniversity.Fundingfor experiment1 wasprovidedby SES8420240to RayBattalio. We havebenefitedfromdiscussionswith Max Bazerman,Daniel Ellsberg,RichardGonzales,RobinHogarth, LindaGinzel,DanielKahneman,andEldarShafir.

6 CHIP HEATH AMOS TVERSKY

indifferentbetweentheprospectof receiving$x if F occurs(andnothingotherwise)andtheprospectof receiving$x if a redball isdrawnfrom aboxthatcontainsaproportionpof redballs.

The Ramseyschemefor measuringbelief and the theoryon which it is basedwerechallengedby Daniel Ellsberg(1961;seealso Fellner,1961) who constructedacompel-ling demonstrationof whathascometobecalledanambiguityeffect, althoughthe termvaguenessmaybe moreappropriate.Thesimplestdemonstrationof this effect involvestwo boxes:onecontains50 redballsand50greenballs,whereasthesecondcontains100redandgreenballsin unknownproportion.Youdrawaball blindly from aboxandguessitscolor. If your guessis correct,you win $20;otherwiseyou get nothing.On which boxwould you ratherbet?Ellsbergarguedthatpeoplepreferto beton the50/50box ratherthanon thebox with the unknowncomposition,eventhoughtheyhaveno colorprefer-encesand so are indifferent betweenbettingon redor on greenin eitherbox. Thispatternof preferences,which was later confirmed in many experiments,violatestheadditivityof subjectiveprobabilitybecauseit impliesthat thesumof the probabilitiesofredandof greenishigherin the50/50box thanin the unknownbox.

Ellsbergsworkhasgenerateda greatdealof interestfor two reasons.First,it providesaninstructivecounterexampleto (subjective)expectedutility theorywithin the contextof gamesof chance.Second,it suggestsa generalhypothesisthatpeopleprefertobetonclearratherthanon vagueevents,at leastfor moderateandhigh probability.Forsmallprobability, Ellsbergsuggested,peoplemay prefervaguenessto clarity. Theseobserva-tions presenta seriousproblem for expectedutility theoryandothermodelsof riskychoicebecause,with thenotableexceptionof gamesof chance,mostdecisionsin the realworld dependon uncertaineventswhoseprobabilitiescannotbe precisely assessed.Ifpeopleschoicesdependnotonly on thedegreeof uncertaintybutalsoon theprecisionwith which it can be assessed,then the applicability of the standardmodelsof riskychoice is severelylimited. Indeed,severalauthorshaveextendedthestandardtheorybyinvoking nonadditive measuresof belief (e.g., Fishburn, 1988; Schmeidler,1989) orsecond-orderprobabilitydistributions(e.g., GardenforsandSahlin, 1982;Skyrm, 1980)inorderto accountfor the effectof ambiguity.Thenormativestatusof thesemodelsis asubjectof lively debate.Severalauthors,notablyEllsberg(1963),maintainthataversionto ambiguitycanbe justifiedon normativegrounds,althoughRaiffa (1961)hasshownthat it leadsto incoherence.

Ellsbergs example,and most of the subsequentexperimentalresearchon the re-sponseto ambiguityor vagueness,wereconfinedto chanceprocesses,suchasdrawingaball from a box, or problemsinwhich the decisionmakeris providedwith a probabilityestimate.The potentialsignificanceof ambiguity, however,stemsfrom its relevancetothe evaluationof evidencein the realworld. Is ambiguityaversionlimited to gamesofchanceandstatedprobabilities,or doesit also hold for judgmentalprobabilities?Wefound no answerto this questionin the literature,but thereis evidencethat castssomedoubton thegeneralityof ambiguityaversion.

Forexample,Budescu,Weinberg,andWallsten(1988)comparedthecashequivalentsgiven by subjectsfor gambleswhoseprobabilitieswereexpressednumerically,graphi-cally, or verbally. In the graphical display,probabilitieswere presentedas the shaded

7PREFERENCEAND BELIEF

areaof a circle. In theverbalform,probabilitiesweredescribedby expressionssuchasvery likely or highly improbable.Becausethe verbalandthe graphicalforms aremoreambiguousthan the numericalform, ambiguityaversionimplies a preferenceforthe numericaldisplay.This predictionwasnot confirmed.Subjectspricedthe gamblesroughlythesamein all threedisplays.In a differentexperimentalparadigm,CohenandHansel(1959) and Howell (1971) investigatedsubjects choicesbetweencompoundgamblesinvolvingbothskill andchancecomponents.For example,in the latterexperi-mentthe subjecthadto hit a targetwith a dart (wherethe subjectsshit rateequaled75%)aswell asspina roulettewheelsothat it would landon a markedsectioncompos-ing 40% of the area.Successinvolvesa 75% skill componentand40% chancecompo-nentwith anoverall probabilityof winningof .75 x .4 .3. Howellvariedthe skill andchancecomponentsof thegambles,holdingthe overallprobabilityof winningconstant.Becausethe chancelevel was known to the subjectwhereasthe skill level was not,ambiguityaversionimpliesthatsubjectswould shift asmuchuncertaintyaspossibletothe chancecomponentof the gamble.In contrast,87% of the choicesreflect aprefer-encefor skill overchance.CohenandHansel(1959)obtainedessentiallythesameresult.

1. The competencehypothesis

The precedingobservationssuggestthattheaversionto ambiguityobservedin a chancesetup(involving aleatoryuncertainty)doesnot readilyextendto judgmentalproblems(involvingespistemicuncertainty).In this article,weinvestigateanalternativeaccountofuncertaintypreferences,calledthecompetencehypothesis,whichappliestobothchanceandevidential problems.We submit that the willingnessto bet on an uncertaineventdependsnot only on the estimatedlikelihood of that eventandthe precisionof thatestimate;it also dependson onesgeneralknowledgeor understandingof the relevantcontext.Morespecifically,we proposethatholdingjudgedprobabilityconstantpeo-ple preferto bet in a contextwheretheyconsiderthemselvesknowledgeableor compe-tent than in a contextwhere they feel ignorantor uninformed.We assumethat ourfeelingof competencetin a givencontextisdeterminedby whatweknowrelativetowhatcanbe known.Thus, it is enhancedby generalknowledge,familiarity, andexperience,and is diminished,for example,by calling attentionto relevantinformation that is notavailableto the decisionmaker,especiallyif it is availabletoothers.

Therearebothcognitive andmotivationalexplanationsfor thecompetencehypothe-sis. Peoplemay havelearnedfrom lifelong experiencethat they generallydo betterinsituationsthey understandthan in situationswherethey havelessknowledge.Thisex-pectationmaycarryovertosituationswherethechancesof winningarenolongerhigherin the familiar thanin theunfamiliarcontext.Perhapsthe majorreasonfor thecompe-tencehypothesisis motivational rather than cognitive. We proposethat the conse-quencesof eachbetinclude,besidesthemonetarypayoffs,thecreditor blameassociatedwith the outcome.Psychicpayoffs of satisfactionor embarrassmentcan result fromself-evaluationor fromanevaluationby others.In eithercase,thecreditand theblame

8 CHIP HEATH/AMOS TVERSKY

associatedwith an outcomedepend,we suggest,on the attributionsfor successandfailure. In thedomainofchance,bothsuccessandfailure areattributedprimarily toluck.The situationis differentwhena personbetson his or herjudgment.If the decisionmakerhaslimited understandingof the problemat hand,failure will be attributedtoignorance,whereassuccessis likely tobeattributedtochance.In cnntrnst if the-decis-ionmakeris an expert, successis attributableto knowledge,whereasfailure cansome-timesbe attributedtochance.

Wedo notwishto denythat in situationswhereexpertsare supposedto know all thefacts, they are probablymore embarrassedby failure than arenovices. However, insituationsthatcall for an educatedguess,expertsare sometimesless vulnerablethannovicesbecausetheycanbetterjustify theirbets,evenif theydonotwin. Inbettingonthewtnnerof a football game,for example,peoplewho considerthemselvesexpertscanclaim creditfor acorrectpredictionandtreatanincorrectprediction.a~an~tpset.Peoplewho donotknow muchaboutfootball,ontheotherhand,cannotclaimmuchcreditfor acorrectprediction (becausethey are guessing),andthey are exposedto blamefor anincorrectprediction(becausetheyare ignorant).

Competenceor expertise,therefore,helpspeopletakecreditwhentheysucceedandsometimesprovidesprotectionagainstblamewhenthey fail. Ignoranceor incompe-tence,on the otherhand,preventspeoplefrom takingcredit for successandexposesthemto blamein caseoffailure. Asa consequence,wepropose,thebalanceof credittoblame is most favorablefor bets in ones areaof expertise,intermediatefor chanceevents,and leastfavorablefor bets in an areawhereonehasonly limited knowledge.This accountprovidesan explanationof the competencehypothesisin termsof theasymmetryof creditandblameinducedby knowledgeorcompetence.

TheprecedinganalysisreadilyappliestoEllsbergsexample.Peopledo not like tobeton theunknownbox,wesuggest,becausethereis information,namelytheproportion.ofredandgreenballsin thebox, that isknowablein principlebutunknownto them.Thepresenceof suchdatamakespeoplefeel lessknowledgeableand less competentandreducesthe attractivenessof the correspondingbet.A closelyrelatedinterpretationofEllsbergsexamplehasbeenofferedby Frisch andBaron(1988).The competencehy-pothesisis alsoconsistentwith thefinding of Curley,Yates,andAbrams(1986)that theaversiontoambiguity isenhancedby anticipationthat thecontentsof theunknownboxwill beshowntoothers.

Essentiallythesameanalysisappliestothepreferenceforbettingon the futureratherthanon thepast.RothbartandSnyder(1970)askedsubjectsto roll a die andbetontheoutcomeeitherbeforethediewasrolledor afterthe diewasrolledbutheforetheresultwasrevealed.The subjectswho predictedthe outcomebefore the die wasrolled ex-pressedgreaterconfidencein their guessesthanthe subjectswhopredictedtheoutcorneafterthe die roll (postdiction). The formergroupalsobetsignificantly moremoneythanthelattergroup.Theauthorsattributedthis phenomenontnmagicalthinkingortheillusionof control,namelythebeliefthatonecanexercisesomecontrolovertheoutcomebefore,butnotafter,the roll of thedie.However, thepreferencetobetonfutureratherthanpasteventsis observedevenwhenthe illusion of controldoesnotprovideaplausi-bleexplanation,asillustratedby thefollowing problemin whichsubjectswerepresentedwith a choicebetweenthetwo bets:

PREFERENCEAND BELIEF 9

1. A stockisselectedat randomfromthe WallSweetJournal.Youguesswhetherit will goup or downtomorrow.If youre right, you win $5.

2. A stockis selectedat randomfrom the WallSfreetJournal.Youguesswhetherit wentup or downyesterday.You cannotcheckthepaper.If youre right, youwin $5.

Sixty-sevenpercentof the subjects(N = 184) preferredtobeton tomorrowsclosingprice. (Tenpercentof theparticipants,selectedat random,actuallyplayedtheir chosenbet.) Becausethe past,unlike the future, is knowablein principle, but not to them,subjectspreferthefuturebetwheretheir relativeignoranceislower. Similarly, BrunandTeigen(1990)observedthatsubjectspreferredto guesstheresultof a die roll, thesexofa child,or theoutcomeof asoccergamebeforetheeventratherthanafterward.Mostofthe subjectsfoundguessingbeforethe eventmoresatisfactory if right andless un-comfortableif wrong. In prediction,only thefuturecanproveyouwrong; inpostdiction,youcouldbewrongrightnow. Thesameargumentappliesto Ellsbergsproblem.In the50/50box, aguesscouldturnouttobewrongonlyafterdrawingtheball. in theunknownbox,ontheotherhand,theguessmay turnoutto bemistakenevenbeforethedrawingoftheballif it turnsoutthat themajorityof ballsin theboxare oftheoppositecolor. It isnoteworthythat thepreferenceto bet on future ratherthanon pasteventscannotbeexplainedin termsof ambiguitybecause,in theseproblems,thefutureisasambiguousasthepast.

Simple chanceevents,suchas drawinga ball from a box with a known compositiontuvolveno ambiguity;thechancesof winningareknownprecisely.If bettingpreferencesbetweenequiprobableeventsaredeterminedby ambiguity,peopleshouldprefertobeton chanceovertheirown vaguejudgments(atleastformoderateandhighprobability).In contrast,the attributionalanalysisdescribedaboveimplies that peoplewill preferbettingon their judgmentovera matchedchanceeventwhentheyfeel knowledgeableandcompetent,butnototherwise.Thispredictionis confirmedby thefindingthatpeo-ple preferbettingon their skill ratherthan on chance.It is also consistentwith theobservationof MarchandShapira(1987) thatmanytopmanagers,who consistentlybetonhighly uncertainbusinesspropositions,resisttheanalogybetweenbusinessdecisionsandgamesof chance.

We havearguedthat thepresentattributionalanalysiscanaccountfor the availableevidenceon uncertaintypreferences,whetheror nottheyinvolve ambiguity.Thesein-clude1) thepreferencefor bettingon the knownratherthanon the unknownbox inEllsbergsproblem,2) thepreferenceto beton futureratherthanon pastevents,and3)thepreferenceforbettingon skill ratherthanon chance.Furthermore,thecompetencehypothesisimpliesa choicejudgmentdiscrepancy,namelya preferencetobetonA ratherthanon B eventhoughB is judgedtobeat leastasprobableasA. In thefollowingseriesof experiments,we testthecompetencehypothesisandinvestigatethechoicejudgmentdiscrepancy.In experiment1 weofferpeoplethechoicebetweenbettingon theirjudgedprobabilitiesforgeneralknowledgeitemsor on a matchedchancelottery.Experiments2and3 extendthe testby studyingreal-worldeventsandelicitinganindependentassess-mentof knowledge.In experiment4,wesort subjectsaccordingtotheir areaof expertiseandcomparetheirwillingnessto betontheir expertcategory,a nonexpertcategory,andchance.Finally, in experiment5,wetestthecompetencehypothesisin apricingtaskthat

CHIP HEATH /AMOS TVERSKY10

doesnot involve probabilityjudgment.Therelationsbetweenbeliefandpreferencearediscussedin thelast sectionof the article.

1.1. Firpenmenf1: Bettingon knowledge

Subjectsanswered30 knowledgequestionsin two differentcategories,suchashistory,geography,or sports.Fouralternativeanswerswerepresentedforeachquestion,andthesubjectfirst selecteda singleanswerand thenratedhisor herconfidencein thatansweron a scalefrom 25% (pureguessing)to 100% (absolutecertainty).Participantsweregiven detailedinstructionsaboutthe useof the scaleand the notion of calibration.Specifically,theywereinstructedtousethescalesothata confidenceratingof 60%,say,would correspondto a hit rateof 60%.Theywerealsotold thattheseratingswould bethe basisfor a money-makinggame,andwarnedthatboth underconfidenceandover-confidencewould reducetheir earnings.

After answeringthequestionsandassessingconfidence,subjectsweregivenan-oppor-tunity tochoosebetweenbettingontheir answersor ona lottery inwhich theprobabilityof winning wasequal to their statedconfidence.For aconfidenceratingof 75%, forexample,thesubjectwasgiven thechoicebetween1) bettingthathisor heranswerwascorrect,or 2)bettingon a 75%lottery,definedby drawinga numberedchip in therange175 froma bagfilled with 100numberedpokerchips.Forhalfof thequestions,lotteriesweredirectlyequatedtoconfidenceratings.Fortheotherhalf of the questions,subjectschosebetweenthe complementof their answer(betting thatan answerotherthan theonethey chooseis correct)or the complementof their confidencerating.Thus, if asubjectchoseanswerA with confidenceof 65%, the subjectcould choosebetweenbettingthatoneof theremaininganswersB, C, or D iscorrect,or bettingon a 100% 65% = 35%lottery.

Two groupsof subjectsparticipatedin theexperiment.Onegroup(N 29) includedpsychologystudentswho receivedcoursecreditfor participation.The secondgroup(N= 26) wasrecruitedfromintroductoryeconomicsclassesandperformedtheexpenmentfor cashearnings.To determinethe subjectspayoffs, ten questionswereselectedatrandom,and the subjectsplayedout the bets they had chosen.If subjectschosetogambleon theiranswer,theycollected$1.50if their answerwascorrect.If subjectschoseto bet on the chancelottery, they drewa chip from the bagandcollected $1.50if thenumberon thechip fell in the properrange.Averageearningsfor the experimentwerearound$8.50.

Paid subjectstook more time thanunpaid subjectsin selectingtheir answersandassessingconfidence;theywereslightly moreaccurate.Both groupsexhibitedoverconfi-dence:the paid subjectsansweredcorrectly 47% of the questionsand their averageconfidencewas60%.Theunpaidsubjectsansweredcorrectly43%of thequestionsandtheir averageconfidencewas53%.Wefirst describethe resultsof the simple lotteries;thecomplementary(disjunctive)lotteriesarediscussedlater.

The resultsaresummarizedby plotting thepercentageof choices(C) that favorthejudgmentbet overthe lottery as a functionof judgedprobability(F). Before discussing


the actualdata,it is instructiveto examineseveralconstrastingpredictions,implied byfive alternativehypotheses,which are displayedin figure 1.

Theupperpanelof figure1 displaysthepredictionsof threehypothesesinwhichC isindependentof P. According to expectedutility theory, decisionmakerswill be indif-ferentbetweenbettingontheirjudgmentorbettingon a chancelottery; henceC shouldequal50% throughout.Ambiguity aversionimpliesthat peoplewill preferto beton achanceeventwhoseprobabilityis well definedratherthanon theirjudgedprobability,which is inevitably vague;henceC shouldfall below 50% everywhere.The oppositehypothesis,calledchanceaversion,predictsthatpeoplewill prefertobeton their judg-mentratherthanon a matchedchancelottery; henceCshouldexceed50%for all P. Incontrastto the flat predictionsdisplayedin the upperpanel, thetwo hypothesisin thelowerpanelimply thatC dependson P. The regressionhypothesisstatesthat the deci-sion weights,which control choice,will be regressiverelativeto statedprobabilities.Thus,Cwill berelativelyhighfor smallprobabilitiesandrelativelylow forhighprobabil-ities. This predictionalso follows from the theoryproposedby Einhorn and Hogarth(1985),who put forth a particularprocessmodel basedon mentalsimulation,adjust-ment,andanchoring.Thepredictionsof this model,however,coincidewith the regres-sionhypothesis.

Finally, the competencehypothesisimpliesthatpeoplewill tend tobeton theirjudg-mentwhentheyfeelknowledgeableandon thechancelotterywhentheyfeelignorant.

0) 100~o ____________________________________ Chance

0)0) 75 Aversion

EC.) ~ 50 Expected

ushity~0)

o ~ 25AmbiguityAversion

25 50 75 100

Judged Probability (P)

~ 1000)o Competence

75 Hypothesis

o ~ 25Regression

O 0 _____________________________________ Hypothesis

10025 50 75


Figure]. Fivecontrastingpredictionsof theresultsofanuncertaintypreferenceexperiment.

12 CHIP HEATH/AMOSTVERSKY

Becausehigherstatedprobability generallyentailshigherknowledge,C will bean in-creasingfunctionofFexceptat 100%wherethe chancelotteryamountsto a surething.

Theresultsof theexperimentaresummarizedin table1 andfigure2. Table1 presents,forthreedifferentrangesof P, thepercentageof paidandnonpaidsubjectswho beton theiranswersratherthan on the matchedlottery. Recall that eachquestionhad four possibleanswers,sothe lowestconfidencelevel is25%.Figure2 displaystheoverallpercentageofchoicesCthat favoredthejudgmentbetoverthe lottery asa function2 of judgedproba-bility P. The graphshowsthat subjectschosethe lottery whenP was low or moderate(below65%)andthat theychosetobeton theiranswerswhenPwashigh:Thepatternofresultswasthe samefor thepaidandfor thenonpaidsubjects,but theeffectwasslightlystrongerfor the lattergroup. Theseresultsconfirm thepredictionof the competencehypothesisandrejectthe four alternativeaccounts,notablythe ambiguityaversionhy-pothesisimpliedby second-orderprobabilitymodels(e.g.,GardenforsandSablin,1982),andtheregressionhypothesisimpliedby the modelof EinhornandHogarth(1985).

To obtaina statisticaltestof thecompetencehypothesis,wecomputed,separatelyforeachsubject,the binary correlation coefficient (~) betweenchoice (judgmentbet vs.lottery) andjudgedprobability(abovemedianvs.belowmedian).Themedianjudgmentwas.65. Seventy-twopercentof thesubjectsyieldedpositivecoefficients,andtheaverage~ was.30, (t(54) = 4.3,p < .01).To investigatetherobustnessof the observedpattern,we replicatedthe experimentwith one majorchange.Insteadof constructingchancelotterieswhoseprobabilitiesmatchedthevaluesstatedby the subjects,we constructedlotteriesin which theprobabilityof winningwaseither6% higheror 6% lowerthan thesubjectsjudgedprobability. Forhigh-knowledgequestions(P = 75%), the majorityofresponses(70%)favoredthejudgmentbetoverthe lottery evenwhen-the-iotte~y-offereda (6%) higherprobabilityof winning.Similarly, for low-confidencequestions(P =50%)themajorityof responses(52%)favoredthelotteryoverthejudgmentbetevenwhentheformeroffereda lower(6%)probabilityof winning.

Figure3 presentsthecalibrationcurve for thedataof experiment1. Thefigureshowsthat,on thewhole,peopleare reasonablywell calibratedfor low probabilitybutexhibitsubstantialoverconfidencefor high probability. The preferencefor thejudgmentbetoverthe lottery for highprobability,therefore,cannotbejustifiedon anactuarialbasis.

Theanalysisof thecomplementarybets,wheresubjectswereaskedin effecttobetthattheir chosenanswerwasincorrect,revealedaverydifferentpattern.Acrosssubjects,thejudgmentbetwasfavored40.5%of the time, indicating a statisticallysignificantprefer-encefor the chancelottery (t(54) = 3.8p < .01).Furthermore,wefoundno systematic

TableI. Percentageof paid and nonpaidsubjectswho preferred thejudgmentbetover the lottery for low,medium,andhighP(the numberof observationsare givenin parentheses)

25=P=50 50

PREFERENCEANDBELIEF 13

80

75

70

65

60

0)C-

)

o 50

(-) 45~ 40

3530

25

20

20 25 30 35 4045 50 55 6065 70 7580 859095100


Figure2. Percentageof choice(C) that favorajudgmentbetover a matchedlotteryasa functionof judgedprobability(P) in experiment1.

relationbetweenC andP, in markedcontrastto the monotonicrelationdisplayedinfigure2. In accordwith ourattributionalaccount,this resultsuggeststhatpeoplepreferto bet on their beliefsratherthan againstthem.Thesedata, however,may alsobeexplainedby thehypothesisthatpeopleprefertobetonsimpleratherthanon disjunctivehypotheses.

1.2 Experiment2: Football andpolitics

Ournextexperimentdiffers from thepreviousonein threerespects.First,it concernsthepredictionof real-worldfutureeventsratherthanthe assessmentof generalknowledge.Second,it dealswith binaryeventssothatthelowestlevel of confidenceis .5 ratherthan.25 asin thepreviousexperiment.Third, in additiontojudgmentsof probability,subjectsalsoratedtheir level of knowledgefor eachprediction.

A groupof 20studentspredictedtheoutcomesof 14 footballgameseachweekfor fiveconsecutiveweeks.Foreachgame,subjectsselectedthe teamthat theythoughtwouldwin the gameandassessedthe probability of their chosenteamwinning.The subjectsalso assessed,on a five-point scale,their knowledgeabouteachgame.Following therating,subjectswereaskedwhethertheypreferredto beton theteam-they-choseor-onamatchedchancelottery.Theresultssummarizedin figure4 confirmthepreviousfinding.

CHIP HEATH !AMOS TVERSKY14

100~ L90 K8580

~. 65~a~ 60rU

.~ 55-

40

35

3025 -

20

20 2530 354045 50 556065707580859095100

Judged Probabflity (P)

Figure3. Calibrationcurvefor experiment1.

For bothhigh andlow knowledge(definedby a mediansplit on the knowledgeratingscale),C wasan increasingfunction of P. Moreover,C wasgreaterfor highknowledgethan for low knowledgeat any P > .5. Only 5% of the subjectsproducednegativecorrelationsbetweenC andF, andtheaveragek coefficientwas.33, (t(77) = 8.7,~,< .01).

We next took the competencehypothesisto the floor of the RepublicanNationalConventionin New Orleansduring Augustof 1988.The participantswerevolunteerworkersat the convention.Theyweregiven a one-pagequestionnairethat containedinstructionsandan answersheet.Thirteenstateswereselectedto representa crosssectionof different geographicalareasaswell to include the most importantstatesintermsof electoralvotes.Theparticipants(N 100)ratestheprobabilityof Bushcarryingeachof the 13 statesin the November1988electionon ascalefrom0 (Bushis certaintolose)to 100 (Bushiscertaintowin).As in thefootballexperiment,theparticipantsratedtheir knowledgeof eachstateon a five-point scaleand indicatedwhetherthey wouldratherbeton their predictionor on achancelottery.Theresults,summarizedin figures,show that C increasedwith P forboth levels of knowledge,andthat Cwas greaterforhighknowledgethanfor low knowledgeat all levelsof P. Whenaskedabouttheir homestate,70%of the participantsselectedthejudgmentbetoverthe lottery.Only 5%of thesubjectsyieldednegativecorrelationsbetweenC andF, andthe average4 coefficientwas.42, (t(99) = l3.4,p < .01).

The resultsdisplayedin figures4 and5 support the competencehypothesisin thepredictionof real-worldevents:in both tasksC increaseswithF, asin experiment1. Inthat study, however,probability andknowledgewere perfectlycorrelated;hencethechoicejudgmentdiscrepancycouldbe attributedto a distortionof the-probabilityscale


100

90

80

70

60-

ao 50-0

40-

30 -

20 -

10 -

0

50 60 70 80 90 100

Judged Probab~lity (P)

Figure 4. Percentageof choices(C) that favora judgmentbetovera matchedlotteryas a functionofjudgedprobability(F), for high- andlow-knowledgeitemsin thefootball predictiontask(experiment2).

in the judgementtask.This explanationdoesnot apply to the resultsof the presentexperiment,which exhibitsan independenteffectof ratedknowledge.As seenin figures4 and 5, the preferencefor the judgmentbet over the chancelottery is greaterforhigh-knowledgeitemsthanfor low-knowledgeitemsfor all levelsofjudgedprobability,itis noteworthythat the strategyof betting on judgmentwas less successfulthan thestrategyof bettingon chancein bothdatasets.Theformer strategyyieldedhit ratesof64%and78%for football andelection,respectively,whereasthe latterstrategyyieldedhit ratesof 73%and80%.Theobservedtendencyto selectthejudgmentbet,therefore,doesnotyield betterperformance.

1.3. Experiment3: Longshots

Theprecedingexperimentsshowthatpeopleoftenprefertobeton theirjudgmentthanon a matchedchanceevent,eventhoughthe formeris moreambiguouisthanthelatter.This effect, summarizedin figures 2, 4 and 5, was observedat the high end of theprobability scale.Thesedatacould perhapsbe explainedby the simplehypothesisthatpeoplepreferthe judgmentbet when the probability of winning exceeds.5 andthechancelottery whenthe probabilityof winning is below .5. To test this hypothesis,we

Low Knowledge

16 CHIPHEATH /AM OSTVERSKY

100 -

90 -

80 - High Knowledge

70 -Low Knowledge

60-C

)

w-~ 50-C-)

40-

30 -

20 -

10

0

50 60 70.. 80 90 100


Figure5. Percentageof choices(C) that favor ajudgmentbetover a matchedlotteryasa functionof judgedprobability (F), for high- andlow-knowledgeitemsinexperiment2 (electiondata).

soughthigh-knowledgeitemsin which theprobabilityof winning is low, sothesubjectsbestguessisunlikely to be true.In this case,the abovehypothesisimpliesa preferencefor thechancelottery,whereasthecompetencehypothesisimplies a preferencefor thejudgmentbet.Thesepredictionsaretestedin the following experiment.

Onehundredandeightstudentswerepresentedwith open-ended.questionsabout12futureevents(e.g.,whatmoviewill win thisyearsOscarforbestpicture?Whatfootballteamwill win thenextSuperBowl? In whatclassnextquarterwill you havethehighestgrade?).Theywereaskedto answereachquestion,to estimatethe chancesthat theirguesswill turn outto be correct,and to indicatewhetherthey havehigh or low knowl-edgeof the relevantdomain.The useof open-endedquestionseliminatesthe lowerboundof 50% imposedby the useof dichotomouspredictionsin the previousexperi-ment.After thesubjectscompletedthesetasks,theywereaskedto consider,separatelyfor eachquestion,whethertheywould ratherbet on their predictionor on a matchedchancelottery.

On average,the subjectsanswered10 out of 12 questions.Table2 presentstheper-centage(C) of responsesthat favor thejudgmentbet over thechancelottery for high-andlow-knowledgeitems,andforjudgedprobabilitiesbelowor above.5. Thenumberofresponsesin eachcell isgiven in parentheses.Theresultsshowthat,forhigh-knowledge

PREFERENCEAND BELIEF

Table2. Percentageof choices(C) that favor a judgmentbetover a matchedlottery for high- and low-ratedknowledgeandfor judgedprobability belowandabove.5 (the numberof responsesare givenin parentheses)

Judgedprobability

Ratedknowledge F

CHIP HEATH JAMOSTVERSKY

containing100 numberedchips.Subjectsweretold that they would actuallyplay theirchoicesin eachof thetriples.To encouragecarefulranking,subjectswere told that theywould play 80% of their first choicesand20% of their secondchoices.

The dataaresummarizedin table3 and in figure6, which plots the attractivenessofthethreetypesof bets(meanrankorder)againstjudgedprobability.The resultsshowaclearpreferencefor bettingon the strongcategory.Across all triples, themeanrankswere 1.64for the strongcategory,2.12 for the chancelottery, and 2.23 for the weakcategory.Thedifferenceamongtheranksishighly significant(p < .001)by theWilcoxenranksumtest. In accordwith the competencehypothesis,peoplepreferto bet on theirjudgmentin their areaof competence,but preferto bet on chancein an areain whichtheyare not well informed. As expected,the lottery becamemore popularthan thehigh-knowledgebet only at 100%.Thispatternof resultis inconsistentwith an accountbasedon ambiguityor second-orderprobabilitiesbecauseboththe high-knowledgeandthelow-knowledgebetsarebasedonvaguejudgmentalprobabilitieswhereasthechancelotterieshaveclearprobabilities.Ambiguity aversioncould explainwhy low-knowledgebets are less attractivethan eitherthe high-knowledgebet or the chancebet, but itcannotexplainthe majorfinding of this experimentthat thevaguehigh-knowledgebetsarepreferredto theclearchancebets.

A noteworthyfeatureof figure 6, which distinguishesit from the previousgraphs,isthatpreferencesare essentiallyindependentof P. Evidently, the competenceeffectisfully capturedin this caseby the contrastbetweenthe categories;hencethe addedknowledgeimplied by thejudgedprobabilityhaslittle or no effect on thechoiceamongthebets.

Figure 7 presentsthe averagecalibrationcurvesforexperiment4, separatelyfor thehigh- and low-knowledgecategories.Thesegraphsshow that judgmentsweregenerallyoverconfident:subjectsconfidenceexceededtheir hit rate.Furthermore,the overconfi-dencewasmorepronouncedin the high-knowledgecategorythan in the-low-knowledgecategory.As a consequence,the orderingof betsdid not mirror judgmentalaccuracy.Summingacrossall triples,bettingon the chancelottery would win 69% of the time,bettingon the novicecategorywould win 64% of the time, andbetting on the expertcategorywouldwin only60%of thetime. By bettingon theexpertcategorythereforethesubjectsare losing,in effect, 15%of their expectedearnings.

The preferencefor knowledgeover chanceis observednot only for judgmentsofprobabilityfor categoricalevents(win, loss),but alsofor probabilitydistributionsovernumericalvariables.Subjects(N 93) weregivenanopportunityto set80%confidenceintervalsfor a variety of quantities(e.g., averageSAT scorefor enteringfreshmenat

Table 3. Ranking data for expert study

Typeof bet

Rank

1st 2nd 3rd Meanrank

High-knowledge 192 85 68 1.64Chance 74 155 116 2.12Low-knowledge 79 105 161 2.23

18

PREFERENCEAND BELIEF

1.5

1.6

1.7

1.8

1.9a

cC 2

~ 2.1

2.2

2.3

2.4

0

2.5 -

50 60 70 80 90 100

Judged Probability (P1

Figure6. Rankingdatafor high-knowledge,low-knowledge,andchancebetsasa functionofP in experiment4.

100

90

80

Co

70

60 -

50 -

40 -

High Knowledge

Low Knowledge

40 50 60 70 80 90 100

Judged Probability IP)

Figure 7. Calibrationcurvesfor high- andlow-knowledgecategoriesin experiment4.

Stanford;driving distancefromSan Franciscoto Los Angeles).After settingconfidenceintervals,subjectsweregiven the opportunityto choosebetween1) bettingthat theirconfidenceinterval containedthe true value,or 2) an80% lottery. Subjectspreferredbettingon theconfidenceintervalin themajorityof cases,althoughthis strategypaidoffonly 69% of the timebecausethe confidenceintervalstheyset weregenerallytoonar-row. Again, subjectspaida premiumof nearly15%to beton their judgment.

19

0

High Knowledgeo Chance Law Knowledge

20 CHIP HEATH/AM OSTVERSKY

1.5. Experiment5: Complementarybets

Theprecedingexperimentsrelyonjudgmentsof probabillity to match-thechancelotteryandthejudgmentbet.Tocontrol forpossiblebiasesin thejudgmentprocess,our lasttestof thecompetencehypothesisisbasedon a pricingtaskthatdoesnot involve probabilityjudgment.Thisexperimentalso providesan estimateof the premiumthatsubjectsarepayingin orderto beton high-knowledgeitems.

Sixty-eight studentswere instructedto statetheir cashequivalent(reservationprice)for eachof 12 bets.They were told that one pairof betswould be chosenand a fewstudents,selectedat random,would play the bet for which they statedthe highercashequivalent.(For a discussionof this payoffscheme,seeTversky,SlovicandKahneman,1990.)All betsin this experimentoffereda prizeof $15 if agivenpropositionweretrue,and nothingotherwise.Complementarypropositionswerepresentedto different sub-jects. For example,half the subjectswere askedto priceabet that paid $15 if the airdistancebetweenNew York andSanFranciscois more2500miles, andnothingother-wise.Thehalfof thesubjectswereaskedtopricethecomplementarybetthatpaid$ISifthe air distancebetweenNew York and San Franciscois less than 2500 miles, andnothingotherwise.

To investigateuncertaintypreferences,wepairedhigh-knowledgeandlow-knowledgepropositions.Forexample,we assumedthat the subjectsknow moreaboutthe air dis-tancebetweenNewYork andSanFranciscothanabouttheair distancebetweenBeijingand Bangkok.We also assumedthat our respondents(Stanfordstudents)know moreaboutthe percentageof undergraduatestudentswho receiveon-campushousingatStanfordthan at the University of Nevada,Las Vegas.As before,we refer to thesepropositionsas high-knowledgeandlow-knowledgeitems, respectively.Note that theselectionof the statedvalueof theuncertainquantity(e.g., air distance,percentageofstudents)controlsa subjectsconfidencein the validity of theproposition in question,independentof hisor hergeneralknowledgeaboutthe subjectmatter.Twelvepairsofcomplementarypropositionswereconstructed,andeachsubjectevaluatedoneof thefourbetsdefinedby eachpair. In theair-distanceproblem,for example,thefourpropo-sitionswered(SF,NY) > 2500,d(SF,NY) < 2500,d(Be,Ba)> 3000,d(Be,Ba)< 3000,whered(SF,NY) andd(Be,Ba)denote,respectively,the distancesbetweenSanFran-ciscoandNew York andbetweenBeijingandBangkok.

Notethat accordingto expectedvalue, theaveragecashequivalentfor eachpair ofcomplementarybets shouldbe $7.50. Summingacrossall 12 pairsof complementarybets,subjectspaidon average$7.12for thehigh-knowledgebetsandonly $5.96for thelow-knowledgebets(p < .01). Thus,peoplewerepaying, in effect, a competencepre-mium of nearly20%in ordertobeton themorefamiliarpropositions.Furthermore,theaveragepricefor the(complementary)high-knowledgebetswasgreaterthanthatfor thelow-knowledgebets in 11 outof 12 problems.Forcomparison,the averagecashequiva-lent for a coin tossto win $15 was$7. In accordwith our previousfindings, thechancelottery is valuedabovethe low-knowledgebetsbutnotabovethehigh-knowledgebets.

We next testthe competencehypothesisagainstexpectedutility theoryLetH andHdenotetwo complimentaryhigh-knowledgepropositions,andlet L andL denotethecorrespondinglow-knowledgepropositions.Supposeadecisionmakerprefersbettingon


H over L and on H over 717. This patternis inconsistentwith expectedutility theorybecauseit impliesthatP(H)>P(L) andP(H)>P(L), contraryto theadditivity assump-tion P(H) P(H) = P(L) + P(L) = 1. If, on theotherhand,high-knowledgebetsarepreferredto low-knowledgebets, sucha patternis likely to arise. Becausethe fourpropositions(H, H,L, L) wereevaluatedby four differentgroupsof subjects,weemploya between-subjecttest of additivity. Let M(H~) be the medianprice for the high-knowledgepropositionH1,etc.Theresponsesinproblemi violateadditivity,in thedirectionimpliedby thecompetencehypothesis,wheneverM(H1) > M(L~) andM(H1) =M(L1).

Five of the 12 pairsof problemsexhibitedthis patternindicatinga preferencefor thehigh-knowledgebets,andnoneof thepairsexhibitedtheoppositepattern.Forexample,themedianpricefor bettingon the propositionmore than85% of undergraduatesatStanfordreceiveon-campushousing was$7.50, andthe mediancashequivalentforbettingon thecomplementarypropositionwas$10.In contrast,the mediancashequiv-alentforbettingonthepropositionmorethan70%ofundergraduatesat UNLV receiveon-campushousingwas$3, and the medianvaluefor the complementarybetwas$7.Themajorityof respondents,therefore,werewilling topaymoreto beton eithersideofa high-knowledgeitemthanon eithersideof a low-knowledgeitem.

Theprecedinganalysis,basedonmedians,canbeextendedasfollows.Foreachpairofpropositions(H1, L1), we computedthe proportionof comparisonsin which the cashequivalentof H1 exceededthe cashequivalentof L1, denotedP(H1 > L1). We alsocomputedP(H~ > L1) for the complementarypropositions.All tieswere excluded.Underexpectedutility theory,

P(H, > L1) + P(HI > L1) = P(L~ > H1) + P(LI > H1) = 1,

becausetheadditivity of probabilityimpliesthatforeverycomparisonthatfavorsH1 overL1, thereshouldbeanothercomparisonthat favorsL1 overH~. Ontheotherhand,if peoplepreferthehigh-knowledgebets,asimpliedby thecompetencehypothesis,-weexpect

P(H~ >L~) + P(HI > L,) > P(LI > H1) + P(LI > H1).

Among the12 pairsof complementarypropositions,theaboveinequalitywassatisfiedin10 cases,theoppositeinequalitywassatisfiedin onecase,andequalitywasobservedinonecase,indicatinga significant violationof additivity in the directionimplied by thecompetencehypothesis(p < .01 by sign test).Thesefindingsconfirm the competencehypothesisin a testthatdoesnot relyonjudgmentsofprobabilityoron a comparison-ofajudgmentbettoa matchedlottery.Hence,thepresentresultscannotbeattributesLtmabiasin thejudgmentprocessor in thematchingof high- andlow-knowledgeitems.

2. Discussion

Theexperimentsreportedin this articleestablisha consistentandpervasivediscrepancybetweenjudgmentsof probabilityandchoicebetweenbets.Experiment1 demonstratesthat thepreferencefor the knowledgebetoverthechancelottery increaseswith judged

CHIP HEATH /AMOS TVERSKY

confidence.Experiments2 and3 replicatethis findingfor future real-worldevents,anddemonstratea knowledgeeffectindependentof judgedprobability.In experiment4, wesort subjectsinto theirstrongandweakareasandshowthatpeoplelike bettingon theirstrongcategoryanddislikebettingontheir weakcategory;thechancebetisintermediatebetweenthe two. This patterncannotbe explainedby ambiguityor by second-orderprobability becausechanceis unambiguous,whereasjudgmentalprobability is vague.Finally, experimentS confirmsthe predictionof the competencehypothesisin a pricingtask that doesnot rely on probability matchings,andshowsthat peopleare paying apremiumof nearly20%for bettingon high-knowledgeitems.

Theseobservationsareconsistentwith our attributionalaccount,which holdsthatknowledgeinducesan asymmetryin the internalbalanceof creditandblame.Compe-tence,we suggest,allows peopleto claim creditwhenthey are right, and its absenceexposespeopleto blamewhenthey are wrong. As a consequence,peoplepreferthehigh-knowledgebetoverthe matchedlottery, and theypreferthe matchedlottery overthe low-knowledgebet.Thisaccountexplainsotherinstancesof uncertaintypreferencesreportedin the literature,notably thepreferencefor clearovervagueprobabilitiesin achancesetup(Ellsberg,1961),thepreferenceto beton the futureoverthe past(Roth-bart and Snyder,1970,Brun andTeigen, 1989), the preferencefor skill overchance(CohenandHansel,1959;Howell, 1971),and theenhancementof ambiguityaversioninthe presenceof knowledgeableothers(Curley, YatesandAbrams, 1986).The robustfinding that, in their areaof competence,peoplepreferto bet on their (vague)beliefsover a matchedchanceeventindicatesthat the impact of knowledgeor competenceoutweighsthe effectof vagueness.

In experiments14 weusedprobabilityjudgmentsto establishbeliefandchoicedatatoestablishpreference.Furthermore,wehaveinterpretedthechoicejudgmentdiscrep-ancyas a preferenceeffect. In contrast,it could be arguedthat the choicejudgmentdiscrepancyis attributableto ajudgmentalbias,namelyunderestimationof the proba-bilities of high-knowledgeitems and an overestimationof the probabilities of low-knowledgeitems.This interpretation,however,is not supportedby the availableevi-dence.First,it implies lessoverconfidencefor high-knowledgethan for low-knowledgeitemscontraryto fact(seefigure7).Second,judgmentsof probabilitycannotbedismissedasinconsequentialbecausein the presenceof a scoringrule, suchasthe oneusedin experi-ment4,thesejudgmentsrepresentanotherform of betting.Finally,ajudgmentalbiascan-not explain the resultsof experiment5, which demonstratespreferencesfor bettingonhigh-knowledgeitemsin apricing taskthatdoesnot involve probabilityjudgment.

The distinction betweenpreferenceandbelief lies at theheartof Bayesiandecisiontheory. Thestandardinterpretationof this theoryassumesthat 1) the expressedbeliefs(i.e., probabililtyjudgments)of an individual areconsistentwith an additive probabilitymeasure,2) the preferencesof an individualare consistentwith the expectationprinci-ple,andhencegive riseto a (subjective)probabilitymeasurederivedfrom choice,and3)thetwo measuresof subjectiveprobabilityobtainedfromjudgmentandfrom choiceareconsistent.Note thatpoints 1 and 2 are logically independent.Allais counterexam-pIe, for instance,violates2 but not 1. Indeed,manyauthorshaveintroducednonadditivedecisionweights,derivedfrom preferences,to accommodatethe observedviolationsofthe expectationprinciple (see,e.g., Kahnemanand Tversky, 1979). Thesedecision


weights, however,neednotreflect the decisionmakersbeliefs.A personmaybelievethat theprobabilityof drawingthe aceof spadesfrom a well-shuffleddeckis 1/52,yet inbettingon this eventheor shemaygive it a higherweight.Similarly, Ellsbergsexampledoesnotprovethatpeopleregardthecleareventasmoreprobable-thanthecorrespond-ingvagueevent;it onlyshowsthatpeopleprefertobetontheclearevent.Unfortunately,thetermsubjectiveprobability hasbeenusedin the literaturetodescribedecisionweightsderivedfrom preferenceas well as direct expressionsof belief. Under the standardinterpretationof the Bayesiantheory, the two conceptscoincide.As wego beyondthistheory, however,it is essentialto distinguishbetweenthetwo.

2.1. Manipulationsofambiguity

Thedistinctionbetweenbeliefandpreferenceisparticularlyimportantfor theinterpre-tationof ambiguityeffects.Severalauthorshaveconcludedthat,whentheprobabilityofwinning is small or whenthe probability of losingis high,peoplepreferambiguity toclarity (Curley andYates,1989;Einhorn andHogarth,1985; HogarthandKunreuther,1989).However, this interpretationcanbechallengedbecause,aswill be shownbelow,the datamay reflect differencesin belief ratherthanuncertaintypreferences.In thissection,we investigatethe experimentalproceduresusedto manipulateambiguityandarguethat theytendtoconfoundambiguitywith perceivedprobability.

Perhapsthe simplestprocedurefor manipulatingambiguity is to vary the decisionmakersconfidencein a givenprobabilityestimate.Hogarthandhis collaboratorshaveusedtwo versionsof this procedure.EinhornandHogarth(1985)presentedthe subjectwith a probability estimate,basedon the judgementof independentobservers,andvariedthe degreeof confidenceattachedto that estimate.Hogarthand Kunreuther(1989)endowedthesubjectwith hisorherbestestimateof theprobabilityof agivenevent,andmanipulatedambiguityby varyingthe degreeof confidenceassociatedwiththis estimate.If wewishto interpretpeopleswillingnesstobeton thesesortsofeventsasambiguityseekingor ambiguityaversion,however,wemust first verify that the manipu-lationof ambiguitydid notaffectthe perceivedprobabilityof the events.

To investigatethis question,wefirst replicatedthemanipulationof ambiguityusedbyHogarthandKunreuther(1989). Onegroupof subjects(N = 62),calledthe highconfi-dencegroup,receivedthefollowing information:

Imaginethat you heada departmentin a large insurancecompany.Theownerof asmall businesscomesto you seekinginsuranceagainsta $100,000loss which couldresultfromclaimsconcerninga defectiveproduct.Youhaveconsideredthemanufac-turingprocess,the reliabilities of the machinesused,andevidencecontainedin thebusinessrecords.After consideringthe evidenceavailabletoyou,your bestestimateof the probability of a defectiveproductis .01. Given the circumstances,you feelconfidentaboutthe precisionof this estimate.Naturallyyou will updateyourestimateasyou think moreaboutthe situationor receiveadditionalinformation.

CHIP HEATH /AMOS TVERSKy

A secondgroupof subjects(N = 64),calledthe low-confidencegroup,receivedthesameinformation,exceptthat thephraseyou feel confidentaboutthe precisionof thisesti-matewasreplacedby you experienceconsiderableuncertaintyabouttheprecisionofthis estimate.All subjectswerethenasked:

Do you expectthat the newestimatewill be (Checkone):Above.01 ______________Below .01 ______________Exactly.01 _______________

Thetwo groupswerealsoaskedtoevaluatea secondcasein which thestatedprobabilityof a losswas.90. If thestatedvalue(.01or .90) is interpretedasthemeanof therespectivesecond-orderprobabilitydistribution,thenasubjectsexpectationfor theupdatedestimateshouldcoincidewith the currentbest estimate.Furthermore,if the manipulationofconfidenceaffectsambiguitybutnotperceivedprobability,thereshouldbeno differencebetweenthe responsesof thehigh-confidenceandthe low-confidencegroups.The datapresentedin table4, underthe headingYourprobability, clearlyviolate theseassump-tions.The distributionsof responsesin the low-confidenceconditionare considerablymoreskewedthanthe distributionsin the high-confidencecondition.Furthermore,theskewnessis positivefor .01 andnegativefor .90. Telling subjectsthat theyexperienceconsiderableuncertaintyabouttheir bestestimateproducesa regressiveshift: the ex-pectedprobabilityof loss is above.01 in the first problemandbelow .90 in the second.The interactionbetweenconfidence(highlow) anddirection(abovebelow)is statisti-cally significant(p < .01).

We also replicatedtheprocedureemployedby EinhornandHogarth(1985)in whichsubjectswere told that independentobservershave statedthat the probability of adefectiveproductis .01. Subjects(N = 52) in the high-confidencegroupweretoldthatyou couldfeel confidentaboutthe estimate,whereassubjects(N = 52) in the low-confidencegroupweretold that you couldexperienceconsiderableuncertaintyabouttheestimate.Both groupswerethenaskedwhethertheirbestguessof theprobabilityofexperiencinga lossis above.01,below.01, or exactly.01. Thetwo groupsalsoevaluateda

Table 4. Subjective assessments of stated probabilities of .01 and .90 under high-confidence and low-confidenceinstructions (the entries are the percentage of subjects who cbose each of the three responses)

Your probability Othersestimate

High Low High LowStatedvalue Response confidence confidence confidence confidence

Above .01 45 75 46 80

.01 Exactly.01 34 11 15 6Below .01 21 14 39 14

Above .90 29 28 42 26.90 Exactly .90 42 14 23 12

Below .90 29 58 35 62

24


secondcasein which the probability of loss was .90. The resultspresentedin table4,under the headingOthers estimate,reveal the patternobservedabove. In the high-confidencecondition, the distributionsof responsesare fairly symmetric,but in thelow-confidenceconditionthe distributionsexhibitpositiveskewnessat .01 andnegativeskewnessat .90. Again, the interactionbetweenconfidence(highlow) anddirection(abovebelow)isthe statisticallysignificant(p < .01).

Theseresultsindicatethat the manipulationsof confidenceinfluencednot only theambiguityof the eventin questionbutalso its perceivedprobability: they increasedtheperceivedprobabilityof thehighly unlikely eventanddecreasedtheperceivedprobabil-ity of the likely event.A regressiveshift of this type is not at all unreasonableandcanevenbe rationalizedby a suitable prior distribution.As a consequenceof the shift inprobability,the betonthevaguerestimateshouldbemoreattractivewhenthe probabil-tty of loss is high(.90) andless attractivewhenthe probabilityof lossis low (.01).This isexactly the patternof preferencesobservedby Einhorn and Hogarth(1985) and byHogarthandKunreuther(1989),but it doesnotentail eitherambiguityseekingor am-biguity aversionbecausetheeventsdiffer inperceivedprobability,notonly in ambiguity.

The resultsof table4 andthe findings of Hogarthandhis collaboratorscanbe ex-plained by the hypothesisthat subjectsinterpretthe statedprobability value as themedian(or the mode) of a second-orderprobability distribution.If the second-orderdistributions associatedwith extremeprobabilitiesareskewedtowards.5, the meanislessextremethanthemedian,andthedifferencebetweenthernisgreaterwhenambigu-tty is high thanwhenit is low. Consequently,themeanof the second-orderprobabilitydistribution,which controlschoice in the Bayesianmodel,will be moreregressive(i.e.,closerto .5)underlow confidencethanunderhighconfidence.

Thepotentialconfoundingof ambiguityanddegreeof beliefarisesevenwhenambi-guity ismanipulatedby informationregardinga chanceprocess.Unlike Ellsbergscom-parisonof the50/50box with theunknownbox,wheresymmetryprecludesa biasin onedirectionor another,similarmanipulationsof ambiguityin asymmetricproblemscouldproducea regressiveshift, as demonstratedin an unpublishedstudyby ParayreandKahneman.3

Theseinvestigatorscomparedaclearevent,definedby the proportionofredballsin abox,withavagueeventdefinedby therangeof ballsof thedesignatedcolor. Foravagueevent{.8, 1], subjectswereinformedthat the proportionof redballscould-beanywherebetween.8 and 1, comparedwith .9for theclearevent.Table5 presentsbothchoiceandjudgmentdatafor threeprobabilitylevels: low, medium,andhigh. In accordwith prevt-ouswork, the choicedatashowthatsubjectspreferredto beton thevagueeventwhenthe probabilityof winningwaslow andwhentheprobabilityof losingwas-high,and theypreferredtobetonthecleareventin all othercases.Thenovelfeatureof theParayreandKahnemanexperimentis theuse of a perceptualrating scalebasedon a judgmentoflength,which providesa nonnumericalassessmentof probability. Using this scale,theinvestigatorsshowedthat thejudgedprobabilitieswereregressive.That is, the vaguelow-probabilityevent[0,.10]wasjudgedasmoreprobablethantheclearevent,.05, andthe vaguehigh-probabilityevent[.8,1]wasjudgedasless probablethan the clearevent,.9. Forthemediumprobability,therewasno significant differencein judgmentbetweenthe vagueevent[0,1] andthe clearevent, .5. Theseresults,like the dataof table4,

CHIP HEATH/AMOSTVERSKY

Table 5. (Basedon Parayre andKahneman).Percentageofsubjectswhofavoredthecleareventandthevagueeventin judgmentand in choice.

Choice

Probability Judgment Win$100 Lose $100(win/lose) N~72 N~58 N=58

Low .05

10,10]

28

47

12

74

66

r)

Medium .5[01]

3822

6026

6021

High .9[.8,1]

5021

5034

2247

Note: The sum of the two values in eachcondition is less than 100%; the remainingresponsesexpressedequivalence.in thechoicetask,thelow probabilitieswere.075 and[0,15]. N denotessamplesize.

demonstratethatthepreferencefor bettingon theambiguousevent(observedat thelowendforpositivebetsandat thehighendfornegativebets)couldreflecta regressive,shiftin the perceptionof probabilityratherthana preferencefor ambiguity.

2.2. Concludingremarks

Thefindings regardingthe effectof competenceandthe relationbetweenpreferencesand beliefs challengethe standardinterpretationof choice modelsthat assumesindependenceof preferenceandbelief. The results are also at variancewith post-Bayesianmodelsthat invoke second-orderbeliefsto explainthe effectsof ambiguityorpartial knowledge.Moreover,our results call into questionthe basicideaof definingbeliefs in terms of preferences.If willingnessto bet on an uncertaineventdependsonmorethantheperceivedlikelihoodof thateventandtheconfidencein thatestimate~it isexceedinglydifficultif not impossibletoderiveunderlyingbeliefs from preferencesbetweenbets.

Besideschallengingexistingmodels,the competencehypothesismight help explainsomepuzzlingaspectsof decisionsunderuncertainty.It couldshedlight on theobserva-tion thatmanydecisionmakersdo notregarda calculatedrisk in their areaof compe-tenceas a gamble(see,e.g.,March andShapira,1987). It might alsohelp explainwhyinvestorsare sometimeswilling to forego theadvantageof diversificationandconcen-trateon a small numberof companies(Blume,Crockett,andFriend,1974)with whichtheyare presumablyfamiliar. Theimplicationsof thecompetencehypothesisto decisionmaking atlargeare left tobe explored.

26


3. Notes

1. We usethetermcompetencein a broadsensethat includesskill, aswell asknowl-edgeor understanding.

2. In this andall subsequentfigures,weplot the isotoneregressionof C onPthatis,thebest-fittingmonotonefunctionin the leastsquaressense(seeBarlow,Bartholomew,BremnerandBrunk, 1972).

3. We aregratefulto ParayreandKahnemanforprovidinguswith thesedata.

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Documents

Tversky - Ambiguity and Competence