Focal points and payoff information in tacit bargaining

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Contents lists available atScienceDirect

Games

and

Economic

Behavior

www.elsevier.com/locate/geb

Focal

points

and

payoff

information

in

tacit

bargaining

Andrea Isoni

a

,

b

,

∗

,

Anders Poulsen

c

,

Robert Sugden

c

,

Kei Tsutsui

d a_Behavioural_Science_Group,_Warwick_Business_School,_Coventry,_UK

b_University_of_Cagliari,_Italy

c_School_of_Economics_and_Centre_for_Behavioural_and_Experimental_Social_Science,_University_of_East_Anglia,_Norwich,_UK d_Department_of_Economics,_University_of_Bath,_Bath,_UK

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received1September2017 Availableonline31January2019

JELclassiﬁcation: C72 C78 C91 Keywords: Focalpoints Tacitbargaining Coordination Conﬂictofinterest Payoffinformation Payoff-irrelevantcue

Schellingproposedthat payoff-irrelevantcuescanaffecttheoutcome oftacitbargaining gamesbycreatingfocalpoints.Testsofthishypothesishavefoundthatconﬂictsofinterest betweenplayersinhibitfocal-pointreasoning.Weinvestigateexperimentallywhetherthis effectisreducedifplayershaveimperfectinformation abouteachother’spayoffs.When playersknowonlytheirownpayoffs,theyfailtoignorethisinformationeven thoughit cannot assistcoordination; the effects of payoff-irrelevant cueson coordination success aresmall.Whennoexactinformationaboutpayoffsisprovided,payoff-irrelevantcuesare morehelpful,butnotasmuchaswhenconﬂictisabsent.

In his famous book TheStrategyofConﬂict, Schelling (1960) develops a theory of focal-pointreasoning ingames with multipleNashequilibria.Thisformofreasoningisfundamentallydifferentfromthebest-responsereasoningthatisstandard ingame-theoreticanalysis.Theessentialideaisthatplayersconcerttheirmutualexpectationsononeequilibrium–thefocal point –byappealingtotheirsharedknowledgeaboutsalientpropertiesofthegame.Insomecases,thefocalpointissalient becauseitpayoff-dominatesotherequilibria,butsalienceoftenderivesfrompayoff-irrelevantfeaturesofthewaystrategies arelabelled.

Itisnowempiricallywell establishedthatrealplayerssuccessfullycoordinateonpayoff-dominantequilibriain coordi-nation gamesinwhichthere arenoconflictsofinterest(e.g.Bacharach,2006; Bardsleyetal.,2010)andthatlabel-salient focalpointsareveryeffectiveinPureCoordinationgames,inwhichallequilibriahavethesamepayoffsforallplayers(e.g. Schelling,1960; Mehtaetal., 1994a, 1994b;Bacharach andBernasconi, 1997; Crawfordetal.,2008; Bardsleyetal., 2010; ParravanoandPoulsen, 2015). Significantly, however,these casesofsuccessfulcoordination occur in gamesin whichthe playershaveasharedrankingoftheequilibria.Schellingarguesthatrationalplayerswouldusethesamekindofreasoning inwhathecallssituationsoftacitbargaining,that is,gamesinwhichtwoplayershaveacommoninterestincoordinating their strategies,buttheir interestsconflictoverhowcoordination shouldbeachieved, andcommunicationisnotpossible.

*

Correspondingauthorat:BehaviouralScienceGroup,WarwickBusinessSchool,Coventry,UK.

E-mailaddresses:a.isoni@warwick.ac.uk(A. Isoni),

a.poulsen@uea.ac.uk

(A. Poulsen),

r.sugden@uea.ac.uk

(R. Sugden),

k.tsutsui@bath.ac.uk

(K. Tsutsui). https://doi.org/10.1016/j.geb.2019.01.008

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He uses the term‘tacit bargaining’ to signify that games ofthis kind can be useful reduced-form models of real-world bargainingsituations,includingsituationsinwhichcommunicationis possible.1

The simplestexamples of tacit bargaininggames havethe 2

×

2 Battleofthe Sexes structure.Like Pure Coordination games,thesegamesaresymmetricalwithrespecttoplayersandstrategies,andsoclassicalgametheorycannotdistinguish between the pure-strategy equilibria (e.g. Harsanyi and Selten, 1988); the difference is that the players have conﬂicting rankings of those equilibria.2 Just as in Pure Coordination games, the players may have common knowledge of salient labelling propertiesthat wouldallowthemtoconcerttheirexpectationsonafocalpoint.Schellingconjecturesthat,when individual interests are secondary tothe primary need forcoordination – that is,when the rewardfromcoordinating is suﬃciently largerelativeto thelossinpayofffromnotcoordinating onone’spreferredequilibrium–rationalplayerscan ‘discipline’themselvestoacceptalessershareifausefulcuepointsthatway(p.286).

It hasbeenfoundthat,consistentlywithSchelling’sconjecture,certainkindsoflabellingasymmetriesareusedascues

forachievingcoordinationintacitbargaininggames.Suchasymmetriesincludethegenderoftheplayers(Holm,2000) and whichplayerisdescribedastheﬁrstmoverinaBattleoftheSexesgamethatisstrategicallyequivalenttoasimultaneous movegame(Cooperetal.,1993; Güthetal.,1998).However,whencoordinationgamesareframedasmatching games–that is,framed sothatcoordinationrequiresbothplayerstochoosethesame strategy–thereissubstantiallylesscoordination onsalientequilibriaifthereisconﬂictofinterestthanifthereisnot(Crawfordetal.,2008; Failloetal.,2017).Inthemost extremecases,suchasthe2

×

2 games inCrawfordetal.’sexperiment,coordinationpayoff differencesofaslittleastwo percentinduceratesofcoordinationlowerthanwouldhaveresultedfromrandombehaviour.Ininterpretingtheseﬁndings, Crawfordetal.propose a versionoflevel-k theory (StahlandWilson,1994;Nagel,1995) inwhichpayoff-irrelevantcues serve onlyastie-breakers,withtheimplicationthatsuchcuesareusedonlyintheveryspecialcaseofPureCoordination games.3

Thisimplicationisclearlytooextreme,evenformatchinggames.In3

×

3 matchinggameswithconﬂictsofinterest,Faillo etal.ﬁndevidenceindicativeoflevel-k reasoningbutincompatiblewithfocal-point reasoning,andevidenceindicativeof theopposite.Therelevantgamescanbedescribedbythethreepairsofpayoffsonthemaindiagonalofthepayoffmatrix, all other payoffs beingzero andno labelbeing salient. For example, in the ‘G6’ game

{(

10,9),

(9,

10),

(9,

8)

}

the third strategy isuniqueby virtueofitspayoffs, butonly5.6percentchosethat strategy,consistentlywithlevel-k reasoningbut contrarytofocal-pointreasoning.Inthe‘G7’game

{(

10,9),

(9,

10),

(9,

9)

}

,69percentchosethethirdstrategy,consistently withfocal-pointreasoning,whenthelevel-k modelimpliesthatnoplayer(exceptat‘level0’)shouldmakethatchoice.

Isoni et al. (2013) investigate coordination games that are framed to incorporate aspects of real-world bargaining – particularly, thatplayersmakeclaims onspecificvaluable objects(withtheimplicationthat coordinationrequiresplayers to choose different objects), that salient labelling cues take the form of relations betweenplayers and objects,and that some oftheavailable surpluscanbe left unclaimed.Inthesegames,thereis significantlymorecoordination ifsuchcues arepresentthaniftheyarenot.Nevertheless,forgivencues,thepresenceofconflictofinterestreducescoordination.This evidence suggeststhat,contrarytoSchelling’shypothesis, givenplayersarelesslikelytousefocal-pointreasoning,oruse it lesseffectively, when they haveconflicting rankings ofcoordination equilibria. Since such conflict islikely to occur in most real-worldsituations of coordinationand bargaining, itis importantto establishunder what conditionsfocal-point reasoningislikelytobeadopted.

The research reportedinthepresentpaperwasprompted by thethoughtthat mostexperiments failtocapturea key aspectofreal-worldsituationsofcoordinationandbargaining.Contrarytowhathappensinthelab,real-worldplayersmay not knowexactlyhowmuchitisworthtothemandtheirco-playerstocoordinateonacertainlabelortoreachacertain agreement. They mayhave better knowledge of their own utilities than their co-players’, orsometimes even be unsure abouttheirown utilities.Bymakingpayoff differencesunambiguousandbyestablishingcommonknowledgeaboutthem, experimental games may overemphasise a kindof information that maybe impossiblefor real-world agents to acquire with a comparable degree of precision. If the presence of conﬂict inhibits focal-point reasoning, the emphasis that lab experimentsputonitmayleadtowrongconclusionsaboutitsrelevanceforreal-worldsettings.Inordertoconductamore realistic testofhowconﬂictsofinterestaffectfocal-pointreasoning,itisimportanttoconsidersituationsinwhichpayoffs arelessthanperfectlyknown.Thisisthemainobjectiveofourpaper.

To addressourresearch question, we needtobe ableto manipulatepayoff informationina setupin whichthereare focalpointsthatreliablyaffectbehaviour,andinwhichtheinhibitingeffectofconﬂictsofinterestsonfocal-pointreasoning hasbeenobservedandcanbeeasilyreplicated.Tothisend,weadaptthebargainingtable designusedbyIsonietal. (2013). Ouradaptationallowsustorepresentthreesimpletypesof2

×

2 gamewithandwithoutpayoff-irrelevantlabellingcues,

1 _{Schelling (}₁₉₆₀_:_{267–272) argues}_that_if_{‘explicit’}_bargaining_takes_place_over_a_ﬁnite_period_of_time_and_if_the_procedure_by_which_players_communicate is‘perfectlymove-symmetrical’,theremustbeaﬁnalperiodinwhichatacitbargainingsubgameisplayed.Ifthesolutiontothissubgameiscommon knowledge,noplayerwillacceptalowerpayoffinearlierroundsofthegame.

2 _The_distinction_between_Pure_Coordination_and_Battle_of_the_Sexes_games_is_often_described_as_that_between_games_with_{symmetric and}_asymmetric payoffs (e.g.Crawfordetal.,

2008

).Becausebothtypesofgameareinvariantwithrespecttorenamingplayersandstrategies,weprefertodistinguish betweentheabsenceandpresenceofconﬂictofinterest.

3 _Although_Crawford_et_al.’s_{level-k model}_can_account_both_for_coordination_success_in_Pure_Coordination_games_and_coordination_failure_in_Battle_of_the Sexes,directinvestigationofreasoninginthosegamesbyvanEltenandPenczynski (2018) revealsmarkeddifferencesbetweenthetwocases,withlevel-k reasoningmorecommoninBattleoftheSexesgamesandSchelling-style‘teamreasoning’inPureCoordinationgames.

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Fig. 1. ACloseness gameviewedbytwomatchedplayers.(Forinterpretationofthecoloursintheﬁgure(s),thereaderisreferredtothewebversionofthis article.)

all withtwo pure-strategyNash equilibria:Pure Coordinationgames, Hi-Logames (i.e.,coordination gamesin whichone pure-strategyequilibriumstrictlyParetodominatestheother),andBattleoftheSexesgames.Conflictofinterestispresent inthe lattergame,butnotin theformertwo. Crucially,inthemodified setup itispossible tovarythe informationthat playershaveaboutthepayoffsofthegame,andhencewhetherconflictsofinterestarecommonknowledge.

Westudygamesthat belongtooneofthreeinformationconditions:aFullInformation condition,inwhichbothplayers knowallpayoffs;anOwnInformation condition,inwhicheachplayerexactlyknowsherpayoffs,butnottheotherplayer’s; andaNoInformation condition,inwhichneitherplayerknowstheexactvaluesofthepayoffs.Inallcases,theinformation condition iscommonknowledge. Inthe FullInformationcondition, it iscommonknowledge whichofthe threetypesof gameisbeingplayed.Intheothertwoconditions,neitherplayerknowswhethertheyareplayingaBattleoftheSexesora Hi-Logame,hencewhetherthereisconﬂictornot.4

Ourinitialconjecturewasthatfocal-pointreasoningwouldbemorelikelytobeused,andhencelabellingcueswouldbe moreeffective,whenconflictsofinterestwerelessobvious.Toformalisethisconjecture,wedevelopamodelinwhicheach playeriscapableofusingboth focal-pointand level-k reasoning(butnotbothatthesametime);whichofthetwomodes ismore likelyto be usedinanygiven gameisinfluenced by thepresence orabsenceof conflictsofinterest. The model organisestheevidence,existingpriortoourexperimentandonwhichourconjecturewasbased,abouttheeffectoflabelling cuesintheFullInformationversionsofPureCoordination,Hi-LoandBattleoftheSexes games.Itmakesnewpredictions fortheOwnandNo Informationgames.IntheOwnInformationcondition,itpredicts bettercoordinationsuccessthanin BattleoftheSexesbutworsethaninPureCoordinationgames.IntheNoInformationcondition,itpredictsthesamelevel ofcoordinationsuccessasinPureCoordinationgames.

Surprisingly,our data provideno supportfor thehypothesis that de-emphasising conflictsof interest facilitates focal-point reasoning in one-shot tacit bargaining games. In the Own Information condition, players seem to find it hard to disregardunhelpfulprivateinformation,andachievelevelsofcoordinationthatare,ifanything,lower thaninBattleofthe Sexesgames.EvenintheNo Informationcondition, coordinationsuccessfallsshortofthelevelsachievedinPure Coordi-nationgamesplayedunderFullInformation.Ourresultsprovidefurtherevidencethatthetensionbetweenfocal-pointand individualbest-responsereasoningsuchaslevel-k thinkingisnoteasilyresolvedinthepresenceofconflictsofinterest.

Theremainder ofthispaperisorganisedasfollows.Section 1describesouradaptationofthebargainingtabledesign. OurmodelisdevelopedinSection 2.InSection 3,thatmodelisused toderivehypothesesforourexperiment.Section4

givesdetailsofhow theexperimentwasimplemented.WepresentourresultsinSections5,6and7.Section8discusses theimplicationsoftheseresultsandconcludes.

1. Experimentaldesign

The left-handside ofFig.1showsone ofthe‘scenarios’ facedby participantsin ourexperiment.Thescenario onthe right-handsideisthesamegameasviewedbythematchedplayer.

Taken together, the two scenarios constitute a tacit bargaining game represented by a 9

×

9 grid. The two squares representthetwoplayers’respectivebases.Eachplayerknowswhichbaseistheirsbecauseitisshowninredatthebottom

4 _Our_focus_on_one-shot_{tacit coordination}_problems_sets_our_contribution_apart_from_the_early_work_by_Roth_and_{Malouf (}₁₉₇₉_),_Roth_et_{al. (}₁₉₈₁_),_and RothandMurnighan (1982),whoexploredtheeffectofvaryingplayers’degreeofpayoffinformationintwo-playerexplicit bargaininggameswithoffers andcounterofferswhichresultedinlotteriesforthetwoplayers,andfoundabroadtendencyforagreementstobebiasedtowardsoutcomesthatequated thetwoplayers’expectedexperimentalearnings.OurgamesalsodiffersigniﬁcantlyfromthematchinggamesstudiedbyAgranovandSchotter (2012),in whichplayerslearntocoordinateonthelabel-salientequilibriumingameswith‘coarse’payoffinformationwhentheseareplayedrepeatedlywithstranger matchingandround-by-roundfeedbackontheactualpayoffconﬁgurationandrealisedoutcomes.InlinewithSchelling’shypothesis,westudysystematic coordinationthatoccurspriortoanylearning.

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ofthetablethey seeandislabelled‘You’.Foreachplayer,itisstraightforwardtoworkouthowthegamelooksfromthe ‘Other’player’sperspective.

There aretwodiscs on thetable,each splitintotwohalves.Thereisa valueineachhalf.The valueonthehalffacing eachbaserepresentsthevalueofthedisc(inUKpounds)totheplayerassignedtothatbase.Forthegameintheexample, eachofthetwodiscsisclosertooneofthetwobases.Foreachplayer,theclose discisworth£10tothemand£11tothe otherplayer,whilethefar discisworth£11tothemand£10totheotherplayer.

The gamewas described to participantsasan ‘opportunityto agreeona division ofthediscs’.Each playerseparately recorded whichdisc(s) she‘proposed totake’ or‘claimed’. Ifnodisc was claimedbyboth players, theplayerswere said to have‘agreed’;each playerearned thetotalvalue toher ofthediscs shehadclaimed.Ifany discwas claimedby both players,therewas‘noagreement’,andneitherplayerearnedanythingfromthegame.5

ForthegameinFig.1,eachplayer’sstrategiescan bedescribedaseither:claimnone ofthediscs,claimtheclose disc

only,claimthefar disconly,orclaimboth discs.Claimingnoneofthediscsisaweaklydominatedstrategy.Ifthisstrategy is eliminated,claiming bothdiscs isweaklydominated.Afteriteratedeliminationofweakly dominatedstrategies,weare leftwitha2

×

2 gamewiththepayoffsshownbelow6:

Other

Close Far

You Close 10, 10 0, 0

Far 0, 0 11, 11

Thedistinctionbetween‘Close’and‘Far’representsadistinctionbetweenthetwoequilibriathatiscommonknowledge be-tweentheplayersandthathasthepotentialtobeusedasameansofcoordination.ThematrixdescribesaHi-Logamewith two Pareto-ranked Nash equilibriainpure strategies([Close,Close] and[Far, Far])anda mixed-strategyNash equilibrium (eachplayerplaysClose withprobability11/21).Giventhatplayersaredescribed as‘You’and‘Other’,theonlyinformation thatwouldallowaplayertodistinguishbetweenthetwopurestrategiesisthedifferencebetweengetting10or11andthe relativesalienceofClose andFar.

Ifourdesignistoidentifythepositiveandnegativeeffectsoflabellingcuesoncoordination,weneedacontrolcondition inwhichsuchcuesareeitherabsentorineffective. Itmightseemthattheobviouswaytoimplementcontrolwouldbeto remove the two player’s bases andshow exactly the samedisplay to both players. While thismay work ina game like theoneintheexample,itwouldbeunsatisfactoryforBattleoftheSexesandPureCoordinationgames,which(apartfrom labelling) areperfectlysymmetricalwithrespect toplayersandstrategies.Insuch an alternativecontrol version ofthese games, the only strategy descriptions available to the playerswould share theproperty that coordination would require them tochoose different strategies.Butwithoutcommunication, playerswho identifythemselvesonlyas‘me’ and‘other’ would have no means ofachieving such coordination. Thus, this control would fundamentally change the nature of the game.7

For ourpurposes, themost usefulcontrol condition isone in whichlabelling cues remain,butare minimally salient. Followingtheprecedentofpreviousbargainingtableexperiments(Isonietal.,2013,2014),wematchgameswithcloseness cueswith‘spatiallyneutral’games.AnexampleofhowsuchagamewouldlookfortwomatchedplayersisshowninFig.2. Thisisessentiallythesameasthepreviousgame,exceptforthepositionsofthetwodiscs,whicharenowequidistant fromthetwobases.Fromthepointofviewofeachplayer,thestrategiescanbedescribedaseither:claimnone ofthediscs, claim thediscmoretotheleft asseenfromyourbase, claimthediscmoretotheright asseenfromyourbase, orclaim bothdiscs.Afteriteratedeliminationofweaklydominatedstrategies,thisgamecanbepresentedinnormalformas:

Other

Left Right

You Left 10, 10 0, 0

Right 0, 0 11, 11

ThispayoffmatrixisidenticaltothatforthegameinFig.1,exceptforthelabels.Likethatgame,thishastwopure-strategy equilibria ([Left, Left] and [Right, Right])and a mixed-strategy Nash equilibrium (each player plays Left withprobability 11/21).

5 _Although_all_our_games_have_just_two_discs,_we_allowed_players_to_also_claim_none_or_both_discs._Isoni_et_{al. (}₂₀₁₃_{) looked}_at_the_effect_of_forcing playerstoclaimexactlyonediscandfoundthat,otherthingsbeingequal,thatsomewhatreducedthesalienceofclosenesscues.Becauseweneedsalient cuestoaddressourresearchquestion,weusedtheversionofthegamesinwhichnoneorbothdiscscouldbeclaimed.Thishasalsotheadvantageof preservingthebargainingfeelofthegame.

6 _As_a_necessary_step_for_experimental_{implementation,}_we_use_the_term_‘payoff’_to_refer_to_{material payoffs}_(expressed_in_money_units),_but_treat_these asif,for eachplayerconsideredseparately,theywerecloseproxiesforutilitiesinthe senseofclassicalgametheory.Asinthattheory,wemakeno assumptionsabouttheinterpersonalcomparabilityofutility.

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Fig. 2. A Spatially Neutral game viewed by two matched players.

Inbothgamesconsideredsofar,therearelabellingcuesthatinprinciplecouldbeusedasameansofcoordinationbut, intuitively,itseemsclearthattheruleofchoosingthediscthatismoretotheleft(ortotheright)islikelytobemuchless conducivetocoordinationthantheruleofchoosingthediscclosertoone’sbase.Andinfact,thereisstrongevidencethat thisisthecase(e.g.,Mehtaetal.,1994b;Isoni,etal.,2013,2014).Ourexperimentaldesignispremisedontheassumption that,whenbothdiscsareequidistantfromthetwobases,playerswhotrytousespatiallocation asacoordination device willdistributetheirchoicesbetweenthediscswithapproximatelyequalprobability.Giventhisassumption,suchgamescan beusedascontrols.Ourdesignallowsustocheckthevalidityofthisassumption.

Inordertoassesstheabilityofpayoff-irrelevantcuestoinﬂuencecoordinationsuccess,wewillcontrastCloseness games

inwhich(asinFig.1)there isonediscclosertoeachbase, andSpatiallyNeutral gamesinwhich (asinFig.2) bothdiscs are equidistant fromthe two bases. Our experimentwas designed to studythe effects ofcloseness cues ina variety of situations,includingdifferentdegreesofconﬂictbetweencoordinationequilibriaandinformationaboutthevalue ofeach disctoeachplayer.

Intheremainderofthispaper,wewillusethewordscenario toindicateagameasseenbyanindividualplayer,andthe wordgame toindicatetheresultoftwomatchingscenarioslikethoseinFigs.1and2.Ineachscenario,thepositionsofthe twodiscswerealwayscommonknowledge,andtherewasadiscvaluepair

{

X

,

Y

}

with0

<

X

≤

Y ,whichwasalsocommon knowledge.Foreachplayer,therewasalwaysonediscworth X andonediscworthY .Theparticularconﬁgurationofwhich discwasworth X andwhichwasworth Y foreachplayerwillbecalledtheassignment ofdiscvalues.When X

=

Y ,thereis justonepossibleassignmentofvaluestodiscs,whichisnecessarilycommonknowledge.ThisproducesaPureCoordination game.When X

<

Y ,thereare fourpossibleassignments ofdiscvalues.Twooftheseproduce Hi-Logames, inwhichthe players’interests arefullyaligned;theothertwo produceBattleoftheSexes games,inwhichtheplayershaveconﬂicting rankings ofthetwoequilibria. Indifferentinformationconditions,playershaddifferentdegreesofinformationaboutthis assignment.Whenplayerswerenotfullyinformedabouttheactualassignment,eachofthefourpossibleassignmentshad thesamepriorprobability.

Thescenariosusedintheexperimentcanbegroupedintofourclasses,deﬁnedintermsofdiscvaluesandinformation conditions.

InPureCoordination (PC)scenarios,it iscommonknowledgethat X

=

Y ,andhencethat theplayersarefacing aPure Coordinationgame.Thesescenariosallow usto verifythat, intheabsenceofconﬂictofinterest,closenesscuesaremore salientthanothercues.AnexampleofaPureCoordinationscenarioisshowninFig.3a.8

Intheotherthreeclassesofscenarios,itiscommonknowledgethat X

<

Y .InFullInformation (FI)scenarios,the assign-mentof discvaluesiscommonknowledge.Thus, itis commonknowledgethat theplayersare facinga Hi-Logame ora BattleoftheSexesgame.IfthegameisHi-Lo,itiscommonknowledgeifclosenessiscongruent orincongruent withpayoffs –i.e.,whetherornotthepayoffdominantequilibriumhasasalientlabel.IfthegameisBattleoftheSexesandifthereis onedisconeachplayer’sside ofthetable,the‘You’playercaneitherbefavoured bycloseness(scenarioC5)orunfavoured

(scenarioC4).AnexampleofanFIscenarioisshowninFig.3b;thisispartofaHi-Logame.

In OwnInformation (OI)scenarios, each player knows the value of each discto her but not their values to the other player; thisiscommonknowledge. Inourdisplays, the ‘Other’player’s discvaluesare replaced byquestion marks. From thesedisplaysandfromtheinformationatherdisposal,eachplayercanﬁgureoutthatherscenariomaybepartofeither aHi-LogameoraBattleoftheSexesgame.InscenarioC6,closenessisbad forthe‘You’player(theclosediscisworth X ),

inscenarioC7itisgood (theclosediscisworthY ).AnexampleofanOIscenarioisshowninFig.3c.

8 _The_numbers_that_appear_next_to_the_left_and_bottom_edge_of_the_bargaining_tables_in_Fig.₃_are_coordinates_that_uniquely_identify_the_positions_of_the twodiscs.Thesewerenotshowntoparticipants,butwillbeusedinSection4todescribetheexact‘layouts’ofdiscs.Intheexperiment,thediscvalues wereprecededbythe‘£’symbol.

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Fig. 3. Examples of experimental scenarios.

From theviewpointofclassicalgametheory,thesescenarios representgamesofincompleteinformation.Afteriterated eliminationofweakly dominatedstrategies,eachofthesegameshasthree pure-strategyBayesian Nashequilibria:one in whichbothplayersalwayschoosetheclose(respectively,left)disc,oneinwhichbothplayersalwayschoosethefar(right) disc,andoneinwhicheachplayeralways choosesthediscthatisworth Y toher.Theﬁrsttwooftheseequilibriaresult inperfectcoordination, whilethelastresultsinsuccessfulcoordinationonly ﬁftypercentofthetime. Thereare alsotwo mixed-strategyequilibria,inoneofwhichtheclose(respectively,left)disc,andintheotherthefar(respectively,right)disc isclaimedwithprobability1whenitisworthY andwithprobability

(

Y

−

X

)/(

X

+

Y

)

whenitisworth X .9

InNoInformation (NI)scenarios,theassignmentiscompletelyunknowntobothplayers;thisiscommonknowledge.In ourdisplays,alldiscvaluesarereplacedbyquestionmarks.Fromthesedisplaysandfromtheinformationattheirdisposal, each playercanﬁgureoutthatherscenariomaybe partofeitheroneoftwoHi-Logamesorbe oneofthetwoscenarios makingupaBattleoftheSexesgame.AnexampleofanNIscenarioisshowninFig.3d.

The scenariosused intheexperimentcanalsobeclassiﬁedaccordingtothespatial locationsofthediscs.InCloseness

scenarios,suchasthoseshowninFigs.3aand3d,onediscisclosertothe‘You’baseandtheotherisclosertothe‘Other’ base.InSpatiallyNeutral scenarios,suchasthoseshowninFigs.3band3c,bothdiscsareequidistantfromthetwobases; necessarily,onediscismoretotheleft,asviewedfromthe‘You’base.

Tomaintain symmetryinourdesign,everyClosenessscenariohasacorresponding SpatiallyNeutralscenario,andvice versa. Ineachpairofcorresponding scenarios,thetwo scenariosbelong tothesameclass(PC, FI,OIorNI)andhavethe same discvalue pair.Recallthatina ClosenessscenarioofclassFIorOI,the ‘You’playerknowswhetherthediscthatis closerto herbaseis moreorlessvaluabletoher thanthe discthatis furtheraway.Inacorresponding Spatially Neutral scenario, the ‘You’ playerknows whetherthe disc that ismore to her left is moreor lessvaluable toher thanthe disc that ismoretoherright. Weadopttheconventionthat aClosenessscenarioinwhichthemorevaluabledisciscloserto

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Table 1

Typesofscenariousedintheexperiment. Class

(game)

Spatially neutral scenarios Closeness scenarios

Conﬁguration Match Conﬁguration Match Closeness cue

PC N1=|(X, X) (X, X)| N1 C1=(X, X)| |(X, X) C1 Sole cue

NI N8=|(?, ?) (?, ?)| N8 C8=(?, ?)| |(?, ?) C8 Sole cue

Notes: PC=PureCoordination;FI=FullInformation;OI=OwnInformation;NI=NoInformation;HL–I=Hi-LowithIncongruentcues;HL–C=Hi-Lo withCongruent cues;BS=BattleoftheSexes.

the‘You’basecorrespondswithaSpatiallyNeutralscenarioinwhichthemorevaluablediscismoretotheleft.Givenour premisethatneitherleftnorrightissalient,thisconventionisinconsequentialtoouranalysis.10

Table1presentsanoverviewofthetypesofscenariosthatparticipantsfacedinourexperiment.Indescribingscenarios, weusethefollowingnotation.Discsareshownbytwo entriesinparentheses, inwhichtheﬁrstentryisthevalue ofthe disctotheparticipant(‘You’),andthesecondisthevaluetotheplayertheyarefacing(‘Other’).Forexample,

(

X

,

Y

)

denotes adiscworth X to‘You’andY to‘Other’;

(

Y

,

?)isadiscworth Y to‘You’andeither X orY to‘Other’;(?,?)isadiscworth either X or Y bothto‘You’ and‘Other’.Thetwo verticalbars| |are usedtoidentifythemiddlerowofthetable asseen by theparticipant. InClosenessgames, theclosediscisshowntothe leftof| | andthe fardisctothe right.In Spatially Neutralgames, bothdiscsare betweenthetwo bars,theﬁrst discindicating theleftmostdiscasseenby theparticipant, thesecondindicatingtherightmost disc.Forexample,thescenarioontheleft-handsideofFig.1isC2

= (

X

,

Y

)|

|(Y

,

X

),

withX

=

10 andY

=

11.Thescenarioontheright-handsideofFig.2isN2

= |(

X

,

Y

)(

Y

,

X

)

|

,alsowithX

=

10 andY

=

11. Table1presentstheeightSpatiallyNeutralscenariosusedintheexperiment(N1toN8)andthecorrespondingCloseness scenarios(C1toC8).Eachrowdescribesapairofcorrespondingscenarios.Theﬁrstcolumnshowswhetherthesescenarios are ofclassPC,FI,OIorNI(andthe correspondinggame fortheFIscenarios).The‘Match’ columnsidentifythescenario faced by ‘Other’ when the relevant scenario is faced by ‘You’. Thismatching is a matter of logical necessity,because in anygivenscenarioaplayercanworkoutwhatscenario(or,inthecaseoftheOwnInformationcondition,whatscenarios) ‘Other’is(mightbe)facing.Inmostcases,thematchedplayerfacesthesamescenario,buttherearesomeexceptions.For example,scenarioC5ispartofaBattleoftheSexesgameinwhichthebestequilibriumfor‘You’istheoneinwhicheach playerclaims thediscclosertoher base.The matchedplayernecessarilyfacesscenarioC4,inwhichthebestequilibrium for‘You’istheoneinwhicheachplayerclaimsthediscfurtherfromherbase.Thelastcolumndescribeshowthecloseness cuerelatestopayoffs.

2. Asimplemodelofmultiplemodesofreasoning

Inordertoderivehypothesesforthegamesinourexperiment,we developamodelwhichorganizesthemainﬁndings ofpreviousexperimentsonPureCoordination,BattleoftheSexesandHi-Logames.Webeginbymodellingplayers’choices ofpurestrategiesin(fullinformation)2

×

2 diagonalcoordinationgames inwhichplayerlabelsaresymmetric(e.g.‘You’and ‘Other’,asinFig.1andFig.2).Ina2

×

2 diagonalcoordinationgame,eachplayeri

=

1,2 hasstrategies j

=

1,2,whereeach strategy j has adistinct payoff-irrelevantlabellj knownto bothplayers. Ifboth playerschoosethe samestrategy j,the

payoffstoplayers1and2are

π

1 jand

π

2 j,with

π

1 j,

π

2 j

>

0;otherwise(i.e.,offthemaindiagonalofthepayoffmatrix), both players’ payoffsare zero.Payoffs are commonknowledge. Anysuch game hastwo pure-strategy Nash equilibria, in eachofwhichtheplayers’strategieshavethesamelabel,andamixed-strategyequilibriuminwhichbothplayers’expected payoffsarelowerthanineitherpure-strategyequilibrium.

Wemodeleachplayerasbeingcapableofusingtwoalternativemodesofreasoning:focal-point reasoning,astheorisedby Schelling (1960) anddevelopedby Bacharach (2006), andlevel-kreasoning,asmodelledbyCrawfordetal. (2008).Ineach game,themodesofreasoningusedbytheplayersaredeterminedbytwoindependentrealisations(oneforeachplayer)of arandommechanism, usingprobabilitiesthatarethesameforbothplayersbutmightvaryaccordingtothespeciﬁcation ofthegame.Thismodellingstrategyisjustiﬁedbyourobjectivetomatchtheevidence,summarisedearlier,thateveninthe sameexperiment behaviourinsome gamesismoreinlinewithone formorreasoning,whilebehaviour inothergamesis morecompatiblewiththeother(e.g.Failloetal., 2017).Amodelthattreatedeachmodeofreasoningasanunconditional propertyofadistinctplayer‘type’wouldnotbeabletoexplainthisevidence.

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Weassumethateachmodeofreasoningisself-contained –thatis,aplayerwhousesitactsasifbelievingthattheother player uses ittoo. Thisassumptionappears the mostpsychologicallyplausible, giventhe boundedrationality ofordinary humanplayers:reasoningaboutthegameinmultiplewaysismuchmorecognitivelydemandingthanusingasinglemode ofreasoning.11Inaddition,theevidencethatsalientlabellingcuescanbeineffectiveinBattleoftheSexesgameswouldbe diﬃculttoexplainifplayerswhousedonemodeofreasoningwereallowedtobelievethattheirco-playersmightbeusing the other.The diﬃcultyhereisthat,underreasonable assumptions,playerswhomixmodesofreasoninginthiswaycan bepredictedtochoosethesaliently-labelledstrategyinaBattleoftheSexesgame.

The mostnatural wayto representtheidea that level-k reasoners believe that their co-players mightbe using focal-pointreasoningistoassumethatasigniﬁcantproportionoflevel-0players(i.e.,thosewhoareusingfocal-pointreasoning) choosethestrategywiththemoresalientlabel.ButinBattleoftheSexesgameswithsmallpayoffdifferences, all higher-level playerswouldthenchoosethesalientlylabelledstrategyandsocoordinatesuccessfully.Themirror-imageidea,that focal-point reasonersbelievethattheir co-playersmightbeusinglevel-k reasoning,canbemodelledusingtheconceptof

circumspectteamreasoning (Bacharach,1999).Roughly, theidea is thatsophisticated playerslookfor astrategy combina-tion thatisoptimalfortheplayerstakentogether,treatingthe behaviourofnaïveplayers(in thiscase, level-k reasoners) as a constraint. But (aswe will show) theoverall effect oflevel-k reasoning in Battle ofthe Sexes games is to produce

discoordination.Takingthisbehaviourasgiven,circumspectteamreasonerswouldchoosethesalientlylabelledstrategyso astocoordinatewithoneanother.

Playerswhousefocal-pointreasoningtrytodiscriminatebetweenpure-strategyNashequilibriabyusingonly informa-tion that iscommonknowledge.When describingfocal-point reasoning,Schelling (1960,pp. 83,96,106,163,298) often uses the metaphor ofa ‘meeting of minds’between theplayers. The suggestion is that, inchoosing their strategies, the playersimaginethemselvesreasoningtogether abouthowtocoordinatetheirbehaviour.12_A_natural_implication_of_this_idea

isthat,intheirreasoning,playersuseonlyinformationthatiscommonknowledgebetweenthem.Itemsofsuchknowledge that can be used in thiswaywill be called cues. Playerswill concentrate on cues that are salient – i.e., that can easily be recognisedby both players– anddiscriminating – i.e., that can identify one ofthe Nash equilibriaas thesolution of the game.Given that player labelsare symmetric,there are onlytwo plausible typesofcue that can pick out a speciﬁc pure-strategy equilibrium. There isa cue oflabelsalience ifone equilibriumstands out relativeto the other by having a moresalientlabelattachedtothecorrespondingstrategy.Thereisacueofjoint-payoffsalience ifoneequilibriumstandsout by havingapairofpayoffsthatisbetterfortheplayerscollectively,accordingtosomesalientcriterionof‘betterness’that treatstheplayerssymmetrically.Forourpurposes,wecanrestrictthecriteriontopayoffdominance.

Our model of level-k reasoning is taken from Crawford et al. (2008). Each player reasons at one the levels 0, 1, 2,

. . .

.Eachplayer’sreasoninglevelisanindependentdrawfromanexogenouslygivendistribution,inwhichlevel0haszero probability.AplayeratanylevelL

≥

1 believesthatherco-playerisatlevelL

−

1,andusesiteratedbest-responsereasoning toformabeliefaboutwhatthatco-playerwillchoose.Thisreasoningisanchoredonbeliefsaboutthebehaviouroflevel-0 players.13 _A_level-0 _player_is _believed_to _have_a _probability _distribution _over _strategies_which _shows_a _payoff_{bias (i.e.,}_it

assigns higherprobabilitytostrategieswhoseequilibriahavehigherownpayoff). Payoffbiasisindependentofthesizeof the payoff difference.Labels are irrelevantforlevel-0 behaviourexcept whenall equilibriahavethesame ownpayoff, in whichcasestrategieswithmoresalientlabelsarechosenwithhigherprobability.

Following Bacharach’s (2006) theoryof teamreasoning, we assume that the probability that a player uses focal-point reasoningdependsonthelikelihoodthattheyidentify withthegroup madeofthemselvesandtheirco-player.InBacharach’s theory, groupidentification isfacilitatedby a varietyof factors,includingbeingmembers ofthesame pre-existing social group orad-hoccategory (Tajfel, 1970), beingexposed tothepronouns ‘we’,‘our’ orsimilar (Perdueetal., 1990), having common interest or common fate (Rabbie and Horwitz, 1969), sharing experiences (Prentice and Miller, 1992), making face-to-facecontact(Dawesetal.,1988),orbeinginterdependent(Sherifetal.,1961).Forourpurposes,weassumethatthe probability thatplayersusefocal-pointreasoningdependsonlyontheplayers’commonknowledgeaboutthepresenceor absence(andifpresent,thedegree)ofconflictofinterest.Thereisconflictofinterestiftheplayershaveopposingpreferences betweenthetwopure-strategyNashequilibria.

We nowapply thismodeltothegames inourexperiment, afteriteratedeliminationofdominatedstrategies.Initially, we consider Pure Coordination games and Full Information games (i.e., Hi-Lo andBattle of the Sexes). These are 2

×

2 diagonalcoordinationgameswithlj

=

Close,Far intheClosenessversionorlj

=

Left,Right intheSpatiallyNeutralversion,

and

π

i j

∈ {

X

,

Y

}

(with 0

<

X

≤

Y ). We assume that the [Close, Close] equilibriumis label-salient inall Closeness games,

thatneitherequilibriumislabel-salientinSpatiallyNeutralgames,andthatthe

[

Y

,

Y

]

equilibriumhasjoint-payoffsalience

11 _Costa-Gomes_and_{Weizsäcker (}₂₀₀₈_{) found}_that_subjects_play_games_as_if_attributing_less_rationality_to_their_opponents_than_to_themselves,_but_when statingtheirbeliefsabouttheiropponents’strategychoices,‘theyputthemselvesintheshoesoftheiropponent’(p.757),andreasonabouttheiropponents’ decisionsasiftheyweretheirs.

12 _This_aspect_of_{Schelling’s}_theory_of_focal_points_is_examined_by_Sugden_and_{Zamarrón (}₂₀₀₆_)._The_concept_of_players_reasoning_together_has_been developedtheoreticallyinthetheoriesofteamreasoning (e.g.Sugden,

1993

; Bacharach,

2006

)andvirtualbargaining (MisyakandChater,

2014

).

13 _An_alternative_{best-response}_model_based_on_limited_levels_of_reasoning_is_Camerer_et_al.’s₍₂₀₀₄_{) cognitive}_hierarchy_model,_in_which_players_at_higher levelsbestrespondtosomedistributionoflowerlevels.Keepingtheassumptionsaboutthelevel-0playerunchanged,usingacognitivehierarchy speciﬁ-cationwouldnotalterthequalitativepredictionssummarisedinourhypotheses:seefootnote

17

.

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in Hi-Lo games.Notice that,by virtueof the symmetry propertiesof PureCoordination andBattle of theSexes, neither equilibriuminthosegamescanhavejoint-payoffsalience.

ThereisconﬂictofinterestinBattleoftheSexesbutnotinPureCoordinationorHi-Lo.InBattleoftheSexes,thedegree ofconﬂictofinterestdependsonthevaluesof X andY .Theprobabilitythataplayerusesfocal-pointreasoningisgivenby afunction

ϕ

(

X

,

Y

),

with0

<

ϕ

(

X

,

Y

)

<

1.14Weassumethatthereissomeprobability

ϕ

0 suchthat

ϕ

(

X

,

Y

)

=

ϕ

0ifthereis noconﬂictofinterestand

ϕ

(

X

,

Y

)

<

ϕ

0otherwise.Wealsoassumethat

ϕ

isweaklyincreasinginX andweaklydecreasing inY (i.e.,focal-pointreasoningisweaklylesslikely,thegreaterthedegreeofconﬂictofinterest).Bynotassuming

ϕ

tobe continuous,we allowthepossibilitythatevensmallconﬂictsofinterestmaysigniﬁcantlyinhibitfocal-pointreasoning,as existingevidencesuggests(Crawfordetal.,2008; Isonietal.,2013; Failloetal.,2017).

Inthe Closenessversions ofPureCoordinationandBattle oftheSexes, a playerwho usesfocal-point reasoningis as-sumedto chooseClose with probability1.In theCloseness versionsofHi-Logames, thereare cuesofbothlabelsalience andjoint-payoffsalience.Weassumethat,conditionalonhavingadoptedfocal-pointreasoninginsuchaHi-Logame,each playerhasanindependentprobability

σ

(

X

,

Y

)

ofbeingguidedbyjoint-payoffsalience(andthereforechoosingthestrategy leading tothe payoff-dominantequilibrium); otherwise,sheis guidedby labelsalience (andthereforechooses the label-salientstrategy). We assume that 0

<

σ

(

X

,

Y

)

<

1 for all X , Y ,andthat

σ

(

X

,

Y

)

is weakly increasing in Y and weakly decreasingin X .(Intuitively,thegreaterthevalueofY relativeto X ,themoresalientisthecuethatpoints tothe payoff-dominantequilibrium.)IntheSpatiallyNeutralversionsofHi-Logames,aplayer whousesfocal-pointreasoninghasonly thejoint-payoffsaliencecueavailable,andthereforechooses thestrategyleading tothepayoff-dominantequilibriumwith probability 1.In the remaining Spatially Neutral games, which do not havejoint-payoff salient cues, a player who uses focal-pointreasoningchooseseachstrategywithprobability0.5.

Nowconsiderthebehaviourofaplayerwhouseslevel-k reasoning.Becauselevel-0playerstendtouselabelsalienceas atie-breaker,andbecausethisleveloccurswithzeroprobability,level-k reasoningimpliesthatClose ischosenwith prob-ability1intheClosenessversion ofPureCoordination,irrespective ofthedistributionofreasoninglevels.IntheSpatially Neutral version ofPureCoordination, level-k reasoning doesnot discriminatebetweenthe twostrategies, andsoeach is chosenwithprobability0.5.Becauselabelsalienceisusedonly asatie-breaker,thedistinctionbetweentheClosenessand SpatiallyNeutralversionsofHi-LoandBattleoftheSexesisirrelevantforlevel-k reasoning.InHi-Lo,level-k reasoning im-pliesthattheY strategy(i.e.,thestrategyleadingtotheNashequilibriuminwhichtheplayer’spayoffisY )ischosenwith probability1.InBattleoftheSexes,theimplicationsoflevel-k reasoningdependonthedistributionofreasoninglevels.For ourpurposes,however,itissuﬃcienttotreatthisdistributionasexogenous,andtodeﬁne

ρ

(

X

,

Y

)

astheprobabilitywith which,giventhisdistribution,arandomly-selectedlevel-k reasonerchoosesherY strategy.15

WenowextendourmodeltoOwnInformationandNoInformationgames.First,weconsiderOwnInformationgames. Recallthat,after iteratedeliminationofweaklydominantstrategies, thesegames havethree pure-strategyBayesian Nash equilibria–twoequilibriainwhicheach player’sexpectedpayoff is

(

X

+

Y

)/2 and

oneinwhichitis Y

/2.

Thus,no equi-libriumstandsout ashavinguniquelybestjointpayoffs.Wethereforetreatthesegamesashavingnocuesofjoint-payoff salience.TheClosenessversionofthegamehasacueoflabelsalience;theSpatiallyNeutralversiondoesnot.Wetherefore assume that,forfocal-pointreasoners,Close is chosenwithprobability1 intheClosenessversion ofthegameandLeft is

chosenwithprobability0.5intheSpatiallyNeutralversion.16Level-k reasoninghasverydifferentimplications.Foralevel-0 player,payoffbiaswillfavourthechoiceofthediscthatgivesherY .But,sincealevel-1playerdoesnotknowwhichdisc thisis,hisbestresponseistochoosethediscthat giveshim Y ;andsimilarlyforhigherlevelsofreasoning.Thus,inboth versionsofthegame,level-k reasonerschoosetheY strategywithprobability1.

Finally,weconsidertheNoInformationgames.Afteriteratedeliminationofdominatedstrategies,thesegamesare equiv-alent to PureCoordination games in which theexpected payoff fromcoordination is

(

X

+

Y

)/2 to

both playersin both pure-strategy equilibria.Thus, asinPureCoordinationgames, allplayerschooseClose with probability1 intheCloseness versionofthegameandLeft withprobability0.5intheSpatiallyNeutralversion.

Table2summarisesthekeypropertiesofourmodel,andits predictionsforthesixtypesofgames usedinour experi-ment.Eachhorizontalpanel refersto theClosenessandSpatiallyNeutralversionsofagivengametype.Foreachversion, weﬁrstreporttheprobabilitydistributionoverpossibletypesofplayersresultingfromtherandomdrawwhichdetermines their modeofreasoning(recallthat,exante,eachplayerhasapositive probabilityofadoptingeithermodeofreasoning): focal-point reasoners who use the cue of label salience (type F), focal-point reasoners who use the cue of joint-payoff salience(typeFP,onlyforHi-Logames),andlevel-k reasoners(typeK).Foreachtype,weshowtheprobabilitythataplayer claims the closedisc (in Closeness games)orthe left disc (in Spatially Neutralgames); whereapplicable, we report the

14 _Given_our_objective_to_produce_a_model_that_matches_the_evidence_available_before_our_experiment,_and_given_that,_as_noted_earlier,_neither_focal-point reasoningnorlevel-k reasoningcaninisolationexplainalltheﬁndings,weneedthatbothtypesofreasoningoccurwithpositiveprobability.

15 _It_is_possible_for_ρ₍_X_,_Y₎_to_be_less_than_0.5._If_the_degree_of_conﬂict_of_interest_is_suﬃciently_small,_level-1_players_respond_to_the_payoff_bias_of level-0playersbychoosingtheX strategy;level-2playersrespondbychoosingtheY strategy,andsoon.Thus,whetherρ(X,Y)isgreaterorlessthan0.5 dependsonthedistributionofreasoninglevels.

16 _In_the_Own_Information_games,_the_three_{pure-strategy}_equilibria_give_each_player_a_payoff_of

(X+Y)/2,(X+Y)/2 andY/2 respectively.Sophisticated playersmightreasonthatchoosingonthebasisofequilibriumpayoffsalonewouldmaketheexpectedpayofffromchoosingatrandombetweenthetwo identicalequilibria(X+Y)/4,andthereforetheequilibriumwithapayoffofY/2 wouldbesuperior.Ourmodelassumesamorenaïveapproachtounique joint-payoffsalience,inlinewithempiricalevidence(e.g.Failloetal.,

2017

).

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Table 2

ModelPredictions.

Game Closeness game Spatially neutral game

Pr(Case) Pr(Close) Pr(Y) Pr(Coord) Pr(Case) Pr(Left) Pr(Y) Pr(Coord)

Pure Coordination F ϕ0 1 n/a – ϕ0 0.5 n/a –

K (1 –ϕ0) 1 n/a – (1−ϕ0) 0.5 n/a –

Label vs. Label 1 – – 1 1 – – 0.5

Hi-Lo Incongruent F α=ϕ0[1−σ(X,Y)] 1 0 – 0 n/a n/a –

FP β=ϕ0[σ(X,Y)] 0 1 – ϕ0 0 1 – K γ= (1−ϕ0) 0 1 – (1−ϕ0) 0 1 – Label vs. Label α2 _– _– ₁ _ϕ2 0 – – 1 Payoff vs. Payoff (β+γ)2 _– _– ₁ _{(1 –}_ϕ 0)2 – – 1 Label vs. Payoff 2α(β+γ) – – 0 2ϕ0(1−ϕ0) – – 1

Hi-Lo Congruent F α=ϕ0[1− −σ(X,Y)] 1 1 – 0 n/a n/a –

FP β=ϕ0[σ(X,Y)] 1 1 – ϕ0 1 1 – K γ= (1−ϕ0) 1 1 – (1 –ϕ0) 1 1 – Label vs. Label α2 _– _– ₁ _ϕ2 0 – – 1 Payoff vs. Payoff (β+γ)2 _– _– ₁ _{(1 –}_ϕ 0)2 – – 1 Label vs. Payoff 2α(β+γ) – – 1 2ϕ0(1 –ϕ0) – – 1

Battle of the Sexes F ϕ(X,Y) 1 0.5 – ϕ(X,Y) 0.5 0.5 –

K [1−ϕ(X,Y)] 0.5 ρ(X,Y) – [1−ϕ(X,Y)] 0.5 ρ(X,Y) – Label vs. Label [ϕ(.)]2 – – 1 [ϕ(.)]2 – – 0.5 Level-k vs. Level-k [1−ϕ(.)]2 _– _– _ρ_{∗ (}_X_,_Y₎ _[₁₋_ϕ_(.)_]2 _– _– _ρ_{∗ (}_X_,_Y₎ Label vs. Level-k 2[ϕ(.)][1−ϕ(.)] – – 0.5 2[ϕ(.)][1−ϕ(.)] – – 0.5 Own Information F ϕ0 1 0.5 – ϕ0 0.5 0.5 – K (1−ϕ0) 0.5 1 – (1−ϕ0) 0.5 1 – Label vs. Label ϕ2 0 – – 1 ϕ 2 0 – – 0.5 Payoff vs. Payoff (1 -ϕ0)2 – – 0.5 (1−ϕ0)2 – – 0.5 Label vs. Payoff 2ϕ0(1−ϕ0) – – 0.5 2ϕ0(1−ϕ0) – – 0.5

No Information F ϕ0 1 n/a – ϕ0 0.5 n/a –

K (1−ϕ0) 1 n/a – (1−ϕ0) 0.5 n/a –

Label vs. Label 1 – – 1 1 – – 0.5

Notes: F =focal-pointreasonerusinglabelsalience;FP =focal-pointreasonerusingjoint-payoffsalience;K=level-k reasoner.ϕ0 =probabilityof focal-pointreasoningabsentconﬂictofinterest;ϕ(X,Y)=probabilityoffocal-pointreasoningwhenthereisconﬂictofinterestandpayoffpairis{X,Y};

σ(X,Y)=probabilitythatafocal-pointreasonerusesjoint-payoffsalienceinHi-Logames;ρ(X,Y)=probabilitythatalevel-k reasonerclaimsthedisc worthY inBattleoftheSexes;ρ∗(X,Y)=2ρ(X,Y)[1−ρ(X,Y)];n/a=notapplicable.

probability thata playerclaimsthediscworth Y toher.So,forexample,inPureCoordinationandNo Informationgames theprobabilities thataplayerusesfocal-point reasoningorlevel-k reasoning are

ϕ

0 and

(1

−

ϕ

0

)

respectively. Bothtypes chosethelabel-salientstrategywithprobability1,soPr(Close)equals1forbothtypes.InHi-Logames(witheither congru-entorincongruentcues),theprobabilitythatafocal-pointreasonerchoosesonthebasisoflabelsalienceis

ϕ

0[1

−

σ

(

X

,

Y

)

]

, the probabilitythat afocal-point reasonerusesjoint-payoffsalienceis

ϕ

0[

σ

(

X

,

Y

)

]

,andtheprobability oflevel-k reason-ing is

(1

−

ϕ

0

).

In thesegames,all playerschoose the Y strategywithprobability 1,except fortheF types,who choose the X strategy withprobability 1 inthe Closenessversion of Hi-Lowithincongruent cues. InBattle ofthe Sexes games, the distributionoftypesdependsontheconﬂict ofinterestinduced bythevaluesof X andY ,accordingtothe function

ϕ

(

X

,

Y

).

We canusethesechoice probabilitiestolookatcoordination success–i.e.,theprobabilitythat twoplayers,chosen at random,coordinatewitheachother.Wedothisby consideringthespeciﬁcbehaviour ofeachtype ineachgame.InPure CoordinationandNoInformationgames,allplayerschooseaccordingtolabelsalience,sotheprobabilityofcoordinationis necessarily1.Inallother games,irrespectiveofthenumberofdifferentplayertypes,thereare justtwodifferenttypesof behaviours. Inall cases,exceptBattleofthe Sexes,playerscaneitherchoose thelabel-salientstrategy (‘Label’inTable 2) orthestrategy witha betterownpayoff (‘Payoff’).(Noticethat,althoughthereare threetypesofplayersinHi-Lo games, FP playersbehave like level-k playersand choosethe Y strategy.)In Battleof the Sexes, a player caneither choose the label-salientstrategyorplayasalevel-k reasoner(inwhichcasetheiractualchoicedependsontheirlevel).Two randomly-chosen playerscaneitherusethesamebehaviouralruleordifferentones.Twoindependentdrawsoftwoplayerswhouse one of two rules(i.e., LabelandPayoff) mayresultin one ofthreepossible combinations(i.e., Labelvs. Label, Payoffvs. Payoff, Labelvs. Payoff). The probability ofeach combination foreach gameis shownin each panel ofTable 2,together with thecorresponding probability of coordination Pr(Coord).Note that ifone player chooses atrandom, theprobability of successfulcoordinationis necessarily0.5, irrespective ofthe behaviourof herco-player. Ina Battleofthe Sexes game betweentworandomly-selectedlevel-k reasoners,theprobabilitythattheycoordinate(i.e.,thatonegetsapayoffofY and

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theothergets X )isgivenby

ρ

∗

(

X

,

Y

)

=

2

ρ

(

X

,

Y

)

[

1

−

ρ

(

X

,

Y

)

]

.Noticethat

ρ

∗

(

X

,

Y

)

≤

0.5:unlessthe X and Y strategies

arechosenwithexactlyequalprobability,level-k reasoninginducesdiscoordination.17

3. Hypotheses

We nowuse theinformation inTable 2to deriveour formal hypotheses aboutcoordinationsuccess inour games.In interpreting our hypotheses, one should bear in mind that, in the interests of simplicity,our model assumesaway the mistakesplayerscould makewhenimplementingtheir strategies(or,equivalently,theexistenceofplayerswhochooseat random).Thus,thespeciﬁcvaluesofcoordinationsuccessgeneratedbythemodelshouldnotbetreatedasﬁrmpredictions aboutthebehaviourofrealplayers.However, thequalitativenatureofourpredictionsshouldnotbeaffectedbymoderate degreesofnoiseinbehaviour, whichareinevitablypresentinexperimentaldata.Forthisreason,ourhypotheseswillfocus onthedirectionofthedifferencebetweencoordinationsuccessindifferenttypesofgame.

Webeginwithcomparisons withinthe panelsofTable2,andlookatthe effectsofsalientlabelskeepingthe typeof gameconstant.ThePureCoordinationgames(facedinscenariosC1andN1)canbeusedtotesttheunderlyingassumption of ourdesign that closeness isa much stronger cue than anycue contained inSpatially Neutral games. In the Spatially Neutralversionweexpect coordinationsuccesstoapproximatethatimpliedby randombehaviour,andtobemuchhigher intheClosenessversion.Sincethisdirectionaleffectisessentiallyaprerequisitefortherestofouranalysis, wewillcallit Hypothesis0:

Hypothesis0(PCgames).WithfullinformationandX

=

Y ,coordinationsuccessishigherinClosenessgamesthaninSpatiallyNeutral games.

OurpredictionsconcerningtheeffectofclosenessinFullInformationgames aresummarisedinHypothesis 1.InHi-Lo games with incongruent cues, our model makes the yet unexplored prediction that labelling cues reduce coordination success.InHi-Logameswithcongruentcues,all ourtypeschoosethestrategyleadingtothepayoff-dominantequilibrium irrespective ofthepresenceoflabelling cues.InBattle oftheSexesgames,the mixofmodesofreasoningissuﬃcientto makecoordinationmorelikelyintheClosenessversionthanintheSpatiallyNeutralversion.ThisreplicationofIsonietal. (2013) iscentraltoourinvestigation,asitmotivatesourinterestintheOwnandNoInformationgames.

Hypothesis1(FIgames).WithfullinformationandX

<

Y :

a) (Hi-Logameswithincongruentcues): whenplayershavethesamepreferencesbetweenequilibriaandclosenessisincongruent withpayoffs,coordinationsuccessislowerinClosenessgamesthaninSpatiallyNeutralgames.

b) (Hi-Logameswithcongruentcues): whenplayershavethesamepreferencesbetweenequilibriaandclosenessiscongruent withpayoffs,coordinationsuccessisequalinClosenessgamesandSpatiallyNeutralgames.

c) (BattleoftheSexesgames): whenplayershaveconﬂictingpreferencesoverequilibria,coordinationsuccessishigherinCloseness gamesthaninSpatiallyNeutralgames.

InOwnInformationgames,accordingtoourmodel,thepresenceoffocal-pointreasonersissuﬃcienttoimprove coor-dinationsuccessintheClosenessversionrelativetotheSpatiallyNeutralbenchmark.Hence:

Hypothesis2(OIgames).WithOwnInformationand X

<

Y ,coordinationsuccessishigherinClosenessgamesthaninSpatially Neutralgames.

Ourmodel treats No Information games exactlyas ifthey were Pure Coordinationgames. The fact that each player’s payoff fromcoordination in the No Information case is a binary lottery giving X and Y with equal probability has no inﬂuenceonouranalysis.Thus:

Hypothesis3(NIgames).WithNoInformationandX

<

Y ,coordinationsuccessishigherinClosenessgamesthaninSpatiallyNeutral games.

We nowturnto comparisonsbetweenCloseness games. Inthesecomparisons,we focuson howcoordination success variesbetweentypesofgameswithlabelsaliencebutnotjoint-payoffsalience.Theﬁrstimportantcomparisonisbetween Pure CoordinationandBattle ofthe Sexes.Because ofthe wayinwhich level-k reasoningdeals with conﬂictofinterest,

17 _For_all_games_except_Battle_of_the_Sexes,_all_the_predicted_{probabilities}_shown_in_Table₂_would_be_unchanged_if_(as_suggested_in_footnote

₁₃

₎_we_used_a cognitivehierarchymodelinplaceofalevel-k one,providedthatlevel0isassumedtohavezeroprobability.Inallthesegames,thepredictedbehaviourof level-k reasonersisthesameateachlevel1,2,. . .,andsothebestresponsetoanyoneoftheselevelsisthesameasthebestresponsetoanyprobability mixofthem.Toarriveatcognitive-hierarchypredictionsforBattleoftheSexes,allweneedtodoistore-interpretρ(X,Y)astheprobabilitythata randomly-selectedcognitive-hierarchyreasonerchoosesherY strategy.Theconclusionthatρ∗(X,Y)≤0.5 isunaffected.

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our modelpredicts thatcoordination successwill belower inBattle oftheSexes thanin PureCoordination, inlinewith previous ﬁndings (e.g.,Crawfordetal.,2008; Isonietal., 2013). Thiseffecthastwo maincauses:level-k reasonershavea tendencytodiscoordinatewitheachother(recall,

ρ

∗

(

X

,

Y

)

≤

0.5),andthelikelihoodthatplayersusefocal-pointreasoning

ϕ

(

X

,

Y

)

isweaklydecreasingintheextentoftheconﬂictofinterestinducedbythevaluesofX andY .

Hypothesis4(Conﬂictofinterest).WithClosenesscues,coordinationsuccessislowerinBattleoftheSexesgamesthaninPure Coor-dinationgames.

Thenextinterestingcomparisonsarebetweeninformationconditions.OurmodelmakespredictionsabouthowOwnand No InformationgamescomparewithPureCoordinationandBattleoftheSexes: intermsofcoordinationsuccess,theOwn Informationgame shouldbeintermediate betweenthetwo,whiletheNo Informationgameshouldbe equivalenttoPure Coordination.Theseareourﬁnaltwohypotheses.

Hypothesis5(BattleoftheSexesvs.OIgamesvs.PCgames).InOwnInformationgameswithclosenesscues:

a) coordinationsuccessishigherthaninBattleoftheSexesgames;

b) coordinationsuccessislowerthaninPureCoordinationgames.

Hypothesis6(NIgamesvs.PCgames).InNoInformationgameswithclosenesscues,coordinationsuccessisthesameasinPure Coordinationgames.

Aﬁnalaspectofourmodelthatisrelevanttoourdesignistheeffectofchangingthepayoffpair

{

X

,

Y

}

.Thismattersin Hi-Logamesviathefunction

σ

(

X

,

Y

),

andinBattleoftheSexesgamesvia

ϕ

(

X

,

Y

)

and

ρ

(

X

,

Y

).

Theseeffectsarebestseen inrelationtoparticipants’tendenciestoclaimthecloseorthemorevaluabledisc.InHi-Logames,increasingthedifference between X and Y makesit more likelythat focal-point reasoners usejoint-payoff salience,and soclaim the discworth

Y to them.Thisonly matterswhenlabelsalience isincongruent withjoint-payoff salience,reducing theprobability that theclosediscisclaimed.InBattleoftheSexes,increasingthedifference between X andY hastheeffectofreducing the likelihoodthatplayersadoptfocal-pointreasoning,andhenceclaimtheclosedisc.However,theeffectonthebehaviourof level-k reasonerscan goineitherdirection, dependingon thedistributionoflevels.So, theoveralleffectofchangingthe payoffpairinBattleoftheSexesgamesisnotunivocal.

4. Implementation

Werecruited118participantsfromthegeneralstudentpopulationoftheUniversityofEastAnglia(UK)usingtheORSEE onlinerecruitmentsystem(Greiner,2015).Participantswhohadalreadytakenpartinsimilarexperimentswerenotallowed to sign up.The experiment was programmedin zTree(Fischbacher, 2007). Sessionstook between60and90 minutesto complete.Theaveragepaymentwas£11.26,includinga£5participationfee.

Allparticipantsfacedthesamethirtyscenarios,inasequencethat wasrandomlydeterminedforeachindividual.They were toldthat,forthedurationoftheexperiment, theyhadbeenmatchedwithananonymous‘otherperson’intheroom whoseidentity wouldneverbe disclosed.Theyknewthat oneofthethirtyscenarioswouldbeselectedattheendofthe experiment,andeachplayerwouldbepaidaccordingtoherdecisionandthatofthematchedpersonintheresultinggame, plustheparticipationfee.Whiletheinstructionswerereadaloud,participantswereguidedthroughanumberofpractices abouthow tomake andcanceltheir claims (see below),were shownavariety ofexamples illustratingwhat each player knew ineachoftheinformationconditions,andwereaskedtoansweracomprehension questionnairetoensuretheirfull understanding of the experimental procedures. The experiment started after all participants had responded correctly to everyquestionandanyoutstandingquerieshadbeenansweredbyanexperimenter.18

In each ofthe thirty scenarios, participantswent through a sequence ofsteps. Theywere first shownthe location of the discsonthe bargainingtable(withtheir baseshownatthebottom, asinFigs. 1to 3),withbothhalvesofthe discs beingempty, andtoldthevaluesof X and Y (Step1).If X and Y weredifferent,they werenext shownthefourpossible configurations ofdiscvalues(Step 2).One oftheseconfigurationswould thenbe selectedby thecomputer tobe played, andpresentedinaseparatescreenwithallhalvesofthetwodiscscoveredbyquestionmarks(Step3).19_The_information

condition was then revealed (Step 4). In the Full Information condition, all question marks were replaced by the corre-sponding discvalues.In the OwnInformation condition,each player’s own valueswere disclosed. Inthe No Information condition,allquestionmarksstayedonthediscs.Claimscouldthenbemade(Step5).20_If_{X and}_{Y were}_equal,_participants

moveddirectlyfromStep1toStep4,andthecorrespondingPureCoordinationscenariowasshownstraightaway.Untilthe

18 _The_full_text_of_the_instructions_can_be_found_in_the_Appendix.

19 _For_the_Full_Information_scenarios,_the_selection_ensured_that_each_participant_faced_all_the_scenarios_implied_by_each_disc_value_pair._For_the_Own InformationandNoInformationscenarios,thecomputerrandomlyassignedthevaluesofX andY toeachdiscforeachplayer.

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Table 3

DisccoordinatesinClosenessandSpatiallyNeutrallayouts.

Closeness (C) layouts Close disc Far disc Match

Column Row Column Row

LL −2 −2 −2 2 RR RR 2 −2 2 2 LL LR −2 −2 2 2 LR RL 2 −2 −2 2 RL Spatiallyneutral(N) layouts

Leftmost disc Rightmost disc Match

Column Row Column Row

LL −3 0 −1 0 RR

RR 1 0 3 0 LL

LR −2 0 2 0 LR

claimsforthecurrentscenarioweresubmitted,participantscouldmovebackandforthbetweenStep1andStep5asthey wished.Afterthedecisionsweresubmitted,thewholeprocesswasrepeatedforthenext scenariointheseries.Therewas nofeedbackbetweenscenarios.

Ineach scenario, participantscould claimdiscs by clicking onthem.A claimon a discwas representedby a redline connectingittothe‘You’base,andbychangingthedisccolourfromwhitetored.Anyclaimcouldbecancelledbyclicking again onthe disc, which wouldturn the disc whiteagain anddisconnect it fromthe ‘You’ base. Because there was no feedbackduringtheexperiment,eachplayercouldseeonlyherownclaims.

Afterallparticipantscompletedthethirtyscenarios,eachpairofmatchedparticipants’‘real’scenario,pickedatrandom by the computer, was displayed on the screen together with both players’ claims andactual disc values. These claims determinedtheparticipants’earnings.

The thirty scenarios used in theexperiment were constructed using threepairs of discvalues:

{

10,10

}

,

{

10,11

}

and

{

6,15

}

.ThereweretwoPureCoordinationscenarios(C1andN1)forthe

{

10,10

}

pair,andfourteenFullInformation,Own InformationorNoInformationscenarios(C2toC8,andN2toN8)foreachofthe

{

10,11

}

and

{

6,15

}

pairs.

Becauseallscenarioshavejusttwo discs,we introducedvarietyinthesetofgamesfacedbyeachparticipantbyusing differentlayouts ofeachscenario.ThesearedetailedinTable3,whichreportsthecoordinatesofeachofthediscsineach ClosenessandSpatiallyNeutralscenario,aswellasthelayoutseenbythe‘Other’player(inthe‘Match’column).Examples oftheselayoutsareshowninFig.3above;thefullsetofgameconﬁgurationsisreportedintheAppendix.

In Closeness scenarios, the close discwas always in row

−

2 ofthe bargaining table (negative rownumbers indicate rowsonthe‘You’sideofthetable),andthefardiscwasalwaysinrow2.Wevariedwhetherthetwodiscswerelocated inthesamecolumntothe leftofthe‘You’ base(theLLlayout), orinthesamecolumntotherightofthat base(theRR layout),oronetotheleftandonetotheright(theLRandRLlayouts).NoticethateachLLscenariocanbematchedwitha correspondingRRscenario,andviceversa.LRscenariosarematchesforeachother,asareRLscenarios.InSpatiallyNeutral scenarios,bothdiscsarenecessarilyinrow0(themiddlerowofthetable).Wevariedwhetherthetwodiscswerelocated to theleft ofthe ‘You’base (the LLlayout), tothe right(the RR layout)or one tothe left andone to the right(the LR layout).TheLLlayoutismatchedwiththeRRlayout,andviceversa.TheLRlayoutisamatchforitself.

Wecounterbalancedtheassignmentoflayoutstoplayerssothateachlayoutwasfacedbyapproximatelythesame num-berofparticipants.Eachplayerexperiencedallpossiblelayouts.Ourexpectationwas thatthesevariationsinthepositions ofthediscswouldnothavesystematiceffectsontherelativestrengthofourspatialcues.Thisexpectationwasconﬁrmed (seeSection5).

5. Results:summarystatisticsandcoordinationsuccessmetrics

The claims that players madein each scenario are summarised in Table 4.Since we found no systematic differences betweenresponses to thedifferent layoutsofgivenscenarios, we pool across layoutswhenpresenting ourresults.21 _For

each ofthe threedisc value pairs,

{

10,10

}

,

{

10,11

}

and

{

6,15

}

, thetable contains a rowforeach pair ofcorresponding ClosenessandSpatiallyNeutralscenarios.Theclassofeachpairofscenarios(andthecorrespondinggameforFull Informa-tionscenarios)isreportedinthe‘Class’column,withtheinformationconditionalsoreﬂectedbythequestionmarksinthe scenarionotation. FortheCloseness scenarios,we report thefrequencies ofcasesinwhicha participantclaimednoneof thediscs,bothdiscs,onlytheclosediscoronlythefardisc,aswellasthecorrespondingpercentages(inparentheses).For

21 _We_used_Fisher’s_exact_test_to_check_whether_different_layouts_of_the_scenarios_resulted_in_different_propensities_to_claim_the_close_disc_(in_Closeness scenarios)ortheleftmostdisc(inSpatiallyNeutralscenarios).Wefoundsigniﬁcanteffects(p<0.05)injusttwoofthirtycomparisons(seetheAppendix fordetails).