morerealistic)choiceproblemforwhichitisnotpossibletocomputeoptimalchoi - PDF document

morerealistic)choiceproblemforwhichitisnotpossibletocomputeoptimalchoices,anddemonstratethatthedirectedcognitionmodelpredictsaggregatesubjectbehavior.SectionIdescribesthesetupandtheresultsfortheÞrst(ÒsimpleÓ)experiment.SectionIIdescribesthesetupforthesecond(ÒcomplexÓ)experiment.SectionIIIdescribestheimplemen-tationofthedirectedcognitionalgorithminthecomplexexperiment.SectionIVsummarizestheresultsofthesecondexperimentandcom-paresthoseresultstothepredictionsofthemodel.SectionVconcludes.I.FirstExperiment:ChoiceamongThreeSimpleGoodsConsiderthefollowingdecisionproblem,whichisaspecialcaseoftheclassofproblemsinvestigatedbyJohnC.Gittins(1979)andMartinWeitzman(1979).Anagentchoosesamongthreeuncorrelatedprojects.Thethreeprojectshavere-spective(stochastic)payoffs,and.IfisaÒwinnerÓthen;otherwise,0.ProjectisawinnerwithprobabilityTheagentcansequentiallyinvestigateprojectsthatarenotknownwinners,therebyrevealingtheirstate.Suchinformationacquisi-tioncostsperproject.Theagentmaystopacquiringinformationatanytimeandchooseoneprojectamongthewinners,whichwewillrefertoasÒtakingÓaproject.Forexample,inoneofourexperimentalgamesweadopttheparametersreportedinTa-ble1(withinformationacquisitioncostforallprojects).Beforewederivetheoptimalstrategy,imag-inehowatypicalsubjectwouldstarttoplaythisgame.Wouldthesubjectbeginbyproject1(theonlyknownwinneratthemo-ment),bypayingadollarto2,orbypayingadollartoproject3?Theoptimalsequenceofinformationacqui-sitioncanbederivedusingaGittins-Weitzman(GW)index(Gittins,1979;andWeitzman,1979).Assumingriskneutrality,theGWindexisprojectÕsreservationvalue,suchIntuitively,theGWindexisthevalueofaoutsideoptionthatmakestheagentjustwillingtopaycosttorevealthetruevalueinsteadofimmediatelytakingtheoutsideForanuncertainprojectinoursetting,theGWindexvalueis,i.e., Afteraprojecthasbeeninvestigated,0andiseitherzeroorone.TheGWvaluewillbeforaknownwinnerand0foraknownloser.Gittins(1979)andWeitzman(1979)showthattheoptimalsequenceofactionstrackstheprojectwiththehighestvalueof.Ifthehighestvalueofcorrespondstoaprojectwithanunknownpayoff,thentheagentshouldacquireinformationaboutthatproject.Wehavealsoimplementedtheanalysisforrisk-averseandloss-aversepreferences.Usingstandardcalibrationsforriskaversion,lossaversion,andnarrowframing(e.g.,AmosTverskyandDanielKahneman,1992)doesnotchangeourtheoreticalpredictionsandourexperimentalÞndings.Inotherwords,inourexperiment,thepredictionsoftherationalmodel andtheDCmodeldonotchangeoncetheassumptionofriskneutralityisreplacedwithcalibratedlevelsofcurvatureintheutilityfunction.Iftheagentdoesnotexplorethevalueof,herpayoffistheÞctitiousoutsideoption.TheexpectedbeneÞtfromexplorationisEquatingthisexpectedbeneÞttothecostofsearch,yieldstheGittins-Weitzmanindex. PayoffinwinningstateProbabilityofwinningstateProject1Project2Project3Asubjectchoosesamongthreeuncorrelatedprojects.IfisaÒwinnerÓ(whichhappenswithprobability),itspayoffis;otherwise,itspayoffis0.Thesubjectcansequentiallyinvestigateprojectsthatarenotknownwinners,therebyrevealingtheirstate.Suchinformationacquisitioncosts$1perproject.Thesubjectmaystopacquiringinformationatanytimeandchooseoneprojectamongthewinners.Wouldasubjectbeginbytakingproject1(theonlyknownwinneratthemoment),bypaying$1toinvestigateproject2,orbypaying$1toinvestigateproject3?1044THEAMERICANECONOMICREVIEWSEPTEMBER2006 Ifthehighestvalueofcorrespondstoawinningproject,thentheagentshouldtakethatinvestmentproject,therebyendingthatForthegameabove,theagentshouldacquireinformationaboutproject2(thelongshot),tak-ingthatprojectifitturnsouttobeawinner.Otherwise,theagentshouldacquireinformationaboutproject3,takingthatprojectifitturnsouttobeawinnerandtakingproject1asalastresort.(SeeTable2.)Studyingthisclassofgamesenablesustorunanempiricalhorseracebetweentheoptimalsearchmodel(i.e.,theGWalgorithm)andthedirectedcognitionmodel(GabaixandLaibson,ApplyingtheDirectedCognitionAlgorithmThedirectedcognition(DC)model,ÒsolvesÓproblemsbysearchingasifsearchoperationwerethelastsearchopera-tion.Toapplydirectedcognition,wecalcu-latetheexpectedbeneÞtandcostofeachavailablesearchoperationasifthisoperationwerethelastoneexecutedbeforeaÞnalin-vestmentprojectistaken.bethevalueofthebestknownwinningprojectattime.The(myopic)expectedbeneÞtfrominvestigatingprojectThecostofthissearchoperationis1.DCselectsthesearchoperatorwiththehighestgainÑthedifferencebetweenbeneÞtsandcost:Thisformulaalsodescribesthemyopicgainfromtakingaknownwinner:0,and1.Thedirectedcognitionalgorithminvestigatesortakestheinvestmentwiththehighestvalue.Thealgorithmiteratesifaninvestmenthasnotyetbeentaken.Forthegameabove,thedirectedcognitionalgorithmpredictsthattheagentwillacquireinformationaboutproject3,takingthatproj-ectifitturnsouttobeawinner.Otherwise,theagentwillacquireinformationaboutproject2,takingthatprojectifitturnsouttobeawinnerandtakingproject1asalastresort.(SeeTable2.)TheDCalgorithmrecognizestheoptionvalueofbeingabletorejectthenextinvesti-gatedprojectifthatprojectdoesnotturnouttobeproÞtable.Butthealgorithmdoesnotrecognizetheoptionvalueofbeingabletoinvestigatesequencesofuncertainprojects.Anoptimalsearchcalculationneedstoeval-uatesuchsequences:ÒProject3hasahigherexpectedvaluethanproject2,butproject2hasasmallchanceofahighpayoff.IfIinvestigateproject2Þrst,anditisnotawinner,thenIcanproceedtoinvestigateproject3.ÓBecausedirectedcognitionismy-opicitcannotrecognizesuchchainsofrea-soning.Hence,gameslikethoseinTable1arewellsuitedtodistinguishbetweenoptimalsearchanddirectedcognition.Foranygame,letthelow-probabilityprojecthaveprobabilityofbeingawinnerandvaluewhenitisawinner.Letthehigh-probabilityprojecthaveprobabilityofbeingawinnerandvaluewhenitisawinner.Consideraparameterizationinwhich pipjVj
c LGORITHMSFOR Gittins-WeitzmanDirectedcognitionInvestigateproject2Investigateproject31Takeproject2ifwinnerTakeproject3ifwinner2Elseinvestigateproject3Elseinvestigateproject23Takeproject3ifwinnerTakeproject2ifwinner4Elsetakeproject1Elsetakeproject1VOL.96NO.4GABAIXETAL.:EXPERIMENTALANALYSISOFABOUNDEDLYRATIONALMODEL ThentheGWalgorithmbeginswithaninves-tigationofthelow-probabilityprojectandtheDCalgorithmbeginswithaninvestigationofhigh-probabilityproject.WestudyÞvegames(AÐE)withsuchnon-congruentpredictionsandÞveothergameswithcongruentpredictions(FÐJ).Fromaheuristicperspective,thenoncongruentgamesareonesinwhichthelow-probabilityinvestmenthashighvarianceandalowexpectedvalue.ExperimentalResultsOnehundredtwenty-ninesubjectsreceivedameantotalpayoffof$26.51,withastandardde-viationof$13.40.Payoffsrangedfrom$10to$63.Subjectsplayedtenrandomlyorderedinvest-mentgamesliketheoneinTable1.Eachofthegamesincludesalow-probabilityproject,ahigh-probabilityproject,andasurething.Theexperi-mentalprotocolandaWeb-basedsimulationoftheactualexperimentareavailableathttp://Fiveoftheteninvestmentgames(gamesAÐE)haveaGWstrategythatdiffersfromthedirectedcognitionstrategy,andthisdifferenceappearsintheÞrstmove.TheÞrstpanelinTable3reportstheproportionsofÞrstmovesineachofthesegamesthatmatchtheGWstrategyandtheproportionsofÞrstmovesthatmatchthedirectedcognitionstrategy.ForalloftheÞvegames,theGWproportionsarebelowtheDCproportions.WerejectthenullhypothesisthattheGWandDCproportionsareequalforfouroftheÞvegames.AveragingoverallÞvegames,34percentofthemovesfollowtheGWstrategy,and63per-centofthemovesfollowthedirectedcognitionstrategy.Werejectthenullhypothesisthatthesemeansareequivalentwithavaluelowerthan.Whenthepredictionsofthetwomodelsdiffer,directedcognitionpredictssubjectsÕchoicesbetterthanoptimalsearch.TheprobabilitiesdonotsumtoonebecauseneitherGWnorDCpredictsselectionofthesure-thinginvestmentsontheÞrstmove.Asmallnumberofsubjectsmadethischoice.Fiveoftheteninvestmentgames(gamesFÐJ)haveaGWstrategythatmatchestheDCstrat-egy.Table4reportstheproportionofÞrstmovesineachofthesegamesthatmatchestheGWandDCstrategy.Onaverage,74percentofthemovesfollowtheGWandDCstrategy.ThepayoffsandprobabilitiesoftheriskyprojectsingamesAÐJare:gameA:($21,0.09)and($10,0.76);gameB:($19,0.11)and($10,0.79);gameC:($23,0.09)and($13,0.72);gameD:($18,0.12)and($10,0.81);gameE:($20,0.12)and0.85);gameF:($22,0.48)and($11,0.74);gameG:($24,0.34)and($9,0.70);gameH:0.52)and($11,0.74);gameI:($25,0.39)and($9,0.70);gameJ:($10,0.09)and($8,0.85).Inallgames,thesurethingis$1.SubjectsareHarvardundergraduatesandHarvardsummerschoolstudents.Ofthetotal,61percentreporthavingtakenatleastonestatisticscourse.Ourgamblesmaybespecial.Forinstance,supposethattheriskyprojectswere($10,0.8)and($70,anexamplesuggestedtousbyareferee.Forthisgame,subjectsmightÞrstexplorethelow-probabilityprize,contradictingdirectedcognition.Sucharesultcould,however,beduetoprobability-reweighting.ProspecttheoryRACTIONOFEITZMANAND PercentageGWandDCGameF73GameG67GameH73GameI74GameJ83FÐJaverage74RACTIONOFLAYTHEOVEACCORDINGTOEITZMANORACCORDINGTO PercentageGWPercentageDCDifferenceGameA3365323.800.0001GameB3563283.300.0010GameC4257161.800.0726GameD2870425.300.0000GameE3462283.320.0009AÐEaverage3463294.550.00001046THEAMERICANECONOMICREVIEWSEPTEMBER2006 Wenowturntoananalysisofallmoves(notjusttheÞrstmove).Table5reportsthefractionofmovesthatarecorrectlypre-dictedbytheGWandDCstrategiesforgamesAÐE.Werecalculatestrategypredic-tionsaftereverymove,conditionalontheplayerÕsotherselections.WhilethestrategiespredictdifferentÞrstmoves,theyoftenpre-dictthesameplayforsubsequentmoves.Becauseofthisoverlap,thestrategiesarenotmutuallyexclusive,andprobabilitiesdonotsumtoone.ThisanalysisdoesnotdistinguishthestrategiesascleanlyastheÞrst-moveanalysispresentedabove.Wepresentthere-sultshereasarobustnesscheck.ForallÞvegames,theGWproportionsarebelowtheDCproportions.WerejectthenullhypothesisthatbothmodelsÕproportionsareequalforfouroftheÞvegames.AveragingoverallÞvegames,67percentofthemovesfollowtheGWstrategy,and79percentofthemovesfollowtheDCstrategy.Werejectthenullhypothesisthatthesemeansareequiva-lentwithavaluelessthan10Table6reportsthefractionofmoves(82percent)thatfollowtheGWandDCstrategyforgamesFÐJ.Wenowcalculatethenumberofgamesinwhichasubjectfollowedofthemovesprescribedbyoneofthetwoalgorithms(i.e.,GWorDC).SpeciÞcally,wecalculatethenumberofgamesinwhicheachsubjectfol-lowedaparticularalgorithmperfectlyfromstarttoÞnish,includingtheÞnalchoice.breakourgamesintotwosubgroups:thenon-congruentgames(AÐE),forwhichthetwoalgorithmsÕpredictionsdiverge,andthecon-gruentgames(FÐJ).Withrespecttothenoncongruentgames(AÐE),47percentofsubjectsdidnotfollowtheGWalgorithminoftheÞvegames,whileonly16percentofsubjectsfailedtofollowtheDCalgorithminoftheÞvegames.More-over,only11percentofsubjectsfollowtheGWalgorithminÞvegames,while22percentofsubjectsfollowtheDCalgorithmingames.Onaverage,subjectsplay1.6noncon-gruentgames(AÐE)exactlyfollowingtheGWalgorithmand2.6noncongruentgamesexactlyfollowingtheDCalgorithm.Theentirefre-quencydistributionisreportedinTable7(forboththecongruentandnoncongruentgames).Usingthesedata,theDCalgorithmoutperformsGW(WehavealsoanalyzedtheabilityofthemodelstopredicttheÞnaloutcomesoftheexperiment.Usingsuchoutcomesasthefocusofthestudy,theGWandDCmodelsaresta-tisticallyindistinguishable.Thelackofresolu-tionispartiallyduetothenoisinessofthe predictsthatsmallprobabilityeventsareoverweightedinIfasubjectfollowsthesearchstrategyofaparticularalgorithm,hewouldmakeabizarremistakeifhedidnÕtalsofollowtheÞnalchoicepredictions.FinalchoicesaremadeamongÒsure-thingÓpayoffs.Accordingly,mistakesonÞnal choicesareveryrare:theyhappenin1.9percentofthegamesthatareplayed.RACTIONOFEITZMANOR PercentageGWPercentageDCDifferenceGameA6679133.670.0002GameB6880113.210.0013GameC7076061.760.0785GameD6583175.000.0000GameE6678113.230.0012AÐEaverage6779124.390.0000RACTIONOFEITZMANAND MatchGW/DCGameF83%GameG78%GameH83%GameI83%GameJ86%FÐJaverage82%VOL.96NO.4GABAIXETAL.:EXPERIMENTALANALYSISOFABOUNDEDLYRATIONALMODEL lotteries.ThisanalysisisreportedintheAERWebAppendix.II.SecondSetofExperiments:ChoiceamongComplexGoodsThesearchexperimentabovestudiesaprob-lemthatisenoughtoadmitananalyticoptimalsolution.Realworldproblems,how-ever,tendtobefarmorecomplex.Indeed,manyrealworldproblemsdonothaveananalyticoptimalsolutionorevenanoptimalsolutionthatcanbepracticallycalculatednu-merically.Wewouldliketohavemodelsthatcansuccessfullypredictbehaviorinsuchcomplexenvironments.Withthesegoalsinmind,weanalyzeasec-ondexperimentthatcapturessomeofthecom-plexfactorsthatariseinrealworldproblems.InthisÒcomplexÓexperiment,subjectschooseonegoodfromasetofgoods,eachofwhichhasnumerousattributes.Inthiscomplexexperiment,decisiontimeisascarceresource,andinformationacquisitionismeasuredcontinuously.Wemaketimescarceintwodifferentways.First,wegivesubjectsanamountoftimetochooseonegoodfromasetofgoodsÑachoiceproblemwithanexogenoustimebudget.Herewemeasurehowsubjectsallocatetimeastheyacquireinforma-tionabouteachgoodÕsattributesbeforemakingaÞnalselection.Thenwegivethesubjectsanopen-endedsequenceofchoiceproblemsliketheoneabove.Inthistreatment,thesubjectskeepfacingdif-ferentchoiceproblemsuntilatotalbudgetoftimerunsout.Theamountoftimeasubjectallocatestoeachchoiceproblemisnowanvariable.Becausepayoffsarecu-mulativeandeachchoiceproblemhasapositiveexpectedvalue,subjectshaveanincentivetomovethroughthechoiceproblemsquickly.ButmovingtooquicklyreducesthequalityoftheirFollowingothereconomists(ColinF.Cam-ereretal.,1993;MiguelA.Costa-Gomesetal.,2001;EricJ.Johnsonetal.,2002;andCosta-GomesandVincentP.Crawford,forthcoming),weusetheÒMouselabÓprogramminglan-guagetomeasuresubjectsÕinformationac-InformationishiddenÒbehindÓboxesonacomputerscreen.Subjectsusethecomputermousetoopentheboxes.Mouselabrecordstheorderanddurationofinformationacquisition.Sinceweallowonlyonescreenboxtobeopenatanypointintime,theMouselabsoftwareenablesustopinpointwhatinformationthesubjectisacquiringonaJohnW.Payneetal.(1993)developedtheMouselablanguageinthe1970s.MouselabisoneofmanyÒprocesstracingÓmethods.Forexample,Payneetal.(1978)elicitmentalprocessesbyaskingsubjectstoÒthinkaloud.ÓJ.EdwardRusso(1978)recordseyemovements.REQUENCIESOFTHEUMBEROFAMESINWHICHAREDICTSTHETRATEGYANDTHE NumberofgamesGamesAÐENumberofgamesGamesFÐJFrequencyFrequencyGWstrategyDCstrategyGW/DCstrategy047%16%05%112%19%19%29%16%212%311%12%319%411%16%422%511%22%533%Mean1.62.59Mean3.45Median13Median4Standarddeviation1.831.81Standarddeviation1.51Thetabledisplaysthenumberofgamesinwhichthefullstrategyoftheplayer,includingsearchpatternandÞnalchoices,coincidedwiththepredictionsoftheGWversusDCmodel.ThosepredictionsaredifferentingamesAÐE,andthesameingamesFÐJ.DCmakesasuccessfulpredictiononagreaternumberofgamesthanGW(1048THEAMERICANECONOMICREVIEWSEPTEMBER2006 second-by-secondbasisthroughouttheTheDetailsofanN-GoodChoiceTask-goodgameisan-rowbymatrixofboxes(Figure1).Eachboxcontainsarandompayoff(inunitsofcents)generatedwithnormaldensityandzeromean.Afteranalyzing-goodgame,thesubjectmakesaÞnalse-lectionandÒconsumesÓasinglerowfromthatgame.Thesubjectispaidthesumoftheboxesintheconsumedrow.Consumingarowrepresentsanabstractionfromaverywideclassofchoiceproblems.Wecallthisprobleman-goodgame,sincerowsconceptuallyrepresentThecolumnsrepresentdifferentattributes.Forexample,considerashopperwhohasdecidedtogotoWal-Marttoselectandbuyatelevision.TheconsumerfacesaÞxednumberoftelevisionsetsatWal-Mart(differentTVÕsfromwhichtochoose).ThetelevisionsetshavedifferentattributesÑsize,price,remotecon-trol,warranty,etc.Byanalogy,theTVÕsaretherowsofFigure1,andtheattributes(inautilitymetric)appearinthecolumnsofeachInourexperiment,theimportanceorvari-abilityoftheattributesdeclinesasthecol-umnsmovefromlefttoright.Inparticular,thevariancedecrementsacrosscolumnsequalone-tenthofthevarianceincolumnone.Forexample,ifthevarianceusedtogeneratecol-umnoneis1,000(squaredcents),thenthevari-anceforcolumn2is900,andsoon,endingwithavarianceforcolumn10of100.Socolumnsontheleftrepresenttheattributeswiththemost(utility-metric)variance,likescreensizeorpriceinourTVexample.MouselabhasthedrawbackthatitusesanartiÞcialdecisionenvironment,butseveralstudieshaveshownthattheMouselabenvironmentdistortsÞnalchoicesovergoods/actionsonlyminimally(e.g.,Costa-Gomesetal.,2001;Costa-GomesandCrawford,forthcoming).MouselabÕsin-terfacedoesgenerateÒupper-leftÓandÒleft-to-rightÓsearchbiases,whichwediscussinSectionIIBbelow. 1.SAMEWITHIntheactualexperiment,thesubjectsseethevalueofonlyoneboxatatime(seeFigure2).Valuesineachcolumnaredrawnindependentlyfromanormaldistributionwiththesamevariance,withvariancesdeclininglinearlyfromlefttoright.Inthissample,theleft-mostcolumnisgeneratedwithastandarddeviationof30.6cents,whichisexplainedtosubjectsasa95-percentconÞdenceintervalof60to60cents.VOL.96NO.4GABAIXETAL.:EXPERIMENTALANALYSISOFABOUNDEDLYRATIONALMODEL Columnsontherightrepresenttheattributeswiththeleast(utility-metric)variance,likeminorclausesinthewarranty.Sofarourgamesoundssimple:ÒConsumethebestgood(i.e.,row).ÓTomeasureinforma-tionacquisition,wemaskthecontentsofboxesincolumns2through.Subjectsareshownonlytheboxvaluesincolumn1.Asubjectcan,however,left-clickonamaskedboxincolumns2throughtounmaskthevalueofthatbox(Figure2).boxfromcolumns2throughbeunmaskedatatime.Thisprocedureenablesustorecordexactlywhatinformationthesub-jectisobservingateverypointintime.vealingthecontentsofaboxdoesnotimplythatthesubjectconsumesthatbox.Note,too,thatifarowispickedforconsumption,thenallboxesinthatrowareconsumed,whetherornottheyhavebeenpreviouslyunmasked.Weintroducetimepressure,sothatsubjectswillnotbeabletounmaskÑorwillnotchoosetounmaskÑalloftheboxesinthegame.Mouselabrecordswhichofthe1)maskedboxesthesubjectsunmask.Ofcourse,wealsorecordwhichrowsthesub-jectschoose/consume.WestudyasettingthatreßectsrealisticÑi.e.,highÑlevelsofdecisioncomplexity.Thiscom-plexityforcessubjectstoconfronttradeoffs.Realconsumersinrealmarketsfrequentlyfacedecisionsthataremuchmorecomplex.Maskedboxesandtimepressurecaptureim-portantaspectsofourWal-MartshopperÕsex-perience.TheWal-MartshopperselectivelyattendstoinformationabouttheattributesoftheInourexperiment,alloftheattributeshavebeendeWerevealthevalueofcolumn1becauseithelpssubjectsrememberwhichrowiswhich.Inaddition,reveal-ingcolumn1initializesthegamebybreakingtheeight-waytiethatwouldexistifsubjectsbeganwiththeexpectationthatallrowshadthesamevalue(zero).Whenwedesignedtheexperiment,weconsideredbutdidnotadoptadesignthatpermanentlykeepsboxesopenoncetheyhavebeenselectedbythesubject.Thisalternativeapproachhastheadvantagethatsubjectsfaceareducedmemoryburden.Ontheotherhand,ifboxesstayopen permanently,subjectshavetheoptiontoquicklyÑandme-chanicallyÑopenmanyboxes,andonlyafterwardanalyzetheircontent.Hence,leavingboxesopenimpliesthatwelosetheabilitycrediblytoinferthesubjectÕsattentionateachpointintime. 2.SAMEWITHThisishowasamplegamewouldappeartosubjects,withvaluesconcealedbyboxes.Subjectscanusethemousetoopenoneboxatatime.Inthisgamethesubjectfacesasettimelimit;theclockintheupper-rightcornerrevealsthefractionoftimeremaining.1050THEAMERICANECONOMICREVIEWSEPTEMBER2006 TVsamongwhichsheispicking.Theshoppermayalsofacesometimepressure,eitherbe-causeshehasaÞxedamountoftimetobuyaTV,orbecauseshehasothertasksshecandointhestoreifsheselectsherTVquickly.Weexplorebothtypesofcasesinourexperiment.GameswithExogenousandEndogenousTimeBudgetsInourexperiment,subjectsplaytwodifferenttypesof-goodgames:gameswithexogenoustimebudgetsandgameswithendogenoustimebudgets.WewillrefertotheseasÒexogenousÓandÒendogenousÓgames.Foreachexogenousgameagame-speciÞctimebudgetisgeneratedfromtheuniformdis-tributionovertheinterval[10seconds,49sec-onds].Aclockshowsthesubjecttheamountoftimeremainingforeachexogenoustimegame(seeclockinFigure2).ThisisthecaseofaWal-MartshopperwithaÞxedamountoftimetobuyagood.Inendogenousgames,subjectshaveaÞxedbudgetoftimeÑ25minutesÑinwhichtoplayasmanydifferent-goodgamesastheychoose.Inthisdesign,adjacent-goodgamesaresep-aratedby20-secondbufferscreens,whichcounttowardthetotalbudgetof25minutes.Subjectsarefreetospendaslittleorasmuchtimeastheywantoneachgame,sotimespentoneachgamebecomesanendogenouschoicevariable.ThisisthecaseofaWal-MartshopperwhocanmoveontootherpurchasesifsheselectsherTVWestudybothexogenoustimegamesandendogenoustimegamesbecausethesetwoclassesofproblemscommonlyariseintherealworldandanycognitionmodelshouldbeabletohandlebothsituationsrobustly.Bothtypesofproblemsenableustostudywithin-problemattentionallocationdecisions.Inaddition,theendogenoustimegamesprovideanaturalframeworkforstudyingstoppingrules,i.e.,thedecisiontoallocatelessanalysistothecurrentgame,freeingtimeforsubsequentgames.ExperimentalLogisticsSubjectsreceiveprintedinstructionsexplain-ingthestructureofan-goodgameandthesetupfortheexogenousandendogenousgames.SubjectsarethengivenalaptoponwhichtheyreadinstructionsthatexplaintheMouselabin-terface.Subjectsplaythreetestgames,whichdonotcounttowardtheirpayoffs.Thensubjectsplay12gameswithseparateexogenoustimebudgets.Finally,subjectsplayasetofendogenousgameswithajoint25-minutetimebudget.Forhalfofthesubjectswereversetheorderoftheexogenousandendog-enousgames.Attheendoftheexperiment,subjectsanswerdemographicanddebrieÞngSubjectsarepaidthecumulativesumofallrowsthattheyconsume.Aftereverygame,feedbackreportstherunningcumulativevalueoftheconsumedrows.III.ApplicationoftheDirectedCognitionApplicationofthedirectedcognitionmodelcanbebrokendownintothreeiterativesteps,whichweÞrstsummarizeandthendescribeinStep1:Usingamyopicplanninghorizon,calculatetheexpectedeconomicbeneÞtsandcostsofdifferentpotentialsearchoperations.SpeciÞ-cally,evaluateeachincrementalsearchoperationasifitwerethelastsearchoperation.Step2:ExecutethesearchoperationwiththehighestratioofexpectedbeneÞttocost.Forexample,ifexplorationoftheÒnextÓtwoboxesinthesixthrowhasthehighestexpectedratioofbeneÞttocost,thenunmaskthosetwoboxes.Step3:Returntostep1unlesstimehasrunout(inexogenoustimegames)oruntiltheratioofexpectedbeneÞttocostfallsbelowsomethresholdvalue(inendogenoustimegames).Sinceourgamesallhaveeightrows(goods),welabeltherows,...,.Weuselower-caseWeimaginethatthissteprecruitsbothconsciousandunconsciousmentalprocessinganddonottakeapositionontheirrelativecontributions.DanielM.Wegner(2002)andThomasGilovichetal.(2002)arguethatunconscious/au-tomatedmentalprocessesplayacentralroleinmanybe-havioralchoices.Muchoftheactivityofthelimbicsystemappearstobeunavailabletoconsciousness,butneverthelessiscriticalforrewardvaluation,approach/avoidance,andattentionallocation.VOL.96NO.4GABAIXETAL.:EXPERIMENTALANALYSISOFABOUNDEDLYRATIONALMODEL lettersÑ,...,ÑtotrackasubjectÕsexpecta-tionsofthevaluesoftherespectiverows.Thesubjectknowsthevaluesofallboxesincolumn1whenthegamebegins.Thus,atthebeginningofthegame,therowexpectationswillequalthevalueofthepayoffintheleft-mostcellofeachrow.Forexample,ifrowhasa23initsÞrstcell,thenattimezeroIfthesubjectunmasksthesecondandthirdcellsinrow,revealingcellvaluesof17andwouldbeupdatedto2911.Wenowdescribethethreestepsindetail.Step1Instep1theagentcalculatestheexpectedeconomicbeneÞtsandcostsofdifferentsearchoperations.Forourapplication,asearchopera-tionisapartialorcompleteunmasking/analysisofboxesinaparticularrowofthematrix.Suchanoperationenablesthedecisionmakertoim-proveherforecastoftheexpectedvalueofthatrow.Inournotation,representstheoperationÒopenadditionalboxesinrow.ÓBe-causetractabilityconcernsleadustolimitagentsÕplanninghorizonstoonlyasinglesearchoperationatatime,weassumethatin-dividualsearchoperationsthemselvescanin-cludeoneormoreboxopenings,thoughmultipleboxopeningsmustbesequentialandinthesamerow.Multipleboxopeningsincreasetheamountofinformationrevealedbyasinglesearchoperator,increasetheoptionvalueofinformationrevealedbythatsearchoperator,andmakethe(partiallymyopic)modelmoreTheoperatorselectstheboxestobeopenedusingamaximal-informationrule.Inotherwords,theoperatorwouldselecttheunopenedboxes(inrow)thathavethehighestvariances(i.e.,withthemostinformation).Inourgame,thiscorrespondswithlefttorightboxopenings(skippinganyboxesthatmayhavebeenopenedalready).Weassumethatanoperatorwhoopensboxeshascost,whereisthecostofunmaskingasinglebox.Wetakethiscosttoincludemanycomponents,includingthetimeinvolvedinopeningtheboxwiththemouse,readingthecontentsofthebox,andupdatingTheexpectedbeneÞt(i.e.,optionvalue)ofasearchoperationisgivenby 
x x representsthestandardnormaldensityrepresentstheassociatedcumula-tivedistributionfunction,istheestimatedvaluegapbetweentherowthatisunderconsid-erationanditsnextbestalternative,andisthestandarddeviationofthepayoffinformationthatwouldberevealedbythesearchoperator.Figure3plotsthisbeneÞtfunction.Wemoti-vateequation(3)below,butÞrstpresentanexamplecalculation.InthegameshowninFigure2,considerasearchoperatorthatexploresthreeboxesin.Theinitialexpectedvalueof28.Thebestcurrentalternativeisrowwhichhasacurrentpayoffof23.SotheTogainintuitionforthiseffect,considertwogoods,with0.SupposethateachoftworemainingboxesinrowAcantakethevalue1or1.Aftertheinformationinasingleboxhasbeenrevealed,.Hence,apartiallymyopicagentwillnotseethebeneÞtofopeningonebox,sincenomatterwhathap-.If,however,theagentconsidersopeningbothremainingboxes,thereisachancethatwillfallbelow0,implyingthatgatheringtheinformationfromthetwoboxeswouldbeusefultotheagent. ThisÞgureplotstheexpectedbeneÞtfromcontinuedsearchonanalternativeifthedifferencebetweenthevalueofthesearchedalternativeandthebestoftheotheralter-nativesis,andthestandarddeviationoftheinformationgainedis1asdeÞnedinequation(3).Thisishomogeneousofdegreeone.1052THEAMERICANECONOMICREVIEWSEPTEMBER2006 estimatedvaluegapbetweenandthebestalternativeisAboxincolumnwillrevealapayoffvariance(40.8)/10),andtheupdatedvalueofafterthethreeboxeshavebeenopenedwillbeHencethevarianceoftheinformationrevealedbythesearchoperatoris 8 7 63.2.SothebeneÞtofthesearchoperatoris7.5,anditscostisWenowmotivateequation(3).ToÞxideas,consideranewgame.Supposethatthedecisionmakerisanalyzingrowandwillthenimme-diatelyusethatinformationtochoosearow.Assumethatrowwouldbetheleadingrowifwereeliminated,sorowisthenextbestalternativetorowTheagentisconsideringlearningmoreaboutbyexecutingasearchoperator.Executingthesearchoperatorwillenabletheagenttoupdatetheexpectedpayoffofrow,whereisthesumofthevaluesinthenewlyunmaskedboxesinrowIftheagentdoesnotexecutethesearchop-erator,herexpectedpayoffwillbeIftheagentplanstoexecutethesearchoperator,herexpectedpayoffwillbeThisexpectationcapturestheoptionvaluegen-eratedbybeingabletopickeitherroworrowontheinformationrevealedbysearchoperator.Thevalueofexecutingthesearchoperatoristhedifferencebetweentheprevioustwoexpressions:Thisvaluecanberepresentedwithasimpleexpression.Letrepresentthestandarddevia-tionofthechangeintheestimateresultingfromapplyingthesearchoperatorThevalueofthesearchoperatorisTodevelopintuitionforthefunction,Fig-ure3plots,1).Inthegeneralcase,,1).Theoptionvalueframeworkcapturestwofundamentalcomparativestatics.First,thevalueofarowexplorationdecreasesthelargerthegapbetweentheactiverowandthenextbest)isdecreasingin.Second,thevalueofarowexplorationincreaseswiththevariabilityoftheinformationthatwillbeob-)isincreasingin.Inotherwords,themoreinformationthatislikelytobeThisresultassumesGaussianinnovations,whichisthedensityusedtogeneratethegamesinourexperiment.Toderiveequation(5),beginbyassumingthat.Inthis,0).isdrawnfromaNormal(0,)distribution,wegettheright-handsideofequation(5): d ab a
b d ab  d 
a
b1
a
b a
b 
a
b a
b ).ThecalculationissimilarwhenVOL.96NO.4GABAIXETAL.:EXPERIMENTALANALYSISOFABOUNDEDLYRATIONALMODEL revealedbyarowexploration,themorevalu-ablesuchanexplorationbecomes.Step2Step2executesthesearchoperationwiththehighestratioofexpectedbeneÞttocost.RecallthattheexpectedbeneÞtofanoperatorisgivenbythe)functionandthattheimplemen-tationcostofanoperatorisproportionaltothenumberofboxesthatitunmasks.ThesubjectexecutesthesearchoperatorwiththegreatestbeneÞt/costratio, isthecostofunmaskingasinglebox.isconstant,thesubjectexecutesthesearchoperatorargmax TheWebappendixtothisarticlecontainsanexampleofsuchacalculation.Step3Step3isastoppingrule.Ingameswithanexogenoustimebudget,thesubjectkeepsre-turningtostep1untiltimerunsout.Ingameswithanendogenoustimebudget,thesubjectkeepsreturninguntilfallsbelowthemarginalvalueoftime,whichmustbecalibrated.CalibrationoftheModelWeusetwodifferentmethodstocalibratethevalueoftimeduringthetimegames.First,weestimatethemarginalvalueoftimeasperceivedbyoursubjects.Advertisementsfortheexperimentimpliedthatsubjectswouldbepaidabout$20fortheirpar-ticipation,whichwouldtakeaboutanhour.Inaddition,subjectsweretoldthattheexperimentwouldbedividedintotwohalves,andthattheywereguaranteeda$5show-upfee.Usingthisinformation,wecalculatethesub-jectsÕanticipatedmarginalpayoffperunittimeduringgameswithendogenoustimebudgets.Thismarginalpayoffperunittimeistherelevantop-portunitycostoftimeduringtheendogenoustimegames.Sincesubjectswerepromised$5ofguar-anteedpayoffs,theirexpectedpayofffortheirchoicesduringtheexperimentwasabout$15.Dividingthisinhalfimpliesanexpectationofabout$7.50ofmarginalpayoffsfortheendog-enoustimegames.Sincetheendogenoustimegameswerebudgetedtotake25minutes,whichwasknowntothesubjects,theperceivedmarginalpayoffpersecondoftimeintheexperimentwas750cents 25minutes60seconds/minutecents/second.Sincesubjectstookonaverage0.98secondstoopeneachbox,weendupwithanimpliedmarginalshadowcostperboxopeningofseconds/boxcents/box.Wealsoexploreaone-parameterversionofthedirectedcognitionmodel,inwhichthecostofcognitionÑÑischosentomakethemodelpartiallyÞtthedata.Calibratingthemodelsoitmatchestheaveragenumberofboxesexploredintheendogenousgamesimpliescents/box.Hereischosenonlytomatchtheaverageamountofsearchperendogenoustimegame,nottomatchtheorderofsearchorthedistributionofsearchacrossgames.Wepostulatethattheagentpickstheoperatormaximizestheratioratherthanthedifference,whereisamarginalcostoftime.TheratiocriteriondoesnotrequirethesubjecttocalculateHence,theratioformulationismorefrugalthanthediffer-enceformulation.InthesimpleexperimentofSectionI,usingaratioordifferenceformulationleadstoidenticalpredictions,asbothriskyprojectsrequirethesamesearchcost.Ingeneral,however,theratioanddifferenceap-proachesyielddifferentpredictions.Whenthecostoftimeisknown,thedifferenceformulationispreferable.OurmodelthusgivesacrudebutcompactwaytoaddresstheÒaccuracyversussimplicityÓtrade-offincog-nitiveprocessing.SeeEnriquetaAragonesetal.(2005).1054THEAMERICANECONOMICREVIEWSEPTEMBER2006 ConceptualIssuesThismodeliseasytoanalyzeandiscompu-tationallytractable,implyingthatitcanbeem-piricallytested.Thesimplicityofthemodelfollowsfromthreespecialassumptions.First,themodelassumesthattheagentcalculatesonlyapartiallymyopicexpectedgainfromexecut-ingeachsearchoperator.ThisassumptionadoptstheapproachtakenbyPhilippeJehiel(1995),whoassumesaconstrainedplanninghorizoninagame-theorycontext.Second,theDCmodelassumesthattheagentusesaÞxedpositiveshadowvalueoftime.Thisshadowvalueoftimeenablestheagenttotradeoffcurrentopportunitieswithfutureopportunities.Third,theDCmodelavoidstheinÞniteregressproblem(i.e.,thecostsofthinkingaboutthink-ingaboutthinking,etc.)byusingamyopicoptionvaluecalculationassumedtobecostlesstotheagent.Insomeevolutionaryrelevantdo-mains,thebrainmayhaveevolvedspecializedalgorithmstosolvetheinÞniteregressproblem(e.g.,intuition).ButgiventheÒevolutionaryirrelevanceÓoftheexperimentaltask,itisun-likelythatsophisticatedevolvedsolutionswouldbeapplicabletothecurrentproblem.Hence,acrudemyopicsolutionÑliketheoneweadoptÑmaybereasonable.Ourassumptionthatthemyopicoption-valuecalculationiscost-lessisnotrealistic,however,andismadeonlywithmodelingconvenienceinmind.Withoutsomeversionofthesethreesimpli-fyingassumptions,themodelwouldnotbeuse-fulinpractice.Withoutsomepartialmyopia(i.e.,alimitedevaluationhorizonforoptionvaluecalculations),theproblemcouldnotbesolvedeitheranalyticallyorcomputationally.Withoutthepositiveshadowvalueoftime,theagentwouldnotbeabletotradeoffhercurrentactivitywithunspeciÞedfutureactivitiesandwouldneverÞnishanendogenoustimegamewithoutÞrst(counterfactually)openingupalloftheboxes.Finally,withouteliminatingcogni-tioncostsatsomeprimitivestageofreasoning,maximizationmodelsarenotwelldeÞned.WereturnnowtotheÞrstofthethreepointslistedinthepreviousparagraph:theperfectlyrationalsearchmodelisnotsolvableinourcontext.Anexactsolutionoftheperfectratio-nalitymodelrequiresthecalculationofavaluefunctionwith17statevariables:oneexpectedvalueforeachoftheeightrows,onestandarddeviationofunexploredinformationineachoftheeightrows,andÞnallythetimeremaininginthegame.Thisdynamicprogrammingproblemsuffersfromthecurseofdimensionalityandwouldoverwhelmmodernsupercomput-Bycontrast,thedirectedcognitionmodelisequivalenttoeightcompletelyproblems,eachofwhichhasonlytwostate,thedifferencebetweenthecurrentexpectedvalueoftherowandthecurrentex-pectedvalueofthenextbestalternativerow;,thestandarddeviationofunexploredinformationintherow.SotheÒdimensionalityÓoftheDCmodelisonly2(comparedto17forthemodelofperfectrationality).Wecancomparetheperformanceofthepar-tiallymyopicDCmodelandtheperformanceoftheperfectlyrationalmodel.LiketheDCmodel,theperfectlyrationalmodelassumesthatexamininganewboxiscostlyandthatcalculatingtheoptimalsearchiscost-less(analogoustoourassumptionthatsolving*iscostless).TheWebAppendixgiveslowerboundsonthepayoffsoftheDCmodelrelativetothepayoffsoftheperfectlyrationalmodel.DCdoesatleast91percentaswellasperfectrationalityforexogenoustimegamesGabaixandLaibson(2006)proposethatconsumermyopiaexplainsaseriesofmarketphenomena.SeeConlisk(1996)foradescriptionoftheinÞniteregressproblemandanexplanationofwhyitplaguesalldecisioncostmodels.WefollowConliskinadvocatingexogenoustruncationoftheinÞniteregressofthinking.Approximatingalgorithmscouldbedeveloped,butafterconsultingwithexpertsinoperationsresearch,weconcludedthatexistingapproximationalgorithmscannotbeusedwithoutaprohibitivecomputationalburden.Ourprob-lemisdiscrete(eightrows,tenboxes),andsothestate-spaceislarge(sevencontinuousvariablesfortherelativecurrentpayoffs,pluseightdiscretevariablesforthedepthofoperations),andthereisnoclearwaytosimplifythestate-space.Thestandardsimplifyingtools(DimitriP.BertsekasandJohnN.Tsitsiklis,1996)donotapplyhere.TheGittinsindex(Gittins,1979;Weitzman,1979)doesnotapplyhereeither,formuchthesamereasonitdoesnotapplytomostdynamicproblems.InGittinsÕsframework,itiscrucialthatonecandoonlyonethingtoarow(i.e.,anÒarmÓ)ataparticularpointintime.Incontrast,inourgame,asubjectcandomorethanonethingwitharow.Shecanexploreitfurther,ortakeitandendthegame.Hence,ourgamedoesnotÞtintoGittinsÕsframework.Weexploredseveralmod-iÞcationsoftheGittinsindex,buttheyprovedunfruitfulatbreakingthecurseofdimensionality.VOL.96NO.4GABAIXETAL.:EXPERIMENTALANALYSISOFABOUNDEDLYRATIONALMODEL andatleast71percentaswellasperfectration-alityforendogenoustimegames.Inotherwords,usingtheDCalgorithminsteadofper-fectrationalitywouldreducepayoffsbynomorethan8percentand29percent,respec-tively,intheexogenousandendogenoustimeOtherDecisionAlgorithmsInthissubsection,wedescribeseveralnaivebenchmarkswhichwecomparetothedirectedcognitionmodel.Thesebenchmarksaremoti-vatedbyeconomistsÕinterestindeterminingwhetherothersimplemodelscanorganizeourexperimentaldata.ThebenchmarksarealsorelatedtoprototypicalmodelsinthepsychologyColumnModel.columnmodelmasksalltheboxesincolumn2(toptobottom),thenincolumn3,...,etc.Inexogenousgames,thiscolumn-by-columnunmaskingcontinuesuntilthesimulationhasexploredthesamenum-berofboxesasaÒyokedÓsubject.Inendog-enoustimegames,theunmaskingcontinuesuntilarowhasbeenrevealedwithanestimatedvaluegreaterthanorequaltoColumnmodel,anaspirationorsatisÞcinglevel.Thisaspirationlevel(andthosediscussedbelow)areallchosensothatthesimulationsgenerateanaveragenumberofsimulatedboxopeningsthatmatchestheaveragenumberofempiricalboxopenings(26boxespergame).RowModel.rowmodelunmasksboxesrowbyrow,sequencingtheunmaskingaccordingtothevaluesincolumn1.Inexogenousgames,thisrow-by-rowunmaskingcontinuesuntilthesimulationhasexploredthesamenumberofboxesasayokedsubject(seepreviousfootnote).Inendogenousgames,theunmaskingcontinuesuntilarowhasbeenrevealedwithanestimatedvaluegreaterthanorequaltoRowmodelDirectedCognitionwithSatisÞcing.ÑInen-dogenousgames,wealsoanalyzeaDCmodelwithasatisÞcingstoppingrule.Inthismodel,theDCmodelisiterateduntilarowhasbeenrevealedwithanestimatedvaluegreaterthanorequaltoDCmodelEliminationbyAspects.ÑAchoicealgorithmEliminationbyAspects(EBA)hasbeenwidelystudiedinthepsychologyliterature(e.g.,Tversky,1972;andPayneetal.,1993).WeuseEBAtoanalyzegameswithendogenoustimebudgets.EachrowisagoodwithtendifferentattributesorÒaspectsÓrepresentedbythetendifferentboxesoftherow.TheEBAalgorithmproceedsaspectbyaspect(i.e.,columnbycol-umn)fromlefttoright,eliminatinggoods(i.e.,rows)withanaspectthatfallsbelowsomeaspirationvalue.Thiseliminationcontinues,stoppingatthepointwheretheinationwouldeliminateallremainingrows.Atthisstoppingpoint,EBApickstheremainingrowwiththehighestestimatedvalue.IV.ResultsOur388subjectsreceivedameantotalpay-offof$29.23,withastandarddeviationof$5.49.Payoffsrangedfrom$13.07to$46.69.Allsubjectsplayed12gameswithexogenoustimes.Onaverage,subjectschosetoplay28.7gamesundertheendogenoustimelimit,withastandarddeviationof7.9.Thenumberofgamesplayed(endogenousandexogenousgamescombined)rangedfrom21to65.OurAERWebAppendixprovidesdetailsabout(negative)testsforsubjectlearningeffects,reportsevidencethatsubjectshaveimperfectmemory,andde-scribesour(standard)bootstrapmethodologyforcalculatingstandarderrors.TheAppendixalsocontainstheexperimentalprotocolandaWeb-basedsimulationoftheactualexperiment.Ofcourse,theseboundsdonotapplytodifferentclassesofgames.ItispossibletoconstructspeciÞcgamesinwhichthelossesapproach100percent.Partiallyrationalalgorithmswillalwaysfailinspecialcircumstancesthataredesignedtoexploitthealgorithm.SeeCamererandTeck-HuaHo(1999),IdoErevandAlvinE.Roth(1998);andGerdGigerenzeretal.(1999).Insuchayoking,thesimulationistiedtoaparticularsubject.IftheempiricalsubjectopensboxesingametheyokedsimulationopensboxesingameSubjectsareHarvardUniversityundergraduates.Ofthem,55percentreporthavingtakenatleastonestatisticsPayoffsdonotvarywithsubjectdemographics(e.g.,statisticalcoursework).Weadoptthesimplifyingapproxi-mationthatallsubjectshaveidenticalstrategies.1056THEAMERICANECONOMICREVIEWSEPTEMBER2006 GameswithExogenousTimeBudgetsWecomparetheempiricalpatternsofboxopeningstothepatternsofboxopeningspre-dictedbythedirectedcognitionmodel.Webe-ginwithatrivialpredictionofourtheory:subjectsshouldalwaysopenboxesfromlefttoright,followingadecliningvariancerule.Inourexperimentaldata,subjectsfollowthedecliningvariancerule91.0percentofthetime(s.e.0.8SpeciÞcally,whensubjectsopenapreviouslyunopenedboxinagivenrow,91.0percentofthetimethatboxhasthehighestvarianceoftheas-yet-unopenedboxesinthatForreasonsthatweexplainbelow,suchleft-to-rightboxopeningsmayarisebecauseofspatialbiasesinsteadoftheinformationpro-cessingreasonsimpliedbyourtheory.Nowweconsiderthepatternofsearchacrosscolumnsandrows.Figure4reportstheaveragenumberofboxesopenedincolumns2Ð10.Wereporttheaveragenumberofboxesunmasked,columnbycolumn,forboththesubjectdataandthemodelpredictions.TheempiricalproÞleiscalculatedbyaverag-ingtogethersubjectresponsesonalltheexog-enousgamesthatwereplayed.SpeciÞcally,eachofour388subjectsplayed12exogenousgames,yieldingatotalof388exogenousgamesplayed.Eachsubjectwasas-signedasubsetof12gamesfromasetof160uniquegames.Hence,eachofthe160gameswasplayedabout4,656/16030timesintheexogenoustimeportionoftheexperiment.Figure4alsoplotsthetheoreticalpredictionsgeneratedbyyokedsimulationsofourmodel.SpeciÞcally,thesepredictionsarecalculatedbysimulatingtheDCmodelontheexactsetof4,656gamesplayedbythesubjects.Wesimu-latethemodeloneachgamefromthissetof4,656gamesandinstructthecomputertoun-Theunitsthroughoutrefertopercentagepoints,nottoapercentageofthepointestimate.Inourendogenousgames,subjectsfollowthedecliningvariancerule92.6percentofthetime(s.e.0.7percent). 4.CROFILESFORAMESWITHThisÞgureplotsthemeannumberofboxesopenedineachcolumnbysubjectsandbytheDCmodel,forgamesinwhichtimebudgetsareimposedexogenously.Dottedlinesshowthebootstrapped95-percentconÞdenceintervalsforthedata.VOL.96NO.4GABAIXETAL.:EXPERIMENTALANALYSISOFABOUNDEDLYRATIONALMODEL maskthesamenumberofboxesthatwasun-maskedbythesubjectwhoplayedeachrespectivegame.Theanalysiscomparestheparticularboxesopenedbythesubjecttotheparticularboxesopenedbytheyokedsimulationofthemodel.Figure4reportsanmeasure,whichcapturestheextenttowhichtheempiricaldatamatchthetheoreticalpredictions.Thismeasureissimplyfromthefollowing)representstheagenumberofboxesunmaskedincolumnrepresentstheagenumberofboxesopenedincolumnNotethatvariesfrom2to10,sincetheboxesincolumn1arealwaysunmasked.Thisstatisticisboundedbelowby(sincethecoefÞcientonisconstrainedequaltounity)andboundedabovebyone(aperfectÞt).Intuitively,thestatisticrepresentsthefractionofsquareddeviationsaroundthemeanexplainedbythemodel.Forthecolumnpredictions,thestatisticis86.6percent(s.e.1.7percent),implyingaveryclosematchbetweenthedataandthepredictionsoftheFigure5reportsanalogouscalculationsbyrowandreportsthenumberofboxesopenedonaveragebyrow,withtherowsrankedbytheirvalueincolumnone.Wereportthenumberofboxesopenedonaveragebyrowforboththesubjectdataandthemodelpredictions.Asabove,themodelpredictionsarecalculatedus-ingyokedsimulations.Figure5alsoreportsanmeasureanalogoustotheonedescribedabove.Theonlydif-ferenceisthatnowthevariableofinterestis),theempiricalaveragenumberofboxesopenedinrow.Forourrowpredic-tionsourmeasureis16.1percent(s.e.9.2percent),implyingapoormatchbetweenthedataandthepredictionsofthemodel.ThedataInotherwords, representsempiricalmeans.Inthissectionofthepaper,theconstantisredundant,sincethedependentvariablehasthesamemeanastheindependentvariable.Inthenextsubsection,however,wewillconsidercasesinwhichthisequivalencedoesnothold,necessitatingthepresenceoftheconstant. 5.RROFILESFORAMESWITHThisÞgureplotsthemeannumberofboxesopenedineachrowbysubjectsandbytheDCmodelforgamesinwhichtimebudgetsareimposedexogenously.Dottedlinesshowthebootstrapped95-percentconÞdenceintervalsforthedata.TherowsareorderedaccordingtothevaluesintheÞrstcolumn.1058THEAMERICANECONOMICREVIEWSEPTEMBER2006 showthatsubjectsunmaskatleasttwotothreeboxesonaverageineachrow.Themodelpre-dictsanaverage,however,ofbetweenzeroandoneunmaskingsintherowsthathavethetwolowestvaluesintheÞrstcolumn(rows7and8intheÞgure).Thesubjectsarelessselectivethanthemodel.TheisnegativebecauseweconstrainthecoefÞcientonsimulatedboxestobeunity.ThisÞgurecapturesthemostproblem-aticpredictionsthatthemodelmakes.Figure6reportssimilarcalculationsusinganalternativewayoforderingrows.Itreportsthenumberofboxesopenedonaveragebyrow,withtherowsrankedbytheirrevealedvaluesatofsearchineachgame.Inotherwords,weranktherowsbasedontheboxesthathavebeenrevealedduringthecourseofinformationacquisitionineachgame.Thisrankingrulepro-ducesverydifferentpatternsthantherankingrulebasedonthecolumn1values(i.e.,inFigure5).Forthisalternativeranking,themodelpredictionshaveanstatisticof86.7percent(s.e.1.4percent).Wealsoreportaperformancemeasurethatisbasedonindividual-leveldata.Weaskwhetherthemodelcanpredictthenextboxopeningatthetimeofarowchange.Byconditioningonarowchange,weeliminateleft-to-rightbiases.Atthetimeofarowchange,theDCalgorithmcorrectlypredictsthenextboxopening37per-centofthetime.WealsocalculatetheDCrankingofrowstowhichsubjectsswitch.Therankingsarenor-malizedsothatanindexscoreofzerorepresentstheworstDCrankingandanindexscoreofonerepresentsthebestDCranking.WeÞndthatrowstowhichsubjectsswitchhaveameanDCrankindexof0.69andamedianrankindexof0.80,implyingthattheDCmodelplacesarel-ativelyhighsearchvalueontherowstowhichsubjectsactuallydoswitch.Finally,wedeterminewhethertheDCalgo-rithmsuccessfullypredictsÞnalconsumptionchoicesusingonlyexanteinformationabouteachgame(i.e.,conditioningonlyontheinfor-mationinthe80boxesofeachgameandnotusinganyofthesubjectchoices).WesimulatetheDCalgorithmoneachgame.WeÞndthattheÞnalrowsconsumedbyoursubjectsmatchtherowschosenbytheDCalgorithm54percentofthetimeinexogenoustimegames(arandomalgorithmwouldhavea13-percentsuccessrate).Likewise,therowspickedbysubjectsmatchoneofthetoptworowschosenbytheDCalgorithm73percentofthetime(25percentforarandomalgorithm).Finally,therowspickedbysubjectsmatchoneofthetopthreerowschosenbytheDCalgorithm82percentofthetime(38percentforarandomalgorithm).GameswithEndogenousTimeBudgetsWerepeattheanalysisabovefortheendog-enousgames.AsdiscussedinSectionIII,weconsidertwovariantsofthedirectedcognitionmodelwhenanalyzingtheendogenousgames.WecalibrateonevariantbyexogenouslysettingtomatchthesubjectsÕanticipatedearningsperunittimeintheendogenousgames:cents/boxopened(seecalibrationdiscussioninSectionIII).Withthiscalibration,subjectsarepredictedtoopen15.57boxespergame(s.e.0.01).Inthedata,however,subjectsopen26.06boxespergame(s.e.0.57).Tomatchthisfre-quencyofboxopening,weconsiderasecondcalibrationwith0.18.Withthislowerlevel,themodelopenstheempiricallyÒrightÓnumberofboxes.Figure7reportstheaveragenumberofboxesConsideranillustrativeexampleinwhichallsubjectsrandomlypickone(andonlyone)row,andthenexploretheninecoveredboxesinthatrow.ThentheproÞleofboxopeningsinFigure5wouldbeßatandequaltoforeveryrow(eachrowhasachanceofbeingchosenfor9openings).Bycontrast,theproÞleinFigure6wouldbeU-shapedwithpeaksslightlybelowforboththebestrowandtheworstrow.Intuitively,therowthatwasrandomlychosenforexplorationislikelytoendupbeingeitherthebestorworstrowwhentherankingismadeincludingthenineboxesthatareopenedinthecourseofsearch;upnineboxesincreasesvariance!Soinanyparticulargame,theextremerowshaveaboutachanceofhavingnineboxopeningsandaboutachanceofhavingzeroboxopen-ings,implyingaboutboxopeningsonaveragefortherowwiththebestrankingandtherowwiththeworstranking.Whenwedonotconditiononrowchanges,themodelpredictsthenextboxopening14percentofthetime.Thispercentageisbiaseddownbytheleft-to-rightbiasinbe-havioraldata.Modelsbasedoncost-beneÞtanalysiswillnotreproducetheleft-to-rightsearchpattern,unlessthemodelassumesthatleft-to-rightmovementsarelesscostlythanothersearchoperations.Theformulafortherankindexis(isthenumberofrowstowhichthesubjectcouldswitchwithoutimplementingarepeatboxopeningandistheDCrank(1to)oftherowtowhichthesubjectswitches(higherranksbeingbetter).VOL.96NO.4GABAIXETAL.:EXPERIMENTALANALYSISOFABOUNDEDLYRATIONALMODEL unmaskedincolumns2to10intheendogenousgames.Wereporttheaveragenumberofboxesunmaskedbycolumnforthesubjectdataandforthemodelpredictionswith0.49and0.18.Togeneratethetheoreticalpredic-tions,weusethedirectedcognitionmodeltosimulateplayofthe11,124endogenousgamesthatthesubjectsactuallyplayed.Themodelgeneratesitsownstoppingrule(sowenolongeryokethespeciÞcnumberofboxopenings).Fortheseendogenousgames,thecolumnis96.3percent(s.e.1.4percent)forand73.1percent(s.e.4.2percent)forFigure8reportsthenumberofboxesopenedonaveragebyrow,withtherowsrankedbytheirvaluesincolumnone.Therowticsare85.3percent(s.e.1.3percent)for0.49and64.6percent(s.e.2.4percent)for0.18.Figure9reportssimilarcalculationsusingthealternativewayoforderingrows,withtherowsrankedbytheirvaluesattheofeachgame.Thealternativerowstatisticsare91.8percent(s.e.0.5percent)for0.49and84.5percent(s.e.0.8percent)forTheseÞguresshowthatthemodelexplainsalargefractionofthevariationinattentionacrossrowsandcolumns.Someoftheseresultsareconfounded,however,byasubjectbiasthatCosta-Gomesetal.(2001)haveidentiÞed.Inparticular,subjectswhousetheMouselabinter-facetendtohaveabiastowardselectingcellsintheupper-leftcornerofthescreenandtransi-tioningfromlefttorightastheyexplorethescreen.Theleft-rightbiasaffectsourcolumnresults(Figures4and7),sinceinformationwithgreatereconomicrelevanceislocatedtowardtheleft-handsideofthescreeninourWealsoreportthemodelÕsabilitytopredictthenextboxopeningatthetimeofarowchange.Byconditioningonarowchange,weeliminateleft-to-rightbiases.Atthetimeofarowchange,theDCalgorithmcorrectlypredictsthenextboxopening38percentofthetime.WealsocalculatetheDCrankingofrowstowhichsubjectsswitch.Therankingsarenor-malizedbetweenzero(low)andone(high)asexplainedabove.WeÞndthatrowstowhichsubjectsswitchhaveameanDCrankindexof0.70andamedianrankindexof0.83.Finally,wedeterminewhethertheDCalgo-rithmsuccessfullypredictsÞnalconsumptionTheup-downbiasdoesnotaffectourresults,sinceourrowsarerandomlyorderedwithrespecttotheirrespectivepayoffs.Neithertheup-downnortheleft-rightbiasesinßu-enceouranalysesofeitherrowopenings(above)orendog-enousstoppingdecisions(below). 6.RThisÞgureplotsthemeannumberofboxesopenedineachrowbysubjectsandbytheDCmodel,forgamesinwhichtimebudgetsareimposedexogenously.Dottedlinesshowthebootstrapped95-percentconÞdenceintervalsforthedata.Therowsareorderedaccordingtoalltheinformationavailabletosubjectsaftertheyhaveconcludedtheirsearch,justpriortomakingachoice.1060THEAMERICANECONOMICREVIEWSEPTEMBER2006 choicesusingonlyexanteinformation.WesimulatetheDCalgorithmwith0.18.TheÞnalrowsconsumedbyoursubjectsmatchtherowschosenbytheDCalgorithm57percentofthetime.Likewise,therowspickedbysubjectsmatchoneofthetoptwo(three)rowschosenbytheDCalgorithm76percent(85percent)ofthe 7.CROFILESFORAMESWITHThisÞgureplotsthemeannumberofboxesopenedineachcolumnbysubjectsandbybothcalibrationsoftheDCmodel,forgamesinwhichagentschoosehowmuchtimetoallocatefromaÞxedtimebudget.Dottedlinesshowthebootstrapped95-percentconÞdenceintervalsforthedata. 8.RROFILESFORAMESWITHThisÞgureplotsthemeannumberofboxesopenedineachrowbysubjectsandbybothcalibrationsoftheDCmodel,forgamesinwhichagentschoosehowmuchtimetoallocatefromaÞxedtimebudget.Dottedlinesshowthebootstrapped95-percentconÞdenceintervalsforthedata.Therowsareorderedaccordingtothevaluesintheinitialcolumn.VOL.96NO.4GABAIXETAL.:EXPERIMENTALANALYSISOFABOUNDEDLYRATIONALMODEL StoppingDecisionsinEndogenousGamesAlmostalloftheanalysisabovereportsgamevariationininformationacquisition.Theanalysisaboveshowsthatsubjectsallocatemostoftheirattentiontoeconomicallyrelevantcol-umnsandrowswithinagame,matchingthepat-ternspredictedbythedirectedcognitionmodel.Ourexperimentaldesignalsoenablesustoeval-uatehowsubjectsallocatesearchtimegames.Inthissubsectionwefocusonseveralmeasuresofsuchbetween-gamevariation.Mostimportantly,weaskwhetherthemodelcancorrectlypredictwhichgamesreceivedthemostattentionfromoursubjects.Ourexperi-mentutilized160uniquegames,thoughnosin-glesubjectplayedall160games.Letrepresenttheaveragenumberofboxesopenedbysubjectswhoplayedgame.Letrepresenttheaveragenumberofboxesopenedbythemodelwhenplayinggame.InthissubsectionweanalyzetheÞrstandsecondmo-mentsoftheempiricalsample{andthesimulatedsample.Notethattheserespectivevectorseachhave160el-ements,sinceweareanalyzinggame-speciÞcWebeginbycomparingÞrstmoments.Theempirical(equallyweighted)meanofis26.06(s.e.0.57).Bycontrast,the0-parameterversionofourmodel(with0.49)generatesapredictedmeanof15.57(s.e.0.01).Hence,unlesswepicktomatchtheempiricalmean0.18),ourmodelonlycrudelyapprox-imatestheaveragenumberofboxesopenedperWeturnnowtosecondmoments.Theempir-icalstandarddeviationof)is6.32(s.e.0.14),whilethe0-parameterversionofourmodel(with0.49)generatesapredictedstandarddeviationof12.05(s.e.0.03).More-over,whenweset0.18tomatchtheaver- 9.RThisÞgureplotsthemeannumberofboxesopenedineachrowbysubjectsandbybothcalibrationsoftheDCmodel,forgamesinwhichagentschoosehowmuchtimetoallocatefromaÞxedtimebudget.Dottedlinesshowthebootstrapped95-percentconÞdenceintervalsforthedata.Therowsareorderedaccordingtoalltheinformationavailabletosubjectsaftertheyhaveconcludedtheirsearch,justpriortomakingachoice.1062THEAMERICANECONOMICREVIEWSEPTEMBER2006 ageboxespergame,thestandarddeviationrisesto15.85(s.e.0.03).Therelativelyhighstandarddeviationsfromthesimulationsreßectthemod-elÕssophisticatedsearchstrategy.Themodelcontinuouslyadjustsitssearchstrategiesinre-sponsetoinstantaneousvariationintheeco-nomicincentivesthatthesubjectsface.Bycontrast,thesubjectsarelesssensitivetohighfrequencyvariationineconomicincentives.Despitetheseshortcomings,themodelsuc-cessfullypredictsthepatternofempiricalvari-ationinthenumberofboxesopenedineachgame.Thecorrelationbetween)andis0.66(s.e.0.02)whenweset0.49.Similarly,thecorrelationis0.61(s.e.0.02)whenweset0.18.SeeFigure10foraplotofthe160datapointsforthecase.Thesehighcorrelationsimplythatthemodeldoesagoodjobpredictingwhichgamesthesubjectswillanalyzemostthoroughly.Themodelalsopredictsthestatisticalrela-tionshipbetweeneconomicincentivesanddepthofanalysis.Figure11reportstherelation-shipbetweenÑthemyopicestimateofthemarginalbeneÞtofincrementalsearchtheaverageamountoftimethesubject(oral-gorithm)continuestoplaythegame.Therela-tionshipiscalculated(usinganonparametrickernelestimator)forbothsimulateddataandtheexperimentaldata.Toconstructthesimu-latedrelationship,weusethemodeltoÒplayÓeachofourgamesandthenusethesimulatedmove-by-movedatatoestimatetherelationshipandtheamountofsimulatedtimeremaining.Toconstructtheexperimentalrela-tionship,weusetheexperimentalmove-by-movedatatoestimatetherelationshipbetweenandtheempiricalamountoftimeremaining.Thedashedlinerepresentstherelationshipestimatedwiththesimulateddata(with0.18).Thesolidlinerepresentstherelationshipestimatedwiththeexperimentaldata.TheÞgurealsoshowsbootstrapestimatesofthe95-percentconÞdenceintervals.FormostlevelsofbeneÞt-costratio),themodelÕspredictionsareclosetothepatterninthesubjectdata.Subjectsdomoreanalysis(i.e.,openupmoreboxes)whentheeconomicincentivestodosoarehigh.Moreover,thefunctionalformofthisrelation-shiproughlymatchestheformpredictedbytheWealsoevaluatetheDCmodelbyaskingpredictswhensubjectsdecidetostopworkingonthecurrentgameandmoveontothenextgame.Werunastoppinglogitstop,0continue)withexplanatoryvariablesthatincludethemeasureoftheeco-nomicvalueofcontinuedsearch(or),thenumberofdifferentboxesopenedtodateinthecurrentgame(),theexpectedvalueincentsoftheleadingrowinthecurrent-goodgame(),andsubjectÞxedef-fects.NotethatsatisÞcingmodelspredictthatshouldnothavepredictivepowerinthislogitregression,butthatahighervalueoftheleadervariablewillincreasethestop-pingprobability.Eachobservationforthislogitisadecision-node(i.e.,achoiceovermouseclicks)inourSeeequation(6):).Recallisthegapbetweentheexpectedvalueofthecurrentrowandtheexpectedvalueofthenextbestrow,isthestandarddeviationoftheinformationrevealedbymental,andisthenumberofboxesopenedbymentaloperator 10.BPENEDBYThisÞguretakeseachofthe160game-typesplayedbysubjectsandcomparesthenumberofboxesopenedbysubjectstothenumberofboxopeningspredictedbythemodel,forthegamesinwhichagentschoosetheircross-gametimeallocations.Thecorrelationbetweenthemodelandthedatais0.61.VOL.96NO.4GABAIXETAL.:EXPERIMENTALANALYSISOFABOUNDEDLYRATIONALMODEL endogenousgames,generating330,873obser-ThecoefÞcienton(standarderror0.0081and-statof45);thecoefÞcientonis0.0404(standarderror0.0008and-statof50);thecoefÞcientonis0.0064(standarderror0.0002and-statof32).Atthemeanvaluesoftheex-planatoryvariables,aone-standard-deviationreductioninthevalueofmorethandoublestheprobabilityofstopping.Aone-standard-deviationincreaseinthevalueofhasaneffectroughlyaslargeandaone-standard-deviationincreaseinthevalueofaneffectlessthanaslarge.Althoughhasthegreatestisthemostimportantpredictorofthedecisiontostopsearching(variationinhasabiggerimpactthanvari-ationin).IncrementallogitregressionsalsoconÞrmthathasmorepredictivepowerthanThelogisticanalysisshowsthattheeconomicvalueofinformationÑÑisthemostimpor-tantpredictorofthedecisiontostopsearching.Oursubjectsarealsousingotherinformation,however,asshownmostimportantlybythestrongpredictivepowerofthevariable.IfsubjectsplacepartialweightontheÒi.e.,increasingthepropensitytostopan-alyzingthecurrent-goodgameasmoreandmoreboxesareopenedÑtheywillbelesslikelytospendalongtimeonanyonegame.Foranun-sophisticatedplayerwhocanonlyimperfectlycal-,theheuristicisausefuladditionaldecisioninput.WeviewoursubjectsÕpartialreli-Inotherterms,weestimate:probabilityofcontinuacontinuaexp( x)],whereisavectorofdecision-nodeattributes,andisthevectorofestimatedWhenastoppinglogitisestimatedwithonlyaconstant,ln48,026,whereisthelikelihoodfunction.Whenastoppinglogitisestimatedwithaconstantand47,403.Whenastoppinglogitisestimatedwithaconstantand,ln45,353.Whenastoppinglogitisestimatedwithaconstant,,ln45,264.Hence,includinginthemodelincreaseslnfourtimesmorethanincluding 11.EEASUREOFThisÞgureplotsnonparametric(kernel)estimatesoftheexpectednumberofadditionalboxopeningsinagame,conditionalonthecurrentÒÓvalue.ÒÓisthebeneÞt-to-costratioofmarginalanalysisintheDCmodel,asdeÞnedinequation(6).1064THEAMERICANECONOMICREVIEWSEPTEMBER2006 anceonasanexampleofsensibleÑperhapsconstrainedoptimalÑdecision-making.ThepredictivepowerofthesupportsexperimentalresearchonÒsystemne-glectÓbyCamererandDanLovallo(1999)andCadeMasseyandGeorgeWu(2005).TheseauthorsÞndthatsubjectsusesensiblerules,butfailtoadjustthoserulesadequatelytothepar-ticularproblemathand.Theheuristicisagoodgeneralrule,butitisnottheÞrst-bestrulesinceitneglectstheidiosyncraticincentivesgeneratedbyeachspeciÞc-goodgame.Theheuristicisatypeofimperfectlysophis-ticatedsearch.Theheuristicisalsocon-sistentwithÞndingsinthesequentialsearchliteraturethatsubjectsareoftenoverlysensitivetothetotalsearchcostspaid(CarlKogut,1990;PhilipMoonandAndrewMartin,1990;JoepSonnemans,1998).Finally,ourexperimentrevealsanotherbiasthatwehavenotemphasizedbecauseweareuncertainaboutitsgeneralizabilitytootherclassesofgames.SpeciÞcally,oursubjectsal-locatedtoomuchtimepergameintheendog-enousgames.Subjectpayoffswouldhavebeenhigheriftheyhadgenerallycollectedlessinfor-mationineachgame,therebyenablingthemtoplaymoregames.Thisresultiscontrarytothesequentialsearchliterature,inwhichsubjectsoftensearchlessthanpredicted(Kogut,1990;Sonnemans,1998).ExploringtherobustnessofthisÞndingisagoalforfutureresearch.ComparisonswithOtherModelsTable8reportsmeasuresforthealternativemodelssummarizedinSectionIIIG.Fortheexogenousgames,theDCmodelhasanaveragevalueof52.4percent(withastandarddeviationof3.4percentagepoints).Thecolumnandrowmodelshaverespectiveaveragesof18.8percent(s.e.6.4percent)andpercent(s.e.13.9percent).Fortheendogenousgames,thezero-parameterversionoftheDCmodelhasanaverageof91.2percent(s.e.0.7percent).ThecolumnmodelwithasatisÞc-ingstoppingrulehasanaverageof42.9percent(s.e.0.7percent).TherowmodelwithasatisÞcingstoppingrulehasanaverage55.4percent(s.e.0.4percent).Theeliminationbyaspectsmodelhasanaveragepercent(s.e.6.2percent).WealsoevaluatethedifferentmodelsÕabilitytoforecastgame-by-gamevariationinaverageVALUATIONOF GameswithexogenoustimebudgetsGameswithendogenoustimebudgetsRowmodelbyaspectsFittedparameterNoneNoneforcolumn107.7%39.3%96.3%73.1%79.7%34.5%75.3%(1.7%)(18.7%)(0.93%)(1.4%)(4.2%)(1.8%)(1.0%)(1.5%)(18.4%)forrow85.3%64.6%25.1%88.6%65.6%(9.2%)(0.55%)(0.40%)(1.3%)(2.4%)(0.44%)(0.69%)(1.0%)(0.66%)foralt.row25.0%89.1%91.8%84.5%23.8%43.0%53.3%82.5%(1.4%)(0.42%)(1.5%)(0.55%)(0.83%)(0.39%)(0.59%)(0.77%)(0.79%)91.2%74.1%42.9%55.4%64.7%(3.4%)(6.4%)(13.9%)(0.67%)(1.7%)(0.73%)(0.41%)(0.62%)(6.2%)Correlationwithnumberof0.660.610.0050.0200.026ÑÑÑ(0.19)(0.018)(0.024)(0.023)(0.024)(0.021)AgreementwithÞnalchoices51.2%52.9%55.7%57.4%50.7%47.1%49.0%47.1%(0.73%)(0.73%)(0.73%)(0.47%)(0.47%)(0.47%)(0.47%)(0.47%)(0.47%)Standarderrorsinparentheses.referstothestatisticfromaregressionoftheempiricalnumberofboxesopenedbysubjectsineachroworcolumnonaconstant,plusthenumberofboxespredictedbythemodels,wherethecoefÞcientonthepredictionsisÞxedtoequal1.ÒCorrelationwithempiricalnumberofboxesÓreferstothecorrelationbetweenthenumberofboxesopenedbysubjectsineachofthe160gamesandthenumberofopeningspredictedbythemodels.ÒAgreementwithÞnalchoicesÓrecordsthepercentageofthegamesplayedbysubjectsinwhichtheirÞnalchoicesmatchthepredictionsofeachmodel,whenthemodelissimulatedwithnoknowledgeofsubjectboxopenings.VOL.96NO.4GABAIXETAL.:EXPERIMENTALANALYSISOFABOUNDEDLYRATIONALMODEL timeallocation.Table8showsthatallofthesatisÞcingmodelsyieldeffectivelynocorrela-tionbetweenthegame-by-gameaveragenum-berofboxopeningspredictedbythesemodelsandthegame-by-gameaveragenumberofboxopeningsinthedata.Bycontrast,thezero-parameterdirectedcognitionmodelgeneratespredictionsthatarehighlycorrelated(0.66)withtheempiricalaverageboxesopenedperV.ConclusionEconomistsassumethatagentsactasiftheywerechoosingcognitiveoperationsoptimally.Wehaveexperimentallyanalyzedaparsimoni-ouscognitionmodelbasedonpartiallymyopiccost-beneÞtcalculations:theDCmodel.Thepartialmyopiamakesthemodelcomputation-allytractable,andourexperimentalresultsdem-onstratethatthepartialmyopiaimprovesthemodelÕsempiricalÞt.OurÞrstexperimentstudiesaclassofsim-pledecisionproblems.Theseproblemsadmitarationalsearchsolution,donotposespatialbiases,anddonotrequiresubjectmemoriza-tion.Inthesesimpleproblems,wealsoas-sumeanexogenousÞnancialsearchcost.WeÞndthattheDCmodelsuccessfullyexplainsthepatternsofinformationacquisition.In-deed,whentheDCmodelandthefullyra-tionalmodelmakedifferentpredictions,theDCmodeldoesajobofmatchingthelaboratoryevidence.Oursecondsetofexperimentsstudiesahighlycomplexchoiceproblem.TheDCmodelcanbeusedtostudysuchcomplexsettingsbecausethemodelsimpliÞesanalysis(forthesubjectandtheresearcher).Incontrast,thera-tionalactormodeldoesnotadmitacomputa-tionallytractablesolutioninsuchproblems.Intheseexperiments,subjectsanalyzedeightgoods,eachofwhichhadninehiddenattributesthatcouldberevealedwithsomeeffort,thoughwithoutÞnancialcost.Subjectsdecidedwhichattributestorevealbeforechoosingtoconsumeoneoftheeightgoods.Subjectsfacedendoge-nouscognitioncostsbecausetheywereundertimepressure.Wecreatedthistimepressureintwoways:Þrst,bygivingsubjectsaÞxedamountoftimeforeachgame,and,second,bygivingsubjectsaÞxedamountoftimeforasetofgames.TheDCmodelsuccessfullypredictstheaggregateempiricalregularitiesofinforma-tionacquisitioninthisexperiment,includingthepatternofinformationacquisitionwithineachgameandthepatternofinformationac-quisitionacrossgames.Thissecondsetofex-perimentsillustratedtheapplicabilityoftheDCproblemtohighlycomplexsettingsinwhichtheclassicalrationalactormodelisanalyticallyandcomputationallyintractable.Understandinghowdecisionmakersthinkaboutproblemsisarelativelynewfrontierineconomicscience.Strippingbackdecision-makingtothelevelofcognitiveprocessesposesfundamentalresearchchallenges.Inthispaperwehavestudiedtheinformationacquisitionprocess.Thisisoneexampleofawidersetofattentionallocationprocesses.Actorsallocateattentiontoacquirenewinformationandtoanalyzealreadyavailableinformation(e.g.,ChristopherA.Sims,2003).Thestudyofcognitiveprocessesineconomicdecision-makingisstillinitsinfancy.Butweareoptimisticthatsuchprocess-basedresearchmaypavethewayforascienceofdecision-makingwithmuchgreaterpredictivepowerthantheclassicalÒasifÓmodelingthattreatscognitionasaclosedblackbox.Thispaperjoinsagrowingbodyofworkthatpriesopenthatbox(e.g.,Camereretal.,1993;Costa-Gomesetal.,2001).Welookforwardtofutureprocess-basedresearchthatwillopentheboxmorecompletely.Aragones,Enriqueta;Gilboa,Itzhak;Postlewaite,AndrewandSchmeidler,David.AmericanEconomicReview,2005,(5),pp.1355Ð68.Recallthatthisanalysisusestheaveragenumberofboxesopenedpergameforthe160uniquegamesinthedataset.Thesedataarecompiledfromtheendogenoustimesegmentoftheexperiment.Wedo,however,ÞndsomesupportforasatisÞcingframework.Ahigherpayoffinthecurrentdecisionprob-lemleadstoaslightlyhigherintensityofsearchinsubsequentproblems.Forexample,a$10higherrealizedpayoffinthecurrentgameleadsto0.16additionalboxopeningsinthesubsequentgame(controllingforsubjectÞxedeffects).AsatisÞcingmodel(withongoingupdatesofthesatisÞcingthreshold)wouldpredictarelationshipwiththissign.1066THEAMERICANECONOMICREVIEWSEPTEMBER2006 Bertsekas,DimitriP.andTsitsiklis,JohnN.Neuro-dynamicprogramming.Belmont,MA:AthenaScientiÞc,1996.Camerer,ColinandHo,Teck-Hua.WeightedAttractionLearninginNormalFormGames.Ó,1999,pp.827Ð74.Camerer,ColinF.;Johnson,Eric;Rymon,TaliaandSen,Senkar.ÒCognitionandFraminginSequentialBargainingforGainsandLosses,ÓinKenBinmore,AlanKirman,andPieroTani,eds.,Frontiersofgame.Cambridge,MA:MITPress,1993,pp.27Ð47.Camerer,ColinF.andLovallo,Dan.ÞdenceandExcessEntry:AnExperimentalAmericanEconomicReview(1),pp.306Ð18.Conlisk,John.ÒWhyBoundedRationality?ÓJournalofEconomicLiterature,1996,pp.669Ð700.Costa-Gomes,MiguelA.andCrawford,VincentP.ÒCognitionandBehaviorinTwo-PersonGuessingGames:AnExperimentalStudy.ÓAmericanEconomicReviewCosta-Gomes,MiguelA.;Crawford,VincentP.andBroseta,Bruno.ÒCognitionandBehav-iorinNormal-FormGames:AnExperi-mentalStudy.Ó,2001,pp.1193Ð1235.Erev,IdoandRoth,AlvinE.ÒPredictingHowPeoplePlayGames:ReinforcementLearninginExperimentalGameswithUnique,MixedStrategyEquilibria.ÓAmericanEconomicRe-,1998,(4),pp.848Ð81.Gabaix,XavierandLaibson,David.RationalityandDirectedCognition.ÓUnpub-lishedPaper,2005.Gabaix,XavierandLaibson,David.Attributes,ConsumerMyopia,andInforma-tionSuppressioninCompetitiveMarkets.ÓQuarterlyJournalofEconomics,2006,(2),pp.505Ð40.Gigerenzer,Gerd;Todd,PeterandABCRe-searchGroup.Simpleheuristicsthatmakeus.Oxford:OxfordUniversityPress,Gilovich,Thomas;GrifÞn,DaleandKahneman,Heuristicsandbiases:Thepsychol-ogyofintuitivejudgment.Cambridge:Cam-bridgeUniversityPress,2002.Gittins,JohnC.ÒBanditProcessesandDynamicAllocationIndices.ÓJournaloftheRoyalSta-tisticalSociety,SeriesB,1979,(2),pp.148Ð77.Jehiel,Philippe.ÒLimitedHorizonForecastinRepeatedAlternateGames.ÓJournalofEco-nomicTheory,1995,(2),pp.497Ð519.Johnson,EricJ.;Camerer,Colin;Sen,SankarandRymon,Talia.ÒDetectingFailuresofBackwardInduction:MonitoringInforma-tionSearchinSequentialBargaining.ÓJournalofEconomicTheory,2002,pp.16Ð47.Kogut,CarlA.ÒConsumerSearchBehaviorandSunkCosts.ÓJournalofEconomicBehaviorandOrganization,1990,(3),pp.381Ð92.Lipman,BartonL.ÒHowtoDecideHowtoDecideHowTo...:ModelingLimitedRatio-,1991,(4),pp.Massey,CadeandWu,George.ÒDetectingRe-gimeShifts:ThePsychologyofUnder-andManagementScience,2005,(6),pp.932Ð47.Moon,PhilipandMartin,Andrew.ÒBetterHeu-risticsforEconomicSearch:ExperimentalandSimulationEvidence.ÓJournalofBehav-ioralDecisionMaking,1990,(3),pp.175Ð93.Payne,JohnW.;Bettman,JamesR.andJohnson,EricJ.Theadaptivedecisionmaker.Cam-bridge:CambridgeUniversityPress,1993.Payne,JohnW.;Braunstein,MyronL.andCar-roll,JohnS.ÒExploringPredecisionalBehav-ior:AnAlternativeApproachtoDecisionOrganizationalBehaviorandHu-manPerformance,1978,(1),pp.17Ð44.Russo,J.Edward.ÒEyeFixationsCanSavetheWorld:ACriticalEvaluationandaCompar-isonbetweenEyeFixationsandOtherInfor-mationProcessingMethodologies,ÓinH.KeithHunt,ed.,Advancesinconsumerre-.Vol.5.AnnArbor:AssociationforConsumerResearch,1978,pp.561Ð70.Simon,HerbertA.ÒABehavioralModelofRa-tionalChoice.ÓQuarterlyJournalofEco-,1955,(1),pp.99Ð118.Sims,ChristopherA.ÒImplicationsofRationalJournalofMonetaryEconom-,2003,(3),pp.665Ð90.Sonnemans,Joep.ÒStrategiesofSearch.ÓnalofEconomicBehaviorandOrganization(3),pp.309Ð32.Tversky,Amos.ÒEliminationbyAspects:ATheoryofChoice.ÓPsychologicalReview(4),pp.281Ð99.VOL.96NO.4GABAIXETAL.:EXPERIMENTALANALYSISOFABOUNDEDLYRATIONALMODEL Tversky,AmosandKahneman,Daniel.vancesinProspectTheory:CumulativeRepresentationofUncertainty.ÓJournalofRiskandUncertainty,1992,(4),pp.297ÐWegner,DanielM.Theillusionofconscious.Cambridge,MA:MITPress,2002.Weitzman,MartinL.ÒOptimalSearchfortheBestAlternative.Ó,1979,pp.641Ð54.1068THEAMERICANECONOMICREVIEWSEPTEMBER2006 CostlyInformationAcquisition:ExperimentalAnalysisofaBoundedlyRationalModelThedirectedcognitionmodelassumesthatagentsusepartiallymyopicoption-value