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Cognitivehierarchiesandemotions
inbehavioralgametheory
ColinF.Camerer1,2
AlecSmith1
1DivisionofHumanitiesandSocialSciences
2ComputationandNeuralSystems
CaliforniaInstituteofTechnology
4/25/1111:15am.Commentswelcome.PreparedforOxfordHandbookofThinkingand
Reasoning(K.J.Holyoak&R.G.Morrison,Editors).ThisresearchwassupportedbyThe
BettyandGordonMooreFoundationandbyNationalScienceFoundationgrantNSF‐SES
0850840.Correspondenceto:[email protected].
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ABSTRACT
Untilrecently,gametheorywasnotfocussedoncognitively‐plausiblemodelsofchoicesin
humanstrategicinteractions.Thischapterdescribestwonewapproachesthatdoso.The
firstapproach,cognitivehierarchymodeling,assumesthatplayershavedifferentlevelsof
partiallyaccuraterepresentationsofwhatothersarelikelytodo,whichvaryfromheuristic
andnaïvetohighlysophisticatedandaccurate.Thereisreasonableevidencethatthis
approachexplainschoices(betterthantraditionalequilibriumanalysis)indozensof
experimentalgamesandsomenaturally‐occurringgames(e.g.,aSwedishlottery,auctions,
andconsumerreactionstoundisclosedqualityinformationaboutmovies).Measurementof
eyetrackingandfMRIactivityduringgamesisalsosuggestiveofacognitive.Thesecond
approach,psychologicalgames,allowsvaluetodependuponchoiceconsequencesandon
beliefsaboutwhatwillhappen.Thismodelingframeworkcanlinkcognitionandemotion,
andexpresssocialemotionssuchas“guilt”.Inapsychologicalgame,guiltismodeledasthe
negativeemotionofknowingthatanotherpersonisunpleasantlysurprisedthatyour
choicedidnotbenefitthem(astheyhadexpected).Ourhopeisthatthesenew
developmentsinatraditionallycognitivefield(gametheory)willengageinterestof
psychologistsandothersinterestedinthinkingandsocialcognition.
KEYWORDS
Boundedrationality,cognitivehierarchy,emotions,gametheory,psychologicalgames,
strategicneuroscience
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I.Introduction
Thischapterisaboutcognitiveprocessesinstrategicthinking.Thetheoryofgames
providesthemostcomprehensiveframeworkforthinkingaboutthevaluedoutcomesthat
resultfromstrategicinteractions.Thetheoryspecifieshow“players”(that’sgametheory
jargon)mightchoosehigh‐valuestrategiestoguesslikelychoicesofotherplayers.
Traditionally,gametheoryhasbeenfocusedonfinding“solutions”togamesbasedon
highlymathematicalconceptionsofrationalforecastingandchoice.Morerecently(starting
withCamerer,1990),behavioralgametheorymodelshaveextendedtherationaltheories
toincludestochasticresponse,limitsoninferringcorrectlywhatotherplayerswilldo,
socialemotionsandconsiderationssuchasguilt,anger,reciprocity,orsocialimage,and
modulatingfactorsincludinginferencesaboutothers’intentions.Twogeneralbehavioral
modelsthatmightinterestcognitivepsychologistsarethefocusofthischapteri:Cognitive
hierarchymodeling,andpsychologicalgametheory.
Conventionalgametheoryistypicallyabstract,mathematicallyintimidating,
computationallyimplausible,andalgorithmicallyincomplete.Itisthereforenotsurprising
thatconventionaltoolshavenotgainedtractionincognitivepsychology.Ourhopeisthat
themorepsychologicallyplausiblebehavioralvariantscouldinterestcognitive
psychologists.Oncelimitedstrategicthinkingisthefocus,questionsofcognitive
representation,categorizationofdifferentstrategicstructures,andthenatureofsocial
cognition,andhowcooperationisachievedallbecomemoreinterestingresearchable
questions.Thequestionofwhetherornotpeopleareusingthedecision‐makingalgorithms
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proposedbythesebehavioralmodelscanalsobeaddressedwithobservables(suchas
responsetimesandeyetrackingofvisualattention)familiarincognitivepsychology.
Numericalmeasuresofvalueandbeliefderivedinthesetheoriescanalsobeusedas
parametricregressorstoidentifycandidatebraincircuitsthatappeartoencodethose
measures.Thisgeneralapproachhasbeenquitesuccessfulinstudyingsimpler
nonstrategicchoicedecisions(Glimcher,Camerer,Fehr,&Poldrack,2008)buthasbeen
appliedinfrequentlytogames(seeBhatt&Camerer,inpress).
Whatisagame?
Gametheoryisthemathematicalanalysisofstrategicinteraction.Ithasbecomea
standardtoolineconomicsandtheoreticalbiology,andisincreasinglyusedinpolitical
science,sociology,andcomputerscience.Agameismathematicallydefinedasasetof
players,descriptionsoftheirinformation,afixedorderofthesequenceofchoicesby
differentplayers,andafunctionmappingplayers’choicesandinformationtooutcomes.
Outcomesmayincludetangibleslikecorporateprofitsorpokerwinnings,aswellas
intangibleslikepoliticalgain,status,orreproductiveopportunities(inbiologicaland
evolutionarypsychologymodels).Thespecificationofagameiscompletedbyapayoff
functionthatattachesanumericalvalueor“utility”toeachoutcome.
Thestandardapproachtotheanalysisofgamesistocomputeanequilibriumpoint,
asetofstrategiesforeachplayerwhicharesimultaneouslybestresponsestooneanother..
ThisapproachisdueoriginallytoJohnNash(1950),buildingonearlierworkbyVon
NeumannandMorgenstern(1947).Solvingforequilibriummathematicallyrequires
solvingsimultaneousequationsinwhicheachplayer'sstrategyisaninputtotheother
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player'scalculationofexpectedpayoff.Thesolutionisacollectionofstrategies,onefor
eachplayer,whereeachplayer’sstrategymaximizeshisexpectedpayoffgiventhe
strategiesoftheotherplayers.
Fromthebeginningofgametheory,howequilibriummightarisehasbeenthe
subjectofongoingdiscussion.Nashhimselfsuggestedthatequilibriumbeliefsmight
resolvefromchangesin“massaction”aspopulationslearnaboutwhatothersdoand
adjusttheirstrategiestowardoptimization.ii
MorerecentlygametheoristshaveconsideredtheepistemicrequirementsforNash
equilibriumbytreatinggamesasinteractivedecisionproblems(cf.Brandenburger1992).
ItturnsoutthatNashequilibriumforn‐playergamesrequiresverystrongassumptions
abouttheplayers’mutualknowledge:thatallplayersshareacommonpriorbeliefabout
chanceevents,knowthatallplayersarerational,andknowthattheirbeliefsarecommon
knowledge(Aumann&Brandenburger1995).iiiThelatterrequirementimpliesthat
rationalplayersbeabletocomputebeliefsaboutthestrategiesofcoplayersandallstates
oftheworld,beliefsaboutbeliefs,andsoon,adinfinitum.
TwoBehavioralApproaches:CognitiveHierarchyandPsychologicalGames
Cognitivehierarchy(CH)andpsychologicalgames(PG)modelsbothmodify
assumptionsfromgametheorytocapturebehaviormorerealistically.
TheCHapproachassumesthatboundedlyrationalplayersarelimitedinthe
numberofinterpersonaliterationsofstrategicreasoningtheycan(orchoose)todo.There
arefiveelementstoanyCHpredictivemodel:
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1. Adistributionofthefrequencyofleveltypesf(k)
2. Actionsoflevel0players;
3. Beliefsoflevel‐kplayers(fork=1,2,…)aboutotherplayers;
4. Assessingexpectedpayoffsbasedonbeliefsin(3).
5. Astochasticchoiceresponsefunctionbasedontheexpectedpayoffsin(4)
Thetypicalapproachistomakepreciseassumptionsaboutelements(1‐5)andsee
howwellthatspecificmodelfitsexperimentaldatafromdifferentgames.Justasintesting
acookingrecipe,ifthemodelfailsbadlythenitcanbeextendedandimproved.
InCamerer,HoandChong(2004),thedistributionoflevelktypesisassumedto
followaPoissondistributionwithameanvalueτ.Oncethevalueofτischosen,the
completedistributionisknown.ThePoissondistributionhasthesensiblepropertythatthe
frequenciesofveryhighleveltypeskdropsoffquicklyforhighervaluesofk.(Forexample,
iftheaveragenumberofthinkingstepsτ=1.5,thenlessthan2%ofplayersareexpectedto
dofiveormorestepsofthinking.)
Tofurtherspecifythemodel,level0typesareusuallyassumedtochooseeach
strategyequallyoften.ivIntheCHapproach,levelkplayersknowthecorrectproportions
oflower‐levelplayers,butdonotrealizethereareotherevenhigher‐levelplayers(perhaps
reflectingoverconfidenceinrelativeability).Analternativeassumption(called“levelk”
modeling)isthatalevelkplayerthinksallotherplayersareatlevelk‐1.
Undertheseassumptions,eachlevelofplayerinahierarchycanthencomputethe
expectedpayoffstodifferentstrategies:Level1’scomputetheirexpectedpayoff(knowing
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whatlevel0’swilldo);level2’scomputetheexpectedpayoffgiventheirguessaboutwhat
level1’sand0’sdo,andhowfrequentthoseleveltypesare;andsoforth.Inthesimplest
formofthemodel,playerschoosethestrategywiththehighestexpectedpayoff(the“best
response”);butitisalsoeasytousealogisticorpowerstochastic“betterresponse”
function(e.g.,Luce,1959).Becausethetheoryishierarchical,itiseasytoprogramand
solvenumericallyusinga“loop”.
Psychologicalgamesmodelsassumethatplayersarerationalinthesensethatthey
maximizetheirexpectedutilitygivenbeliefsandtheutilityfunctionsoftheotherplayers.
However,inpsychologicalgamesmodels,payoffsareallowedtodependdirectlyupon
player’sbeliefs,theirbeliefsabouttheircoplayers’beliefs,andsoon,adependencethatis
ruledoutinstandardgametheory.Theincorporationofbelief‐dependentmotivations
makesitpossibletocaptureconcernsaboutintentions,socialimage,orevenemotionsina
game‐theoreticframework.Forexample,inpsychologicalgamesoneperson,Conor(C)
mightbedelightedtobesurprisedbytheactionofanotherplayer,Lexie(L).Thisis
modeledmathematicallyasClikingwhenL’sstrategyisdifferentthanwhathe(C)
expectedLtodo.Someofthesemotivationsarenaturallyconstruedassocialemotions,
suchasguilt(e.g.,apersonfeelsbadchoosingastrategywhichharmedanotherpersonP
whodidnotexpectit,andfeelslessbadifPdidexpectit).
Ofthetwoapproaches,CHandlevel‐kmodelingareeasytouseandapplyto
empiricalsettings.Psychologicalgamesaremoregeneral,applyingtoabroaderclassof
games,butaremoredifficulttoadapttoempiricalwork.
II.TheCHmodel
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Thenextsectionwillgivessomemotivatingempiricalexamplesofthewidescopeof
gamestowhichthetheoryhasbeenappliedwithsomesuccess(includingtwokindsoffield
data),andconsistencywithdataonvisualfixationandfMRI.TheCHapproachisappealing
asapotentialcognitivealgorithmforfourreasons:
1. Itappearstofitalotofexperimentaldatafrommanydifferentgamesbetter
thanequilibriumpredictionsdo(e.g.,Camereretal.,2004;Crawford,Costa‐
Gomes,&Iriberri,2010).
2. Thespecificationofhowthinkingworksandcreateschoicesinvites
measurementofthethinkingprocesswithresponsetimes,visualfixationson
certainpayoffs,andtransitionsbetweenparticularpayoffs.
3. TheCHapproachintroducesaconceptofskillintobehavioralgametheory.
IntheCHmodel,theplayerswiththehighestthinkinglevels(higherk)and
mostresponsivechoices(higherλ)areimplicitlymoreskilled.(In
equilibriummodels,allplayersareperfectlyandequallyskilled.)
NextwewilldescribeseveralempiricalgamesthatillustratehowCHreasoning
works.
Example1:p‐beautycontest
AsimplegamethatillustratesapparentCHthinkinghascometobecalledthe“p‐
beautycontestgame”(orPBC).ThenamecomesfromafamouspassageinJohnMaynard
Keynes’sbookTheGeneralTheoryofEmployment,InterestandMoney.Keyneswrote:
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“Professionalinvestmentmaybelikenedtothosenewspapercompetitionsinwhich
thecompetitorshavetopickoutthesixprettiestfacesfromahundredphotographs,
theprizebeingawardedtothecompetitorwhosechoicemostnearlycorrespondsto
theaveragepreferencesofthecompetitorsasawhole;sothateachcompetitorhas
topick,notthosefaceswhichhehimselffindsprettiest,butthosewhichhethinks
likeliesttocatchthefancyoftheothercompetitors,allofwhomarelookingatthe
problemfromthesamepointofview.Itisnotacaseofchoosingthosewhich,tothe
bestofone'sjudgment,arereallytheprettiest,noreventhosewhichaverage
opiniongenuinelythinkstheprettiest.Wehavereachedthethirddegreewherewe
devoteourintelligencestoanticipatingwhataverageopinionexpectstheaverage
opiniontobe.Andtherearesome,Ibelieve,whopractisethefourth,fifthandhigher
degrees.”
IntheexperimentalPBCgamepeoplechoosenumbersfrom0to100simultaneously
withouttalking.vThepersonwhosenumberisclosesttoptimestheaveragewinsafixed
price.
Atypicalinterestingvalueofpis2/3.Thenthewinnerwantstobetwo‐thirdsofthe
waybetweentheaverageandzero.Butofcourse,theplayersallknowtheotherplayers
wanttopick2/3oftheaverage.InaNashequilibrium,everyoneaccuratelyforecaststhat
theaveragewillbeX,andalsochoosesanumberwhichis(2/3)X.ThisimpliesX=(2/3)Xor
X*=0.
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Intuitively,supposeyouhadnoideawhatotherpeoplewoulddo,soyouchose2/3
of50=33.Thisisareasonablechoicebutisnotanequilibrium,sincechoosing33while
anticipating50leavesagapbetweenexpectedbehaviorofothersandlikelybehaviorby
oneself.Soapersonwhothinks“Hey!I’llpick33”shouldthenthink(toadheretothe
equilibriummath)“Hey!They’llpick33”andthenpick22.Thisprocessofimagining,
choosingandrevisingdoesnotstopuntileveryoneexpects0tobechosen,andalsopicks0.
Figure1showssomedatafromthisgameplayedwithexperimentalsubjectsandin
newspaperandmagazinecontests(wherelargegroupsplayforasinglelargeprize).There
issomeevidenceof“spikes”innumberscorrespondingto50p,50p2andsoon.
Example2:Bettingonselfishrationalityofothers
AnothersimpleillustrationoftheCHtheoryisshowninTable1.Inthisgamearow
andcolumnplayerchoosefromoneoftwostrategies,TorB(forrow)orLorR(for
column).Thecolumnplayeralwaysgets20forchoosingLand18forchoosingR.Therow
playergetseither30or10fromT,andasure20fromB.
Ifthecolumnplayeristryingtogetthelargestpayoff,sheshouldalwayschooseL(it
guarantees20insteadof18).ThestrategyLiscalleda“strictlydominantstrategy”because
ithasthehighestpayoffforeverypossiblechoicebytherowplayer.
Therowplayer’schoiceisalittletrickier.Shecanget20forsurebychoosingB.
ChoosingTistakingasocialgamble.Ifsheisconfidentthecolumnplayerwilltrytoget20
andchooseL,sheshouldinferthatP(L)ishigh.ThentheexpectedvalueofTishighandshe
shouldchooseT.However,thisinferenceisessentiallyabetontheselfishrationalityofthe
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otherplayer.Therowplayermightthinkthecolumnplayerwillmakeamistake,oris
spiteful(andprefersthe(10,18)cellbecauseshegetslessabsolutepayoffbutahigher
relativepayoffcomparedtotherowplayer).Thereisacrucialcognitivedifferencein
playingL—whichistherightstrategyifyouwantthemostmoney—andplayingT—which
istherightstrategyifyouarewillingtobetthatotherplayersareverylikelytochooseL
becausetheywanttoearnthemostmoney.
WhatdoestheCHapproachpredicthere?Supposelevel0playersrandomize
betweenthetwostrategies.Ifτ=1.5,thenf(0|τ=1.5)=.22.Thenhalfofthelevel0players
willchoosecolumnRandrowB,whichis.11%ofthewholegroup.
Level1playersalwayschooseweaklydominantstrategies,sotheypickcolumnL
(infact,allhigherlevelcolumnplayersdotoo).Sincelevel1rowplayersthinkLandR
choicesareequallylikely,theirexpectedpayofffromTis30(.5)+10(.5)=20,whichisthe
sameastheBpayoff;soweassumetheyrandomizeequallybetweenTandB.Since
f(1|τ=1.5)=.33,thismeanstheunconditionaltotalfrequencyofBplayforthefirsttwo
levelsis.11+.33/2=.27.
Level2rowplayersthinktherelativeproportionsoflowertypesare
g2(0)=.22/(.22+.33)=.40andg2(1)=.33/(.22+.33)=.60.Theyalsothinkthelevel0’splay
eitherLorR,butthelevel1’schooseLforsure.Together,thisimpliesthattheybelieve
thereisa.20chancetheotherpersonwillchooseR(=.5(.40)+0(.60))andan.80chance
theywillchooseL.Withtheseodds,theyprefertochooseT.Thatis,theyaresufficiently
confidenttheotherplayerwill“figureitout”andchoosetheself‐servingLthatTbecomesa
goodbettoyieldthehigherpayoffof30.
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Puttingtogetherallthefrequenciesf(k)andchoicepercentages,theoverall
expectedproportionofcolumnRplayis.11androwBplayis.27.Notethatthese
proportionsgointhedirectionoftheNashprediction(whichiszeroforboth),butaccount
morepreciselyforthechanceofmistakesandmisperceptions.Importantly,choicesofR
shouldbelesscommonthanchoicesofB.Rchoicesarejustcareless,whileBchoicesmight
becarelessormightbesensibleresponsestothinkingtherearealotofcarelessplayers.
Table1showsthatsome(unpublished)datafromCaltechundergraduateclassroom
games(formoney)overthreeyearsaregenerallyclosetotheCHprediction.TheRandB
choicefrequenciesaresmall(asbothNashandCHpredict)butBismorecommonthanR.
[InsertTable1abouthere]
OnepotentialadvantageofCHmodelingisthatthesamegeneralprocesscould
applytogameswithdifferenteconomicstructures.Inbothofthetwoexamplesabove,a
Nashequilibriumchoicecanbederivedbyrepeatedapplicationoftheprincipleof
eliminating“weaklydominated”strategies(i.e.,strategieswhichareneverbetterthan
anotherdominatingstrategy,forallchoicesbyotherpeople,andisactuallyworseforsome
choicesbyothers).Hence,thesearecalled“dominancesolvable”games.Indeed,the
beauty‐contestexampleisamongthosethatmotivatedCHmodelinginthefirstplace,since
eachstepofreasoningcorrespondstoonemorestepindeletionofdominatedstrategies.
Hereisanentirelydifferenttypeofgame,called“asymmetricmatchingpennies”.In
thisgametherowplayerearnspointsifthechoicesmatch(H,H)or(T,T).Thecolumn
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playerwinsiftheymismatch.Thereisnopairofstrategiesthatarebestresponsestoeach
other,sotheequilibriumrequireschoosingaprobabilistic“mixture”ofstrategies.Here,
equilibriumanalysismakesabizarreprediction:TherowplayershouldchooseHandT
equallyoften,whilethecolumnplayershouldshyawayfromH(asifpreventingRowfrom
gettingthebiggerpayoffof2)andchooseT2/3ofthetime.(Evenmorestrangely:Ifthe2
payoffisx>1ingeneral,thenthemixtureisalways50‐50fortherowplayer,andisx/(x+1)
onTforthecolumnplayer!Thatis,intheorychangingthepayoffof2onlyaffectsthe
columnplayer,anddoesnotaffecttherowplayerwhomightearnthatpayoff.
TheCHapproachworksdifferentlyvi.Thelowerlevelrowplayers(1‐2)areattracted
tothepossiblepayoffof2,andchooseH.However,thelowlevelcolumnplayersswitchto
T,andhigherlevelrowplayers(levels3‐4)figurethisoutandswitchtoT.Thepredicted
mixture(fortau=1.5)isactuallyratherclosetotheNashpredictionforthecolumnplayer
(P(T)=.74comparedtoNash.67),sincethehigher‐leveltypeschooseTmoreandnotH.
Andindeed,datafromcolumnplayerchoicesinexperimentsareclosetobothpredictions.
TheCHmixtureofrowplay,averagedacrosstypefrequencies,isP(H)=.68,closetothedata
averageof.72.Thus,thereasonablepartoftheNashprediction,whichislopsidedplayofT
andHbycolumnplayers,isreproducedbyCHandisconsistentwiththedata.The
unreasonablepartoftheNashprediction,thatrowplayerschooseHandTequallyoften,is
notreproducedandthedifferingCHpredictionismoreempiricallyaccurate.
[InsertTable2abouthere]
Entrygames
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Insimple“entry”games,Nplayerssimultaneouslychoosewhethertoentera
marketwithdemandC,ornot.Iftheystayout,theyearnafixedpayoff($.50).Iftheyenter,
thenalltheentrantsearn$1ifthereareCorfewerentrants,andearn0iftherearemore
thanCentrants.ItiseasytoseethattheequilibriumpatternofplayisforexactlyCpeople
toenter;thentheyeachearn$1andthosewhostayoutearn$.50.Ifoneofthestayer‐
outersswitchedandentered,shewouldtipthemarketandcausetheC+1entrantstoearn
0.Sincethiswouldlowerherownpayoff,shewillstayput.Sothepatternisanequilibrium.
However,thereisaproblemremaining(it’sacommononeingametheory):How
doesthegroupcollectivelydecide,withouttalking,whichoftheCpeopleenterandearn
$1?EverybodywouldliketobeintheselectgroupofCentrantsiftheycan;butiftoomany
entertheyallsuffer.viiThisisafamiliarproblemof“coordinating”toreachoneofmany
differentequilibria.
Thefirstexperimentsonthistypeofentrygameweredonebyateamofeconomists
(JamesBranderandRichardThaler)andapsychologist,DanielKahneman.Theywere
neverfullypublishedbutweredescribedinachapterbyKahneman(1988).Kahneman
saystheywereamazedhowclosethenumberoftotalentrantswastotheannounced
demandC(whichvariedovertrials).“Toapsychologist”,hewrote,“itlookedlikemagic”.
Sincethen,acoupleofdozenstudieshaveexploredvariantsofthesegamesandreported
similardegreesofcoordination(e.g.,Duffy&Hopkins,2005).
Let’sseeifcognitivehierarchycanproducethemagic.Supposelevel0playersenter
andstayoutequallyoften,andignoreC.Iflevel1playersanticipatethis,theywillthink
therearetoomanyentrantsforC<(N/2)andtoofewifC>(N/2)‐1.Level1playerswill
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thereforeenterathighvaluesofC.Noticethatlevel1playersarehelpingthegroupmove
towardtheequilibrium.Level1’sundothedamagedonebythelevel0’s,whoover‐enterat
lowC,bystayingoutwhichreducestheoverallentryrateforlowC.Theyalsoexploitthe
opportunitythatremainsforhighC,byentering,whichincreasestheoverallentryrate.
Combiningthetwolevels,therewillbelessentryatlowCandmoreentryathighC(itwill
looklikeastepfunction;seeCamereretal.,2004).
Furthermore,itturnsoutthataddinghigher‐levelthinkerscontinuestopushthe
populationprofiletowardanoverallentrylevelthatisclosetoC.Thetheorymakesthree
sharppredictions:(1)Plottingentryrates(asa%ofN)againstC/Nshouldyielda
regressivelinewhichcrossesat(.5,.5).(2)EntryratesshouldbetoohighforC/N<.5and
toolowforC/N>.5.(3)EntryshouldbeincreasinginC,andrelativelyclose,evenwithout
anylearningatall!(e.g.,inthefirstperiodofthegame).
Figure3illustratesaCHmodelpredictionwithτ=1.25,single‐perioddatawithno
feedbackfromCamereretal.(2004),andtheequilibrium(a45‐degreeline).Exceptfor
somenonmonotonicdipsintheexperimentaldata(easilyaccountedforbysampling
error),thepredictionsareroughlyaccurate.
Thepointofthisexampleisthatapproximateequilibrationcanbeproduced,asifby
“magic”,purelyfromcognitivehierarchythinkingwithoutanylearningorcommunication
needed.Thesedataarenotsolidproofthatcognitivehierarchyreasoningisoccurringin
thisgame,butdoesshowhow,inprinciple,thecognitivehierarchyapproachcanexplain
bothdeviationsfromNashequilibrium(inthebeautycontest,betting,andmatching
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penniesgamesthatweredescribedabove),andalsosurprisingconformitytoNash
equilibrium(inthisentrygame).
Privateinformation
Thetrickiestclassofgameswewilldiscuss,briefly,involve“privateinformation”.
Thestandardmodelingapproachistoassumethereisahiddenvariable,X,whichhasa
possibledistributionp(X)thatiscommonlyknowntobothplayersviii.TheinformedplayerI
knowstheexactvaluexfromthedistributionandbothplayersknowthatonlyIknowsthe
value.Forexample,incardgameslikepoker,playersknowthepossiblesetofcardstheir
opponentmighthave,andknowthattheopponentknowsexactlywhatthecardsare.
Thecognitivechallengethatisspecialtoprivateinformationgamesistoinferwhat
aplayer’sactions,whethertheyareactuallytakenorhypothetical,mightrevealabouttheir
information.Variousexperimentalandfielddataindicatethatsomeplayersarenotvery
goodatinferringhiddeninformationfromobservedaction(oranticipatingtheinferable
information).
Asimpleandpowerfulexampleisthe“acquire‐a‐company”problemintroducedin
economicsbyAkerlof(1970)andstudiedempiricallybyBazermanandSamuelson(1983).
Inthisgame,aprivately‐heldcompanyhasavaluewhichisperceivedbyoutsiderstobe
uniformlydistributedfrom0to100(i.e.,allvaluesinthatrangeareequallylikely).The
companyknowsitsexactvalue,andoutsidersknowthatthecompanyknows(duetothe
commonpriorassumption).
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Abiddercanoperatethecompanymuchbetter,sothatwhateverthehiddenvalueV
is,itisworth1.5Vtothem.Theymakeatake‐it‐or‐leave‐it(Boulwarean)offerofapriceP.
Thebargainingcouldhardlybesimpler:ThecompanysellsifthepricePisabove“hidden”
valueV—whichthebidderknowsthatthecompanyknows—andkeepsthecompany
otherwise.Thebidderwantstomaximizetheexpected“surplus”gainbetweentheaverage
ofthevalues1.5Vtheyarelikelytoreceiveandtheprice.
Whatwouldyoubid?Theoptimalbidissurprising,thoughthealgebrabehindthe
answerisnottoohard.ThechanceofgettingthecompanyisthechancethatVislessthan
P,whichisP/100(e.g.,ifP=60then60%ofthetimethevalueisbelowPandthecompany
changeshands).Ifthecompanyissold,thenthevaluemustbebelowP,sotheexpected,
valuetotheselleristheaverageofthevaluesintheinterval[0,P],whichisP/2.Thenet
expectedvalueistherefore(P/100)timesexpectedprofitifsold,whichis1.5*(P/2)‐P=
‐1/4P.Thereisnowaytomakeaprofitonaverage.Theoptimalbidiszero!
However,typicaldistributionsofbidsarebetween50and75.Thisresultsina
“winner’scurse”inwhichbidders“win”thecompany,butfailtoaccountforthefactthat
theyonlywonbecausethecompanyhadalowvalue.Thisphenomenonwasfirstobserved
infieldstudiesofoil‐leasebidding(Capenetal1971)andhasbeenshowninmanylaband
fielddatasetssincethen.Thegeneralprinciplethatpeoplehaveahardtimeguessingthe
implicationsofprivateinformationforactionsotherswilltakeshowsupinmanyeconomic
settings(akindofstrategicnaivete;e.g.Brocasetal.,2009).
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TheCHapproachcaneasilyexplainstrategicnaiveteasaconsequenceoflevel1
behavior.Iflevel1playersthinkthatlevel0players’choicesdonotdependonprivate
information,thentheywillignorethelinkbetweenchoicesandinformation.
Eyetrackingevidence
Apotentialadvantageofcognitivehierarchyapproachesisthatcognitivemeasures
associatedwiththealgorithmicstepsplayersareassumedtouse,intheory,couldbe
collectedalongwithchoices.Forpsychologiststhisisobviousbut,amazingly,itisarather
radicalpositionineconomicsandmostareasofgametheory!
Theeasiestandcheapestmethodistorecordwhatinformationpeoplearelooking
atastheyplaygames.Eyetrackingmeasuresvisualfixationsusingvideo‐basedeyetracking,
typicallyevery5‐50msec.Cameraslookintotheeyeandadjustforheadmotiontoguess
wheretheeyesarelooking(usuallywithexcellentprecision).Mosteyetrackersrecord
pupildilationaswell,whichisusefulasameasureofcognitivedifficultyorarousal.
Sincegametheoryisaboutinteractionsamongtwoormorepeople,itisespecially
usefultohavearecordingtechnologythatscalesuptoenablerecordingofseveralpeople
atthesametime.Onewidely‐usedmethodiscalled“Mouselab”.InMouselab,information
thatisusedinstrategiccomputations,intheory,ishiddeninlabeledboxes,which“open
up”whenamouseismovedintothem.ix
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Severalstudieshaveshownthatlookuppatternsoftencorrespondroughly,and
sometimesquiteclosely,todifferentnumbersofstepsofthinking.We’llpresentone
example(seealsoCrawfordetal.,2010).
Example1:Alternating‐offerbargaining
Apopularapproachtomodelingbargainingistoassumethatplayersbargainovera
knownsumofjointgain(sometimescalled“surplus”,likethevaluablegapbetweenthe
highestpriceabuyerwillpayandthelowestpriceasellerwillaccept).However,astime
passestheamountofjointgain“shrinks”duetoimpatienceorothercosts.Players
alternatemakingoffersbackandforth(Rubinstein,1982).
Athree‐periodversionofthisgamehasbeenstudiedinmanyexperiments.The
amountdividedinthefirstroundis$5,whichthenshrinksto$2.50,$1.25,and0inlater
rounds(thelastroundisan“ultimatumgame”).Ifplayersareselfishandmaximizetheir
ownpayoffs,andbelievethatothersaretoo,the“subgameperfect”equilibrium(SPE)offer
bythefirstpersonwhooffers(player1),toplayer2,shouldbe$1.25.However,deriving
thisoffereitherrequiressomeprocessoflearningorcommunication,orananalysisusing
“backwardinduction”todeducewhatofferswouldbemadeandacceptedinallfuture
rounds,thenworkingbacktothefirstround.Earlyexperimentsshowedconflictingresults
inthisgame.Neelinetal.(1988)foundthataverageofferswerearound$2,andmanywere
equalsplitsof$2.50each.Earlier,Binmoreetal.(1985)foundsimilarresultsinthefirst
roundofchoices,butalsofoundthatasmallamountofexperiencewith“rolereversal”
(player2’sswitchingtotheplayer1first‐offerposition)movedofferssharplytowardthe
SPEofferof$1.25.Otherevidencefromsimplerultimatumgamesshowedthatpeopleseem
19
tocareaboutfairness,andarewillingtorejecta$2offeroutof$10abouthalfthetime,to
punishabargainingpartnertheythinkhasbeenunfairandgreedy(Camerer,2003).
Soaretheoffersaround$2duetocorrectanticipationoffairness‐influenced
behavior,ortolimitedunderstandingofhowthefutureroundsofbargainingmightshape
reactionsinthefirstround?Tofindout,Camereretal.(1993)andJohnsonetal.(2002)did
thesametypeofexperiment,buthidtheamountsbeingbargainedoverineachroundin
boxesthatcouldbeopened,or“lookedup”,bymovingamouseintothoseboxes(an
impoverishedexperimenter’sversionofvideo‐basedeyetracking).Theyfoundthatmost
peoplewhoofferedamountsbetweentheequalsplitof$2.50andtheSPEof$1.25werenot
lookingaheadatpossiblefuturepayoffsasbackwardinductionrequires.Infact,in10‐20%
ofthetrialstheround2andround3boxeswerenotopenedatall!
Figure4illustratesthebasicresults.Thetoprectangular“icongraphs”visually
representtherelativeamountsoftimebargainersspentlookingateachpayoffbox(the
shadedarea)andnumbersofdifferentlookups(rectanglewidth).Theboldarrowsindicate
therelativenumberoftransitionsfromoneboxtothenext(withaveragesoflessthanone
transitionomitted).
Eachcolumnrepresentsagroupoftrialsthatarepre‐classifiedbylookuppatterns.
Thefirstcolumn(N=129trials)averagespeoplewholookedmoreoftenattheperiod1box
thanatthefutureperiodboxes(indicating“level‐0”planning).Thesecondcolumn(N=84)
indicatespeoplewholookedlongeratthesecondboxthanthefirstandthird(indicating
“levelOne”planningwithsubstantialfocusonestepahead).Thethirdcolumn(N=27)
indicatesthesmallernumberof“equilibrium”trialsinwhichthethirdboxislookedatthe
20
most.NotethatinthelevelOneandEquilibriumtrials,therearealsomanytransitions
betweenboxesoneandtwo,andboxestwoandthree,respectively.Finally,thefourthand
lastcolumnshowspeoplewhowerebrieflytrainedinbackwardinduction,thenplayeda
computerizedopponentthat(theyweretold)plannedahead,actedselfishlyandexpected
thesamefromitsopponents.
Themainpatterntonoticeisthatofferdistributions(shownatthebottomofeach
column)shiftfromright(fairer,indicatedbytherightdottedline)toleft(closertoselfish
SPE,theleftdottedline)asplayersliterallylookaheadmore.Thelinkbetweenlookupsand
higher‐than‐predictedoffersclearlyshowsthatoffersabovetheSPE,inthedirectionof
equalsplitsofthefirstroundamount,arepartlyduetolimitsonattentionandcomputation
aboutfuturevalues.Eveninthefewequilibriumtrials,offersarebimodal,clusteredaround
$1.25and$2.20.However,offersarerathertightlyclusteredaroundtheSPEpredictionof
$1.25inthe“trained”condition.Thisresultindicates,importantly,thatbackwardinduction
isnotactuallythatcognitivelychallengingtoexecute(afterinstruction,theycaneasilydo
it),butinsteadisanunnaturalheuristicthatdoesnotreadilyspringtothemindsofeven
analyticalcollegestudents.
fMRIevidence
Severalneuralstudieshaveexploredwhichbrainregionsaremostactivein
differenttypesofstrategicthinking.Theearlieststudiesshoweddifferentialactivation
whenplayingagameagainstacomputercomparedtoarandomizedopponent(e.g.,
(Gallagheretal.,2002;McCabeetal.,2001;Coricelli&Nagel,2009).
21
Oneofthecleanestresults,andanexemplarofthecompositepictureemergingfrom
otherstudies,isfromCoricelli&Nagel’s(2009)studyofthebeautycontestgame.Their
subjectsplayed13differentgameswithdifferenttargetmultipliersp(e.g.,p=2/3,1,3/2
etc.).Oneachtrial,subjectschosenumbersintheinterval[0,100]playingagainsteither
humansubjectsoragainstarandomcomputeropponent.Usingbehavioralchoices,most
subjectscanbeclassifiedintoeitherlevel1(n=10;choosingptimes50)orlevel2(n=7;
choosingptimesptimes50,asifanticipatingtheplayoflevel1opponents).
Figure7showsbrainareasthatweredifferentiallyactivewhenplayinghuman
opponentscomparedtocomputeropponents,andinwhichthathuman‐computer
differentialislargerinlevel2playerscomparedtolevel1players.Thecrucialareasare
bilateraltemporo‐parietaljunction(TPJ),MPFC/paracingulateandVMPFC.xTheseregions
arethoughttobepartofageneralmentalizingcircuit,alongwithposteriorcingulate
regions(Amodio&Frith,2006).
Inrecentstudies,atleastfourareasarereliablyactivatedinhigher‐levelstrategic
thinking:dorsomedialprefrontalcortex(DMPFC),precuneus/posteriorcingulate,insula,
anddorsolateralprefrontalcortex(DLPFC).Nextwesummarizesomeofthesimplest
results.
DMPFCactivityisevidentinFigure7.Itisalsoactiveinresponsetononequilibrium
choices(wheresubjects’guessesaboutwhatotherswilldoarewrong;Bhatt&Camerer,
2005),anduncertaintyaboutstrategicsophisticationofanopponent(Yoshidaetal.,2009).
Inaddition,DMPFCactivityisrelatedtothe“influencevalue”ofcurrentchoicesonfuture
rewards,filteredthroughtheeffectofaperson’sfuturechoicesonanopponent’sfuture
22
choices(Hampton,Bossaerts,&O’Doherty,2009;a/k/a“strategicteaching”Camerer,Ho
andChong2002).Amodio&Frith(2006)suggestanintriguinghypothesis:that
mentalizing‐valueactivationforsimplertomorecomplexactionvaluecomputationsare
differentiallylocatedalongaposterior‐to‐anterior(back‐to‐front)gradientinDMPFC.
Indeed,thelatterthreestudiesshowactivationroughlyinaposterior‐to‐anteriorgradient
(Tailarachy=36,48,63;andy=48inCoricelli&Nagel,2009)thatcorrespondstoincreasing
complexity.
Activityintheprecuneus(adjacenttoposteriorcingulate)isassociatedwith
economicperformanceingames(“strategicIQ”;Bhatt&Camerer2005)anddifficultyof
strategiccalculations(Kuoetal.2009).Precuneusisabusyregion,withreciprocal
connectionstoMPFC,cingulate,andDLPFC.Itisalsoactivatedbyawidevarietyofhigher‐
ordercognitions,includingperspective‐takingandattentionalcontrol(aswellasthe
“defaultnetwork”activeatrest;seeBhatt&Camerer,inpress).Itislikelythatprecuneusis
notactivatedinstrategicthinking,perse,butonlyinspecialtypesofthinkingwhich
requiretakingunusualperspectives(e.g.,thinkingaboutwhatotherpeoplewilldo)and
shiftingmentalattentionbackandforth.
Theinsulaisknowntobeinvolvedininteroceptiveintegrationofbodilysignalsand
cognition.Disgust,physicalpain,empathyforothersinpain,andpainfromsocialrejection
activateinsula(Eisenbergeretal.2003,Krossetal.2011).Financialuncertainty
(Preuschoff,Quartz,Bossaerts,2008),interpersonalunfairness(Sanfeyetal.,2003;Hsuet
al.,2008),avoidanceofguiltintrustgames(Changetal.,2011),and“coaxing”orsecond‐
trysignalsintrustgamesalsoactivateinsula.Instrategicstudies,BhattandCamerer
23
(2005)foundthathigherinsulaactivityisassociatedwithlowerstrategicIQ
(performance).
TheDLPFCisinvolvedinworkingmemory,goalmaintenance,andinhibitionof
automaticprepotentresponses.Differentialactivitythereisalsoassociatedwiththelevel
ofstrategicthinking(Yoshidaetal.,2009)withstrongerresponsetohumanopponentsin
higher‐levelstrategicthinkers(Coricelli&Nagel,2009),andwithmaintaininglevel‐2
deceptioninbargaininggames(Bhattetal.,2009).xi
Dothinkingstepsvarywithpeopleorgames?
Towhatextentdostepsofthinkingvarysystematicallyacrosspeopleorgame
structures?Fromacognitivepointofview,itislikelythatthereissomeintrapersonal
stabilitybecauseofdifferencesinworkingmemory,strategicsavvy,exposuretogame
theory,experienceinsportsbettingorpoker,taskmotivation,etc.However,itisalsolikely
thattherearedifferencesinthedegreeofsophistication(measuredbyτ)acrossgames
becauseofaninteractionbetweengamecomplexityandworkingmemory,orhowwellthe
surfacegamestructuremapsontoevolutionarilyfamiliargamesxii.
Todate,thesesourcesofleveldifferenceshavenotbeenexploredverymuch.Chong,
HoandCamerer(2005)notesomeeducationaldifferences(Caltechstudentsareestimated
todo.5stepsofthinkingmorethansubjectsfromanearbycommunitycollege)andan
absenceofagendereffect.Otherstudieshaveshowedmodestassociations(r=.3)between
strategiclevelsandworkingmemory(digitspan;Devetag&Warglien,2003)andthe“eyes
ofthemind”testofemotiondetection(Georganas,Healy,&Weber2010).
24
Manypapershavereportedsomedegreeofcross‐gametypestabilityinlevel
classification.Studiesthatcompareachoiceinonegamewithonedifferentgamereport
lowstability(Georganasetal.,2010;Burchardi&Penczynski2010).However,asiswell‐
knowninpersonalitypsychologyandpsychometrics,intrapersonalreliabilitytypically
increaseswiththenumberofitemsusedtoconstructascale.Otherstudiesusingmore
gamechoicestoclassifyreportmuchhighercorrelations(comparabletoBig5personality
measures)(Bhui&Camerer,2011).
Asoneillustrationofpotentialtype‐stability,Figure5belowshowsestimatedtypes
forindividualsusingthefirst11gamesina22‐gameseries(x‐axis)andtypesforthesame
individualsusingthelast11games.Thecorrelationisquitehigh(r=.61).Thereisalsoa
slightupwarddriftacrossthegames(theaveragelevelishigherinthelast11games
comparedtothefirst),consistentwithatransferorpracticeeffect,eventhoughthereisno
feedbackduringthe22games(seealsoWeber,2003).
Fielddata
Sincecontrolledexperimentationcamelatetoeconomics(c.1960)comparedto
psychology,thereisalong‐standingskepticismaboutwhethertheoriesthatworkinsimple
labsettingsgeneralizetonaturally‐ocurringeconomicactivity.Fivestudieshaveapplied
CHorlevel‐kmodelingtoauctions(Gillen,2009),strategicthinkinginmanagerialchoices
(GoldfarbandYang,2009;GoldfarbandXiao,inpress),andboxofficereactionwhen
moviesarenotshowntocriticsbeforerelease(Brown,Camerer&Lovallo,2011).
Onestudyisdescribedhereasanexample(Ostlingetal.,2011).In2007the
SwedishLotterycreatedagameinwhichpeoplepay1eurotoenteralottery.Eachpaying
25
entrantchoosesaninteger1‐99,999.Thelowestuniquepositiveinteger(hence,the
acronymLUPI)winsalargeprize.
Thesymmetricequilibriumisaprobabilisticprofileofhowoftendifferentnumbers
arechosen(a“mixed”equilibrium).Thelowestnumbersarealwayschosenmoreoften
(e.g.,1ischosenmostoften);therateofdeclineinthefrequencyofchoiceisaccelerating
uptoasharpinflectionpoint(number5513);andtherateofdeclineslowsdownafter
5513.
Figure6showsthedatafromonlythelowest10%ofthenumberrange,from1‐
10,000(highernumberchoicesarerare,asthetheorypredicts).ThepredictedNash
equilibriumisshownbyadottedline—aflat“shelf”ofchoiceprobabilityfrom1to5513,
thenasharpdrop.AfittedversionoftheCHmodelisindicatedbythesolidline.CHcan
explainthelargefrequencyoflownumberchoices(below1500),sincethesecorrespondto
lowlevelsofstrategicthinking(i.e.,peopledon’trealizeeveryoneelseischoosinglow
numberstoo).Sincelevel‐0typesrandomize,theirbehaviorproducestoomanyhigh
numbers(above5000).Sincethelowestandhighestnumbersarechosentoooften
accordingtoCH,comparedtotheequilibriummixture,CHalsoimpliesagapbetween
predictedandactualchoicesintherange2500‐5000.Thisbasicpatternwasreplicatedina
labexperimentwithasimilarstructure.WhiletherearecleardeviationsfromNash
equilibrium,consistentwithevidenceoflimitedstrategicthinking,inourviewtheNash
theorypredictionisnotbadconsideringthatusesnofreeparameters,andcomesfroman
equationwhichiselegantinstructurebutdifficulttoderiveandsolve.
26
TheLUPIgamewasplayedinSwedenfor49daysinarow,andresultswere
broadcastonanightlyTVshow.Analysisindicatesanimitate‐the‐winnerfictivelearning
process,sincechoicesononedaymoveinthedirectionof600‐numberrangearoundthe
previousday’swinner.Theresultofthisimitationisthateverystatisticalfeatureofthe
numberschosenmovestowardtheequilibriumacrossthesevenweeks.Forexample,in
thelastweektheaveragenumberis2484,within4%ofthepredictedvalueof2595.
IIIPsychologicalgames
Inmanystrategicinteractions,ourownbeliefsorbeliefsofotherpeopleseemto
influencehowwevalueconsequences.Forexample,surprisingapersonwithawonderful
giftthatisperfectforthemismorefunforeveryonethanifthepersonhadaskedforit.
Someofthatpleasurecomesfromthesurpriseitself.
Thistypeofpatterncanbemodeledasa“psychologicalgame”(Geanakoplos,Pearce
&Stacchetti,1989andBattigalli&Dufwenberg,2009).PGsareanextensionofstandard
gamesinwhichtheutilityevaluationsofoutcomescandependonbeliefsaboutwhatwas
thoughttobelikelytohappen(aswellastypicalmaterialconsequences).Thisapproach
requiresthinkingandreasoningsincethebeliefisderivedfromanalysisoftheother
person’smotives.Togetherthesepapersprovidetoolsforincorporatingmotivationssuch
asintentions,socialnorms,andemotionsintogame‐theoreticmodels.
Emotionsareanimportantbeliefdependentmotivation.Anxiety,
disappointment,elation,frustration,guilt,joy,regret,andshame,amongother
emotions,canallbeconceivedofasbelief‐dependentincentivesormotivationsand
27
incorporatedintomodelsofbehaviorusingtoolsfrompsychologicalgametheory.
Oneexampleisguilt:Baumeister,Stillwell,andHeatherton(1994)write:“If
peoplefeelguiltyforhurtingtheirpartners…andforfailingtoliveuptotheir
expectations,theywillaltertheirbehavior(toavoidguilt).”BattigalliandDufwenberg
(2007)operationalizethenotionthatpeoplewillanticipateandavoidguiltintheir
modelofguiltaversion.Intheirmodel,playersderivepositiveutilityfromboth
materialpayoffsandnegativeutilityfromguilt.Playersfeelguiltyiftheirbehavior
disappointsaco‐playerrelativetohisexpectations.xiii
ConsiderFigure8,whichillustratesasimpletrustgame.Player1maychoose
either“Trust”or“Don’t.”Inthefirstcaseplayer1getsthemove,whileafterachoiceof
“Don’t”thegameendsandeachplayergetspayoff1.Ifplayer2getsthemove,she
choosesbetween“Grab”and“Share.”ThepayoffstoGrabare0forplayer1and4for
player2.
ThesubgameperfectequilibriumofthisgameforselfishplayersisforPlayer2to
chooseGrabifshegetsthemovesinceitresultsinahigherpayoffforherthanchoosing
Share.Player1anticipatesthisbehaviorandchoosesDon’ttoavoidreceiving0.Both
playersreceiveapayoffof1,whichisinefficient.
NowsupposethatPlayer2isguiltaverse.Thenherutilitydependsnotonlyonher
materialpayoff,butalsoonhowmuchshe“letsdown”player1relativetohisexpectations.
Letpbetheprobabilitythatplayer1assignsto“Share.”Letp’representPlayer2’s(point)
beliefregardingp,andsupposethat2’spayofffromGrabisthen4‐θp,wheretheta
28
representsplayer2’ssensitivitytoguilt.Ifplayer1choosesTrustitmustbethatpis
greaterthan½‐otherwiseplayer1wouldchooseDon’t.Thenifθ≥2,player2willchoose
SharetoavoidtheguiltfromlettingdownPlayer1.Knowingthis,player1willchoose
Trust.Inthisoutcomebothplayersreceive2(insteadof1intheselfishsubgameperfect
equilibrium),illustratinghowguiltaversioncanfostertrustandcooperationwhereselfish
behaviorleadstoinefficiency.
Anumberofexperimentshavestudiedguiltaversioninthecontextoftrustgames,
includingDufwenberg&Gneezy(2000),Charness&Dufwenberg(2006,2011),and
Reubenetal.(2009).Allofthesepapersfindevidencethatadesiretoavoidguiltmotivates
playerstobehaveunselfishlybyreciprocatingtrust(foracontraryopinionseeEllingsenet
al.,2010).RecentfMRIevidence(Changetal,inpress)suggeststhatavoidingguiltintrust
gamesisassociatedwithincreasedactivityintheanteriorinsula.
Psychologicalgametheoryalsomaybeemployedtomodelothersocialemotions
suchasshame(Tadelis,2008)oranger(Smith,2009)ortoimportexistingmodelsof
emotionssuchasdisappointment,elation,regret,andrejoicing(Bell,1982,1986;
Loomes&Sugden,1982,1985)intogames.xivBattigalli&Dufwenberg(2009)provide
someexamplesoftheseapplications.Thesemodelsarejustaglimpseofthepotential
applicationsofpsychologicalgametheorytotheinteractionofemotionandcognitionin
socialinteractions.
Anotherimportantapplicationofpsychologicalgametheoryissociological
concerns,suchasreciprocity(whichmaybedrivenbyemotions).Inanimportantwork,
Rabin(1993)modelsreciprocityviafunctionsthatcaptureaplayer’s“kindness”tohis
29
coplayerandtheotherplayer’skindnesstohim.Thesekindnessfunctionsdependonthe
players’beliefsregardingeachother’sactions,andtheirbeliefsabouteachother’sbeliefs.
Dufwenberg&Kirchsteiger(2004)andFalk&Fischbacher(2006)extendRabin’smodelto
sequentialgames.
Psychologicalgametheoryprovidesausefultoolkitforincorporatingpsychological,
social,andculturalfactorsintoformalmodelsofdecision‐makingandsocialinteractions.
Manyapplicationsremaintobediscoveredandtestedviaexperiment.
Conclusions
Comparedtoitsimpactonotherdisciplines,gametheoryhashadlessimpactin
cognitivepsychologysofar.Thisislikelybecausemanyoftheanalyticalconceptsusedto
derivepredictionsabouthumanbehaviordonotseemtocorrespondcloselytocognitive
mechanisms.Somegametheoristshavealsocomplainedaboutthisunrealism.EricVan
Damme(1999)wrote:
Withouthavingabroadsetoffactsonwhichtotheorize,thereisacertaindangerof
spendingtoomuchtimeonmodelsthataremathematicallyelegant,yethavelittle
connectiontoactualbehavior.Atpresentourempiricalknowledgeisinadequate
anditisaninterestingquestionwhygametheoristshavenotturnedmore
frequentlytopsychologistsforinformationaboutthelearningandinformation
processesusedbyhumans.
30
Butrecently,anapproachcalledbehavioralgametheoryhasbeendevelopedwhich
usespsychologicalideastoexplainbothchoicesinmanydifferentgames,andassociated
cognitiveandbiological(Camerer2003;Bhatt&Camerer,2011)
Thischapterdiscussedtwoelementsofbehavioralgametheorythatmightbeof
mostinteresttocognitivepsychologists:Thecognitivehierarchyapproach;and
psychologicalgamesinwhichoutcomevaluescandependonbeliefs,oftenaccompaniedby
emotions(e.g.,alowbargainingoffercouldcreateangerifyouexpectedmore,orjoyifyou
expectedless).
Thecognitivehierarchyapproachassumesthatsomeplayerschooserapidlyand
heuristically(“level0”)andhigher‐levelplayerscorrectlyanticipatewhatlower‐level
playersdo.Thetheoryhasbeenusedtoexplainbehaviorinlabgameswhichisbothfar
fromandclosetoequilibjriumindifferentgames,issupportedbyevidencefromvisual
eyetrackingandMouselab,isevidentin“theoryofmind”circuitryduringfMRI,andalsocan
explainsomepatternsinfielddata(suchastheSwedishLUPIlottery).
Researchonpsychologicalgamesislesswelldevelopedempirically,buthasmuch
promiseforunderstandingphenomenalike“socialimage”,normenforcement,how
emotionsarecreatedbysurprises,andtherelationshipbetweenemotion,cognition,and
strategicbehavior.
FutureDirections
Therearealotofopenresearchquestionsinwhichcombiningcognitivescienceand
gametheorywouldbeuseful.Hereareafew:
31
1. Canthedistributionofleveltypesbederivedendogeneouslyfrommorebasic
principlesofcognitivedifficultyandperceivedbenefit,orperhapsfrom
evolutionaryconstraintonworkingmemoryandtheoryofmind(e.g.Stahl,1993).
2. CHmodelshavethepotentialtodescribedifferencesinskillorexperience.Skill
arisesineverydaydiscussionsaboutevensimplegameslikerock,paperand
scissors,ingameswithprivateinformationsuchaspoker,andgamesthattax
workingmemorysuchaschess.Areskilldifferencesgeneralordomain‐specific?
Canskillbetaught?Howdoesskilldevelopmentchangecognitionandneural
activity?
3. Thecomputationalapproachtostrategicthinkinginbehavioralgametheorycould
beusefulforunderstandingthesymptoms,etiologyandtreatmentofsome
psychiatricdisorders.Disorderscouldbeconceptualizedasfailurestocorrectly
anticipatewhatotherpeopledoandfeelinsocialinteractions,ortomakegood
choicesgivensensiblebeliefs.Forexample,inrepeatedtrustgamesKing‐Casaset
al.,(2008)foundthatborderlinepersonalitydisorder(BPD)didnothavetypical
activityininsulacortexinresponsetobeingmistrusted,andearnedlessmoney
becauseoftheinabilitytomaintainsteadyreciprocaltrustbehaviorally.Chiu
(2008)foundthatautismpatientshadlessactivityinaregionofanteriorcingulate
thattypicallyencodessignalsofvaluationduringone’sownstrategicchoices
(comparedtochoicesofothers).
4. Asmallemergingapproachinthestudyofliteraturefocusesonthenumberof
mentalstatesthatreaderscantrackandtheireffect(e.g.,Zunshine,2006).One
theoryisthatthreementalstatesareasociallyimportantreasonablenumber(e.g.,
32
lovetriangles)andarethereforenarrativelyengaging.Workoncognitiveschemas,
socialcategorization,computationallinguistics,andgametheorycouldthereforebe
ofinterestinthestudyofliterature.
5. Formalmodelsconnectingemotionswithbeliefs,actions,andpayoffscanilluminate
therelationshipsbetweenaffectivestatesandbehavior.Theutilityfunction
approachtomodelingemotionsmakesclearthatemotionsinfluencebehavioronly
whenthehedonicbenefitsofemotionalbehavioroutweighthecosts.Thisapproach,
whichconsidersevenemotion‐drivenbehaviorastheoutcomeofanoptimization
problem(perhapssculptedbyhumanevolutionratherthanconsciouscost‐benefit,
ofcourse).,promisestoopenupnewavenuesofresearchstudyingtherelationship
betweenemotionandstrategicchoices.
33
Tablecaptions
Table1:Payoffsinbettinggame,predictions(NashandCH),andresultsfromclassroom
demonstrationsin2006‐08.UpperleftistheuniqueNashequilibrium.
Table2:PayoffsfromHandTchoiceina“matchingpennies”game,predictions,anddata.
Table1:
predictions Data
L R Nash CH 2006+07+08 Average
T 30, 20 10, 18 1.00 .73 .81+.86+.78 .82
B 20, 20 20, 18 .00 .27 .19+.14+.22 .18
Nash 1.00 0
CH .89 .11
2006+07+08 .95+.95+.75 .05+.05+.25
average .88 .12
Table2:
predictions
H T Nash CH Levels
1-2
Levels
3-4
data
H 2,0 0,1 .50 .68 1 0 .72
T 0,1 1,0 .50 .32 0 1 .28
34
Nash .33 .67
CH .26 .74
data .33 .67
Figurecaptions
Figure1:Choicesin“2/3oftheaverage”game(Nagel,2005?)
Figure2:Predictedandobservedbehaviorinentrygames
Figure3:ThegameboardfromHeddenandZhang(2002)
Figure4:Anicongraphofvisualattentioninthreeroundsofbargaining(1,2and3)and
correspondingdistributionsofoffers.Eachcolumnrepresentsadifferent“type”ofperson‐
trialclassifiedbyvisualattention.
Figure5:Estimatedstrategicleveltypesforeachindividualintwosetsof11different
games(Chong,Camerer,Ho&Chong,2005).Estimatedtypesarecorrelatedintwosets
(r=.61)
Figure6:Numberschoseninweek1ofSwedishLUPIlottery(Napproximately350,000).
DottedlineindicatesmixedNashequilibrium.Solidlineindicatestochasticcognitive
hierarchy(CH)modelwithtwofreeparameters.Best‐fittingaveragestepsofthinkingisτ
=1.80andλ=.0043(logitresponse).
35
Figure7:Brainregionsmoreactiveinlevel2reasonerscomparedtolevel1reasoners
(classifiedbychoices),differentiallyinplayinghumancomparedtocomputeropponents
(fromCoricelliandNagel,2009,FigureS2a).
Figure8:Asimpletrustgame(DufwenbergandGneezy,2008)
36
Figure1
Beauty contest results (Expansion, Financial Times, Spektrum)
0.000.050.100.150.20
numbers
rela
tiv
e
fre
qu
en
cie
s
22 50 10033
average 23.07
0
37
Figure2
How entry varies with demand (D), experimental data and thinking model
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Demand (as % of # of players)
% e
ntry
entry=demandexperimental data!=1.25
Figure3
38
Figure4
39
Figure5
40
Figure6
41
Figure7
Figure8
42
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iAnotherimportantcomponentofbehavioralgametheoryislearningfromrepeatedplay(perhapsusingreinforcementrulesaswellasmodel‐based“fictivelearning”(Camerer&Ho,1999).Learningmodelsarewidelystudiedbutliebeyondthescopeofthischapter(seee.g.Fudenberg&Levine,1998;Camerer,2003,chapter6).
iiOthermechanismsthatcouldproduceequilibrationincludelearningfromobservation,introspection,calculation(suchasfirmshiringconsultantstoadviseonhowtobidonauctions),imitationofattention‐gettingorsuccessfulstrategiesorpeople,oraprocessofpre‐playtalkingaboutfuturechoices.Thelearningliteratureiswelldeveloped(e.g.,Camerer,2003,chapter6)butthestudyofimitationandpre‐playtalkingcouldcertainlyusemorecollaborationbetweengametheoristsandpsychologists.
iiiCommonknowledgerequires,fortwoplayers,thatAknowsthatBknowsthatAknows…adinfinitum.
ivAmoregeneralviewisthatlevel0’schooseintuitivelyor“heuristically”(perhapsbasedonvisuallysalientstrategiesorpayoffs,or“luckynumbers”),butthattopichasnotbeenexploredverymuch.
vRestrictingcommunicationisnotmeanttoberealisticandcertainlyisnot.Insteadcommunicationisrestrictedbecausechoosingwhattosayisitselfa“strategy”choicewhichcomplicatesanalysisofthegame—itopensaPandora’sboxofpossibleeffectsthatlieoutsidethescopeofstandardgametheory.However,gametheoristsarewellawareofthepossiblepowerfuleffectsofcommunicationandhavebeguntostudyitinsimpleways.InherthesisNagel(1995)reportssomesubjectdebriefingwhichareillustrativeofCHthinking,andSbriglia(2008)reportssomeprotocolstoo.BurchardiandPenczynski(2010)alsousedchatmessagingandteamchoicetostudycommunicationandreportevidencelargelyconsistentwithCHreasoning.
viThisanalysisassumesτ=1.5butthegeneralpointholdsmorewidely.
viiNotethatthisisacloserelativeofa“thresholdpublicgoods”game.Inthatgame,apublicgoodiscreated,whichbenefitseveryone,ifTpeoplecontribute,butifevenoneperson
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doesnotethepublicgoodisnotproduced.Inthatcase,everyonewouldliketobeintheN‐Tgroupofpeoplewhobenefitwithoutpaying.
viiip(X)isa“commonprior.”Anexampleisagameofcards,inwhicheveryoneknowsthecontentsofthecarddeck,donotknowwhatface‐downcardsotherplayersareholding,butalsoknowthattheotherplayersdoknowtheirownface‐downcards.
ixThereareseveralsubtlevariants.IntheoriginalMouselab,boxesopenandcloseautomaticallywhenthemouseentersandexits.Costa‐Gomesetal(2001)wantedmoredeliberateattentionsotheycreatedaversioninwhichaclickisrequiredtoopenabox.Brocasetal(2010)createdaversionthatrequiresthemousebuttontobehelddowntoviewboxcontents(ifthebuttonpressishaltedtheinformationdisappears).
xTherostralACC,labeledrACC,ismoreactiveinlevel1thaninlevel2playersinthehuman‐computercontrast.
xiDLPFCisalsoinvolvedincognitiveregulationofemotions(e.g.,Ochsneretal.,2009)
xiiWhatwehaveinmindhereissimilartoHolyoakandCheng(1985),Fiddick,CosmidesandTooby(2000)argumentsaboutthedifferencebetweenabstractlogicperformanceandcontextualizedperformance.Forexample,gamesthatresemblehidingfoodandguardinghiddenlocationsmightmaproughlyontosomethinglikepoker,whereasalotofgamesconstructedforchallengeandentertainment,suchaschess,donothaveclearcounterpartsinancestraladaptiveenvironments.
xiiiOthermodelsofbelief‐dependentutilitycanbeplacedinthegeneralframeworkofBattigalliandDufwenberg(2009).Forexample,CaplinandLeahy(2004)modeldoctor‐patientinteractionswhereuncertaintymaycausepatientanxiety.Thedoctorisconcernedaboutthepatient’swellbeingandmustdecidewhetherornottoprovide(potentially)anxiety‐causingdiagnosticinformation.Bernheim(1994)proposesamodelofconformitywhereplayerscareaboutthebeliefstheircoplayershaveregardingtheirpreferences.Themodelcanproducefadsandadherencetosocialnorms.RelatedworkbyBenabouandTirole(2006)modelsplayerswhoarealtruistic,andalsocareaboutother’sinferencesabouthowaltruistictheyare.GillandStone(2010)modelplayerswhocareaboutwhattheyfeeltheydeserveintwo‐playertournaments.Theplayers’perceivedentitlementsdependupontheirowneffortlevelandtheeffortsofothers.
xivA(singleperson)decisionprobleminvolvinganyoftheseemotionsmaybemodeledasapsychologicalgamewithoneplayerandmovesbynature.
