Luca Polonio
a,
b,
∗
,
Sibilla Di Guida
c,
Giorgio Coricelli
b,
daDepartment of Economics, University of Minnesota, 4-101 Hanson Hall, Minneapolis, MN 55455, United States
bCenter for Mind/Brain Sciences, CIMeC, University of Trento, Palazzo Fedrigotti – Corso Bettini 31, 38068 Rovereto, Trento, Italy cDepartment of Business and Economics, COHERE, Syddansk Universitet, Campusvej 55, 5230 Odense M, Denmark
dDepartment of Economics, University of Southern California, 3620 S Vermont Ave 300, Los Angeles, CA 90089, United States
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Article history:Received25April2014 Availableonline3October2015
JEL classification: C91 D83 C71 C72 C83 Keywords: Gametheory Strategicsophistication Socialpreferences Attention Eye-tracking
Weusedeye-trackingtomeasurethe dynamicpatterns ofvisualinformationacquisition
intwo-player normal-form games. Participants played one-shot gamesin whicheither,
neither,oronlyoneoftheplayershadadominantstrategy.First,weperformedamixture
modelsclusteranalysistogroupparticipantsintotypesaccordingtothepatternofvisual
informationacquisition observed inasingle classof games.Then, wepredicted agents’
choicesin different classes of games and observed that patterns of visual information
acquisitionweregameinvariant.Ourmethodallowedustopredictwhetherthedecision
process would lead to equilibrium choices or not, and to attribute out-of-equilibrium
responsestolimitedcognitivecapacitiesorsocialmotives.Ourresultssuggesttheexistence
ofindividuallyheterogeneous-but-stablepatternsofvisual informationacquisition based
onsubjectivelevelsofstrategicsophisticationandsocialpreferences.
©2015TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCC
BY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
Whiletraditionalgametheoryassumesagentstobefullyrationalandself-interested,recentresearchinbehavioraland experimental economicshasattemptedtorelax bothoftheseassumptions,drawing increasingattentiontowardstheidea thattherearedifferenttypesofplayers,bothwithregardstocognitionandtomotivation(Camereretal.,2011).
Forinstance,Level-k (Crawford,2003; Nagel,1995;
Stahl
andWilson,1994,1995)andCognitiveHierarchy (Camereret al.,2004; Hoetal.,1998) modelsrelaxtheassumptionoffullrationalityandexplainout-of-equilibriumoutcomesassuming individualsbeingboundedlyrationalandperformingdifferentandlimitedlevelsofiterativestrategicthinkingduetolimited cognitivecapacities.Ontheotherhand,relaxingthenotionofself-interest,theoriesofsocialpreferencesaccountforthefactthatindividuals oftenhavepositiveornegativeattitudestowardsothers,andmaygooutoftheirway–thusoutofequilibrium–toharmor benefitthem.Undersuchassumptions,agentsmaydifferintheirmotivestowardsaltruism,reciprocity,andinequity(Bolton
*
Correspondingauthorat:CenterforMind/BrainSciences,CIMeC,UniversityofTrento,PalazzoFedrigotti–CorsoBettini31,38068Rovereto,Trento, Italy.E-mail addresses:polonio.luca@gmail.com(L. Polonio),sidg@sam.sdu.dk(S. Di Guida),giorgio.coricelli@usc.edu(G. Coricelli). http://dx.doi.org/10.1016/j.geb.2015.09.003
0899-8256/©2015TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).
andOckenfels,2000; CharnessandRabin,2002; Coxetal.,2008; FehrandCamerer,2007; FehrandSchmidt,1999),rather thanenvyorcompetition(CoricelliandRustichini,2010; Kirchsteiger,1994; Maccheronietal.,2012).
Sofar,the twostreams ofresearchinvestigatingcognitive andmotivationalaspects havebeendevelopedmostly sepa-rately,thoughitislikelythatbothaspectsco-existandinteract.Apromisingwaytoevaluatedifferenttheoriesofstrategic choicecomesfrommethodsthatallow investigatingtheprocessesunderlyingchoice.Theaforementionedresearch mostly focused onchoice data,however,methods like mouse-tracking(Brocasetal., 2010; Costa-Gomes etal., 2001; Johnsonet al.,2002),eye-tracking(Arielietal.,2011; Krajbichetal.,2010; Reutskajaetal.,2011),andfMRI(BhattandCamerer,2005; Coricelli andNagel,2009) allowobservationof howdecisions aredeveloped, permittingresearchers to compareandtest multipletheories,andexploringalternativeexplanationsmoreefficiently(CamererandJohnson,2004).
Inthepresentstudy,weanalyzeeye-trackingdatatotestthecontributionsofbothmotivationandcognitioninstrategic decision-making.Startingfromboththeoriesofsocialpreferencesandlimitedcognition,wemakespecificpredictionsasto whichpatternsofvisual analysiswouldbeobservedbydifferentplayertypes.Forinstance,a playermotivatedbyfairness would observe the payoffs of the other player, regardless of whether thesepayoffs are strategically relevant (as with a strictly dominantstrategy). Conversely, theoriesoflimited thinking,such aslevel-k theories, predict that agentsmaynot play in equilibriumbecause they fail to process relevant information (Camerer andJohnson, 2004; Johnson etal., 2002; Rubinstein,1999).Therelationshipbetweeninformationsearchpatternsandstrategicbehaviorhasbeenstudiedindifferent settings,andattentionaldatahavebeenusedtoclassifyagentsintodifferentstrategictypes.
Costa-Gomes
etal. (2001)used mouse-trackingtoinvestigatetheinformationsearchprocessinnormalformgameswithdifferentstrategicstructures.They identify nine strategic types, and observethat mostof their participants exhibit look-ups anddecisions consistent with level-k models. Devetag etal. (2015) manipulate gamefeatures (suchas presenceof focal points and strategy riskiness) in 3×
3 normal form gamesanduse eyemovementsto test howthose features impact not onlysubjects’behavior, but alsotheirinformationsearchpatterns.Theyobservestabilityininformationsearchbothacrossgamesanddespitefeatures manipulation,buttheyreport adaptationinsubjects’strategic behavior.Chen
etal.(2009) useeyemovementsto identify participants’level of strategicsophistication in a spatial beautycontest. Theirresultsshow that lookups-based typesare better models than choice-based types. Funaki et al. (2011) observe a strong correspondence between participants’ eye movementsandchoicesinsimplethree-persondistributionexperiments,suggestingthatsubjects’socialpreferencescanbe inferred fromthe typeofanalysisofthe gamethat theyperform.Hristova
andGrinberg (2004)show that,when playing prisoner’sdilemmas,agentswithdifferentstrategicbehavioranalyzethegamemarkedlydifferently;whileinFiedler
etal. (2013),differencesininformationsearchturnouttobecorrelatedwiththeindividualSocialValueOrientation.While it is now well established that there is a strict correlation between information search patterns and strategic behavior,therehavebeennofinalresultsthat tellusclearlyhowtoexploit eyemovementstoidentifyplayers’typesand predicttheir behaviorindifferentinteractive situations.The aforementionedarticleseitherfocusonsingle typesofgames orfit data(bothchoice data andinformationsearch data) acrossgames, without predicting them.Inthis paper,we will insteaduseasubsetofourinformationsearchdata–collected ina specificclassofgames–topredictsubjects’strategic behaviorindifferentclassesofgames.Themethodwedevelop forclusteringparticipantsbasedontheirpatternsofvisual analysisallowustomakecognitionvisibleandtoidentify,withahighlevelofprecision,thedecisionruleadopted.
Wewillalsoanswersomeopenquestionsconcerningtherelationbetweencognitionandstrategicbehavior.Forexample: is the individual pattern of informationacquisition constant across different classesof games? Is it possible to identify, independentlyby thegametype, awell-defined temporalpatternofinformationacquisition thatleads toequilibrium(or leads to a certain equilibrium in games with multiple equilibria)? Is it the pattern of information search adopted that determinesthedecisionstrategy(bottom-uphypothesis)or,vice-versa,theinformationsearchpatternthatisdeterminedby theplayer’stype(top-downhypothesis)?Inthispaperwewilltrytoanswertothesequestionsbylinkingbasicattentional processestotheunderlyingcognitiveandmotivationaldrivers.
The games we useto test the role ofmotivation andcognitionare two-by-twoone-shot normalform games.To test the levelof thinking reachedby the agents,we introduce games having differentlevel ofstrategic complexity, inwhich either,neither,oronlyoneofthetwoplayershasastrictly dominantstrategy. Totesttheroleofmotivation(cooperation aswell ascompetition)weintroduceinallourgamesbothacellwithsymmetricpayoffsthatalsomaximizedthesumof thetwoplayers’payoffs(cooperation),andacellinwhichthedifferencebetweenplayer’sownpayoffandthepayoffofthe counterpartwasmaximal,attheplayer’sownadvantage(competition).
Wehypothesizethatpeople’sbehaviorinone-shotgamesisguidedbytheirsocialmotivesandbytheabilitytoperform iterativestepsofreasoning.Inthosesituationswherethereisnoopportunitytolearn,socialpreferencesaswellascognitive abilitiescanbeconsidered asrelatively stabletraitswithin individuals.Therefore,we hypothesizethat differentbehaviors (inone-shotgames)aredetermined,possiblywitherror,bydifferentdecisionrulesortypes,witheachtype ofplayerthat remainsconstantacrossgames(asin
Costa-Gomes
etal.,2001).Toidentifythedecisionrulesadoptedbytheplayers,weemployaneye-trackingtechnique(seeMethodsection),which allows foraccurately measuring fixation timesandsaccades (eye-movementsfromone target toanother) withthe same levelofprecisionusedinpsychophysiologicalstudiesonperceptionandattention.
Usingamixturemodelsclusteranalysis,weassociateeachplayerwithaspecificdecisionruleonthebasisofthepattern ofvisualanalysisexhibitedinaspecificclassofgames(gameswithastrictlydominantstrategyfortheotherplayer).Then foreachtype,we predictagents’choicesindifferentgameshavingunique(e.g.,prisonerdilemmas)ormultipleequilibria (e.g.,staghunt).We alsoobservethatpatternsofinformationacquisitionarestableacrossdifferentclassesofgames. Our
that canbe correctlydetectedby analyzingtheVIApatternthatthey adopt.Forexample,we identifyaspecifictemporal patternofVisualInformationAcquisitionthatleadstoequilibriumindependentlyoftheclassofgames.Ourdatashowthat is possibletopredict players’choicesjustbyobservingthe patternofvisual analysisthatthey exhibitedina singleclass ofgames.Finally,ourresultsstronglysupporttheroleofbothsocialmotivesandlevelofstrategicthinkingindetermining players’behaviorinone-shotgames.
2. Experimentalprocedure
2.1. Participants
Participants were90undergraduatestudentsfromtheUniversityofTrento,Italy(41 males,49females,meanage21.2, SD 2.46).Thestudywas approvedbythe localethicscommitteeandall participantsgaveinformedconsent. Eye-tracking datafromfouroftheparticipantswereexcludedduetopoorcalibration.
Participantstookpartintheexperimentindividually(asin:
Bhatt
andCamerer,2005; Knoepfleetal.,2009).After enter-ingthelab,eachparticipantwasrandomlyassignedtothe“eye-trackinggroup”(60participants)ortothe“non-eye-tracking group” (30 participants),andthen totherole of“rowplayer”or“columnplayer”.Participantswere equallysplitbetween thetworoles.Participantswereinstructedintheprocedureandtherulesoftheexperiment.Controlquestionswere admin-isteredbeforetheexperimentstartedtoverifythat rulesandprocedures ofpaymentwere understood.Whenparticipants failedtoanswercontrol questions,instructionswere repeated.Itwas madeclearthat thenumbersinthematrices repre-sentedpayoffsineuroandthat participantswouldbepaid accordingtotheoutcome ofonerandomlychosen gameplus a show-up fee. Participants played 32 games dividedinto four blocksof 8trials each (with two games fromeach class, asdescribed below). Wekept theorderof theblocksfixed butrandomizedthe gameswithin each block. Ineach round, players selected their choice by typing the corresponding key number on the keyboard. Before starting the experiment, participantsplayedfourpracticegames.Aswewereinterestedininitialbehavioronly,nofeedbackwasprovided,andeach gamewas playedonlyonce.Participants camebackforthepaymentstwoweeksaftertheendoftheexperiment.Agame anda counterpart were randomlyselected,andparticipants were paidaccordingto their choiceandthat ofthe selected counterpart.Participantsearnedbetween1euroand9eurosinadditiontothe5eurosshow-upfees.2.2. Thegames
We selectedfourclassesofgames withdifferentequilibriumstructures, andcreatedeighttwo-by two-gamesforeach class,varyingthesizeofthepayoffsbutkeepingconstantthedifferenceamongthem.
Thefourclassesofgames(presentedin
Fig. 1
,fromtheperspectiveofarowplayer)were:(1)dominancesolvable“self ” games (DSS),in whichonly the eye-trackedparticipant hada strictly dominantstrategy; (2) dominance solvable“other” games(DSO),inwhichonlythenon-eye-trackedparticipanthadastrictlydominantstrategy;(3)“prisoner’sdilemma”games (PD),inwhichbothplayershadastrictlydominantstrategy;(4)“staghunt”games(SH),acoordinationgameinwhichthere were nodominantstrategiesandboth playerscould choosebetweenasafe-low returnchoice anda high-riskhigh-return one.Thedominance-solvablegameshadauniqueNashequilibriuminpurestrategies,whereasthecoordinationgamehad twopurestrategiesNashequilibria(oneofwhichisPareto-efficient)andoneequilibriuminmixedstrategies.ToreachtheequilibriuminDSSandPDgames,onlyonestepofiteratedeliminationofdominatedstrategiesisrequired, whileinDSOgamestwostepsarerequired(i.e.,firsttheeliminationofthedominatedstrategyofthecounterpart,thenthe eliminationofowndominatedstrategy).
2.3. Eye-trackingprocedure
Participants inthe“eye-trackinggroup” wereseatedinachairwithasoftheadrestrainttoensure aviewingdistance of60 cm from themonitor.Presentationofthestimuliwasperformedusingacustom madeprogram implementedusing theMatlab Psychophysicaltoolbox.EyemovementsweremonitoredandrecordedusinganEyelinkII system(SR.Research
Fig. 1. Thegamesusedintheexperiment,groupedbyclass(fromtheperspectiveofarowplayer).Whentheparticipantswereassignedtotheroleof columnplayer,thematriceswerepresentedtransposed.Nashequilibriuminpurestrategiesishighlightedingrey.Dindicatestheactionsconsistentwith dominance;PE,thePareto-efficientequilibrium;RD,theriskdominantequilibrium.
OntarioCanada)withasamplingrateof500Hz.Afixationwasdefinedasanintervalinwhichgazewasfocusedwithin1◦ ofvisualangleforatleast100ms(ManorandGordon,2003).Anine-pointcalibrationwasperformedatthebeginningof eachblock. Thecalibrationphasewasrepeateduntilthedifferencebetweenthepositionsofthepointsonthescreenand thecorrespondingeyelocationswas lessthan1◦.Afterthecalibrationphase,anine-pointvalidationphasewasperformed (similartothecalibrationphase)tomakesurethatthecalibrationwasaccurate.Recalibrationswereperformedifnecessary, andeye-trackinginterruptedifthesewereunsuccessful.Beforethebeginningofeachtrialadriftcorrectionwasperformed
Fig. 2. Ascreenshotshowinghowthematrixwaspresentedtotheparticipants.Rowplayer’spayoffsarelocatedatthebottom-leftofeachcell,while column’sonthetop-right.Thecircles(notvisibletotheparticipants)representtheAOIs,whilethenumbersinitalicthecorrespondinglabels.Inbold,the proportionoffixationtimeofeachAOI,averagedamongparticipants.
(except forthefirsttrialofeachblock).Afterthedrift correction,afixationpoint waspresented,locatedoutsidethearea covered by the matrix and between the two possible choices, to minimize biasesrelated to the starting fixation point.
The game matrix was presented after the fixation point was fixated for 300 milliseconds and remained on the screen
untilaresponsewasmade.Eyemovementswererecordedduringthegamematrixdisplay.Tominimizenoise,information displayedonthemonitorwaslimitedtopayoffsandparticipant’sstrategylabels.Inaddition,thepayoffswerepositionedat an optimaldistancefromeachother(calibratedinapilotstudy)todistinguishfixationsandsaccadesbetweenthem,with rowandcolumnplayerpayoffsatdifferentlatitudesandindifferentcolors.
Inordertoanalyzetheeye-movementsdata,wedefined8areasofinterest(AOIs),centeredinthepayoffs.AlltheAOIs hada circularshape withasize of36 000pixels (Fig. 2). The AOIscovered only23% ofthegame matrixarea andnever overlapped. Allthe fixationsthatwere not locatedinsidethe AOIswere discarded.However, althougha largepart ofthe matrixwasnotincludedinanyAOI,thelargemajorityoffixations(85%)fellinsidetheAOIs.
Fourtypesofvariableswererecordedbytheeye-trackerineachroundforeachparticipant:(1)thetimespentlooking within anAOI(fixationtime),(2)thenumberoftimesa participantlookedinsidean AOI(fixationcount),(3)thenumber oftimesaparticipantreturnedtolookatthesameareaofinterestduringatrial(numberofruns),and(4)thenumberand type ofsaccades (definedastheeye movementsfromone AOIto thenext).Since thefirst threevariables (fixationtime, fixationcount,andnumberofruns)arestronglycorrelated,inouranalysiswewillmostlyrefertothefirstvariable(fixation time).
3. Results
Wewillfirstpresentanoverviewofchoiceandresponsetimedata,andthenanalyzedataoneyemovements.
3.1. Behavioraldata
We used a Kolmogorov–Smirnov test to evaluate possible effects dueto the useof the eye-tracking apparatus orto the role of the player. Even ifpayoffs in the two roles are slightly different, we did not find any significant difference betweenthetwogroups(“eye-trackinggroup”vs.“non-eye-trackinggroup”)ontheproportionofequilibriumresponsesin dominance-solvablegames(D
=
0.112,p= .
97,Kolmogorov–Smirnovtwo-sampletest).Thesameanalysiswasperformedto testpossibledifferencesinresponsetimes,butagainnodifferencewasfoundbetweenthetwogroups(D=
0.147,p= .
36, Kolmogorov–Smirnovtwo-sampletest).Basedontheseresults,wefocusouranalysisonthe“eye-trackinggroup”only.We performed a Kolmogorov–Smirnov test to analyze a possible “display effect” (whether thegames were presented from the perspective ofrow orofcolumn player)on the proportionof equilibriumresponses and onresponse times in the “eye-trackinggroup”. Nodifferenceswerefound inbothequilibriumresponserates(D
=
0.133, p= .
95,Kolmogorov– Smirnov two-sample test) and response times (D=
0.267, p= .
24, Kolmogorov–Smirnov two-sample test). Given these results,wemergedthedata.WeusedaFriedmantest toevaluateforpossibleeffectsoftheclassofgames(DSS, DSO,andPD)ontheproportionof equilibriumresponses.Resultsshowedasignificanteffectoftheclassofgames(
χ
2=
48.03,p< .001).
Subsequentpost-hoc Wilcoxonpairedtest(using Holm’scorrection)revealedalowerrateofequilibriumresponsesforDSO gamescomparedto DSS games (p< .001),
and PD games (p< .001).
No significant differences were observed between DSS and PD games (p= .
105).We did nottest fordifferencesin equilibriumresponse rateswiththe SH gamessince, inthistype of game, bothactionsrepresentanequilibriumplay. AFriedmantestshowedamarginallysignificanteffectoftheclassofgameson response time(χ
2=
6.54,p= .
09).Theaverageproportionofequilibriumresponsesandtheresponsetimesarereported inTable 1
.Table 1
Proportionofequilibriumresponses(Paretoequilibriuminstaghuntgames)andaverageresponsetimesinthefourclassesofgames(standarddeviation inparenthesis). DSS DSO PD SH Equilibrium responses 0.70 0.35 0.68 0.45 (S.D.) (0.35) (0.34) (0.39) (0.37) Response times 8765 ms 8565 ms 8353 ms 7679 ms (S.D.) (7736 ms) (6638 ms) (7285 ms) (6167 ms)
Dividing theanalysisofeach experimentalsession infourblocks(of 8trialseach),we could excludea learning effect by performinga Friedman testwiththe nullhypothesis that thenumberofequilibriumresponses ingames withunique equilibriumisthesameineach blockofgames(
χ
2=
3.16,p= .
37).Moreover,themeanresponsetimedidnotdecrease acrossthefourblocks(χ
2=
2.78,p= .
42).Inaddition,weevaluated thecorrelationsintheproportionofequilibriumresponsesinthefourclassesofgames.The correlationsbetweenDSO–DSSandDSO–PDgameswerebothpositiveandsignificant(DSO–DSS:Pearson’sr
= .
42,p< .001;
DSO–PD:Pearson’sr= .
43,p< .001).
Conversely,theproportionofequilibriumresponsesinDSS,PD,andDSOgameswere significantly negatively correlated with the proportion of Pareto-equilibrium responses in SH games (DSS–SH: Pearson’sr
= −.
68, p< .001;
PD–SH: r= −.
69, p< .001;
DSO–SH: r= −.
46, p< .001).
Lastly, there was a positive correlation betweenDSSandPDgames(Pearson’sr= .
91,p< .001).
Thenumberofequilibriumresponsesandthecorrelationsacrosssubjectssuggestthatstrategicbehaviorsinthevarious classesofgamesarestronglyrelatedandmostlyconsistent.Inthefollowinganalysisoftheeye-trackingdatawewill eval-uatewhetherdifferentchoiceswithineachclassofgamesarerelatedtodifferentattentionalpatternsofvisualinformation acquisition.
3.2. Analysisoffixations
Only fixationslonger than 100 milliseconds were considered for the analysis, since this duration has beenshown to be an accurate threshold to discriminate between fixations and other ocular activities (Manor and Gordon, 2003). The proportionoftimespentbyplayerslookingateachAOIisreportedin
Fig. 2
.Weevaluatepossiblestatisticaldifferencesin themeanpercentageoftimespentby participantstowardstheir ownandtheir counterpart’spayoffsbyusingaWilcoxon paired test. Results show that players spent significantly more time fixating their own payoffscompared to the payoffs of the counterpart (Wilcoxon paired test, V=
2071, p= .
003). These results are in accordance with those obtainedin previousstudiesthathaveusedmouse-trackingandeye-trackingtechniques(Costa-Gomesetal.,2001; Devetagetal.,2015; Wangetal.,2010).ThelevelofattentiontowardeachAOIwas alsoinfluencedbythespatiallocationofthepayoffsonthe screen(see onlineAppendixF),however,equilibriumandfocal cellswere both evenlydistributedacrossthe fourcellsof thematrix(top-left,top-right,bottom-left,bottom-right)amonggames.Therefore,thetendencytofixatemorethosepayoffs locatedclosertothecenterofthescreenshouldnothaveaffectedtheresults.Mostof theparticipants starteddirecting their attention fromthe fixation pointto the top left corner ofthe matrix, showinganaturaltendencytoprocessvisualstimuliwitheyemovementsgoingfromlefttorightandfromtoptobottom, awell-knownbiasassociatedwiththewesternwritingconvention(Abed,1991; Chuaetal.,2005; Ishiietal.,2011).
3.3. Dynamicpatternsofinformationacquisition:analysisofthesaccades
Saccadesare movementsfromone fixationto another.We includedin ouranalysisonly thosesaccadesthat occurred betweenfixationslastinglongerthan100ms.
Our main objectivewhen analyzing the saccades was to identify which algorithms of visual analysis playersused to acquireinformationaboutthestructureofthegameswhenfacingthetwo-by-twomatrices. Intotal,consideringthateach pairofareasofinterestcanbeconnectedbytwosaccades(i.e.,from“AOI_1”to“AOI_2”andfrom“AOI_2”to“AOI_1”),there are 56possible saccadesthat could beobserved foreach game.However, we considered asrelevantonly thosesaccades usefulforcapturing pieces ofinformationthat are necessary to: (1) identify thepresence ofdominant strategiesforthe player; (2) identify the presence of dominant strategies for the counterpart; (3) identify the strategy with the highest average payoff amongthose of the player; (4)identify the strategy withthe highestaverage payoff among thoseof the counterpart;(5)comparethepayoffsofbothplayerswithinthesamecell.
Weidentified24saccadesthatmeettheconditionslistedabove.However,ourinterestconcernsthetypeofinformation that can be obtained by each type of saccade, regardless of their starting spatial location and direction. Therefore, we reducedthesampletotwelvesaccadesbyconsideringasequivalentallthoseconnectingthesametwoAOIs(Fig. 3,Panel A). Weshowedthatthereisnoeffectoftherole inplayers’choices(whethertheyactasroworcolumnplayer).Here,we testedwhetherthepatternsofsaccadeswerealsounaffectedbytherole.Achi-squaretestrevealednosignificanteffectof theroleontherateofthefiveclassesofsaccades(Pearson’sChi-squaretest,3.73, p
=
0.444).ThenextstepistodetermineifsomeoftheseinformativesaccadesmightbegroupedaspartofthesameVisualPattern
Fig. 3. PanelA)Classificationofthe12relevantsaccadesforrowplayers.Blackcirclesindicaterow-playerpayoffs,andgreycirclesindicatecolumn-player payoffs.Arrowsindicatesaccades.PanelB)Divisiveclusteringtree(ontheleft)andcorrelationmapforthe12saccades(ontheright).Differentshades indicatedifferentlevelsofcorrelation,fromwhite(−1)toblack(1).
neededtoextractspecific informationaboutthepayoffstructure ofthe game.We assumedthat,tobe consideredaspart ofthesameVIApattern,twosaccadeshadtosatisfythefollowingconditions:
i. Co-occurrence:fora givenplayerinagivengame,thenumberoftimesthattheyoccurredhadtobe highlyand posi-tivelycorrelated.
ii. Sequentiality1:foragivenplayerinagivengame,theymusthaveahighprobabilityofbeingperformedsequentially. Inordertoestablishwhichsaccadessatisfiedco-occurrenceweuseda seriationmethod,whichisanexploratory com-binatorial dataanalysis technique that produces a formal arrangement ofunits. The seriation methodis used to reorder objectsintoasequencealongaone-dimensionalcontinuumsothatitbestrevealsregularityandpatternsamongthewhole series(Liiv,2010; Marquardt,1978; O’BrienandLyman,1999).First,wedeterminedforeachpairofsaccadesthenumber oftimesthattheyoccurredforagivengameinagiventrial.Then,weappliedtheseriationmethodtoreorderthe12types of saccadesaccording totheir level ofcorrelation. The greater thepositive correlation betweentwo saccades,the higher the probability they co-occurred.Conversely, the greaterthe negative correlation,the lower theprobability that saccades co-occurred2.As
Fig. 3
(Panel B)shows, theseriationmethodassignedthe12saccadesintothreegroupsoffoursaccadeseach. Saccadeswerepositivelycorrelatedwithin eachgroup (darkshadedcells)andmostlynegatively correlatedbetween groups(faircells).Thethreesub-groupsare:
i. ownpayoffssaccades:saccadesbetweenaplayer’sownpayoffs; ii. otherpayoffssaccades:saccadesbetweenthecounterpart’spayoffs;
iii. intra-cellsaccades:saccadesbetweenthepayoffsofthetwoplayers,withinthesamecell.
Ownpayoffs saccades included saccades that were necessary to identify the presence of own dominant choices, and
ownstrategieswiththehighestaveragepayoff.Otherpayoffssaccades includedsaccadesthatwerenecessarytoidentifythe counterpart’s dominantstrategy andstrategies withthe highest averagepayoff. Intra-cellsaccades included saccadesthat werenecessarytocomparethetwoplayers’payoffswithinaspecificcell.
However, inordertobe consideredaspartofthesameVIApattern, visualsaccadesofthesamegroupalsohadtobe executedsequentially.
1 Tocreatecomplexinformationstructures(e.g.identifyadominantstrategy),multipleelements(payoffs)havetobeconsideredandintegratedamong
them.Thisintegrationprocessrequiresthatallelementsofnecessaryinformationbecollectedsequentially;thereforeweassumedthat,tobepartofthe sameVIApattern,saccadeshavetobeexecutedinasequentialorderwithahighprobability.
2 Seriationmethodiscloselyrelatedtoclusteringmethod.Bothmethodsincludeaclusteringwithoptimalleafordering;howeverseriationmethoddoes
notidentifyclusterboundariesbutsimplyprovidesanoptimalorderingandrearrangementoftheobjects(foranextensivereviewofthemethodseeLiiv, 2010).Inourcase,theobjectswerethe12differenttypesofrelevantsaccades.Thenumberoftimesthateachtypeofsaccadeoccurredinagiventrialwas calculatedandthecorrelationforallpairsofsaccadescomputed.Theseriationmethodsimplyre-orderedthe12saccades,puttingtogetherthosesaccades havingahigherpositivecorrelationanddividingthosesaccadeshavingnegativecorrelation.
Table 2
Probabilityforeachofthe12saccadestobefollowedbyasaccadeclassifiedas“ownpayoffs”,“otherpayoffs”,or“intra-cell”saccade.Thestarsrefertothe levelofsignificance:onestar<0.05,twostars<0.01andthreestars<0.001.
Own payoffs saccades Other payoffs saccades Intra-cell saccades
1. Own within choice payoffs saccade I 0.74∗∗∗ 0.04 0.22
2. Own within choice payoffs saccade II 0.72∗∗∗ 0.07 0.21 3. Own between choice payoffs saccade I 0.66∗∗∗ 0.08 0.26 4. Own between choice payoffs saccade II 0.64∗∗∗ 0.08 0.28 5. Other within choice payoffs saccade I 0.07 0.54∗∗∗ 0.38 6. Other within choice payoffs saccade II 0.13 0.48∗∗ 0.39 7. Other between choice payoffs saccade I 0.12 0.56∗∗∗ 0.32 8. Other between choice payoffs saccade II 0.13 0.53∗∗∗ 0.33
9. First cell payoffs saccade 0.23 0.18 0.58∗∗∗
10. Second cell payoffs saccade 0.27 0.19 0.53∗∗
11. Third cell payoffs saccade 0.23 0.17 0.59∗∗∗
12. Fourth cell payoffs saccade 0.26 0.19 0.54∗∗
Table 3
Generalizedlinearmixedmodels.Dependentvariable:equilibriumchoice(orParetoequilibriuminSHgames)=1;independentvariables:numberofown payoffs,otherpayoffs,andintra-cellsaccades.Regressionswereperformedseparatelyforthefourclassesofgames.
Games Predictor variables Estimates Z value p-Value
DSS Own payoffs saccades 0.219 3.603 p<0.001
Other payoffs saccades 0.078 1.105 p=0.269
Intra-cell saccades −0.169 −2.961 p=0.003
N◦of obs.=442 AIC=343 BIC=368 log L= −166
DSO Own payoffs saccades −0.080 −1.766 p=0.077
Other payoffs saccades 0.366 5.228 p<0.001
Intra-cell saccades 0.018 0.510 p=0.610
N◦of obs.=442 AIC=418 BIC=443 log L= −203
PD Own payoffs saccades 0.332 3.300 p<0.001
Other payoffs saccades 0.153 1.621 p=0.105
Intra-cell saccades −0.166 −2.480 p=0.013
N◦of obs.=439 AIC=300 BIC=325 log L= −144
SH Own payoffs saccades −0.227 −3.700 p<0.001
Other payoffs saccades −0.076 −1.152 p=0.249
Intra-cell saccades 0.085 1.887 p=0.059
N◦of obs.=440 AIC=430 BIC=454 log L= −208
Thus, we determinedthe probability foreach saccadetobe followedby a saccadeofthe samegroup. Accordingto a one-tailedbinomialtest(Siegel andCastellan,1988),theprobabilitythateachsaccadewas followedbyanothersaccadeof thesamegroupwasalwayssignificantlylargerthanwhatwouldhavebeenpredictedbychance,i.e.0.33(Table 2).
Based on theseresults, we maintained the classification obtainedwith theseriation method,treating each ofthe 12 visualsaccadesaspartofoneofthethreeidentifiedVIApatterns.
WeexpectedtheanalysisoftheVIApatternstogive usimportantinsightsabouttheactuallevelofstrategicreasoning underpinningplayers’ behavior.Thus,we testedtherelationshipbetweenthepreviouslydefinedVIApatternsandplayers’
choices in the four classes of games using a generalized linear mixed model (GLMM). We assumed a binomial family
distributionwherethedependentvariablewasequaltooneifthestrategychosenwasanequilibriumone(Pareto-efficient equilibriuminSHgames)andzerootherwise(Risk-dominantequilibriuminSHgames).Theindependentvariableswerethe numberofownpayoffs,otherpayoffs,andintra-cellsaccades.Participantsandtrialsweretreatedasrandomeffects.
WeranaseparateregressionforeachofthefourgamestoevaluatethehypothesisthatdifferentVIApatternspromote differentchoicesdependingonthestructureofthegame.Forexample,equilibriumchoicesinDSOgamesshouldbelinked tothelevelofattentionto theotherplayer’spayoffs. Conversely,equilibriumchoicesinDSSgames shouldbesensitiveto thelevelofattentiontowardtheplayer’sownpayoffs.
Table 3 reportsthe results forthe fourclasses of games. The equilibriumresponses in both DSS andPD games –in whichtheplayershadadominantstrategy–werepositivelycorrelatedwiththenumberofsaccadesamongplayers’own payoffs,andnegativelycorrelatedwiththenumberofintra-cellsaccades(p
< .001 and
p= .
003 forDSSgames; p< .001
and p= .
013 for PD games). InDSO games, equilibriumchoiceswere positively correlated withthe numberofsaccades betweenotherplayers’payoffs(p< .001),
andmarginallynegativelycorrelatedwiththenumberofsaccadesbetweenthe player’sownpayoffs(p= .
077).InSHgames,thechoicescorrespondingtothePareto-efficientequilibriumwerenegatively correlatedwiththenumberofsaccadesbetweenplayer’sownpayoffs(p< .001)
andmarginallypositivelycorrelatedwith thenumberofintra-cellsaccades(p= .
059).Fig. 4. Panel A) Bayesian information criterion of the 10 models. Panel B) Distribution of the three clusters.
Overall, theregressions showedthat equilibriumandout-of-equilibriumresponses were strictly relatedtothe levelof attentiontodifferenttypesofinformation.However,we cannotsaymuchyetaboutthealgorithmsofvisualanalysisused bytheplayers;inparticular,wedonotknowifthebehaviorisdetermined,aswehypothesized,bydifferentdecisionrules ortypesandifplayersbehavedconsistentlyacrossdifferentgames.
3.4. Clusteranalysisbasedonsaccades
Inthissection,we soughtto answertotwoquestions:(1)Do playersapply similaralgorithms ofvisual analysiswhen acquiringinformationwithinaspecificclassofgames–or,alternatively,isthereheterogeneityinthealgorithmsthatthey use – and(2)do thealgorithms ofvisual analysisused by playersremain the samewhen they play differentclasses of games?
Toanswerthefirstquestion,weclassifiedplayersbasedontheirVIApatterns.Specifically,weclusteredplayersbasedon theirproportionofownpayoffs,otherpayoffs,andintra-cellsaccadesoverthetotalofsaccadesinDSOgames(i.e.,including also thenon-classifiedsaccadesinanyofthesethreeclasses).We choseDSO gamesbecausethisclassofgames required participantstofocusonthepayoffsoftheotherplayer,andtoperformiterativethinkingtodetectequilibriumplay.
Toidentifytheclusters, weadoptedthe mixturemodelsclusteranalysisusedby
Brocas
etal. (2010)andproposed byFraley andRaftery (2002, 2006). The advantage of using thisclustering method is that the numberof clusters and the clusteringcriterionarenotdeterminedapriori,rathertheyareoptimizedbythemethoditself.Inthisway,wecantestthe hypothesis oftheheterogeneityofplayers’ behaviorbyevaluating thenumberandcharacteristicsoftheclustersresulting fromtheclusteringprocedure.
Mixture models consider each cluster like a component probability distribution; then a Bayesian statisticalapproach is used tochoose betweendifferentnumbers ofclusters anddifferentstatisticalmethods. As inBrocas etal. (2010),we consideredamaximumofnineclustersforupto10differentmodels,choosingthecombinationthatmaximizestheBayesian InformationCriterion(BIC).Forourdata,theBayesianInformationCriterionwasmaximizedat355byanellipsoidalmodel withequalvolumeandshape,yieldingthreeclusters(Fig. 4,Panels A–B).
Thethreeclustersrepresentplayerswhoanalyzedthegamesinsimilarways.Thefirstclusterconsistsof23playerswho mademorethan60%ofintra-cellsaccades,androughly20%ofownand20%ofother payoffssaccades.Thesecondcluster counts 17playerswho mademore than80% of ownpayoffssaccades. Thethird cluster consistsof16 playerswho have their attentionequallydistributedamongthethreetypesofsaccades;theyusedonaverage37%ofownpayoffs,36%other payoffs,and27%ofintra-cellsaccades.
Fig. 5(PanelA)summarizesthealgorithmofvisualanalysisusedthemostbyeachcluster,whilePanelBshowsan ex-ampleoftheobservedpatternsforeachcluster.Theseresultsareconsistentwiththehypothesisofindividualheterogeneity inplayers’behaviorandgiveusapictureofthedifferentmodelsbuiltbyplayerstorepresentthisspecificclassofgames. It isremarkable to note that more than two-thirds ofthe players (cluster1 andcluster 2) focused their attention on a singletypeofinformation(intra-cellsaccadeforcluster1,andownpayoffssaccadeforcluster2),neglectingotherpiecesof informationthatwererelevanttobestrespondtotheotherplayer’sstrategy.Welabeledtheclustersafterthesaccadethat theplayersusedthemost:playersfocusedon intra-cellcomparisons (cluster1),playersfocusedonownpayoffs (cluster2)and playerswithdistributedattention (cluster3).
Wethentestedwhetherplayersmaintainedthesametypeofanalysiswhendealingwithgameshavingdifferentstrategic structures. Fig. 6reports thenormalized distribution of own payoffs, other payoffs, and intra-cell saccadesfor the three clustersinthefourgamesandshowsthatthedistributionsofthethreesaccadetypesremainedfixedwithinclustersacross games.Ownfocused playersmainlymadesaccadesbetweentheirownpayoffs,whileintra-cell playersmadesaccadeswithin
Fig. 5. PanelA)Preferredalgorithmsofvisualanalysisforthethreeclusters(fromtheperspectiveofacolumnplayer).PanelB)Examplesofanalysis performedbythreecolumnplayersclassifiedas
players focused on intra-cell saccades,
playersfocused on own payoffs and players with distributed attention.
Linesindicatethesaccades;circles,thefixationlocation.Fig. 6. Proportion of each type of saccade for the three clusters in the four classes of games.
the fourcells. Lastly, playerswithdistributedattention always took intoaccount the other player’s payoffs, even inthose gameswherethiswasnotnecessarytodeterminethedominantchoice.
From theseresults,we canconcludethat playersdidnot adapttheir informationsearchpatternto differentclassesof games.
3.5. Temporalpatternsofvisualinformationacquisition
To better understand the algorithms of visual analysisresulting fromthe clustering, we analyzed the individual VIA patterns overtime, for each cluster. Specifically, we evaluated how theproportion ofown, other,andintra-cell saccades evolvesthroughtime.Insteadofarbitrarilychoosingafixedtimespanforouranalysis,wedecidedtocalibrateitonplayers’ behavior.Foreachcluster,wecomputedthenumberofsaccadesneededby75%oftheparticipantstoreachadecision,and usedthatnumberascluster-specifictimespan.
Fig. 7 showsthe temporal distribution of own, other,and intra-cell saccades forthe three clusters inthe four game classes.It isinteresting to note that,for allthe three clusters,the temporalpattern ofvisual analysisremainedconstant regardlessofthegametype.
Players in the distributedattention cluster exploited all the saccades types within the decision time. In addition, the patterntheyfollowed wasthesamein allclassesofgames: i.e.,they startedby consideringtheir ownpayoffs;afterfour orfivesaccades(dependingontheclassofgame),theproportionofother payoffssaccadesincreaseduptoexceedingthe proportionofownpayoffssaccades.Lastly,andbeforemakingadecision,ownpayoffssaccadesalwaysreturnedtobemore
(a Poisson distribution) thatcharacterizesthe distributionofplayersbelongingtoeach level.Thedistributionisdescribed byonlyoneparameter(
τ
),correspondingtothemeanandthevariance.According to CH theory,playersshould adoptiterativedecisionrules that are based ona step-by-step reasoning pro-cedure.
Bhatt
andCamerer (2005) definetheinformationrequiredby playersexhibitingdifferentlevelsofthinking, when playingnormalformgames.Asarguedbytheauthors,playersthatperform1-stepofthinkingbelievetheyarefacing0-steps players, therefore,“theydonotneedto lookatthe otherplayer’s payoffsatall sincethey donot usethisinformationto refine their guessaboutwhat others willdo” (page 426).Conversely, playerswho perform two-steps ofthinking believe they areplayingamixtureof0and1-stepplayers.Level-2players“workharderatformingabelief,lookatotherplayers’ payoffs,andusetheirbelieftopickanoptimalchoice”(page426).Considering ourdata,playersclassifiedashavingdistributedattention couldbe associatedwithlevel-2 players.In fact, theytookintoaccountthecounterpart’spayoffs,andexhibitedatemporalpatternofvisualanalysisthatwasinaccordance withaniterativestep-by-stepprocedure(i.e.theylookedfirstattheirownpayoffs,thenatthepayoffsofthecounterpart, then again attheir own). Therefore,we expected forthiscluster a parameter
τ
closeto 2.Players who were focusedonownpayoffs could be associatedwithlevel-1 thinkers, becausethey hardly looked atthe counterpart’s payoffs, andtheir
temporal patternofvisual analysisstoppedafter consideringtheir own payoffs.For them,theparameter shouldbe close to1.Thetemporalpatternofvisualanalysisexhibitedbyplayersfocusedonintra-celltransitions didnotreflectanyiterative decisionruleandcouldnotbeassociatedwithanyspecifick-stepofthinking.Then,wecouldnotpredictwhichvalueof
τ
couldcapturethedata.
Weestimatedthevalueoftheparameter
τ
foreachofourclusters(distributedattention,ownpayoffsfocused andintra-cellfocused) inall gamestogether.The parameters estimatedwere
τ
=
1.7 for participantsinthe distributedattention cluster,and
τ
=
0.82 forparticipantsintheownpayoffcluster.Inbothcases,thevaluesoftheparameterswereconsistentwithour interpretationsofplayers’behavior.The
τ
estimatedforparticipantswhofocusedonintra-cellsaccades wasequalto0,indicatingthataccordingtothemodel theseplayerswereassumedtochooserandomly.However,consideringthesystematicandconsistentVIApatternobserved by these participants, they can hardly be considered level-0 players. In what follows we will interpret the VIA pattern exhibitedbyplayersfocusedonintra-cellsaccades intermsofotherregardingpreferences.A playermotivatedbyfairnessorcompetitionmaylookatthepayoffsofthecounterpart,regardlessofwhetherthese payoffs are strategicallyrelevant or not (Johnson and Camerer, 2004, 2004b). On the one hand cooperative players aim to maximize thesumoftheir own andtheirpartners’ payoffs. Ontheother,competitive playersintendto maximizethe difference betweentheirown andtheiropponent’s gain3 (VanLange, 1999). Inordertodoso,both playertypesneed to
comparetheir ownpayoffswiththoseoftheircounterpart,foreachpossiblegameoutcome(cell).Bothofthesestrategies areconsistentwiththeVIApatternexhibitedbyplayersfocusedonintra-cellsaccades.
Inallourgamestherewas acellwithsymmetricpayoffsthat alsomaximizedthesumofthetwoplayers’payoffs.We assume cooperativeplayersastheonesfocusingonthatcell.Conversely,competitiveplayersshouldhavefocusedmoreon thecellinwhichthedifferencebetweenplayer’sownpayoffandthepayoffofthecounterpartwasmaximal,attheplayer’s ownadvantage.Wecalledthesecells:“cooperativecell”and“competitivecell”,respectively.
Foreachplayerfocusedonintra-cellsaccadeswecalculatedtheSearchIndex(henceforthSI.
Glöckner
andBetsch,2008; Norman and Schulte-Mecklendeck, 2010), obtained by subtracting the time ratio spent looking at the AOIs located in correspondence withthe competitive cell fromthe time ratio spent looking at AOIslocated in correspondencewith the cooperativecell,anddividingthisdifferencebythesumofthem.SIrangesfrom−
1 to+
1;negativevaluesindicatemore time spent looking atthe payoffslocatedin thecompetitive cell, whilepositive indicatemore time spent lookingat the payoffslocatedinthecooperativecell.Playerswithaparametervaluegreaterthanzerowereclassifiedascooperative(17players)andthosewithaparameter valuesmallerthanzeroascompetitive(6players).
3 Competitiveplayersassumethattheircounterpartischoosingeachactionwithequalprobability,thereforethismodelimposesnorationalityonthe
et a l. / G ames and E conomic B eha v ior 9 4 (2015) 80–96 91
Fig. 8. Proportionofeachtypeofsaccadesmadebytrainedparticipantswhenapplyingthefourdecisionrules(Cooperative,Competitive,L1andL2),inthe fourclassesofgames.
Tosummarize,weassociatedplayerswithdistributedattention tolevel-2thinkers(16players),playersfocusedontheirown
payoffs tolevel-1thinkers(17players),andplayersfocusedonintra-cellsaccades weredividedincooperative(17players)and
competitive(6players)typesbasedontheirsearchindex.
3.7. Patternsofvisualinformationacquisitionadoptedbytrainedparticipants
WhiletheVIApatternsobservedinourclusters(see
Fig. 6
) appearconsistentwiththetypesweidentify(albeitnoisy), onemightwonderhowclosedtheyaretothosewewouldobserveifparticipantsweretrainedtolookforspecificstrategies. FollowingFunaki
etal. (2011),werunaseparateexperimentwhereparticipantsweretrainedtolookforfourstrategies(one atatime),whenplayingthegamespresentedinFig. 1
.32 additionalsubjects participatedin thisexperiment.Participants were askedto apply each ofthe fourstrategies to the 32 games for a total of 128 trials per subject4. We collected participants’ saccades for each strategy and built the
clustersaccordingly.Thefourstrategiesparticipantsweretrainedonwere:1)level-1(L1):inwhichparticipantswereasked to choose the optionwiththe highestaverage payoff; 2) level-2 (L2):in whichparticipants were askedto best respond to aplayer whochooses theoptionwiththehighestaveragepayoff; 3) Cooperative:inwhichparticipantswere askedto coordinate their actionswiththose ofthecounterpart onthe outcomethat maximize thejoin payoff; 4)Competitive: in whichparticipantswereaskedtochoosetheoptionthatmaximizesthedifferencebetweentheirownpayoffandthepayoff ofthecounterpart,assumingthatthecounterpartwaschoosingrandomly.
Fig. 8 reportstheproportionofeachtype ofsaccadefortrainedparticipants.In Fig. 6 and8,distributions ofsaccades in the three clusters(Intra-cell focused, Own focused, Distributed attention) are very close, although obviously slightly noisierintheclustersobtainedwithuntrainedparticipants.5 Tofurthertesttheaccuracyofourclassification,wecompared
the analysis6 adoptedby each untrainedparticipant(in DSOgames)withthose oftrainedparticipants(again intheDSO
games)andassociateeachsubjecttoadecisionrulebasedontheminimumEuclideandistance.Resultsshow thatonly5 of56participants(8.9%)wouldbeclassifieddifferently.Thisisafirst evidencethatourclusters(obtainedwithuntrained participants)reflectnotonlysubjects’preferenceforaspecificVIApattern,butalsotheirpreferenceforaspecificstrategic behavior. Evenmore,thisevidencesupports thetop-downhypothesis, i.e.that VIApatternsaretheproductofthesearch foraspecificstrategicsolutionofthegame,notviceversa.Inthenextsection,wewillpredictthebehavioroftheplayers basedonourcategorization.
4. Forecastingagents’behavior
Ourfinal stepisto predictthechoicesofplayersinthefourclassesofgames, simplyusingourdefinitionoftypesas proxyof players’strategic behavior. Weremind that clustersandSearchIndex wereboth determined onlyusingdata on saccadesinDSOgames.
Table 4
reportstheexpectedandobservedproportionofequilibriumresponses(Paretoequilibrium instaghuntgames)forthefourtypesinthefourclassesofgames.4 FurtherdetailsontheexperimentalmethodcanbefoundinonlineAppendix E.
5 Intheclusterswithtrainedparticipantsnoiseisminimizedsinceweonlyusethetrialsinwhichsubjectscorrectlyselectedthestrategyweasked
themto,excludingthosetrialsinwhichtheiranswerwaswrong.Theclusterswithuntrainedsubjectswereinsteadbuiltexclusivelybasedonsaccades, withoutconsideringsubjects’actualchoice.
Table 4
Proportionofexpectedandobservedequilibriumresponses(Paretoequilibriuminstaghuntgames)forthefourtypesinthefourclassesofgames(standard deviationinparenthesis).
Game
DSS DSO PD SH
Own focused Exp. 1.00 0.00 1.00 0.00
Obs. 0.91 (0.19) 0.13 (0.20) 0.90 (0.22) 0.28 (0.30)
Distributed attention Exp. 1.00 1.00 1.00 0.00
Obs. 0.93 (0.12) 0.65 (0.25) 0.92 (0.17) 0.23 (0.27) Cooperative Exp. 0.00 0.00 0.00 1.00 Obs. 0.27 (0.25) 0.16 (0.21) 0.19 (0.22) 0.83 (0.17) Competitive Exp. 1.00 1.00 1.00 0.00 Obs. 0.88 (0.25) 0.85 (0.24) 0.88 (0.31) 0.31 (0.44) Table 5
Averagechoicedistributioninthecoordinationgames(SH)amongthethreepossibleoutcomesdividedbytypeofplayer.
Level-2 Level-1 Cooperative Competitive
Coord. Miscoord. Coord. Miscoord. Coord. Miscoord. Coord. Miscoord. Risk dom. Pareto Risk dom. Pareto Risk dom. Pareto Risk dom. Pareto
Level-2 0.59 0.04 0.36 0.55 0.07 0.38 0.13 0.19 0.68 0.53 0.07 0.40
Level-1 0.50 0.07 0.43 0.12 0.24 0.64 0.49 0.09 0.42
Cooperative 0.02 0.70 0.28 0.11 0.26 0.63
Competitive 0.43 0.06 0.51
Based on our definition of types, ownpayoffs participants should behave as level-1 players and play equilibrium in DSS andPD gamesonly. Inthe remaining games,they should bestrespond toa counterpart playingrandomly,therefore they should choose the strategy giving the highest expected payoff (the non-equilibrium strategy in DSO and the Risk DominantinSH).Participantsclassifiedasdistributedattention shouldbehaveaslevel-2playersandplayequilibriuminall the dominantsolvable games,while chasingthe Risk Dominant equilibriumin thestag hunt. Intra-cell participants with
cooperative attitude shouldnot play equilibriumin dominance solvablegames, sincethe cell maximizing thesum ofthe
players’payoffsliesalwaysinthenon-equilibriumstrategy;theyshouldalsobeattractedtothePareto-efficientequilibrium inSHgames.Lastly,intra-cell competitive playersshouldplay alwaysequilibriumduetothefactthatthecellwithrelative higher payoffs for them lies always in the equilibrium strategy in dominant solvable games, and in the risk dominant strategyintheSHgames.
Indominancesolvablegames(DSS,DSO,PD)players’choiceswerecorrectlypredicted84%ofthetimebyourdefinition oftypes,whileonly59%ofthechoiceswereconsistentwiththeNashequilibrium.Incoordinationgames(SH),ourdefinition oftypesexplained77%ofthechoices.Acomparativelylowerpredictivepower(65%)wasfoundforDSOgamesforplayers categorized asdistributedattention. We tried to explain this lack ofpredictability by comparing the temporal patternof visualanalysisinequilibriumandoutofequilibriumresponses.As shownin
Fig. 9
whenplayerscategorizedasdistributedattention choseaccordingtotheequilibrium(Panel A),they startedlookingattheir ownpayoffs,then theyevaluated the
payoffsoftheir counterpart,andfinally,theychose theirbestresponsere-evaluatingtheir ownpayoffs.Conversely, itwas notpossibletoidentifyawell-definedtemporalpatternofvisualanalysis(PanelB)whentheydidnotchooseinaccordance withtheequilibrium.Thusshowingadistinctiveandwell-characterizedpatternforequilibriumplay.
Finally,weusedourclassificationoftypestopredictwhether,andinwhichequilibrium,therewillbecoordinationinthe staghuntgames.Wematchedparticipants’actualchoicesinall SHgames(8games),calculatedeach participant’saverage outcome distributionamong thethree possibleoutcomes (coordinationinthe risk-dominant equilibrium, coordinationin theParetoequilibrium,miscoordination), thenaveragedthedistributionsbytype.Asexpected,participantsclassifiedusing eye-tracking data as level-2, level-1 and competitive players coordinate mostly on the Risk Dominant equilibrium, and miscoordinatewhenmatchedwithcooperativeplayers(Table 5).CooperativeplayerscoordinateontheParetoEquilibrium, butonlywhenmatchedwithanothercooperativeplayer;theymiscoordinateotherwise.
5. Concludingremarks
Inthispaperwestudystrategicinteractioninnormalformgames,testingthehypothesisthatplayers’behaviorisbased ondifferentdecisionrulesortypes,andthatthesetypescanbeinferredbyobservinghowagentsacquireinformation.
Monitoringeyemovement,wecouldidentifythreetypesofplayersintermsofvisualpatternsofinformationacquisition (VIA):playersfocusing ontheir own payoffs,playerswithdistributed attention,andplayersthat analyzed payoffswithin cell.
Playerscategorizedasownpayoffs systematicallyneglectedthepayoffsoftheotherplayer;moreover,theeye-movements were consistent withthose ofa personlooking forher dominantstrategy orthe strategy givingher thehighestaverage payoff.TheyacquiredpartialinformationandtheypurposelyplayedequilibriumonlyinDDSandPDgames,thereforeonly
Fig. 9. First16saccades(meanandstandarderror)inDSOgames,forplayersclusteredas
distributed
attention (level 2),dividedbyequilibriumresponses (PanelA)andoutofequilibriumresponses(PanelB).OntherightsideofeachPanelwereportedtheproportionofown,otherandintra-cellsaccadesat thetimeofchoice(lastsaccade).when they themselves hada dominantstrategy. Theyperformedonly one stepof iterationandthey based their choices on their own rationality, ignoring the rationality ofthe other player. The presence ofplayers of thistype demonstrates non-attendanceofrelevantstrategicinformationandhownon-attendanceleadstoout-ofequilibriumbehavior.
Playerscategorizedasdistributedattention acquiredalltheavailableinformation,thustheyperformedsaccadesand fixa-tions overownandotherplayer’spayoffs.TheirVIApatternshowsthattheyperformedtwo stepsofiteration,thustaking into consideration their own and other player’s rationality.They played the unique equilibriumin the three dominance solvablegamesandtheyplayedriskdominantequilibriuminthestaghunt.Interestingly,theyperformedasystematicand consistent dynamicpatternofinformationacquisitionwhenplayingequilibrium.Thus theysequentiallylookedat(i)their own payoffs,(ii) thepayoffsoftheother player,(iii)they integratedthetwo,and(iv)finally lookedattheir ownpayoffs prior to their response (i.e. the equilibriumchoice). Deviation fromthis specific VIApattern leads to out-of equilibrium choices. Thissuggeststhatthepatternofvisualattentionisstrictlycorrelated withthefinal choice.Also, thisresult indi-cates that,forthistype ofplayers, deviationfromequilibriumisnot duetonon-attendance ofavailableinformation: the trialsinwhichdistributedattention participantsdeviatefromequilibriumarethetrialsinwhichtheylookedmainlyattheir ownpayoffs.Thus,theiroutofequilibriumbehaviorischaracterizedbyaself-referentialbias.
Alsoplayersfocusedonintra-cellsaccades acquiredcompleteinformation.TheirVIAwascharacterized byintra cell sac-cades between own and other player’s payoffs. The VIA patterns of the intra-cell players reflected two types of social preferences: cooperative players search for the cell that minimizes the difference in relative payoffs, while competitive players look for the cell that maximizes the difference between own and other player’s payoffs. This classification was determined usinga Search Index on the eye-tracking data, thus showinghow our method provides a measure ofsocial preferences. FollowingthisVIApattern, they neverplayed equilibriumin thedominance solvablegamesandthey played payoffsdominantequilibriuminthestaghuntgame.Theevidencefromthistypeofplayersshowsagainthatfullattention isnotasufficientconditiontoplayequilibrium.Thusplayersfocusedonintra-cellsaccades werenon-strategiceventhough theyacquiredalltheavailableinformation.
Strategicdeliberationrequirestheplayertoacquireinformationaboutownandotherplayer’spayoffsfollowingaspecific order,suchastheorderofinformationprocessingadoptedbythedistributedattention players.Takentogether,the specifica-tionsoftypessuggestthatVIApatterns(directlymeasuredwitheye-tracking)isasetofinformationprocessingmechanisms associatedwithsocialpreferencesandstrategicsophistication.
Theresultsobtainedinourstudystronglysupportthehypothesisthatparticipantsinone-shotgamesapplydifferent de-cisionrulesthatcanbeaccuratelydetectedbyanalyzingtheVisualPatternofInformationAcquisition thattheyadopted.These
decisionrulescan beeasily described intermsofinteractionbetweensocialmotives andlevelofstrategicsophistication. Wealsoobservedthatplayersdonotchangetheirattentionalpatternofvisual informationacquisitionwhendealing with gameshavingdifferentstructures.
We believe that the algorithms of visual analysis adopted by our participants are actually the result of higher-level processesbasedontheinteractionbetweenmotivationalandcognitiveaspects.Thus,preferencesandcognitionmay char-acterizethevisualattentionpatternthroughpreferentiallooking.
Ourresultsindicatethattheagentshaveapredeterminedandstablewaytoacquireinformation(i.e.representcontexts) ininteractive games,andthatthispatternofvisual attentionis strictlyrelatedtoboth theirpreferences andtheir ability toreasonstrategically.Comparingpatternsofinformationacquisitioninourclusterswiththoseoftrainedsubjectssuggests thatVIApatternsaretheproductofthesearchforaspecificstrategicsolutionofthegame(top-downhypothesis),notvice versa.
Appendix A. Supplementarymaterial
Supplementarymaterialrelatedtothisarticlecanbefoundonlineat
http://dx.doi.org/10.1016/j.geb.2015.09.003
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