Testing the level of consistency between choices and beliefs in games using eye-tracking

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Contents lists available atScienceDirect

Games

and

Economic

Behavior

www.elsevier.com/locate/geb

Testing

the

level

of

consistency

between

choices

and

beliefs

in games

using

eye-tracking

Luca Polonio

a

,

∗

,

Giorgio Coricelli

b

a_Center_for_Mind/Brain_Sciences,_CIMec,_University_of_Trento,_Palazzo_Fedrigotti_–_corso_Bettini_31,₃₈₀₆₈_Rovereto,_Trento,_Italy b_Department_of_Economics,_University_of_Southern_California,₃₆₂₀_S_Vermont_Ave_300,_Los_Angeles,_CA_90089,_United_States

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received22December2015 Availableonline22November2018 JELclassiﬁcation: C72 C51 D84 Keywords: Gametheory Beliefs Boundedrationality Eye-tracking

We use eye-tracking to identify possible causes of inconsistency between choices and beliefs ingames. Participantsplay aseriesof two-player 3×3 one-shot games(choice task) and state theirbeliefs about whichactions theyexpect their counterpartto play (beliefelicitation task).We usea model-basedclustering methodto group participants accordingto thepattern ofvisual analysis theyuseto maketheirdecisions inthe two tasks.Wefindthatheterogeneityinthelookuppatternsreflectstheadoptionofdifferent modelsofchoice.Ourresultssuggest thatthereare two mainreasonswhyparticipants do not best respond to their beliefs in games. First, many ofthem take into account theincentivesofthecounterpartwhenstatingtheirbeliefs,butnotwhenchoosingtheir actions.Second,someparticipantshaveother-regardingpreferencesandattempttofinda cooperativesolutionofthegame.

1. Introduction

In strategic interactions,optimal decisionsdepend onbeliefs aboutother players’ decisions,which inturn depend on others’expectationsaboutone’sowndecisions,andsoon.Beliefsarecrucialformanybehavioralgame-theoretic solution conceptsandalsoformanybehavioralmodelsofgameplay.Standardequilibriummodelsassumethatplayersbestrespond to their beliefsaboutother players’ strategies.Inlevel-k(Stahl andWilson, 1994, 1995; Nagel,1995; Costa-Gomesetal.,

2001; Costa-Gomes andCrawford, 2006) and cognitive hierarchy models (Camerer et al., 2004) it is also assumedthat players bestrespondto their beliefsaboutothers,1 _even_if_they _might _use_limited_cognition _in_forming_beliefs _and_have incorrectassumptionsabouttherationalityofother players.2 Thus,theconsistency betweenchoiceandbeliefs(i.e.,when players bestrespond totheir beliefsabout other players’ choices)is relevant forboth equilibriumtheories andbounded rationalitymodelswithincorrectbeliefs.

Acommonmethodusedinbehavioralgametheorytostudythelevelofconsistency betweenchoicesandbeliefsisto elicit individuals’beliefsabouttheexpectedactionofthecounterpart.3 Beliefelicitationprocedures havebeenextensively appliedtonormalformgameswithﬁniteactionsetsinbothone-shot(Costa-GomesandWeizsäcker,2008; Rey-Biel,2009;

*

Correspondingauthor.

E-mailaddresses:luca.polonio@unitn.it(L. Polonio),

giorgio.coricelli@usc.edu

(G. Coricelli).

1 _See_Crawford_et_{al. (}₂₀₁₃_{) for}_a_survey_of_the_applications_of_level-k_and_cognitive_hierarchy_models_in_economics. 2 _In_this_case_they_would_best_respond_to_{“incorrect”}_beliefs.

3 _See_{Manski (}₂₀₀₄_{) for}_a_survey_on_elicitation_of_{probabilistic}_{expectations.}_See_Armantier_and_{Treich (}₂₀₁₃_),_Manski_and_{Neri (}₂₀₁₃_),_Schotter _and Trevino (2014),andTrautmannandKuilen (2015) forexhaustivediscussionsofdifferentmethodsofbeliefelicitationinthelab.

https://doi.org/10.1016/j.geb.2018.11.003

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Dufwenbergetal.,2011) andﬁnitely-repeatedsettings(NyarkoandSchotter,2002; PalfreyandWang,2009; Rutströmand Wilcox,2009; Danzetal.,2012; Hyndmanetal.,2012),aswellastogameswithcontinuousactionsets(Dufwenberg and Gneezy,2000; Neri,2015).

Itisgenerallyacceptedthat inrepeatedgames,playersformbeliefsovertimeaboutthelikelybehaviorofthe counter-part,andthatthesebeliefscontributetotheprocessofconvergingtoequilibrium. However,experimentalevidenceclearly indicatesthatingameswithoutclearprecedents(one-shotgames),playersoftenmakechoicesthatarenotbestresponses totheirownstatedbeliefs.4 _These_results_raise_the_question_of_the_determinants_of_this_general_{inconsistency.}_In_this_paper, we trytoanswerthisquestionby studyingthedecisionprocessthat leadsplayerstochoosetheir actions andstate their beliefsabouttheexpectedactionofthecounterpart(probabilisticﬁrst-orderbeliefs)inaseriesoftwo-player3

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3 one-shot games.Tostudythedecisionprocess,werecordtheeyemovementsoftheparticipants.Theanalysisofthelookuppatterns allowsustomakethedecisionprocessvisibleandrevealpossiblecausesofinconsistentbehavior.

In light ofprevious studies, whichshow that heterogeneity in lookuppatterns isassociated withthe useof different models ofchoice (for a detailedreview,see Section 3), we hypothesize that players’ behavior inone-shot games is het-erogeneous (Weizsäcker, 2003) and depends on two features: the level of sophistication ofthe decision model adopted andthe motives that driveplayers’ choices. According to level-kandcognitive hierarchymodels, playersdiffer fromone another in their depth ofreasoning: each player’s behavior is determined by her cognitive type andreﬂects the extent to which the player takesthe game structure andother players’ incentives into account.5 _A _second _type _of heterogene-ity is related to the existence of different types of other-regarding preferences (Rabin, 1993; Fehr and Schmidt, 1999; Bolton andOckenfels, 2000; Charness andRabin, 2002; Dufwenberg andKirchsteiger, 2004; Falk and Fischbacher,2006; Coxetal.,2008).Indeed,anumberofstudieshavereportedasigniﬁcantproportionofparticipantswhosebehavioris mo-tivatedbyfairnessconsiderations(Fehretal.,1997; IsaacandWalker,1988; AndreoniandMiller,2002; Fismanetal.,2007; IriberriandRey-Biel, 2013). Startingfromthistheoretical frameworkwe usea model-basedclusteringmethodto classify participantsaccordingtotheirpatternsofvisualanalysisandidentifythedecisionmodelthatmostaccuratelypredictstheir behavior.Finally,wetestwhetherthelevelofconsistencybetweenchoicesandbeliefsdependsonthecoherence between thedecisionmodelusedinthechoicetaskandtheoneattributedtothecounterpartinthebeliefelicitationtask.

Ourresults show that approximately one third of theparticipants devotes a considerableamount of attentionto the incentives ofthe counterpart in both choice andbelief elicitation tasks. Theseparticipants play according tothe Level-2 modelandbelieveitishighly probablethattheir counterpartplaytheLevel-1 action.Asimilar proportionofparticipants focuses their attention predominantly on their own payoffswhen choosing their actions, and almost exclusively on the payoffsof their counterpart when statingtheir beliefs.These participants followthe predictions ofthe Level-1 model in thechoicetask, butexpect thecounterparttoplaytheLevel-1 actioninthebeliefelicitation task.Wealsoﬁndthatthere are participants whose analysis in the two tasks is based on the comparison between their own payoffs and those of the counterpart foreach possibleoutcome ofthe game. Theirchoices andbelief statementsare inaccordance withthe prediction of a model of Prosociality.This model assigns a positive weight to the total payoff of the two players anda negativeweighttotheinequalityinpayoffs.

Ourﬁndings showthatparticipantsallocateaconsiderableamountofattentiontothepayoffsofthecounterpartwhen statingtheirbeliefsbutonlysomeofthembasedtheiranalysisonthepredictionoftheotherplayer’sactioninthechoice task.Weﬁndthatthenumberofbestresponsestoparticipants’ownstatedbeliefsisrelatedtotheamountofattentionthat theypaytothepayoffsofthecounterpartinthechoicetask:themoreparticipantsfocustheirattentionontheincentives ofthecounterpart,themore theybestrespondtotheir beliefs.Wethus observeconsistencybetweenchoicesandbeliefs onlywhen theinformation acquisitionpatternof theparticipantsin thechoice taskiscentered onthe predictionofthe otherplayer’saction.Anydeviationfromthispatternleadstoinconsistencybetweenchoicesandbeliefs.

2. Relatedliteratureonbeliefelicitation

Elicitationofﬁrst-orderbeliefshasrecentlybecomeacommonpracticeinexperimentaleconomicstopredictchoice be-haviorandinferdecisionprocessesininteractivedecisionmaking(Manski,2004).ThisprocedurewasusedbyCosta-Gomes andWeizsäcker (2008) toinvestigatethelevelofconsistencybetweenchoicedataandbeliefdatausing3

×

3 matrixgames.6 Theyestimatedplayers’beliefsfromtheirchoicesinachoicetaskandcomparedthemwiththosereportedinabelief elic-itationtask.Analyzingchoicesgamebygame,theytestedthehypothesisthat behaviorinthetwotaskswasbasedonthe same beliefs.However, their participants best responded to their statedbeliefs only slightlymore than 50% of the time (18%morethan predictedby randombehavior), andtheconsistencyhypothesis was rejectedformostofthegames. Ina subsequent analysis, Costa-GomesandWeizsäckerassumedthe existence ofﬁve differenttypesof players:Nash, Level-2,

Level-1,Dominance-1 andOptimistic.Amongtheseﬁvemodels,theLevel-1 modelprovidesthebestdescriptionofthechoice

data.However,whentheplayersstatedtheirﬁrst-orderbeliefs,theyattributedtheirownlevelofsophisticationtotheother

4 _We_will_discuss_the_related_literature_on_belief_elicitation_methods_in_detail_in_section₂_.

5 _For_example,_in_level-k_models,_{Level-0 types}_choose_randomly_with_uniform_probability_among_the_available_options._{Level-1 players}_best_respond_to thebeliefthatallotherplayersareLevel-0.Level-2 playersbestrespondtothebeliefthatallotherplayersareLevel-1,andsoonforhigherlevels.

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players.Thisbehaviorisinconsistent,becauseifplayersbelievetheircounterparttobeLevel-1,theyshouldbestrespondto thisbelief,actinglikeaLevel-2 player.Theauthorsexplainedthisinconsistencybystatingthattheperceptionofthegames, and/or ofhow players’counterparts play thegames, issigniﬁcantlydifferentwhen playerschoosetheir actionsandstate theirbeliefs.7

AlowlevelofconsistencybetweenchoicesandstatedbeliefswasalsofoundbyNyarkoandSchotter (2002) inaseriesof two-personconstantsumgameexperiments.The authorstestedtheexplanatorypowerofdifferentbelieflearning models andfoundthattheydonotpredictstatedbeliefswell.Ahigherlevelofconsistencybetweenchoicesandstatedbeliefswas foundbyRey-Biel (2009).Healsousedone-shottwo-person3

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3 normalformgamesbutwithasimplerpayoffstructure comparedtoCosta-GomesandWeizsäcker.

Therearealsoanumberofstudieswhichelicitsecond-orderbeliefs:forexample,BhattandCamerer (2005) recordedthe brainactivityofparticipantswhileelicitingﬁrstandsecond-orderbeliefsinone-shotnormalformgames.Theyfoundthat playershavesimilarbrainactivations whentheirchoicesandbeliefstatementsarebothconsistentwithNashequilibrium. Vanberg (2008) elicitedﬁrstandsecond-orderbeliefsinaminidictatorgamewithrandomdictatorship,showingthatpeople donotsimplydislikegoingagainsttheother’spayoffexpectations,butthattheydisliketheactoflyingperse.Bellemareet al. (2011) measuredthewillingnesstopaytoavoidguiltinasimplesequentialgamesimilartothetrustgame.Theyfound that willingnesstopay,estimatedusingstatedbeliefdata,isoverestimatedwhenthecorrelationbetweenpreferencesand beliefsisnotaccountedfor.

In the threestudies we havejustmentioned, thesecond-order beliefs were measured aspointforecasts eventhough subjective expectations in games were usually elicited probabilistically to allow the forecaster to convey her degree of uncertaintyaboutherownforecast(Offermanetal.,1996; NyarkoandSchotter,2002; Costa-GomesandWeizsäcker,2008; Rey-Biel, 2009; Blancoetal., 2010; Ziegelmeyeretal., 2010).8 _Only_recently,_Manski _and_{Neri (}₂₀₁₃_{) proposed}_a _method to probabilistically elicit both ﬁrst- and second-order beliefs ina hide-and-seek game.The objectof their study was to investigatewhethertheobservedchoicescoincidewiththeoptimalone,givenelicitedbeliefs.Theyfound that89%ofthe choicesarebestresponsestoﬁrst-orderbeliefsand75%ofthemarebestresponsestosecond-orderbeliefs.9

In linewiththeliterature onprobabilistic elicitationofsubjective beliefsingames, weelicit ﬁrst-orderbeliefsby ask-ing participants toassign aprobability to eachpossible actionofthe counterpart.Beliefelicitation is incentivizedwitha quadraticscoringrule(describedbelow).

3. Relatedliteratureonprocessdata

Manyprocesstracingstudiessupporttheideathatplayersdifferintheirlevelofsophisticationandhavedifferentbeliefs about the other player’s actions. In this regard, Costa-Gomes et al. (2001) used a computer mouse-tracking method to investigateinformationsearchprocessesinnormalformgames.Theauthorstestedthecognitiveimplicationsofalternative models of choice combining choice data and patterns of information search. In order to explain players’ behavior, nine possible typesofplayerswere speciﬁed apriori. Fivetypes werestrategic andrequiredplayersto formbeliefsabout the expectedactionoftheircounterpart,andtheremainingfourtypeswerenon-strategic(oralternativelyhaddiffusedbeliefs). To describethelink betweendecisionprocess andchoice,theauthors assignedeachplayer typewithone (ormorethan one) search pattern(s). They observed that most of their participants exhibited lookup andchoices consistent with the predictionsofthelevel-kmodel.

Mouse-tracking was used alsoby Costa-GomesandCrawford (2006) ina studywherethey elicitedplayers’ initial re-sponses to a seriesoftwo-person guessinggames. Theauthors assumedthat each player’s behavioris determined (with error) bya uniquetype, whichdeterminestheir informationsearch patternsandguessesinall games.Theyinferred par-ticipants’beliefsfromtheirguessesand/orinformationsearchand,similarlytoCosta-Gomesetal. (2001),classiﬁedplayers intotypesusinganeconometricanalysis.Theyfoundthatdeviationfromtheequilibriumcouldbepredictedandexplained assuming ahierarchyofboundedlyrationaltypes.Inparticular,theyfoundthathalfoftheirparticipantswerefollowinga decisionrule(Level-1,Level-2,Level-3 orEquilibrium)thatincludedaprecisespeciﬁcationofbeliefs.10

Brocas etal. (2014) investigated thelink betweeninformation searchand decisionstrategy ingames withprivate in-formation.Theauthorsusedmouse-trackingtostudystrategicthinkingintwo-personbetting gameswiththreestatesand two-sidedprivateinformation.Theauthorsusedamodel-basedclusteringmethodtogroupparticipantsaccordingtotheir lookuppatternsandchoices.TheyidentiﬁedthreeclusterswhichcorrespondedapproximatelytoLevel-1,Level-2 andLevel-3 thinking, andafourthclusterofplayerswhoanalyzedthe gameexhaustivelybutmadeinferentialmistakes. Theirresults showedthatdeviationsfromNashequilibriumwereassociatedwithafailuretolookatrelevantinformation.

7 _They _also_performed_additional_tests_to_exclude_the_possibility_that_{inconsistency}_among_choice_data_and _belief_data _was_due_to_{risk-aversion}_or other-regardingpreferences.Thesehypotheseswereruledout.

8 _Bhatt_and_{Camerer (}₂₀₀₅_{) measured}_both_{ﬁrst- and}_second-order_beliefs_as_point_forecasts._Other_examples_of_{non-probabilistic}_elicitation_of_ﬁrst-order beliefsareinCroson (2000),DufwenbergandGneezy (2000) andSapienzaetal. (2013).

9 _They_also_found_evidence_of_{heterogeneity}_across_{participants.}

10 _The_behavior_of_the_other_half_of_the_participants_was_still_coherent_(with_only_few_violations_of_dominance_for_most_of_them)_but_less_well_described bytheirspeciﬁcationofpossibletypes.Intheiranalysis,theauthorsrejectedthepossibilitythattheobservedbehaviormightbeexplainedbyotherfactors suchasaltruisticbehavior,riskaversionorconfusion.

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Altogether,theseﬁndingssupporttheideathatindividualsdifferfromoneanotherintheirlevelofsophistication.There isalsoagrowingbodyofevidencethatevensocialpreferencessystematicallyaffectindividualdecisionmakingingames.In Johnsonetal. (2002),theauthorsusedmouse-tracking inthree-stageRubinsteinbargaininggamestotestwhetherplayers deviate from the equilibriumbecause of their limitedcognition or other-regarding preferences. They ran an experiment inwhich playersbargainedwithself-interestedrobotsandone inwhich theybargained withother players.Their results showed that participants’ offers in the two experiments were greater than what would be expected ifthey were using game-theoreticreasoning.Thisisinlinewiththehypothesis thatindividualsareboundedlyrational.However,they found that social preferences alsomatter, because abouta third ofthe difference betweenthe equilibrium prediction andthe observedmeanofferdisappearedwhenparticipantsswitchedfrombargainingwithindividualstobargainingwithrobots.

Polonioetal. (2015) recordedparticipants’eyemovementswhileplayinga seriesoftwo-player 2

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2 one-shotgames. Theauthorsﬁrstgroupedparticipantsintotypesaccordingtotheinformation-searchpatternsobservedinasingleclassof games andthen predictedtheir choicesindifferentclassesbasedon theirtype. Theyfound threegroupsofparticipants: theﬁrstgroupofparticipantsusedaniterativestep-by-stepproceduretoformbeliefsabouttheotherplayers’actionsand bestrespondtothem(Level-2 players).Thesecondgroupofparticipantstookintoaccounttheirownincentives,butlargely neglected theincentivesoftheir counterpart(Level-1 players).The thirdgroup ofparticipantsbasedtheir analysisonthe comparisonbetweentheirownpayoffsandthoseoftheircounterpart(playerswithother-regardingpreferences).

Theroleofsocialpreferencesingameswashighlightedbyanothereye-trackingstudyconductedbyDevetagetal. (2016). Intheirstudy,theauthorsshowedthat,intwo-personthree-by-threeone-shot-games,playersadoptedsimpliﬁedstrategies suchaschoosingtheactionleadingtoanattractiveandsymmetricpayoff.Theyfoundthatmanyplayersdidnottakeinto accounttheotherplayers’incentivesandchoseaccordingtofocalpayoffs,asexpectedforplayerswithsocialpreferences.

Overall, these studies suggest that there are two main causes of heterogeneity in strategic interactions: the level of sophistication of theindividual; andher social preferences. This,in turn,suggests that the levelof consistency between choicesandbeliefsingamesmightdifferbasedontheplayertype.11

4. Experimentaldesign

4.1. Thegames

Our main goal is to shed light on the determinants of the general inconsistency observed in the literature between choices andbeliefs in one-shot games. In particular, we hypothesize that the level of consistency between choicesand beliefs is related to the coherence betweenthe model of choice used in the choice taskand the one attributed to the counterpart in the beliefelicitation task.To test thishypothesis we selected 18games, including the 14games used in Costa-GomesandWeizsäcker (2008), plusfourgameswithmultipleequilibria.10ofourgamesaresolvableintwo,three, orfourstepsofiterated eliminationofdominatedactions, fourgameshaveaunique Nashequilibriumwithoutdominant strategies,andfourgamesareweak-linkgames(a3

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3 versionofthestag-huntgame)withthreeequilibria,oneofwhich is Pareto optimal. In games with a unique pure-strategy equilibrium, there are no salient payoffs. Conversely, weak-link gamesincludepossibleequilibriathat mayact asattractors.Thegameswithauniquepure-strategy equilibrium(games1 to14)areorganizedintopairsofisomorphicgames.Isomorphicgamesareequivalentinthesensethatthesecondgameof eachpairisidenticaltotheﬁrstexceptfortransposingtheplayers’roles, changingtheorderofthethreeactions(forboth players),andaddingor subtractinga smallconstant amountfromthe payoffsofeach game.12 _Using_pairs _of_isomorphic games,allparticipantsfacethesamesetofgamesregardlessoftheroletheyplay(roworcolumn).Incoordinationgames, we donot need tohavepairs ofisomorphic games, becauseeach gamehasa symmetric structure.Fig. 1reportsthe 18 gamesandTable1showstheirstrategicstructure.

Theexperimental designconsistsoftwo treatmentswithtwo sessionseach. Thetreatments aredefinedasfollows:in treatmentAB,participantsfirstchoosetheiractionsinthe18gamesandthenstatetheirfirst-orderbeliefs.IntreatmentBA, playersstatetheir first-orderbeliefsinthe18gamesandthenmaketheirchoices.Participantsarerandomlyassignedtoa role,roworcolumnplayer,andarerandomlypaired.Eyemovementsarerecordedinbothtreatmentsandinbothsessions.

4.2. Experimentalprocedure

AllsessionswererunattheEPLlab(ExperimentalPsychologyLaboratory)oftheUniversityofTrento.Participantswere 72undergraduatestudentsfromtheUniversityofTrento(19males,53females),themeanagewas23(SD2.95). Presenta-tionofthestimuliwasperformedusingacustom-madeprogramimplementedusingMatlabPsychophysicaltoolbox.

11 _Eye_tracking_was_used_also_to_study_{non-interactive}_decision_making._For_example,_Krajbich_et_{al. (}₂₀₁₀_{) developed}_a_{computational}_model_of value-basedbinarychoiceinwhichtheevolutionoftherelativedecisionvaluedependsontheallocationofattention.Arielietal. (2011) showedthatparticipants choosingbetweentwolotteriesoftencompareprizesandprobabilitiesseparately,ratherthantakingintoaccountthewholestructureofeachlottery,as suggestedbyexpectedutilitytheory.Reutskajaetal. (2011) studiedindividualchoiceofconsumptiongoodsundertimepressureandoptionoverload.

12 _The_payoffs_of_Game₂_are_generated_by_subtracting₂_points_from_the_payoffs_of_Game_1._The_payoffs_of_Game₄_are_generated_by_adding₂_points tothepayoffsofGame3.ThepayoffsofGame6aregeneratedbyadding1pointtothepayoffsofGame5.ThepayoffsofGame8aregeneratedby subtracting1pointfromthepayoffsofGame7.ThepayoffsofGame10aregeneratedbyadding2pointstothepayoffsofGame9.ThepayoffsofGame 13aregeneratedbyadding2pointstothepayoffsofGame11.ThepayoffsofGame14aregeneratedbysubtracting3pointsfromthepayoffsofGame 12.

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Fig. 1. Thegamesusedintheexperiment.TheunderlinedpayoffsindicatetheNashequilibriainpurestrategies.Games15to18arecoordinationgames withmultipleequilibria.

Table 1

StructuresofthegamesandtheirNashequilibria(Top=T,Middle=M,Bottom=B,fortherowplayersandLeft=L,Middle=M,Right=Rforthe columnplayers).Thenumberofroundsofeliminationofdominatedactionsrequiredbyrowandcolumnplayerstoreachtheequilibriumisshowninthe thirdrowofthetable.Ingameswithmultipleequilibria(games15to18)wereporttheactionsconsistentwiththeParetooptimalequilibrium.

Game 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 N◦ofequilibria 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 Roundsof elimination (row, column) 2,3 3,2 2,3 3,2 2,3 3,2 2,3 3,2 3,4 4,3 No No No No No No No No EQ(Pareto) (row, column) T, L M, L B, R M, M T, M B, M M, R B, R T, R B, L M, M B, L T, R T, L T, L M, M B, L B, M

Beforetheexperimentstarted,acopyofpreliminaryinstructionsthatdescribeda3

×

3 strategicgamewasgiventothe participantsandreadaloudbytheexperimenter.Controlquestionswereadministeredtoassurethatparticipantsunderstood the rules ofthe choice taskandtherules of thebelief elicitation task.13 When participantsfailed toanswer thecontrol

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questions,instructionswere repeated,andthenthe controlquestionswere administeredagain. Participants whofailedto answer oneof thecontrol questionsforthe second time were excluded fromall analyses.36 participantswere included ineach treatment.14Participants intreatments AB were providedwiththeinstructionsforthechoice taskandweretold how their choices wouldbe incentivized. Then, after they completed the control questions,and theeye-tracking system wascalibrated,participantsunderwenttwopracticegamesandthenplayedthe18games.15_At_the_end_of_the_choice_task, participantswereprovidedwiththeinstructionsonhowtostatetheirbeliefsandhowtheirrewardinthistaskwouldbe determinedbasedonthescoringrule.Next,theyunderwenttwopracticegamesandthenstatedtheirbeliefsin18games withoutknowinginadvancetheoutcomeoftheirchoices.Notimelimitwasimposedonparticipantstogivetheirresponse. IntreatmentBA,we invertedtheorderofthetwotasks.First,participantswereprovided withtheinstructionsforthe choice taskandhowthey wouldbeincentivized. Then,they were providedwiththeinstructionsforthebeliefelicitation taskandwereinformedonhowtheywouldbepaidfortheaccuracyoftheirbeliefs.Next,they statedtheirbeliefsforall 18gamesandﬁnallytheyplayedthegames.

Since one of ourpriorities was to compare thepatterns of visual analysisused by the participants when they chose their actionsandwhenthey statedtheir beliefs,thestructureofthe choicetaskandthebeliefelicitation taskwere kept assimilaraspossible.Inthechoicetask,theparticipantschosetheir actionsbypressing thecorresponding button.Inthe beliefelicitation task, theparticipants were askedto assigna probability to each possible actionof their counterpart.In particular,theyweretoldtothinkabouttheprobabilitiestheywouldassigntothethreeactionswhileobservingthegame matrix.Then, once they haddecided onthe threeprobabilities, they hadtopress thespacebar tomove to theresponse screenwheretheycouldselecttheirprobabilitiesandconﬁrmtheirresponse(seeAppendixAintheonlinesupplementary materialforanexhaustivedescriptionofthebeliefelicitationtask).

Allparticipants were testedindividually.16 At thebeginning oftheexperiment, the experimentertoldthe participants thattheirresponseswouldbepairedwiththoseofarandomlyselectedcounterpart,anditwasmadeclearthat,inthe18 games,theselectedcounterpart wouldremain thesamebothwhenchoosingtheir actionsandwhenstatingtheirbeliefs. Nofeedbackabouttheresultoftheirdecisionswasgiventotheparticipantsuntiltheendofthedatacollection.Attheend ofthedatacollection,participantswereanonymouslyandrandomlypaired,andtheirresponseswerecombined.

Allparticipantsreturnedtoreceivetheirpaymentsafewdaysaftertheendoftheexperiment,andtheyeachdrewtwo tagsfromtwodifferentjars.Eachofthetagsintheﬁrstandsecondjarswasassociatedwithoneofthegamesusedduring theexperiment.Thetagextractedfromtheﬁrstjardeterminedthepaymentforthechoicetask,andthetagextractedfrom thesecondjardeterminedthepaymentforthebeliefelicitationtask.

For the choice task participants were paid at an exchange rate of 10 cents per point. For the belief elicitation task participantswerepaidaccordingtoaquadraticscoringruledeﬁnedasfollows:let yi_g bethestatedbeliefofplayeri ingame g,where yigisaprobabilitydistributionoverthethreeactions(L, M,andR) ofplayer j (thecounterpart).Theprobability

distributionis yi

g

≡ (

yig,Lyig,Mygi,R

)

,suchthat yig

∈

2

≡ {

yig

∈ R

3

| Σ

c∈{L,M,R}yig,c

=

1

}

.Theactionchosenbyplayer j (the

counterpart)isdeﬁnedasxgj

≡ (

x j g,L

,

x j g,M

,

x j g,R

)

,wherex j

g equals1forthechosenactionandzerootherwise.Thepayoffvg

ofplayeri iscalculatedwiththefollowingquadraticscoringrulevg

=

A

−

c

[(

yig,L

−

x j g,L

)

2

+(

yig,M

−

x j g,M

)

2

+(

yig,R

−

x j g,R

)

2

],

where A and c are constants ( A

=

10 Euros and c

=

5 Euros).17 Before starting the experiment the participants were providedwithacleardescriptionofthescoringruleandseveralexamples.

Participants earned between1.00 and19.9 Euros in addition to a 4 Euro show-up fee. One participant in treatment

AB and one intreatment BA werediscarded fortechnical reasons;thus intotal we collected eye-tracking andbehavioral

datafrom35participantsineachtreatment.TheethicalcommitteeoftheUniversityofTrentoapprovedthestudy,andall participantsgaveinformedconsentpriortoadmissiontothestudy.

5. Results

Inthissection,weprovideadescriptiveanalysisofchoiceandstatedbeliefsdata.Theaveragerateofequilibriumchoices in games with a unique Nash equilibrium is equal to 0.37, which is very close to what would be predicted by chance (rate

=

0

.

33).Importantly, thisproportionisnotaffectedby thetreatment(AB

=

0

.

34,BA

=

0

.

39). Theaverageprobability withwhichparticipantsestimatedthat thecounterpartis playingthe equilibriumstrategy(in gameswithaunique Nash equilibrium)isequalto0.29intreatmentsAB andto0.33intreatmentBA (Table2).Theseresultssuggestthatparticipants donotbelievetheircounterparttoplayaccordingtotheequilibrium,regardlessofthetreatmentgroup.

14 _The_reasons_for_choosing_a_sample_size_of₇₂_participants_is_based_on_previous_results_showing_that_monitoring_eye_movements_in_similar_experimental settingsitispossibletoidentifythreegroupsofparticipantsassociatedwiththreegeneralvisualpatternsofinformationacquisition(seePolonioetal., 2015,andDevetagetal.,

2016

).Thesestudiesshowthatparticipantsareequallydistributedamongthesethreegroups.Basedonthesepreviousresults,we wantedtohaveatleast20participantsforeachgroup.Thereforewedecidedtocollectdatafrom70participants.

15 _An_extensive_description_of_the_eye-tracking_procedure_is_provided_in_online_{supplementary}_material_(see_{Supplementary}_{Fig. 1}_{Appendix C).} 16 _In_order_to_allow_the_experimenter_to_carefully_set_up_and_control_the_eye-tracking_data_acquisition,_a_high_precision_eye-tracking_device_requires_a dedicatedlaboratory.Forthisreason,andalsobecauseofthehighcostofthedevice,ineye-trackingstudies(aswellasinfMRIexperiments)participants aretestedoneatatime.

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Table 2

TheleftsideofthetableshowstheNashequilibriainthechoicetask(Top=T,Middle=M,Bottom=B,fortherowplayersandLeft=L,Middle=

M,Right=Rforthecolumnplayers)andtheaverageproportionofchoicesinaccordancewiththeequilibriumpredictionsbytreatment(AB andBA). TherightsideofthetableshowstheNashequilibriainthebeliefelicitationtaskandtheaverageprobabilityofstatedbeliefsthatmatchtheequilibrium predictionsbytreatment.Notethatthedataarepresentedfromtheperspectiveoftherowplayers.Ingameswithmultipleequilibria(games15to18),the valuesrefertotheParetooptimalequilibrium.Thetotalaverageiscalculatedoverthegameswithuniqueequilibrium(games1to14).

Game Equilibrium model

Choice task Belief elicitation task

Model predictions Pareto Eq. Treatment AB Treatment BA Model predictions Pareto Eq. Treatment AB Treatment BA

1 T, L 0.26 0.40 L, T 0.45 0.53 2 M, L 0.69 0.40 L, M 0.20 0.17 3 B, R 0.26 0.31 R, B 0.20 0.22 4 M, M 0.20 0.14 M, M 0.19 0.21 5 T, M 0.49 0.49 M, T 0.13 0.15 6 B, M 0.03 0.14 M, B 0.24 0.25 7 M, R 0.31 0.43 R, M 0.44 0.62 8 B, R 0.51 0.49 R, B 0.19 0.30 9 T, R 0.77 0.74 R, T 0.24 0.30 10 B, L 0.26 0.57 L, B 0.60 0.70 11 M, M 0.31 0.51 M, M 0.50 0.59 12 B, L 0.14 0.06 L, B 0.28 0.22 13 T, R 0.40 0.57 R, T 0.31 0.31 14 T, L 0.17 0.20 L, T 0.13 0.11 15 T, L 0.34 0.29 L, T 0.43 0.32 16 M, M 0.49 0.31 M, M 0.47 0.28 17 B, L 0.46 0.34 L, B 0.30 0.24 18 B, M 0.34 0.29 M, B 0.43 0.27 Average 0.34 0.39 0.29 0.33 Table 3

Averageproportionofbestresponsesforeachgameandplayer’srole.

Game ID 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Average

Row .44 .50 .44 .78 .61 .83 .53 .50 .47 .50 .42 .61 .47 .47 .72 .78 .69 .81 .59

Column .56 .32 .74 .41 .65 .65 .44 .59 .41 .50 .41 .47 .50 .76 .79 .82 .74 .74 .58

Next, we test the level ofconsistency betweenchoices andbelief statementsat the individual level.To test whether participantsinthetwotreatmentsbestrespondtotheirbeliefsmoreoftenthanthey wouldifchoosingrandomly,weuse Kolmogorov–Smirnov teststocomparethe cumulativefrequencydistributionoftreatments AB andBA with thatexpected from randomchoices. Weﬁnd that bothdistributions differsigniﬁcantly (both p-values

<

0.001)fromwhat wouldhave beenpredictedbychance.However,thefrequenciesofchoicesthatareconsistentwiththeparticipants’statedbeliefsdonot differsigniﬁcantlybetweenthetwotreatments(two-tailed,two-sampleKolmogorov–Smirnov test; D

=

0

.

143,p

=

0

.

87).18 Regardlessofthetreatment,theaverageproportionofbestresponsestothebeliefstatementisequalto0.59.Thisproportion varies from a minimum of 0.32 to a maximum of 0.83depending on the type of game andplayer’s role (Table 3). In particular,theproportionofbestresponsesingameswithauniqueequilibrium(games1to14)ismuchlower(rate

=

0

.

54) thanthatobtainedinweak-linkgames(games15to18,rate

=

0

.

76).

Tosummarize,we ﬁndthat players’choicesaregenerallyinconsistentwiththeir statedbeliefsingameswithaunique equilibriumandthattheproportionofbestresponses tobeliefstatementsdependsonthetype ofgame.19 _Moreover,_our resultsshow thatparticipants’choicesandstatedbeliefsare notwellexplainedby Nashequilibrium.However, alternative modelsmayexplainthebehaviorofourparticipants.Inthisregard,onepossibilityisthatparticipantshaveaboundeddepth of reasoning that dependson their cognitive type. Experimental evidence indicates that the generallevel-kof reasoning achieved by players in strategic interactions is between 1 and 2. Therefore, we test whether Level-1 or Level-2 models explain the participants’observed choicesandbeliefstatements.The two modelsare deﬁnedasfollows: theLevel-1 (L1) modelassumesthatplayersbestrespondtothebeliefthatassignsequalprobabilitytoallcounterpart’sactions.TheLevel-2 (L2)modelassumesthatplayersbestrespondtothebeliefthatthecounterpartplaysaccordingtotheLevel-1 model.

Anotherpossibility isthat thebehavior ofparticipants isdriven byfairness considerations.According tothe modelof other-regarding preferences proposed by Fehr andSchmidt (1999), some individuals are inequity-averse and are willing to cooperate with their counterpart to achieve more equitable outcomes. According to the Fehr and Schmidt model, in a two player game involving players i and j, player i evaluates her payoff according to the following utility function:

18 _{Supplementary}_{Fig. 2,}_in_the_online_Appendix_D,_shows_the_cumulative_frequency_distribution_of_the_number_of_best_responses_to_stated_beliefs_for_each ofthetwotreatments.

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Table 4

Theleftsideofthetableshowsthepredictionsofthefourmodelsinthechoicetask(Top=T,Middle=M,Bottom=B,fortherowplayersandLeft=

L,Middle=M,Right=Rforthecolumnplayers).Therightsideofthetableshowstheproportionofchoicespredictedbyeachmodelreportedfromthe perspectiveoftherowplayers.Dataarepooledacrossthetwotreatments.

Games Choice task

Models’ prediction Models’ accuracy

L2 L1 Inequity aversion Prosociality L2 L1 Inequity aversion Prosociality

1 T, M M, L M, L B, R 0.33 0.40 0.40 0.27 2 T, L M, M M, M B, R 0.29 0.54 0.54 0.17 3 B, M T, M T, R T, M 0.29 0.64 0.64 0.64 4 T, M T, L M, L T, L 0.81 0.81 0.17 0.81 5 T, L B, L B, M B, L 0.49 0.47 0.47 0.47 6 M, M M, R B, R M, R 0.84 0.84 0.09 0.84 7 M, R B, R M, R T, L 0.37 0.30 0.37 0.33 8 B, R B, L B, R M, M 0.50 0.50 0.50 0.33 9 M, R T, L T, R T, R 0.21 0.76 0.76 0.76 10 B, M T, L B, L B, L 0.41 0.36 0.41 0.41 11 M, R B, M M, M M, M 0.41 0.46 0.41 0.41 12 M, M M, R M, M B, L 0.80 0.80 0.80 0.10 13 B, R T, M T, R T, R 0.31 0.49 0.49 0.49 14 M, M B, M M, M T, L 0.41 0.40 0.41 0.19 15 B, R B, R B, R T, L 0.63 0.63 0.63 0.31 16 T, L T, L T, L M, M 0.54 0.54 0.54 0.40 17 M, R M, R M, R B, L 0.59 0.59 0.59 0.40 18 T, R T, R T, R B, M 0.61 0.61 0.61 0.31 Average: 0.49 0.56 0.49 0.43

Ui

(x)

=

xi

−

α

imax

[

xj

−

xi

,

0

]

− β

imax

[

xi

−

xj

,

0

]

where

β

i

≤

α

i and0

≤ β

i

<

1.Theparameters

α

iand

β

imeasuretheloss

fromdisadvantageousandadvantageousinequity,respectively.Themodel(Inequityaversion)assumesthatparticipantshave correctbeliefsaboutthedistributionofotherplayers’actions.Therefore,todeterminethemodel’spredictions,wemultiply theobserved strategyfrequencies withthetransformed payoffstoget theexpectedpayoff ofeachstrategy. Experimental evidencesuggeststhatthemodelachievesagoodlevelofoverallﬁtwithparameters

α

i

=

2 and

β

i

=

0

.

6 (FehrandSchmidt, 2010; Dreberetal.,2014).Therefore,inwhatfollowswetestthemodelusingthiscombinationofparameters.20

Finally, we test the hypothesis that participants use a simple decisionmodel conceived in terms of cooperation and egalitarianism(Van Lange,1999,2000). Thismodel (Prosociality)assigns a positive weight to thetotal payoff ofthe two players and a negative weight to the inequality in payoffs. In particular, each player changes the payoffs of the game accordingto theutility function: Ui

(x)

=

xi

+

xj

− |

xi

−

xj

|

.Then, themodelassumes thateach player choosesan action

thatleadstotheParetosuperioroutcome(inthenewgame)andbelievesthatthecounterpartisdoingthesame.Empirical supportforthisassumptioncomesfromVanLange (1999).Theauthorinvestigatesreciprocityinthecontextofasingle-trial social dilemma in which the participant andthe counterpart madetheir choicessimultaneously. The author found that playerswho aremotivated tomaximize jointoutcomesandequalityin outcomesexpect theircounterpart to reciprocate theirbehavior79.6%ofthetime.

Table4showstheproportionsofchoicesthatarepredictedbythefourmodelsmentionedabove.21Thetableshowsthat themodelwhoseactionpredictions aremostconsistentwiththedataistheLevel-1 model(rate

=

0

.

56).Totest whether thereisasignificantdifferencebetweenthelevelofaccuracyachievedbytheLevel-1 modelandthemodelwiththesecond highestlevelofaccuracy(Level-2 andInequityaversion modelsachievethesameaccuracyrate:0.49)we comparetheirhit rateingameswherethefirst- andthesecond-rankedmodelsmakedifferentpredictions.Werunapermutationtestwhere werandomly assignparticipantstogroupsandcomparetheresultsfromtruelabelswiththedistribution generatedfrom therandomlyassignedones.Resultsshowasignificantdifferencebetweenthefirst- andthesecond-rankedmodel(Level-1

–Level-2:n

=

70, Z

=

2

.

30,p

=

0

.

011,one-tailed permutationtest;Level-1 –Inequityaversion:n

=

70,Z

=

2

.

32,p

=

0

.

010,

one-tailedpermutationtest).

Similarly, lookingatthe averageprobability ofstated beliefs(Table 5), itis mostoftenpredictedthat thecounterpart wouldchooseaccordingtotheLevel-1 model(probability

=

0.54).ThedifferencebetweentheLevel-1 modelandthesecond rankedmodel(Inequityaversion model,probability

=

0.44)issigniﬁcant(n

=

70, Z

=

6

.

46,p

<

0

.

001,one-tailed permuta-tiontest).

TheseresultssuggestthatparticipantsexpecttheircounterparttochoosetheLevel-1 actionwithhighprobability. How-ever, they do not best respond to this belief,because their choices are mainly consistent with the Level-1 model. One

20 _Additional_results_using_different_combination_of_parameters_α

iandβi arereportedinthesupplementarymaterial(seeSupplementaryTable1and SupplementaryTable 2inonlineAppendix E).

21 _{Level-1 and}_Inequity_{aversion models}_make_different_prediction_only_in_six_games_(dominance_solvable_games_and_games_with_unique_Nash_equilibrium withoutdominantactions).Thisisduetotheobserveddistributionofchoicesusedtoestimatetheinequityaversion model.

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Table 5

Beliefsaboutthemodelofchoiceofthecounterpart.Theleftsideofthetableshowsthemodels’predictions(Left=L,Middle=M,Right=R,forthe rowplayersandTop=T,Middle=M,Bottom=Bforthecolumnplayers).Therightsideofthetablesshowsthestatedbeliefs(averageprobabilitieson modelpredictions)aboutthemodelofchoiceofthecounterpart.Dataarereportedfromtheperspectiveoftherowplayers.

Games Belief elicitation task (beliefs about the model of choice of the counterpart)

Models’ prediction Accuracy on model prediction

L2 L1 Inequity aversion Prosociality L2 L1 Inequity aversion Prosociality

1 M, T L, M L, M R, B 0.33 0.49 0.49 0.18 2 L, T M, M M, M R, B 0.18 0.49 0.49 0.33 3 M, B M, T R, T M, T 0.71 0.71 0.21 0.71 4 M, T L, T L, M L, T 0.20 0.62 0.62 0.62 5 L, T L, B M, B L, B 0.74 0.74 0.14 0.74 6 M, M R, M R, B R, M 0.24 0.48 0.48 0.48 7 R, M R, B R, M L, T 0.53 0.53 0.53 0.24 8 R, B L, B R, B M, M 0.24 0.35 0.24 0.40 9 R, M L, T R, T R, T 0.27 0.52 0.27 0.27 10 M, B L, T L, B L, B 0.28 0.65 0.65 0.65 11 R, M M, B M, M M, M 0.21 0.55 0.55 0.55 12 M, M R, M M, M L, B 0.26 0.49 0.26 0.25 13 R, B M, T R, T R, T 0.31 0.45 0.31 0.31 14 M, M M, B M, M L, T 0.59 0.59 0.59 0.12 15 R, B R, B R, B L, T 0.45 0.45 0.45 0.38 16 L, T L, T L, T M, M 0.50 0.50 0.50 0.38 17 R, M R, M R, M L, B 0.54 0.54 0.54 0.27 18 R, T R, T R, T M, B 0.50 0.50 0.50 0.35 Average: 0.39 0.54 0.44 0.40

possibility that can account for theseﬁndings is that the behavior ofthe decisionmakers is heterogeneous. Inthe next section, we ﬁrst provide a detailed analysis of the lookup patterns of the participants at the aggregate level. Then, we groupparticipantsaccordingtothepatternsofvisualanalysistheyusedinthechoiceandbeliefelicitationtasks,underthe assumption that thelookup patternsthey employto acquire informationaboutthepayoff structureofthe gamedisclose their decisionstrategy andtheone attributedtothe counterpart.Ifthisis thecase, wemight beable toprovidea clear descriptionofthebehaviorofourparticipants.Finally,wetestwhetherthelevelofconsistencybetweenchoicesandbeliefs dependsonthecoherence betweenthemodelofchoiceusedinthechoicetaskandtheoneattributedtothecounterpart inthebeliefelicitationtask.

6. Eye-trackingdata

6.1. Fixationanalysis

Inordertoanalyzetheeyemovementdata,we define18areasofinterest(AOIs),centeredonthepayoffs.AlltheAOIs have a circular shape witha size of 36000 pixels (Fig. 2). The AOIscover only 23% of the gamematrix area and never overlap. All the fixations that are not located inside the AOIs are discarded. Although a large partof the matrix is not includedinanyAOI,thevastmajorityoffixations(92%)fallinsidetheAOIs.

Three types ofvariableare recordedby the eye-tracker ineach roundforeach participant: 1)the numberoftimes a participant looks inside an AOI (fixationcount); 2) the time a participant spends looking within an AOI(fixation time); and3) thenumberandtypeoftransitions(defined astheeyemovementsfromoneAOIto thenext).Sincethefirsttwo variables (fixation count and fixation time) are strongly correlated, in our analysis we mostly refer to the first variable (fixationcount).However,ourresultsarethesamewhenusingfixationtime.

We usethe Wilcoxon rank-sum test to evaluate possible effects ofthe two treatments (AB,BA) on the proportion of ﬁxations onplayers’ ownpayoffs(ﬁxation withinAOIs from1 to9for arowplayer participantandfrom10to 18fora column playerparticipant, see Fig.2) during the choice task.Thelevel ofattentiongiven toown payoffsdoesnot differ betweenthetwo treatments (Wilcoxonrank-sumtest, W

=

483, p

=

0

.

129).Next,we usetheWilcoxonrank-sumtest to evaluatepossibleeffectsofthetreatmentsonthelevelofattentiongiventoownpayoffsinthebeliefelicitationtask.Once again, theproportionofown-payoffﬁxationsbetweenthetwotreatmentsdoesnotdiffersigniﬁcantly(Wilcoxonrank-sum test, W

=

565, p

=

0

.

576).

Accordingtolevel-kandcognitivehierarchymodels,whenplayersstatetheirbeliefs,theyshouldattributealowerlevel of sophisticationto their counterpartthan theirown level.At theprocess level,thisshouldbe reflected indifferencesin theallocationofvisualattentioninthetwotasks.Forexample,inthechoicetask,Level-2 playersneedtotakeintoaccount boththepayoffsoftheircounterpart(toidentifytheactionwiththehighestaveragepayoff)andtheirownpayoffs(tobest respond to thisaction). In thebelief elicitation task, it issufficient forthem to focus onthe payoffsof the counterpart. Therefore, wemightexpect that theproportionoffixationsmadeby Level-2 playersontheir counterpart’s payoffsinthe belief elicitation task would be higher than the proportion of fixations made on their own payoffs in the choice task.

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Fig. 2. Thescreenshotshowshowthematrix ispresentedtorow(upperpanel)andcolumn(lowerpanel)playersand thesummaryofthe relevant transitions(describedindetailinSection6.2);thenumbersfrom1to18initalicsrepresentthelabelsofthedifferentAreasOfInterest(AOIs,notshown totheparticipants).Examplesofthe5typesofrelevanttransitionsarerepresentedasfollows:Ownwithin transitions:solidlinearrows.Ownbetween transitions:solidlinecircles.Otherwithintransitions:dottedlinecircles.Otherbetween transitions:dottedlinearrows.Intracell transitions:dashedline arrows.

However, this prediction doesnot hold forLevel-1 players, since they should focustheir attentionon their own payoffs whenchoosingtheiractionandignoretheincentivesofbothplayerswhenstatingtheirbeliefs.22

Totest whetherparticipants changehow they allocatetheir attentioninthe two tasks,we compare theaverage pro-portion offixations onplayers’ ownpayoffs (fixationwithin AOIs from 1to 9 fora rowplayer participant andfrom 10 to 18fora columnplayer participant)when they choose their actionswith theaverage proportionoffixationson other player’spayoffs(fixationwithin AOIsfrom10to18forarowplayerparticipantandfrom1to9foracolumnplayer par-ticipant)whentheystatetheir beliefsinthetwotreatments (foreachparticipanttheaverageistakenoverallthegames). We find a significant difference (Wilcoxon matched-pairs signed-rank test, N

=

70, W

=

1220, p

<

0

.

001). Results show that participants fixatemore on the other player’s payoffswhen statingtheir beliefs than they do on their own payoffs whenchoosingtheir action.InFig.3,wereportthe proportionoffixationsonthe18AOIsforthetwotreatments during thechoice(Fig.3,upperpanel)andthebeliefelicitation(Fig.3,lowerpanel)tasks.Thefigureshowsthatplayersallocate almostthesameamountofattentiontotheirownandtheircounterpart’spayoffswhenchoosingtheiractions.Conversely, theyaremuchmorefocusedontheotherplayer’spayoffswhenstatingtheirbeliefs.23Theseresultssupportthehypothesis thatthebehaviorofsomeparticipantsmightbeexplainedbytheLevel-2 model.

22 _{Level-1 model}_assumes_that_the_counterpart_is_choosing_randomly.

23 _In_{Supplementary}_Fig.₃_(online_Appendix_D),_we_show_that_using_the_proportion_of_ﬁxation_time,_instead_of_the_proportion_of_ﬁxations,_leads_to_very similarresults.

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Fig. 3. Proportionofﬁxationswithinthe18AOIsforthechoice(upperpanel)andthebeliefelicitation(lowerpanel)tasks.Thedashedlinesindicatethe averageproportionofﬁxationsonone’sownandtheotherplayer’spayoffsforthetwotreatments.

6.2. Deﬁnitionofrelevanttransitions

To extractinformation aboutthepayoff structure ofa game,participants needto move their attentionamongthe 18 AOIs. Following Polonio et al. (2015) and Devetag et al. (2016), we deﬁne eye movementsfromone AOIto the next as transitions.ConsideringallpossiblepairsofAOIsandassumingthateachpaircanbeconnectedbytwotransitions(onefor each direction),the numberoftransitionsthat could bepotentially observed equals324,includingtransitions withinthe same AOI.However, ourmainobjective,whenanalyzingthe transitions,is toidentifythepatternsofvisual analysisused by participantstoacquire informationaboutthepayoffstructure ofthegames whenfacingthe3

×

3 matrices.Therefore, we consider only transitions usefulto capture pieces of information that are necessary to: 1) identify the participant’s actionwiththehighestaveragepayoff;2)identifythepresenceofdominantactionsfortheparticipantoridentifythebest response to the expectedactionof thecounterpart; 3) identifythe counterpart’s action withthehighestaverage payoff; 4) identifythepresenceofdominantactions forthecounterpart oridentifyacounterpart’s bestresponsetotheexpected action of theparticipant; 5) compare thepayoff of the participantwiththat of the counterpartwithin thesame matrix cell. Thisinformationcanbe capturedbyconsideringthefollowingﬁvetypesoftransitionwhereAOI-own correspondsto the AOIs of participants’ payoffs(AOIsfrom 1to 9 for a rowplayer participant andfrom10 to 18 fora column player participant),andAOI-other tothecounterpart’spayoffs(AOIsfrom10to18forarowplayerparticipantandfrom1to9for acolumnplayerparticipant):

1. Ownwithin transitions: transitions fromone AOI-own to another AOI-own within the sameaction of the participant (Player’sactions:Top,MiddleorBottomforarowplayerandLeft,MiddleorRightforacolumnplayer;e.g.,transitions from1to2orfrom1to3fortheactionTopofa rowplayerandtransitionsfrom10to13orfrom10to16forthe actionLeftofacolumnplayer). ExamplesofthesetransitionsaremarkedinFig.2withsolidlinesarrows.Transitions thatremainwithinthesameAOIareexcluded.

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2. Ownbetween transitions:transitionsfromoneAOI-own toanotherAOI-own betweendifferentactionsoftheparticipant (e.g.,transitionsfrom1to4orfrom1to7forarowplayerparticipantandtransitionsfrom10to11orfrom10to12 foracolumnplayer participant).ExamplesofthesetransitionsaremarkedinFig.2withsolidlinecircles.Transitions that remain within the same AOI are excluded aswell as transitions that are not useful to detect dominance (e.g. transitionsfrom1to5orfrom1to9forarowplayerparticipantandtransitionsfrom10to14orfrom11to18fora columnplayerparticipant).

3. Otherwithin transitions:transitionsfromoneAOI-other toanotherAOI-other,withinthesameactionofthecounterpart (Counterpart’sactions:Left,MiddleorRightforarowplayerandTop,MiddleorBottomforacolumnplayer;e.g.,from 12to15orfrom12to18fortheactionRightofthecounterpartofarowplayerparticipantandfrom1to2orfrom1 to3fortheactionTopofthecounterpartofacolumnplayerparticipant).Examplesofthesetransitionsaremarkedin Fig.2withdottedlinecircles.TransitionsthatremainwithinthesameAOIareexcluded.

4. Otherbetween transitions:transitionsfromone AOI-other toanotherAOI-other, betweendifferentactionsofthe coun-terpart(e.g.,from16to18orfrom17to18forarowplayerparticipantandfrom1to4orfrom1to7foracolumn playerparticipant).ExamplesofthesetransitionsaremarkedinFig.2withdottedlinearrows.Transitionsthatremain withinthesameAOIareexcluded,aswellastransitionsthatarenotusefultodetectdominance(e.g.from16to15or from17to12).

5. Intracell transitions:transitionsfromanAOI-own toanAOI-other orviceversa,withinthesamecell(e.g.from5to14). ExamplesofthesetransitionsaremarkedinFig.2withdashedlinearrows.

Wewillrefertotheseﬁvecategoriesasrelevanttransitions.

6.3. Clusteranalysisbasedontransitionsinthechoiceandbeliefelicitationtasks

In thissection, we usethe informationsearch patternofthe participants inthe two tasksto identify their modelof choice. To achieve this goal we grouped participants in clusters based on a set of ten variables: the proportion ofown

within, ownbetween, otherwithin, otherbetween, and intracell transitions, in the two tasks. To identify the clusters, we

use a model-based clustering method proposed by Fraley and Raftery (2002) that hasbeen extensively used in process data studies that involve games (Brocas et al., 2014; Devetag etal., 2016; Polonio etal., 2015). The advantage ofusing this clustering method is that the number of clusters and the clustering criterion are not determined a priori, but are endogenouslydeterminedbythemethoditself.Mixturemodelstreateachclusterlikeacomponentprobabilitydistribution. ABayesian statisticalapproach isused to choosebetweendifferentnumbers ofclustersand differentstatisticalmethods (Fraley and Raftery, 2002; Fraley etal., 2012). We consider a maximum of nine clustersfor up to ten differentmodels, choosingthecombinationthatmaximizestheBayesianInformationCriterion(BIC).24_The_BIC_is_a_model_selection_criterion whichaddsapenaltytothelog-likelihoodbasedonthenumberofparameters.Usingthismodel-basedclusteranalysiswe obtainsixclusters(seeSupplementaryFig.4andSupplementaryFig.5inonlineAppendixD).25

Table6showstheproportionsofdifferenttransitionsthatparticipantsbelongingtodifferentclustersemploytoanalyze the games in choice and belief elicitation tasks. The information search patterns are markedly different across clusters. Participants inclusters1and2(n

=

7 and 17,respectively)devoteconsiderableattentionto theother player’spayoffsin the choice task.In particular, the pattern ofvisual analysis of participants in cluster 1 is mainly based on otherwithin transitions (rate

=

0

.

48),whereas thelookup pattern ofparticipants incluster 2 is moreheterogeneous andincludes all typesofrelevanttransitions(seecluster2inTable6).Inthebeliefelicitationtask,participantsinclusters1and2mainly

useotherwithin transitions(rate

=

0

.

74 and0.46,respectively).

Participants inclusters3 and4(n

=

7 and17,respectively) focus their attentionentirelyontheir own payoffsinthe choicetask,usingalmostexclusivelyownwithin transitions(rate

=

0

.

71 and0.46,respectively).Inthebeliefelicitationtask, they focus their attention on theother player’s payoff, usingalmost exclusively otherwithin transitions (rate

=

0

.

72 and 0.49,respectively).Participantsinclusters5and6(n

=

14 and8,respectively)mainlyuseintracell transitionsinbothchoice (rate

=

0

.

40 and0.69, respectively)andbeliefelicitation tasks(rate

=

0

.

33 and0.65,respectively), whichmeans thattheir analysisisfocusedoncomparingtheirownpayoffswiththoseofthecounterpartwithintheninecells.Inthenextsection, wetestwhethertheadoptionofdifferentpatternsofvisualanalysisisassociatedwiththeapplicationofdifferentdecision modelsinthechoiceandbeliefelicitationtasks.26

24 _Fraley_and_{Raftery (}₂₀₀₂_{) and}_Fraley_et_{al. (}₂₀₁₂_{) use}_the_following_deﬁnition_of_BIC:_BIC

M,G=2lM,G(x|Ψ )−vlog(n).WhereΨ includestheparameters ofthemixturemodel.lM,G(x|Ψ )isthelog-likelihoodatthemaximumlikelihoodestimatorΨ forthemodelM withG components.v isthenumberof estimatedparametersandn isthesamplesize.ThemodelM andthenumberofcomponentsG whichmaximizeBICM,Gareselected.SeeScruccaetal. (2016) foranexhaustivedescriptionofthismodel-basedclusteringmethod.

25 _{Supplementary}_Fig.₄_(online_Appendix_D)_reports_a_pairs_plot_of_the₁₀_variables._{Supplementary}_Fig.₅_(online_Appendix_D)_reports_the_BIC_plot_for the10models.

26 _In_{Supplementary}_Fig.₆_(online_Appendix_D),_we_test_whether_the_six_clusters_of_participants_change_their_pattern_of_visual_analysis_when_dealing_with gameswithdifferentstrategicstructures.Wedividethe18gamesintofourcategories:(1)gamessolvableintwostepsofiteratedeliminationofdominated actions(games1,3,5and7forrowplayersandgames2,4,6,and8forcolumnplayers);(2)gamessolvableinthreeorfourstepsofiteratedeliminationof dominatedactions(games2,4,6,8,9and10forrowplayers,andgames1,3,5,7,9and10forcolumnplayers);(3)gameswithuniqueNashequilibrium withoutdominantactions(games11,12,13,14);and(4)coordination(weak-link)games(games15,16,17and18).SupplementaryFig.6showsthat,in

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Table 6

Averageproportionsofrelevanttransitionsinthetwotasks(standarddeviationinparentheses).Dataarecalculatedfromthetotalnumberofrelevant transitionsandreportedbyclusterandtypeoftransition.

Cluster Numberof participants

Own within Own between Other within Other between Intracell Own within Own between Other within Other between Intracell 1 7 0.16 0.16 0.48 0.12 0.08 0.00 0.01 0.74 0.23 0.02 (0.09) (0.04) (0.07) (0.02) (0.03) (0.00) (0.01) (0.04) (0.05) (0.01) 2 17 0.24 0.14 0.27 0.15 0.21 0.11 0.08 0.46 0.19 0.14 (0.07) (0.04) (0.06) (0.04) (0.07) (0.06) (0.04) (0.10) (0.05) (0.05) 3 7 0.71 0.15 0.02 0.05 0.07 0.00 0.00 0.72 0.25 0.03 (0.08) (0.06) (0.02) (0.05) (0.05) (0.00) (0.00) (0.05) (0.04) (0.01) 4 17 0.46 0.24 0.11 0.08 0.11 0.12 0.08 0.49 0.20 0.11 (0.11) (0.11) (0.07) (0.04) (0.05) (0.10) (0.06) (0.16) (0.08) (0.05) 5 14 0.21 0.11 0.16 0.11 0.40 0.13 0.10 0.30 0.14 0.33 (0.08) (0.03) (0.07) (0.02) (0.011) (0.07) (0.04) (0.09) (0.03) (0.08) 6 8 0.10 0.06 0.07 0.08 0.69 0.03 0.07 0.17 0.08 0.65 (0.06) (0.03) (0.04) (0.02) (0.011) (0.02) (0.01) (0.06) (0.04) (0.07) Average 0.18 0.09 0.11 0.06 0.15 0.05 0.04 0.27 0.11 0.12 Standard deviation (0.12) (0.05) (0.09) (0.03) (0.012) (0.05) (0.03) (0.13) (0.05) (0.11)

6.4. Analysisofdecisionsandinformationsearch

In thechoicetask, tobest respondto aLevel-1 player, Level-2 playersneed tocollectinformationaboutthepayoffsof thecounterpart(Costa-Gomesetal.,2001; BhattandCamerer,2005; Brocasetal.,2014; Polonioetal.,2015; Devetagetal.,

2016).Inparticular,they needtouseotherwithin transitionstolookforthecounterpart’s actionwiththehighestaverage payoff,oralternatively,otherbetween transitionstolookatwhethertheother playerhasdominantstrategies.Then, before making a decision, they should useownbetween transitions to compare their own possible outcomesand find the best response tothe expectedactionofthe counterpart.The lookuppatterns ofparticipantsinclusters1 and2inthechoice task are roughly consistent withthat expectedfor Level-2 players.To betterunderstand their patterns ofvisual analysis, we perform an analysis oftransitions over time (Fig. 4). We decided not to use a fixed time spanfor our analysis, but to calibrate it on player behavior. Foreach trial andfor each participantin clusters 1 and 2, we compute the number of transitionsneededbefore makinga decisionanddividethem inten temporallyordered intervals. Wefind thatat the beginning oftheir decisionprocess,thevisual analysisofplayersin cluster1(leftpartofFig. 4)isalmost entirelybased

on otherwithin transitions(windows1to8);onlyattheendoftheir analysis(windows9to10),dotheytake their own

payoffsintoaccount,increasingtheproportionofownwithin andownbetween transitions.Thefrequencyoftheothertypes ofrelevanttransition(otherbetween andintracell transitions)remainslowerthroughoutthewholedecisionmakingprocess. Players incluster 2start their analysislookingattheir own payoffs(rightpartof Fig.4),using ownwithin transitions (windows 1 to 3) to move from one payoff to the next. Then, they evaluate the payoffsof the counterpart using other

within transitions(windows4to9).Finally,beforemakingtheirdecision,theylookagainattheirownpayoffsusingmainly

ownwithin transitions.27 _The _temporal _pattern_of_visual _analysis _of_players_in _cluster ₂_is _not _as_well _deﬁned_as _that _of

participantsincluster1,butisverysimilartothatreportedintheliteratureforLevel-2 players(Polonioetal.,2015). Ingeneral,thetwotemporalinformationsearchpatternsare bothcompatiblewithwhatwouldbeexpectedforLevel-2 players. Acriticaltestwouldbetodemonstratethatthevisualpatternofplayersinclusters1and2isbasedonaspeciﬁc subset ofownbetween transitions. This subset should include transitions that are required to identify thebest response to the expectedactionofa Level-1 counterpart. Totest thisprediction,we ﬁrstdetermine, separately foreach game,the expected actionof a Level-1 counterpart, andthen test whetherthevisual analysis ofparticipants inclusters1 and2 is focused onthesubsetofownbetween transitionsrequiredtobestrespondtothisaction.Wecomparethenumberofown

between transitions incorrespondencewithexpectedandnon-expectedactionsofaLevel-1 counterpart(Fig.5)andﬁnda

signiﬁcantdifference,intheexpecteddirection,forbothclusters(cluster1:Wilcoxonmatched-pairssigned-ranktest,N

=

7, W

=

27,p

=

0

.

031;cluster 2:Wilcoxonmatched-pairssigned-ranktest,N

=

17,W

=

142,p

=

0

.

002).

Inthe beliefelicitation task, theanalysisofparticipantsinclusters1and2ismainlybasedonotherwithin transitions (see Table 6), which indicates that they believe that their counterpart is choosing the Level-1 action. Overall, our data supportthehypothesisthatparticipantsincluster1and2areLevel-2 players.

eachclusterandinbothtasks,thedistributionoftheﬁvetypesoftransitionsdonotchangeingameswithdifferentstrategicstructures.Theseresultsare inagreementwiththehypothesisthatparticipantshaveapredeterminedandstablewayofacquiringinformation,whichdoesnotdependonthetypeof game.

(14)

Fig. 4. Analysisoftransitionsthroughtimeofplayersinclusters1(ontheleft)and2(ontheright).Theproportionsofrelevanttransitionsarenormalized sothateachtemporalwindow(w1,w2,etc. . . )includes1/10 ofthetransitionsmadeineachtrial.

Fig. 5. Averagenumberofownbetween transitions(inparentheses)madebyplayersinclusters1(ﬁgureontheleft)and2(ﬁgureontheright)inthe choicetask,forexpectedandnon-expectedchoicesofaLevel-1 counterpart.Ownbetween transitionsarerepresentedbyverticallines.Thesizeofthelines isproportionaltotheaveragenumberofoccurrencesamongthecorrespondingpayoffs.Dataarepresentedfromtheperspectiveofarowplayer.

Tables7a and7b show theaverageproportionofchoices(leftsideofthetables)andtheaverageprobability ofstated beliefs(rightsideofthetables)inaccordancewiththepredictionsofthe4modelsdescribedabove,forplayersincluster1 and2.Weﬁndthat,onaverage,thechoicesofbothclustersaremainlyconsistentwiththepredictionoftheLevel-2 model (rate

=

0

.

75 and 0.58 for participants in cluster 1 and 2, respectively). To test whether there is a signiﬁcant difference between the level of accuracy achieved by the Level-2 model and the model with the second highest accuracy (Level-1 model) we use a one-tailed permutation test. Results show a signiﬁcant difference for both clusters (cluster 1: n

=

7, Z

=

2

.

31, p

=

0

.

010, one-tailed permutation test; cluster 2: n

=

17, Z

=

2

.

47, p

=

0

.

007, one-tailed permutation test). The averageprobability withwhich participantsin cluster1 and2estimate the counterpart tochoose theLevel-1 action (probability

=

0.62and0.59,respectively) lendsfurthersupportto thehypothesisthattheseparticipantsbestrespondto the belief that the counterpart is a Level-1 player. Evenin the beliefelicitation task, the difference between the Level-1 modeland thesecond ranked model(Inequityaversion model) issigniﬁcant forboth clusters(cluster 1:n

=

7, Z

=

2

.

55, p

=

0

.

005,one-tailedpermutationtest;cluster2:n

=

17, Z

=

3

.

50,p

<

0

.

001,one-tailedpermutationtest).

Tochoose accordingto the Level-1 model, itis suﬃcient fora player to onlytake her own payoffsintoaccount. The pattern of visual analysis used by participants in clusters 3 and 4 in the choice task is almost entirely based on own

within transitions(rate

=

0

.

71 and0.46,respectively)andisthereforecompatiblewiththepredictionoftheLevel-1 model.

However, inthebeliefelicitation task,the visualanalysisoftheseparticipantsismainly basedonotherwithin transitions (rate

=

0

.

72 and0.49,respectively).ThispatternisconsistentwiththatexpectedforLevel-2 playerswhoholdthebeliefthat thecounterpart isaLevel-1 player. Coherentlywiththeeye-tracking data,resultsreportedinTables7cand7d showthat participantsinclusters3and4followthepredictionsoftheLevel-1 modelinthechoicetaskandbelievethatthe counter-partischoosingtheLevel-1 actioninthebeliefelicitationtask(rate

=

0

.

86 and0.70 inthechoicetaskandprobability

=

(15)

Table 7

Consistencybetweenmodels’predictionsandobservedbehaviorinthetwotasksbycluster.Theleftsideofeachtableshowstheaverageproportionof choicesthatareconsistentwiththepredictionsoftheﬁvemodels.Therightsideofthetablesshowsthestatedbeliefs(averageprobabilitiesonmodel predictions)aboutthemodelofchoiceofthecounterpart.Dataarepresentedfromtheperspectiveoftherowplayers.

(a)

Games Cluster 1 (7 participants)

Eq. Pareto L2 L1 Inequity aversion Pros. Eq. Pareto L2 L1 Inequity aversion Pros.

1 0.86 0.86 0.14 0.14 0.00 0.64 0.33 0.64 0.64 0.04 2 0.00 1,00 0.00 0.00 0.00 0.05 0.05 0.59 0.59 0.36 3 0.71 0.71 0.29 0.29 0.29 0.24 0.66 0.66 0.24 0.66 4 0.14 0.86 0.86 0.14 0.86 0.04 0.04 0.65 0.65 0.65 5 0.86 0.86 0.14 0.14 0.14 0.01 0.96 0.96 0.01 0.96 6 0.00 0.86 0.86 0.00 0.86 0.09 0.09 0.50 0.50 0.50 7 0.86 0.86 0.14 0.86 0.00 0.65 0.65 0.65 0.65 0.03 8 0.71 0.71 0.71 0.71 0.00 0.12 0.12 0.40 0.12 0.48 9 0.57 0.43 0.57 0.57 0.57 0.07 0.07 0.79 0.07 0.07 10 0.43 0.43 0.43 0.43 0.43 0.63 0.34 0.63 0.63 0.63 11 0.43 0.43 0.43 0.43 0.43 0.62 0.04 0.62 0.62 0.62 12 0.14 0.86 0.86 0.86 0.14 0.28 0.16 0.56 0.16 0.28 13 0.29 0.57 0.29 0.29 0.29 0.19 0.19 0.46 0.19 0.19 14 0.00 0.71 0.29 0.71 0.00 0.03 0.61 0.61 0.61 0.03 15 0.14 0.86 0.86 0.86 0.14 0.21 0.67 0.67 0.67 0.21 16 0.14 0.86 0.86 0.86 0.14 0.20 0.60 0.60 0.60 0.20 17 0.14 0.86 0.86 0.86 0.14 0.23 0.54 0.54 0.54 0.23 18 0.00 0.86 0.86 0.86 0.00 0.28 0.64 0.64 0.64 0.28 Average 0.43 0.75 0.52 0.50 0.25 0.26 0.38 0.62 0.45 0.36 (b)

Games Cluster 2 (17 participants)

Eq. Pareto L2 L1 Inequity aversion Pros. Eq. Pareto L2 L1 Inequity aversion Pros.

1 0.35 0.35 0.24 0.24 0.41 0.50 0.34 0.50 0.50 0.16 2 0,41 0.41 0.41 0.41 0.18 0.09 0.09 0.62 0.62 0.29 3 0.47 0.47 0.41 0.41 0.41 0.27 0.67 0.67 0.27 0.67 4 0.12 0.88 0.88 0.12 0.88 0.18 0.18 0.68 0.68 0.68 5 0.76 0.76 0.24 0.24 0.24 0.10 0.79 0.79 0.10 0.79 6 0.06 0.82 0.82 0.06 0.82 0.28 0.28 0.48 0.48 0.48 7 0.59 0.59 0.06 0.59 0.35 0.66 0.66 0.66 0.66 0.09 8 0.29 0.29 0.29 0.29 0.53 0.19 0.19 0.45 0.19 0.35 9 0.59 0.41 0.59 0.59 0.59 0.20 0.20 0.61 0.20 0.20 10 0.53 0.53 0.18 0.53 0.53 0.65 0.25 0.65 0.65 0.65 11 0.59 0.59 0.35 0.59 0.59 0.61 0.18 0.61 0.61 0.61 12 0.06 0.88 0.88 0.88 0.06 0.23 0.31 0.46 0.31 0.23 13 0.35 0.41 0.35 0.35 0.35 0.30 0.30 0.49 0.30 0.30 14 0.18 0.59 0.24 0.59 0.18 0.12 0.64 0.64 0.64 0.12 15 0.24 0.71 0.71 0.71 0.24 0.39 0.42 0.42 0.42 0.39 16 0.35 0.59 0.59 0.59 0.35 0.31 0.59 0.59 0.59 0.31 17 0.35 0.65 0.65 0.65 0.35 0.22 0.63 0.63 0.63 0.22 18 0.35 0.59 0.59 0.59 0.35 0.30 0.63 0.63 0.63 0.30 Average 0.38 0.58 0.47 0.47 0.41 0.31 0.41 0.59 0.47 0.38

0.64and0.54inthebeliefelicitation taskforparticipantsincluster 3and4,respectively).Aone-tailed permutationtest showsthat the differencebetweentheLevel-1 model andthesecond ranked modelissigniﬁcant forbothclustersin the choicetask(clusters3:n

=

7, Z

=

2

.

50,p

=

0

.

006,one-tailedpermutationtest;cluster4:n

=

17, Z

=

3

.

36,p

<

0

.

001, one-tailedpermutationtest)andalsointhebeliefelicitationtask(clusters3:n

=

7,Z

=

2

.

37,p

=

0

.

009,one-tailedpermutation test;clusters4:n

=

17,Z

=

3

.

90, p

<

0

.

001,one-tailedpermutationtest).

The modelof Inequityaversion assignsa negative weight to inequality in payoffs. The modelof Prosociality assigns a negative weighttoinequalityinpayoffsandapositive weighttothetotalpayoffofthetwoplayers.Thus,inordertouse one ofthesetwo decisionmodelsitisnecessaryforaparticipanttocompareher ownpayoffwiththatofthecounterpart foreachpossibleoutcomeofthegame.Participantsinclusters5and6useavisualpatternofinformationacquisitionthat is mainlybasedonintracell transitions (rate

=

0

.

40 and0.69, respectively)andthereforecompatiblewiththe adoptionof one ofthese two models. As shownin Table 7e, the choicesof participants incluster 5 are mainly consistent with the predictionsofthemodelofProsociality (rate

=

0

.

65).TheEquilibrium modelachievesthesecondhighesthitrate.Similarly,