Figure1AsimplerepresentationofatwosystemsmodelSystem1isaboveSystem2is - PDF document

1 x,z##"""""""""""""""""""""!"#$%&'(2E[v|x
x,z##"""""""""""""""""""""!"#$%&'(2E[v|x,z Figure1:Asimplerepresentationofatwo-systemsmodel:System1isabove,System2isbelow.Bothsystemsformrationalexpectationsabouttheunobservedvariablev,howeverSystem1receivesonlyx(low-levelinformation),whileSystem2additionallyreceivesz(high-levelinformation).Althoughthisclassofmodelshasbeenusedtogivepersuasiveanalysesofindividualbi-ases,theysu!erfromtwoimportantempirica

2 lproblems:theresponseofbiasestoincentive
lproblems:theresponseofbiasestoincentives,andtheresponseofbiasestojointevaluation.First,themodelpredictsthatbiaseswilldisappearwhenSystem2isactivated,whichshouldoccurwhenevertimeandincentivesaresu"cientlyhigh.Althoughincentivesdotendtoreducethemagnitudeofbiases,itiscommonlyobservedthatmanybiasesremainevenwhenthestakesbecomequitehigh.CamererandHogarth(1999)sayÒnoreplicatedstudyhasmaderationalityviolatio

3 nsdisappearpurelybyraisingincentives.ÓSi
nsdisappearpurelybyraisingincentives.ÓSimilarly,behavioroutsidethelaboratoryoftenseemstobeinßuencedbyirrelevantinformationevenwithveryhighstakes(Postetal.(2008),ThalerandBenartzi(2004)).Asimilarpointistrueforperceptualillusions:spendingalongertimestaringattheillusionmayreducethemagnitudeofthebias,butitrarelyeliminatesit(Predebonetal.(1998)).Thusitbecomesapuzzle2Althoughthetheorieslistedallsharethesameb

4 asicdiagnosisofwhybiasesoccur,theydi!ero
asicdiagnosisofwhybiasesoccur,theydi!eronanumberofotherimportantdimensions,discussedlaterinthepaper.MorerecentlytheSystem1/System2terminologyhasbeenusedtorefertodi betweensituations.Thesetwogeneralizations-thatinconsistenciesareinsensitivetoincentives,butsensitivetojointpresentation-suggestthatourreßectivejudgmentsobeyprinciplesofconsistency,butaredistortedbythesamebiasesthatdistortourautomaticsystem.T

5 hiscouldoccurifSystem2Õsjudgmenttakesint
hiscouldoccurifSystem2ÕsjudgmenttakesintoaccountthejudgmentsthatSystem1makes.Andthis,inturn,wouldberationalifSystem1ÕsjudgmentsincorporatedinformationnotaccessibletoSystem2.Thispaperproposesthatthereasonwemakeinconsistentjudgmentswhenusingourfullreßectivejudgmentisthatindi!erentsituationswereceivedi!erentsignalsfromSystem1(orintuitions),anditisrationaltotakeintoaccountthosesignalsbecausetheycontainvalu

6 ableinformation.Icalltheunderlyingassump
ableinformation.Icalltheunderlyingassumptionimplicitknowledge,becauseitisknowledgethatisprivatetoourautomaticsystem,andthusavailabletoourreßectivesystemonlyindirectly,throughobservingtheautomaticsystemÕsjudgments.Figure2showshowtheformalanalysisdi!ers:System1nowhasaccesstoprivateinformation,!,andSystem2canobserveSystem1Õsposteriorjudgment(E[v|x,!]).System2facesasignalextractionproblemininferring!from |

7 x,!]! !!!!!!!!!!!!!!!!!!!!!x,z##""""" |x
x,!]! !!!!!!!!!!!!!!!!!!!!!x,z##""""" |x,z,E[v|x,!]]!!Figure2:Atwo-systemmodelwithimplicitknowledge:System1nowadditionallyreceivesprivateinformation,!,andSystem2conditionsitsjudgmentsonSystem1Õsexpectation.notbeabletoperfectlyextractthisinformation,andSystem2Õsjudgmentwillnotbethesameasifithadaccessto!.WecanthereforedeÞnetheÒbiasÓofthetwosystemsrelativetothebenchmarkcaseinwhichalltheinformationispooled

8 :4System1Õsbias=E[v|x,!]" x,z,!](2) repr
:4System1Õsbias=E[v|x,!]" x,z,!](2) reproducingthem.Asasimpleexample,mostpeopleÞnditdi"culttoanswerthequestionÒisthereanEnglishwordwhichcontainsÞveconsecutivevowels?Ó,butinstantlyrecognizethatthestatementistruewhentheyareremindedofawordthatÞtsthepattern.7Mostpeoplecaneasilyrecognizewhetherasentenceisgrammatical,buthavedi"cultymakinggeneralizationsaboutthesetofgrammaticalsentences.8Thesedistinctionsinac

9 cessibilitywouldnotexistifourknowledgewa
cessibilitywouldnotexistifourknowledgewasexplicit,i.e.storedasadistributionoverpossiblestatesoftheworld.Thispaperproposesthattheknowledgeweuseinmakingeconomicdecisionsisstoredinasimilarlyimplicitform:thatpeopleareabletomakeconÞdentsnapjudgmentsaboutthevalueofdi!erentalternatives,buttheyhavelimitedinsightintohowthosejudgmentsareformed.Thisseparationofknowledgebetweensystemscanexplainwhyourdecisionsoften

10 violatenormativeprinciplesthatwereßectiv
violatenormativeprinciplesthatwereßectivelyendorse.Themodelmakesavarietyofpredictionsabouthumanjudgment:(1)biaseswilloccurwhenthereisaninteractionbetweenimplicitknowledgeandhigh-levelinformationininfer-ringv(i.e.,between!andz);(2)judgmentswillappearinconsistenttoanoutsideobserverbecauseindi!erentsituationsthereßectivesystemwillhavedi! |Fv(")=v,Fx(")=x,Fz(")=z,F!(")=!}),andconditionaldistributionsderive

11 dfromthat.WecanthendeÞnethefollowingexpe
dfromthat.WecanthendeÞnethefollowingexpectations,whichrepresentrespectivelytheexpecta-tionsaboutvformedbySystem1,bySystem2,andbyahypotheticalagentwhoisabletopoolbothinformationsets:E1=E[v|x,!]E2=E[v|x,z,E[v|x,!]]E System2Õsbias=E2"EPBothE1andE2willhaveazeroaveragebias,i.e.E[E1"EP]=E[E2"EP]=0,however i.e.thebiaswhichsurvivesinapersonÕsreßectivejudgment.WhenIsaythatjudgmentisunbiasedIwillmeanthatforallx#

12 X,!#A,z#Z,E[v|x,z,E[v|x,!]]=E[v|x,z,!]2.
X,!#A,z#Z,E[v|x,z,E[v|x,!]]=E[v|x,z,!]2.1ConditionsforaBiasinReßectiveJudgmentAsimplesu"cientconditionforunbiasednessisthatE[v|x,!]isaone-to-onefunctionof!.IfitwasthenSystem2wouldsimplybeabletoinvertE1toinfer!.Howeverthisconditionisnotnecessary,becauseinmanycasesSystem2canextractalltheinformationitneedsfromE1withoutknowing!.Weareabletogivemoreinterestingconditionsbelow.TherelationshipbetweenE1,E2,andEP

13 canbeillustratedinthefollowingtable:9Amo
canbeillustratedinthefollowingtable:9Amoresophisticatedmodelwouldallowthisdecisiontocond E[v|!]E[v|x,z,!!]E[v E1)mustalsobeidenticalforeverycell.Proposition2.Judgmentwillbeunbiasedifandonlyif,forallx#X,!,!!#A,E[v|x,!]=E[v|x,!!]=)*z#Z,E[v|x,z,!]=E[v|x,z,!!]ToillustrateIgiveanexamplewhereaggregationofinformationfails(xisignoredinthisexample).Example1.Letv,!,z#{0,1},withf(v=1)=12.Supposethatifv=0then!andz

14 areuniformlydistributedandindependent,bu
areuniformlydistributedandindependent,butifv=1thenwithequalprobability!=z=1,or!=z=0.I.e.,forall!,z#{0,1}:f(!,z|v=0)=14f(!,z|v=1)=#$$%$$&12 [v|!,z]=#$$%$$&23,!=z0, v|E[v|!],z]=12Herethepooled-informationexpectationincludesaninteractiontermbetween!andz.Inthiscasewedonotknowwhetherarealizationof!representsgoodnewsorbadnewsabout !,sobothSystem1and2willbebiasedrelativetothepooled-informationbenchmark(i.e.,*

15 !# v|x,z,!]isweaklymonotonicin!whenorder
!# v|x,z,!]isweaklymonotonicin!whenorderedby,x.Intermsofthetableabove, ,andi:R$R,suchthatEP=E[v|x,z,!]=i(g(x,!)+h(x,z))andiisstrictlymonotonic.Proof.Inthiscaseforanyxthereexistsanorderingof foranyz(i.e.,theorderingaccordingtog(x,!)).Judgmentwillthereforebeunbiased,bythepreviousproposition.Theexistenceofbiasissensitivetothedistributionofknowledge:forexample,nobiaswouldoccurifyoursisterkneweverythingabou

16 tthehalfofthebaseballcardsthatarealphabe
tthehalfofthebaseballcardsthatarealphabeticallyÞrst,A-M,andyoukneweverythingaboutthe x,!),notjustE[v|x,!]).TheyalsoshowthatE1willalmostalwaysrevealtheentireposterior,in elements,v=Rm,x=Xm,!=Am,z=Zm,withthejointdistribution, )asasituation,andanelement(xi,zi E[v|x,!]E2=E[v|x,z,E1]EP=E[v|x,z,!]Aswritten,thissetupallowsmanychannelsofinference,soIintroducefurtherassumptionsinordertoconcentratejustonthechann

17 elsofinterest.First,asdiscussed,ourprinc
elsofinterest.First,asdiscussed,ourprincipalinterpretationisthatSystem1Õsprivateinformationrepresentslong-runknowledge,soIassumethatallelementsof!areidentical,andthereforesimplyrefertoitas!.Second,Iwillassumethateachcase(xi,zi wereinformativeabout!thenwewouldexpectjointandseparatejudgmenttodi!erevenwithoutanysignalfromSystem1.Finally,wewillalsoassumethatallobservableinformationabouteachobjectisidiosyn-

18 cratic,i.e.xiandziareinformativeonlyabou
cratic,i.e.xiandziareinformativeonlyaboutvi,notaboutvjforj(=i.Thesethreepointsareincorporatedintothefollowingassumptionaboutthedistributionofinformation:fm(v,x,z,!)=(m)i=1f(vi|xi,zi,!)*f(z|x)f(x)f(!)(A1)WecanÞrstnotethat,withinthisframework,neitherSystem1Õsexpectationnorthepooled-informationexpectationwilldi!erbetweenjointandseparateevaluation,i.e.:Ei1=E[v|xi,!]EiP=E[v|xi,zi,!]HoweverwhenSystem2observe

19 savectorE1,itcanlearnabout!fromtheentire
savectorE1,itcanlearnabout!fromtheentiresetof11Forexamplewithbaseballcards,youmightinferthatmorecommoncardsarelessvaluable,andthiscouldcauseseparateandjointevaluationtodi!erforanoth x#Xm,z#Zm,x!#Xn,z!#Zn,withxi=x!iandzi=z!ifori#{1,...,m},thenforanyj#{1,...,m}Var[Ej2"EjP]'Var[E!j2"E!jP]whereE2=E[v|x,z,E[v|x thedecision-makercanremembertheirintuitionsforprevi ,z!)isevaluatedwithoutanyhistory.Inotherwords

20 ,moreexperienceispredictedtoreducethebia
,moreexperienceispredictedtoreducethebias.Toapplythispredictionaboutsequentialjudgmentsrequiresinterpretationofwhenasetofcasesarepartofthesamesituation.TheimportantassumptionisthatSystem2canrecallallpreviousstimuli( .InsomecasesIwillinterpretthisasautilityfunction,inwhichtheunknownutilityofanobjectisinferredfromitsfeatures.12One x,z)=u(x!,z!)}.Wewillbeinterestedonlyinconvexrestrictions,i.e.:DeÞnition1.

21 |x,z,E[v|x,!]]. theresultwillbemixed:wh
|x,z,E[v|x,!]]. theresultwillbemixed:whenevaluationsaremadeseparately(i.e.,whenconditioningondi!erentsetsx),thentherestrictionmaybeviolated,butwhenevaluationsaremadejointly,withthesameconditioningsetx,theywillalwayssatisfytherestriction.13Forexample,theindi!erencerestriction{u:u(x,z)=u(x",z")}isconvexbecauseanymixturebetween thenforall!#A,x#Xm,�m1,u!,x2(x,z)#UForexampleconsiderhowpeoplewillresp

22 ondtoirrelevantdi!erences.Supposethatour
ondtoirrelevantdi!erences.Supposethatourrestrictionis,asabove,thatforsomex,x!#X,z,z!#Z,{u:u(x,z x,z 1,c!2,c!,x2pickoutthemaximalelementsofthechoicesetaccordingtothefunctionsu!P,u!1,u!2,andu!,x2).Inthecaseofc!,x2Imakethefurtherassumptionthattheconditioningsetxisformedbyelementsofthechoiceset,D "Z,andcorrespondingchoicerestrictionCU-DD,15ifpooled-informationjudgmentsatisÞesU(*!#A,u! )=argmax(x,z)#Au(x,z

23 Proof.Bytheproposition,eachu!,x2belongst
Proof.Bytheproposition,eachu!,x2belongstoU,thereforeitmustsatisfythechoicere-strictionsimpliedbyU.DecisionsmadebySystem1mayviolaterestrictionsaxiomsonchoice,becausethosedecisionswillfailtoconditiononz(putanotherway,inattentivedecisionsmayviolateaxiomsofchoice).Proposition5impliesthatSystem2Õsdecisionswillneverviolateanaxiominagivenchoiceset,althoughdecisionsmadeseparat !!,z) ,!!#A,.x#X,E[v|x,x,v|x,!!]L

24 earnabilityisanaturalrestrictionifwethin
earnabilityisanaturalrestrictionifwethinkofSystem1asanaivelearner:i.e.,if Z,x#X,m#N,x#Xmx!#x0)x!# fislearnable,thereisonlyasingle!#Athatisconsistentwiththispattern,thusE[!|x,E1]=!.ThereforeE2=E[v|x,z,E1]=E[v|x,z,!]=EP. =R2),andE[v|x,z,!1,!2]isassumedtobeincreasinginboth!1and!2.System1observes(!1,!2)andcalculatesanexpectation,E1=E[v|x,!1,!2].System2observesthatexpectation,andthereforelearnsthat!1and!2li

25 eonacertaincurve,whichleadshimtoupdatehi
eonacertaincurve,whichleadshimtoupdatehisposteriorover(!1,!2).Anaturalassumptionwillbethat,whenSystem2observesahigherE1,hisposteriorsoverboth!1and!2increase,inthesenseofhavinghigherexpectedvalues.IfthisistruethentherewillbeaÒspilloverÓe!ect:anincreasein!1willcauseanincreaseinSystem2Õsestimateof!2.Thusinsituationswhere!1isknowntobeirrelevant,itwillneverthelessa!ectSystem2Õsjudgment,andthedirectionofinßu

26 encewillbepredictable. isirrelevant(i.e.
encewillbepredictable. isirrelevant(i.e.,undersomerealizationofxandz,!1isunrelatedtov),neverthelessSystem2Õsjudgmentwillbeamonotonicfunctionof!1.Forexample,whenyouknowthattheBabeRuthcardiscounterfeit,yourbidwillbeneverthelessincreasinginthevalueofthatcard,becauseitsvalueindirectlya!ectsyourjudgmentthroughyoursisterÕsreportofthevalueofthepacket.Asimpleexampleisif IdeÞnethispropertyasunambiguousness,mean

27 ingthatachangein!whichincreasesE1willwea
ingthatachangein!whichincreasesE1willweaklyincreaseE2,forallvaluesofxandz:DeÞnition3. v,nomattertherealizationofz.If! andzareindependent,andE1=E[v|x,!]=n'i=1 |x,z,!]=E[v ?Icallthispropertycongruence:DeÞnition4.Adistributionfiscongruentforsomex,x!#X,andz#Zif, fisunambiguousinthesensedeÞnedabove.TherearetwoqualiÞcations.First,ifSystem2knowsthatx!isassociatedwithahighervthanx,thenitwillalreadydiscountthee

28 !ectonE1.Thereforewearereallyinterestedi
!ectonE1.Thereforewearereallyinterestedinthedi!erencebetweenwhateachSystembelievesabouttherelationshipbetweenxandv.AnaturalbenchmarkiswhenSystem2expectsnodi!erence:i.e.,whenE[v x!!,z,!1,!!2]forallx!!#X,!1#A1,!2,!!2#A2(iv)given! inducesahigherjudgmentthanx,evenwhenpeopleknowthatthedi!erenceisnorma-tivelyirrelevant,then,undertheassumptionsabove,thisimpliesthatE[v|x!,!]�E[v|x,!],i.e.thatonaverage(i

29 gnoringz)x!isassociatedwithhigherv,thoug
gnoringz)x!isassociatedwithhigherv,thoughpeoplemaynotbeconsciouslyawareofthisassociation.Idiscussevidenceforthispredictioninalatersection.3GaussianModelInthissectionIassumeaspeciÞcdistributionforf(v,x,z,!),andsolveexplicitlyforthebias.UnderthisdistributionSystem2Õsproblemcanbeseenasreweightingaweightedaverage.System1Õsestimate,E1,willbeaweightedaverageoftheirprivateinformation(!1,...,!n),withweightsgiv

30 enbythepublicinformation( =E[v|!,z]=n'j=
enbythepublicinformation( =E[v|!,z]=n'j=1!jzjWecanthereforeexpresseachSystemÕsexpectationas:E1=n'j=1! ]=E[!j]+E[ [zj]2$2j:E2=n'j=1E[!j|E1]zj=n'j=1E[!j]zj+n'j=1%j+ !jshouldbegivenlessweightthanusual),thenSystem1willtendtoover-reacttofeaturej,andviceversa.WecanwriteSystem2ÕsÞnalbiasas:E2"EP=n'j=1E[!j]zj+n'j=1%j+k%kE1"E0E[zj]zj"n'j=1!jzj=n'j=1(E[!j]"!j)zj+(E1"E0)+n'j=1zj"E[zj]E[zj]%j+k%k(E1"E0)=n'j=1(E[!j

31 ]"!j)zj+n'j=1(!j"E[!j])E[zj]+n'j=1zj"E[z
]"!j)zj+n'j=1(!j"E[!j])E[zj]+n'j=1zj"E[zj]E[zj]%j+k%k(E1"E0)=n'j=1(E[zj]"zj)((!j"E[!j])"1E[zj]%j+k%k(n'k=1(!k"E[!k])E[ P=n' j+ !j])"%j+k%k'k$=jE[zk]E[zj](!k"E[!k])*TheintuitioncanbebetterunderstoodwithafurthersimpliÞedversion,conditioningjustonsomepair!j,zj, rminedbythedeviationsof!andzfromtheirexpectedvalues.Wecanmakeanumberofobservationsaboutthenatureofthebiases:1.Abiasrequiresthatboth !j],thiswillca

32 usepeopletooverestimatev.Intuitively,the
usepeopletooverestimatev.Intuitively,thesecondSystemdiscountsthesignal,because!jislessimportantthanusual.But!jisbiggerthanexpected,sothereßectivesystemdoesnotdiscountenough.3.ThesizeofSystem2Õsbiasisdecreasingin%j=$2jE[zj]2.Ahigher willbeattributedto!j,andthereforethebiaswillbesmallerbecausethediscountappliedtoE1willbegreater.Inotherwords,ifSystem2islesssureaboutthee!ectof!j,thenitwillbelessinßuencedby

33 changesin!j.Thismodelhasasimpleinterpret
changesin!j.Thismodelhasasimpleinterpretationasthereweightingofaweightedaverage.ForexampletheUSEPAÕsÒcombinedMPGÓfornewcarsiscalculatedasaweightedaverageofcityMPG(55%)andhighwayMPG(45%).Eachcarbuyermayhavetheirownpreferredweights.IfIobserveonlytheEPAÕscombinedMPGforacar,Iwillthenhavetoinfertheunderlyingdatainordertoconstructmypreferredweighting.IfmypriorsaboutthecarÕsunderlyingMPGvariablesareGaussian,t

34 henthemodelinthissectionisanexactdescrip
henthemodelinthissectionisanexactdescriptionoftheproblemIface,withSystem1representingtheEPAandSystem2representingme.MyposteriorcanbedescribedbytheexpressionaboveforE2,andmybiasrelativetothecaseinwhichIknewtheunderlyingdataasE2"EP(heren=2,!willrepresentthecityandhighwayMPGvariables,zaremyidiosyncraticweightings,andtheEPAÕsweightscouldbeinterpretedasE[z],i.e.theaverageweightsusedbycarbuyers).Thepredictio

35 nswillapplytobuyingacar:ifmyweightsdi!er
nswillapplytobuyingacar:ifmyweightsdi!erfromtheEPAÕsweights,andifthecarÕsattributesdi!erfromtheirexpectedvalues,thenmyjudgmentwillbebiasedinapredictableway.Insomecasesmyjudgmentwillbea!ectedbyinformationIknowtobeirrelevant:ifIputa0%weightonhighwayMPG,neverthelessIwillvaluemorehighlyacarwhichhasahigherhighwayMPG,everythingelseequal(becausehighwayMPGa!ectstheEPAÕscombinedMPG,whichinturnaectsmybeliefsabou

36 tcityMPG).3.1JudgingAlternativeswithAttr
tcityMPG).3.1JudgingAlternativeswithAttributesInowextendtheGaussianmodeltoincludepublicinformationwhichcanbeinterpretedasasetofattributes(x#Rn)whichareobservabletobothSystem1andSystem2.Iassumethattheinterpretationoftheseattributesisa!ectedboth [v|x,z,!]=n'j=1xjzj!j(4)Asbefore,inordertoachieveananalyticalsolution,Iassumethateachelementin!isdrawnfromanormaldistribution,i.e.!j1N(E[!j],$2j forE[zj].Thesolu

37 tionsaretherefore:E1=E[v|!,x]=n'j=1 1]=E
tionsaretherefore:E1=E[v|!,x]=n'j=1 1]=E[!j]+E[zj]2x2j$2!j+kE[zk]2x2k$2!kE1"E0x P='j(E[zj !j]),1"&j+k&k-+&j+k&k1E[zj]xjn'k$=j(!k"E[!k])E[zk]xk*where&j=$2jE[zj]2x2j.Asbefore,asimplerwaytounderstandthisresultistoderivetheaveragebiasforsomegiven!j,zj.E[E2"EP|x,zj,!j]="(zj"E[zj])(!j"E[!j])xj,1"&j+k& "&j+k&k-=xj,1"$2jE[zj]2x2j+k$2kE[zk]2x2k-=xj+k$2kE[zk zj]).19Intuitively,thisnon-monotonicitycanbethoughtofl

38 ikethechangeinestimatedsignalduetoachang
ikethechangeinestimatedsignalduetoachangeinthesignal-to-noiseratio.Whentheratioisverylow,youremainatyourprior.Whenitishigh,youlearnthesignalquiteprecisely.Whenitisintermediate,yourestimateofthesignalwillbetendtobebiasedtowardstherealizationofthenoise.Appliedtoperceptionandjudgment,thise!ectcorresponds =E[v|x,!]E !jxijE[zj]Inmatrixnotationthiscanbewritten,E1=x(!3E[z])where(P3Q)ij=Pij+Qij.Forcompactness,

39 andwithoutlossofgenerality,IsetE[zj]=1fo
andwithoutlossofgenerality,IsetE[zj]=1forallj.AsbeforeSystem2wishestoinfer!fromE1,buthenowhasmoreinformation.Becausetheelementsof!aredistributedindependentlyandnormally,theexpectationof!,ö!,willmaximizethelikelihoodfunction,"(ö!"E[!])T"(ö!"E[!])subjecttotheaboveconstraint,where"isadiagonalmatrix,withelements$%21,...,$%2n,andPTisthetransposeofP.TheÞrst-orderconditionofaLagrangiancanbewritten(BoydandVand

enberghe(2004),p304):2(ö!"E[!])T"='xö!=E[!]+12"%1xT'T37 !]+12x"%1xT'T(x"%1xT)%1(E1"xE[!])=12'TThiscanbesubstitutedbackintotheÞrst-orderconditiontoget:ö!=E[!"%1xT(x"%1xT)%1(E1"xE[!])Proposition11.System2ÕsbiascanbeexpressedasE2 Ep=0).Likewise,ifevery!j=E[!j],thenE2=EP.Inthiscase,ifyouobserveasu"cientnumberofobjectsyoucaninferallofSystem1Õsinformation,andthereforebiaswillgotozero.Inparticulariftherearen

41 SimilarargumentsaremadebyMarr(1982)andFo
SimilarargumentsaremadebyMarr(1982)andFodor(1983).22SimilarlyhesaysÒtheconstraints[i.e.,inferencesmadebythevisualsystem]showupeveniftheobserverknowsthatthereareconditionsinacertainscenethatrendertheconstraintsinvalidinthatparticularcase Second,thatassociation-basedjudgmentspersist;hegivesexamplesofreasoningproblemswhichÒcausepeopletobelievebelievetwocontradictoryresponsessimultaneously...[inwhich]theÞr

42 stresponsecontinuestobecompellingirrespe
stresponsecontinuestobecompellingirrespectiveofbeliefinthesecondanswer,irrespectiveevenofcertaintyinthesecondanswer.ÓStanovichandWest(2000)introducedtheÒSystem1ÓandÒSystem2Óterminology,andadditionallyarguedthatSlomanÕsdual-systemmodelcouldgiveagoodaccountofinterper-sonaldi!erencesinsusceptibilitytobias.DanielKahnemanhassinceadoptedtheSystem1/2terminologyasafoundationfortheÒheuristicsandbiasesÓprogram(K

43 ahnemanandFrederick(2005),Kahneman(2011)
ahnemanandFrederick(2005),Kahneman(2011)).Eachofthesewritersusesdi!erentwordstodescribetheknowledgewhichtheautomaticdoesnothaveaccessto(inthelanguageofthispaper, importanttotheinterpretationofthecurrentsituation,butourautomatically-producedintuitionsmaynottakethemintoconsideration.Finallyturningtoeconomicsanumberofrecentmodelshaveproposedthatpeoplemakerationaldecisionsbutusingonlyasubsetoftheavailablei

44 nformation;thisliteratureisoftencalledÒi
nformation;thisliteratureisoftencalledÒinattentionÓorÒrationalinattentionÓ(Sims(2005),Chettyetal.(2007),Woodford(2012),CaplinandMartin(2011),Gabaix(2012)).Thesetheoriesputlessemphasisontheexistenceoftwoseparatesystems,andinsteadassumethattheamountofinformationcanvarycontinuously,thoughinpracticetheyoftenemphasizethatthechoiceofinformationmaybemadeonce,exante,foraseriesofsubsequentdecisions.27Thepartiti

45 onbetween ÓÕDiscussingaparticularexample
onbetween ÓÕDiscussingaparticularexampletheysayÒ[A]lthoughpeoplearecapableofconsciouslycorrectingtheirimpressions...theycommonlyfailtodoso.ÓThemodelinthispaperdi!er Thereisevidencethathavingshortconsiderationtime,orcognitiveload,doestendtoincreasebiases,andthishasoftenbeenregardedasevidenceinsupportofadual-systemmodelofjudgment(Pocheptsovaetal.(2009),Alos-FerrerandStrack(forthcoming)). improvemeandecis

46 ionperformance...incentivesreducebiasesi
ionperformance...incentivesreducebiasesinonlyahandfulofcases.ÓThereisalsoevidencefromoutsideofthelaboratorythatdecisionscanbeinßuencedbynormativelyirrelevantdetailsevenwhenthestakesareveryhigh(Postetal.(2008),ThalerandBenartzi(2004)).Asimilarpointistrueforperceptualillusions:spendingalongertimestaringatanillusionwillsometimesreduceitsmagnitude,butrarelyeliminatesit(Predebonetal.(1998)).5.2BiasesareRati

47 onalgiventheInformationUsedThemodelpredi
onalgiventheInformationUsedThemodelpredictsthatanomaliesinjudgmentreßectoptimalinferencegiventheinformationavailabletoeachsystem.Thishastwoaspects.Underlowincentives(whenonlySystem1operates)judgmentsshouldberationalrelativetothelow- )whichisnormativelyirrelevantinthecurrentcase,then(i)thischangeshouldordinarilybeapositivesignalaboutv,and(ii)peopleshouldnotbeconsciouslyawareofthisassociation(ortheyunder

48 estimateit).Thisis29SeealsoHarrison(2006
estimateit).Thisis29SeealsoHarrison(2006). model.Thispredictionisacommonobservationinthestudyofperception.Inmanycaseslaboratoryexperimentshaveestablishedarelationshipbetweensensationandperceptionthatseemsarbitrary,butislaterfoundtocorrespondtoafactaboutthephysicalworld,i.e.thesensationisdiscoveredtobecorrelatedwiththepropertybeinginferredbyperception.AfamousexampleisEHWeberÕsdiscovery,in1876,thatcoolco

49 insarejudgedtobeheavierthanwarmcoinswhen
insarejudgedtobeheavierthanwarmcoinswhenplacedontheforehead.Weberproposedthatinformationabouttemperatureandpressurearesentthroughthesamenerves,thuscausinginterferencebetweenthesignals.Howeveritwaslaterfoundthatseparatenervescarriedthesignals,andthecommonlyacceptedmodernexplanationofthisphenomenonisthattemperatureisinfactaninformativesignalaboutweight:aheavycoinwillpressmoredeeplyintoyourßesh,soitwillma

50 keyourskincolderthanalightcoin(ifthecoin
keyourskincolderthanalightcoin(ifthecoinisbelowskintemperature).Thuswhenreceivingacoolersensationitisrationaltoinferthattheobjectonyourforeheadisheavier,allelseequal.Supportforthisexplanationcomesfromevidencethatjudgmentofweightis Infact,TverskyandKahnemanÕs1974paperintroducingthephraseÒheuristicsandbi-asesÓgivesamotivatingexamplefromperceptionthatÞtsthemodelinthispaperverywell.Theydiscussevidencethatp

51 eoplesystematicallyover-estimatedistance
eoplesystematicallyover-estimatedistancesonfoggydays.Thestandardexplanationforthisbiasisthatpeoplerationallyuseblurrinessasacueforesti- ToconÞrmthispredictionIranasurvey,askinghowfoga!ectsperceptionofdistance. i]!irepresentthatSystem2underestimatesthisrelationship.Finallyforafoggydayletzi[z ,willbeproportionalto ratelymayviolaterulesofconsistency,butjudgmentselicitedjointlywillalwaysbeconsistent.Noteag

52 ainthatjointly-elicitedjudgmentsneednotb
ainthatjointly-elicitedjudgmentsneednotbeunbiased,justconsistentrelativetosomenormativeconstraint.Notealsothattestingthispredictiondoesnotrequireknowinghowinformationispartitionedbetweenlow-levelhigh-levelaspectsofthecase(xandz).EvidencefromawidevarietyofsourcesseemstoconÞrmtheprediction.33Thesimplestcaseisofframinge!ects,whereanirrelevantchangeinthecontexta! A,E[v|x,z,!]=E[v|x",z",!],butE[v|x,z,E[v|x,

53 !]](=E[v|x",z",E[v|x",!]].35Someframinge
!]](=E[v|x",z",E[v|x",!]].35Someframinge!ectscannotbeeasilybepresentedjointly:forexample,choiceisoftena!ectedbymanipulationsofreferencepoint(asinKahnemanandTversk obabilitywasdistributeduniformlybetween$0and$10.Thedistributionofprobabilitiesisnormativelyirrelevant,byourusualstandards,yetthesubjectsineachconditionstatedasigniÞcantlydi! Finally,thesubjectswerepresentedwithbothdistributionsandaskedtosubmi

54 ttwobids,oneforeachscenario,andtheactual
ttwobids,oneforeachscenario,andtheactualdistributionusedwassubsequentlychosenbycoinßip.Inthisversionthedi!erenceinbidsshrankdramatically:theaveragebidswere$3.09and$3.38,respectively.ThisexperimenthasaclearinterpretationintermsofthispaperÕsmodel:thedistributionsofpricesaresalientfeatures(xandx!),butineachcasehigh-levelinfor-mation(zandz!)tellsusthatthedistributionisnormativelyirrelevant(i.e.,theserepres

55 entinformationthatthedistributionsareran
entinformationthatthedistributionsarerandomlydrawn).Judgmentisneverthelessa! 37 andone-thirdofsubjectsÕswitchÕpreferencesonrepeatedquestions.ÓHoweverastrikingfactisthat,despitethisvolatility,subjectsveryrarelychooseastochasticallydominatedoption(CarboneandHey(1995),LoomesandSugden(1998),Hey(2001)).38Thisisaparadoxformodelsofattention:peopleareinattentiveenoughtomakeinconsistentchoices,butattentiveenoug

56 htoneverchooseadominatedalternative.Howe
htoneverchooseadominatedalternative.Howeverthispatternmatchesamodel,asinthispaper,inwhichsubjectshavenormativepreferences,butreceiveadi!erentsignalaboutthenatureofthosepreferencesineachsituation.39KahnemanandFrederick(2005)discussthedi!erencesbetweenseparateandjointjudg-ment.40TheynotethatmanyjudgmentanomalieswereÞrstdiscovered therepeatedpairswas12percent.Ó39Thefactthatpeopleoftenreversetheirpreferenc

57 escouldbeduetoanyoftheothernoisepickedup
escouldbeduetoanyoftheothernoisepickedupbySystem1:thehistoryofpriorcases,orthetimeofday,orafeelingoffatigue. ,wherep!dominatesp.Thesechoicesareallowedbythemodelinthispaper,butitpredictsthatthepumpwillrunoutifthedecisionsaremadesequentially,i.e.afterbeingexposedtoallthreealternatives(p,p!,q)subjectswillmakesubsequentdecisionswiththesamebeliefsabout!,sowillmakeconsistentdecisions.Thispredictionissupporte

58 dbyChuandChu(1990),whoÞndthatclassicalpr
dbyChuandChu(1990),whoÞndthatclassicalpreferencereversals(LichtensteinandSlovic(1971))decreasesigniÞcantlyafterchoicesaremadesequentially,suchthatsubjectscometofacethepossibilityofamoneypump.Finally,whenappliedtoperceptualbiases,thetheorymakesthecounterfactualpredictionthatinconsistencieswilldisappearinjointpresentations.Thisproblemcanberesolvedifweadditionallyassumethatpeopledonothavedirectreßectiveac

59 cesstotheirownrawsensations,acommonobser
cesstotheirownrawsensations,acommonobservationintheliteratureonperception.Takethecommonillusion tionwillnotholdifSystem1observesadditionalcase-speciÞcinformationwhichisnotaccessibletoSystem2:thenSystem2willnotknowthatthedi!erencebetweenthetworeports(E1andE!1)isdueonlytothebackgroundcolor.Thisassumptionseemsreasonable:thatpeopledonothave43Similarcontraste!ect Finally,Loewensteinetal.(2003)discussfurther

60 evidencethatpeoplearepoorjudgesoftheirfu
evidencethatpeoplearepoorjudgesoftheirfuturepreferences(ÒprojectionbiasÓ).6Discussion6.1RelatedLiteratureThemodelinthispaperissimilartoexistingdual-agentmodelswithineconomics(e.g.,Fu-denbergandLevine(2006),BrocasandCarrillo(2008))inthatbehaviorresultsfromtheinteractionoftwoagentswithinthedecision-maker.Howevertheresultsinthosemodelsaredrivenbytheconßictofpreferences,thispaperdi erentlytoastreamofinform

61 ationdependingontheorderinwhichitisrecei
ationdependingontheorderinwhichitisreceived.48Thesemodelsallcanexplaindecisionsbeinga!ectedbyirrelevantinformation,butonlyirrelevantinformationinthepast,because )modelofÒcoarsethinkingÓwhichcanbeinterpretedasareduced-formversionofthemodelinthispaper.Inthatpaperdecision-makersobserveamessageandasituation,andfromtheseformajudgmentofvalue.Howeversubjectsareassumedtoconßatesomesetofsituations,thustheirreac

62 tiontothemessagecanbebiasedrelativetothe
tiontothemessagecanbebiasedrelativetothecaseinwhichtheydiscriminatedperfectlybetweensituations.50Inparticular,subjectsmaybea!ectedbyamessagewhichisuninformativeinthecurrentsituationwhenthatmessageisinformativeinothersituations.Inanexamplefromthepaper,puttingsilkinashampoobottlecancauseconsumerstovalueitmorehighly,despiteitbeingobjectivelyworthless,becauseconsumersco-categorizethissituationwithoneinwhic

63 hsilkinessisapositivesignalaboutthequali
hsilkinessisapositivesignalaboutthequalityofshampoo(e.g.ifitwassilkinessofthehairbeingtreatedbytheshampoo).51Intermsofthemodelofimplicitknowledge,thesituationisz,themessageisx,andtheinterpretationofthemessages(E[q|m]intheirpaper)is!.Mullainathanetal.(2008)assumesthatpeoplearefundamentallynon-Bayesian.Interpretedusingthemodelinthispaper,peopleareBayesian,buttheyhaveaccesstoanautomaticsystem,anditisthisw

64 hichcausestheirjudgmenttorespondtoirrele
hichcausestheirjudgmenttorespondtoirrelevantinformation.Theimplicitknowledgeinterpretationofcoarsethinkingmakesadditionalpredictionsabouttheirexamples:mostparticularly,itpredictsthatsubjectswillnotbeinßuencedbyanirrelevantattributeinjointevaluation:willingnesstopayforshampoowillbeinßuencedbythepresenceofsilkinabetween-subjectsexperiment,butnotinawithin-subjectsexperiment,inwhichbothproductsareevaluated

65 simultaneously.5249Example2inMueller-Fra
simultaneously.5249Example2inMueller-FrankandArieli(2012)Þtstheassumptionsofthemodelinthispaper:although Themodelofimplicitknowledgecanbeusedtointerprettheframinge!ectsandrelatedanomaliesoftenfoundineconomicdecision-making.Someofthewell-knownanomalouse!ectsoneconomicdecision-makingare(i)thatpeo- Second,itimpliesthatwehavepoorinsightintowhichfeaturesa!ectourautomaticjudgment,orwhate!ecteachfeaturehas.Th

66 isiscommonineverydayvernacular:peoplesay
isiscommonineverydayvernacular:peoplesaytheirdecisionisbasedonaÒhunchÓ,ÒintuitionÓ,Ògutf zisclosertoE zi].Ifthereisonlyonedimensionthatisunusual,i.e.forwhichzi(=E[zi],thenwecangivethesubjectacomparisoncase,(x!,z!),whichisidenticalexceptfordimensioni,i.e.xj=x!jforj(=i.ProvidingthiscomparisonwillallowSystem2toexactlyidentify!i(!i=E!1%E1x!i%xi),andthereforetoexactlyinferEP.55Ineverydayscenarios,thiscouldb

67 einterpretedastryingtoconsideracasefroma
einterpretedastryingtoconsideracasefromadi E2"!ixiE[zi]+!ixizi.56 andsetx"i=xiziE[zi]$1,andz"i=E[zi]zi,thenE"1=+x"i!iE[zi]=+xi!izi=EP,andE"2=E"1=EP.ThesubjectÕsjudgmentofthiscomparisoncasewillbeequaltothepooled-informationjudgmentoftheoriginalcase.Thishasasimpleinterpretationinthebaseball-cardsexample:togetanaccurateestimateofthevalueofapacket,simplygivethesubjectapacketwithallthefakecardsremoved,andta

68 ketheirestimateofthatpacket.Recallthatxi
ketheirestimateofthatpacket.Recallthatxi=1ifthepacketcontainsthatcard,andzi=1ifitisgenuine;sothetheoryproposessettingx"i=0whereverzi=0.Italsoimpliessettingx"i=E[zi]$1fortheothercards,butif JournalofEconomicPsychology.Anderson,B.L.,B.G.Khang,andJ.Kim(2011),ÒUsingcolortounderstandperceivedlight-ness.ÓJournalofVision,11.Arieli,ItaiandManuelMueller-Frank(2013),ÒInferringbeliefsfromactions.ÓAvailableat Jour

69 nalofriskanduncertainty,19,7Ð42.Caplin,A
nalofriskanduncertainty,19,7Ð42.Caplin,A.andD.Martin(2011),ÒAtestabletheoryofimperfectperception.ÓTechnicalreport,NationalBureauofEconomicResearch.Carbone,EnricaandJohnDHey(1995),ÒAcomparisonoftheestimatesofeuandnon-eupreferencefunctionalsusingdatafrompairwisechoiceandcompleterankingexperiments.Ó ,1Ð18.Gneezy,U.,J.A.List,andG.Wu(2006),ÒTheuncertaintye!ect:Whenariskyprospectisvaluedlessthanitsworstpossi

70 bleoutcome.ÓTheQuarterlyJournalofEconomi
bleoutcome.ÓTheQuarterlyJournalofEconomics,121,1283Ð1309.Grether,DavidMandCharlesRPlott(1979),ÒEconomictheoryofchoiceandthepreferencereversalphenomenon.ÓTheAmericanEconomicReview,69,623Ð638.Harrison,GlennW(2006),ÒHypotheticalbiasoveruncertainoutcomes.ÓUsingexperimentalmethodsinenvironmentalandresourceeconomics,41Ð69.Harrison,GlennWandJohnAList(2008),ÒNaturallyoccurringmarketsandexogenouslaboratoryexper

71 iments:AcasestudyofthewinnerÕscurse*.ÓTh
iments:AcasestudyofthewinnerÕscurse*.ÓTheEconomicJournal,118,822Ð843.Hey,JohnD(2001),ÒDoesrepetitionimproveconsistency?ÓExperimentaleconomics,4,5Ð54.Holley,Rose(2009),ÒHowgoodcanitget?analysingandimprovingocraccuracyinlargescalehistoricnewspaperdigitisationprograms.ÓD-LibMagazine,15.Hsee,C.K.(1998),ÒLessisbetter:Whenlow-valueoptionsarevaluedmorehighlythanhigh-valueoptions.ÓJournalofBehavioralDecisionMa

72 king,11,107Ð121.Hsee,C.K.,G.F.Loewenstei
king,11,107Ð121.Hsee,C.K.,G.F.Loewenstein,S.Blount,andM.H.Bazerman(1999),ÒPreferencereversalsbetweenjointandseparateevaluationsofoptions:Areview "E[v|p]])2=E[v2"2vE[v|p]+E[ v|p]2"v2]=Var[E[v|p]]Soequation3canberewrittenas:Var[E[v|p,q Theconditionalvarianceformulastatesthat:Var[X]=E[Var[X|A]]+Var[E[X|A]]Applyingtheconditionalvarianceformula,ourtargetcanbeexpressedas:E[Var[v|p,q]]&E[Var[v|p]]Applyingthec

73 onditionalvarianceformulato!r[v|p]weobta
onditionalvarianceformulato!r[v|p]weobtain:Var[v|p]=E[Var p,q E[Var[v|p,q]]&E[Var[v|p]]ProofofProposition1:Proof.WewishtoshowthatSystem2Õsbiasisalwaysweaklysmaller,i.e.Var[E[v|x,z,E[v|x,!]]]&Var[v"E[v|x,!]]thisfollowsdirectlyfromtheLemmaifwenotethat,bythelawofiteratedexpectations,E[v|x,!]=E[v|x,E[v|x,!]]thusE2canbewrittenasconditioningonastrictlylargerinformationsetthanE1.ProofofProposition2:Proof.Abia

74 sexistsifandonlyifthereissomex#X,!#A,z#Z
sexistsifandonlyifthereissomex#X,!#A,z#Z,suchthatE[v|x,z,!](=E[v|x,z,E[v|x,!]].WecanexpressE2asE2=E[v|x,z,E[v|x,!]]=öø!#AE[v|x,z,ø!]f(dø!|x,z,E[v|x,!])Iftheassumedconditionholds,thenforanyø [v|x,z,!!]E[v|x,z,E[v|x,!]]=E[v|x,z,E[v|x,!!]]BothcasesgiverisetothesameE2,buthavedi!erentEP,thusonecasemustbebiased,i.e.eitherfor!or!!,E2(=EP.ProofofProposition3.Supposethatthereexistedsomexand!,suchthattherewasano

75 n-zeroprobabilityofbias,i.e.:öz#Z1{E[v|
n-zeroprobabilityofbias,i.e.:öz#Z1{E[v| öz#Zf(dz|x)+öz# !!]'E[v|x,z,!]E[v|x,z,!!]'öø!#øAE[v|x,z,ø!]f(dø!|x,z)=E[v|x,z,E[v|x,!]]soforZ,*z#Z,E[v|x,z,!][v|!!]meaningthatif«z#Zf z#Z(E[v|x,z,!!]"E[v|x,z,!])f(dz|x)�0,howeverthiscontradictsourassumptionthat! Proof.TheÞnalexpectationcanbewrittenasanintegraloverexpec ö,övf(dv|x,z,ø!)-f(dø!|x,z,E1)=öE[v|x,z,ø!]f(dø!|x,z,E1 x, 1]=«E[v|x,z,ø!]f(E1|x,ø v|x,z