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Contents lists available atScienceDirect

Games

and

Economic

Behavior

www.elsevier.com/locate/geb

Equilibrium

trust

Luca Anderlini

a

,

,

Daniele Terlizzese

b

aGeorgetownUniversity,UnitedStates bEIEFandBankofItaly,Italy

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received5November2015 Availableonline9March2017 JELclassification: D80 D89 C79 Keywords: Trust Socialnorms Multipleequilibria Enforcement

Trustingbeliefscanbeexploited.Atrustfulplayerwhoischeatedtoooften,shouldstart trustingless,untilherbeliefsarecorrect.Forthisreasonwemodeltrustasanequilibrium phenomenon.Receivers ofanoffer totransactchoosewhetherornot tocheat.Cheating entailsacost,withanidiosyncraticcomponentandasociallydeterminedone,decreasing withthe massofplayerswhocheat. Themodeleitherhasaunique equilibriumlevelof trust(theproportionoftransactionsnot cheatedon), ortwo —one withhigh and one withlowtrust.Differencesintrustcanresultfromdifferentfundamentalsorfromdifferent equilibriabeingrealized.Surprisingly,undercertainconditionsthesetwoalternativesare partiallyidentifiablefromanempiricalpointofview.Ourmodelcanbereinterpretedwith thecostofcheatingarisingfromanenforcementmechanism thatpunishes cheatersina targetedwayusinglimitedresources.

©2017ElsevierInc.Allrightsreserved.

1. Introduction

1.1. Motivationandoverview

Notmanytransactions arecarried outusingrotatingsecurity trayswithmoneyonone sideandthepurchased good ontheother.Yet,withoutthesedevicesorsomeequivalentarrangement,thestandardselfishplayersthatpopulatemodern economicmodelswouldbeunabletotrade,barringrepeatedinteractionorbindingcontracts.Itisinsteadself-evidentthat tradeamongplayersflourisheswaybeyondwhatthishypotheticalworldwouldlooklike.

The lubricantthat makes so manytransactions take placeis trust. This isthe belief that economic playershold that theother sideofthetransactionwillnot behaveinacompletelyopportunisticway,thusimpedingmutuallyadvantageous exchange.1

Ourpurposehereistobuildasimplemodeloftrustasanequilibrium phenomenon.Theplayers’beliefsaboutnotbeing cheated—theirleveloftrustaswejustdefinedit—shouldbeendogenouslydeterminedinequilibrium,andhencecorrect.

WearegratefultoHelenaAten,LarryBlume,MartinDuwfenberg,HüliaEraslan,LeonardoFelli,DinoGerardi,HugoHopenhayn,GuidoKuersteiner,Roger Lagunoff,GeorgeMailath,FacundoPiguillem,AlehTsyvinski,BillZameandseminarparticipantsforhelpfulcomments.Wealsothankthreeanonymous refereesandtheAdvisoryEditorofthisJournalforconstructivecriticism.LucaAnderlinithanksEIEFfortheirgeneroushospitality.Allerrorsremainour own.

*

Correspondingauthorat:DepartmentofEconomics,GeorgetownUniversity,37thandOStreets,Washington,DC20007,USA. E-mailaddress:luca@anderlini.net(L. Anderlini).

1 Wedefinetrustasabelief,inlinewithCharnessandDufwenberg (2006),ratherthanadoptingthebehavioraldefinitionadvocatedbyFehr (2009). Sinceinequilibriumtherewillbeaone-to-onemappingbetweenbeliefsandbehavior,thedistinctionforusislargelyimmaterial.Foreaseofexposition, wepostponeuntilSubsection1.2adiscussionofspecificcontributionsfromthelargeliteratureonthesourcesandeffectsofthepresenceoftrust.

http://dx.doi.org/10.1016/j.geb.2017.02.018 0899-8256/©2017ElsevierInc.Allrightsreserved.

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Wefocusonequilibriumbeliefsbecauseofabasictensionbetween“trustingbeliefs”,andconsequent“trustingbehavior”, andtheincentivestocheatofotherplayersinsociety.Trusting beliefscanbeexploited.However, atrustfulplayershould notbecheatedtoomuch;ifsheis,shewouldchangeher beliefandstarttrustingless.Thistensiondrivesourinterestfor whatanequilibriummodeloftrustcangenerate.

Wearenotthefirsttostresstheneedforanequilibriumanalysisoftrust.Withintheframeworkofpsychologicalgames,

Huang and Wu (1994)and

Dufwenberg (1996, 2002)

analyzeequilibriumtrustinmodelsthatbear manyresemblancesto ours.What isnewinthispaperisananalysisofboth theextensiveandintensivemarginsoftrustingbehavior(notonly whetherto trust butalsohow much)andelements ofheterogeneity (notall agentsdeserve tobe trustedequally).2 The conjunctionofthesetwofeaturesallows ustoexplorenovelcomparativestaticswhichwethink shedlightontheroleof multiplevsuniquetrustequilibriaintheinterpretationofempiricalevidence.

Tomodeltrustasan equilibriumphenomenonweplacea“costofcheating”intheplayers’utilityfunction.Thecostof cheating hastwo components.Onewhich isan exogenous, idiosyncraticcharacteristicofeach player, andanotherwhich issocially determined bythe behaviorof others.Thelesscommoncheating behavior isinsociety, thehigheris thecost ofcheatingforindividual players.Thisfeedbackcomponentofthecostofcheating isacentralingredient ofouranalysis. Weinterpretitasreflectingasocialnorm,determinedbytheaveragebehaviorofothers.Itcanalsobegivenasomewhat differentpsychologicalunderpinning,inthecontextofthetheoryofpsychologicalgames(wepostponeadiscussionofthis pointtoSubsection

1.2

).Theexperimental evidencepresentedin

Charness and Dufwenberg (2006)

offersstrongempirical supportforthepresenceofasocialfeedbackofasimilarnature.Alternatively,wemightthinkofthefeedbackasresulting fromanenforcementtechnology,whoseeffectivenessdepends, forgivenresources,onthe averagebehavior.Wereturnto thisalternativeinterpretationinSection6.

Our model is deliberately kept simple in the extreme. However, as we will argue below, the flavor ofour results is independentofmanyofthestarkfeaturesofourmodel.Twofeaturesofourset-upareworthmentioningattheoutset.

First, ourmodel isnot dynamic, nor should itbe interpreted as a “reducedform” ofa dynamic set up. Repeated in-teractionscanandhavebeenusedsuccessfullytogeneratecooperativeandtrust-likebehavior,together witha verylarge varietyofotherequilibria.However,bothinreallifeandinlaboratoryexperimentstrustseemstoemergeeveninone-shot interactions,insituationsthatwouldseemtocallfor“swivel-traytrading”iftrustwerenotpresent.Tosharpenour under-standingofthesesituations,weworkwithamodelthatavoidsreputationalissuesandmoregenerallyrepeatedinteraction altogether.

Secondly,oursetupisa“one-sided”modeloftrust,withoneplayermakingaproposal,andaresponderwhocandecide whethertocheatornot.Admittedly,trustisoftenrequiredonbothsidesofatransaction,andinthosecasestrustshould bemodeled asa two-sidedphenomenoninwhichbothplayersinamatchhaveachoiceofwhethertocooperateornot. Ourapproach is thereforerestrictive. There is, however,a large andgrowing experimental literature on trust games and reciprocitythatexplorestheemergenceoftrustinsituationsakintotheonewemodel.

Inspite ofits extremesimplicity,we believe oursetup provides a rich enoughframework to addressthe well docu-mented diversityoflevels oftrust indifferentsocieties.Indeed,depending on theconfigurationofpreferences andother parameters, ourmodeleither generatesa unique equilibrium(witha single equilibriumlevel oftrust), ortwo equilibria, one witha higherandanotherwitha lower leveloftrust. Moreover, byvarying the parametersof themodel,higheror lowerlevelsofequilibriumtrustcanbeobtainedwithoutswitchingacrossdifferentequilibria.Giventhisrichsetofpossible equilibriumoutcomes,themodelallowsustoframeinanaturalwaywhatseems akeyquestionconcerningthelevels of trust indifferentsocieties.When weobserve differentlevelsoftrust acrossdifferentsocieties, isthisduetoa difference inthe fundamentalparameters thatunderpinthe differentsocieties,oris itpossible thatthe samefundamentalparameters

giverisetodifferentequilibria?Obviouslythesetwopossibilitieshavevastlydifferentpolicyimplications,anditistherefore importanttohaveaframeworkinwhichtheycanbemadeprecise,andhencedisentangled.

Ofcourse,unless themultiplicity canbe somehowvalidatedempirically, modelswithmultipleequilibriaare nomore thanatheoreticalbenchmarkprovidingan —albeitimportant—onlypotential interpretationofreality.Akeyinsightfrom ourmodelisthatthetwocasesmentionedabove,undersomeconditions,arepartiallyidentifiable:ifdifferentlevelsoftrust resultfrommultipleequilibria,thentheleveloftrustmustbenegativelycorrelatedwiththesizeofindividualtransactions. Apositivecorrelation canonlyemerge ifdifferentlevels oftrustwereto resultfromdifferencesintheparametersofthe model.3Toourknowledge,thepossibilitytoempiricallydisentanglemultiplevs.uniqueequilibriumregimesisnew.

Aswementionedabove,ourmodelcanbereinterpretedsothatthesocialfeedbackcomponentofthecostofcheating comesinsteadfromanenforcementtechnologythatpunishes cheatingusinglimitedresources ina targetedway.Wefind that,inthemultipleequilibriaregime,aninfinitesimalincreaseintheresourcesdevotedtoenforcementcanyielda discon-tinuousincrease intheleveloftotalactivityintheeconomy.Inthesingleequilibriumregime,instead,thelevelofactivity changescontinuouslywiththeresourcesspentintheenforcementtechnology.

2 HeterogeneityisalsoconsideredbyAttanasietal. (2016).Seefootnote8.

3 Sincetheterm“partialidentification”hasbeenusedbeforePhillips (1989),itisusefultobepreciseastothemeaningwegiveithere.Theidentification ispartial inthesensethatwecannotruleoutthattwoequilibria,correspondingtotwosetsofparameters,entailanegativecorrelationbetweentrust levelsandsizeofindividualtransactions.Therefore,whileapositivecorrelationexcludesthatthedifferentlevelsoftrustresultsfrommultipleequilibria withunchangedparameters,theobservationofanegativecorrelationisinconclusive.

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1.2. Relatedliterature

Thestartingpointofthispaper—anelementoftrustisneededformosttransactionstotakeplace—isuncontroversial. Long ago Arrow (1972, p. 357)noted that “Virtually every commercial transactionhas within itself an element of trust, certainlyanytransactionconductedoveraperiodoftime.”Intheabsenceof“instantaneousexchange,”anelementoftrust isrequired.

Arrow basedhiscommentonthepath-breaking studyby

Banfield (1958)

ofthedevastatingeffectsofthelackoftrust on a “backwards” small community in southern Italy pervaded by “amoral familism.” Following Banfield (1958) a large literaturehasblossomedontherootsandeffectsofthelack,orpresence,oftrust.

Theliteratureiswaytoolargeandvariedtoattemptevenareasonedoutlinehere,letaloneasurvey.Weonlyrecallhow

Putnam (1993) documentstheheterogeneouslevels of“social capital”indifferentregions ofItalyandits roleinfostering growth.Acoupleofyearslater,

Fukuyama (1995)

publishedaninfluentialmonographconcerningthepositive roleoftrust in largefirmsandhenceeconomicgrowth.4 Bohnet et al. (2010) documentcrucialdifferencesintrustingbehavior across Gulf andWestern countries. Morerecently

Bigoni

et al. (2016)findthat the differencesincooperativebehavior between northernandsouthernItalycannotbeexplainedbyproxiesforeithersocialcapitaloramoralfamilism.Theyconcludethat persistentdifferencesinsocialnormsareresponsiblefortheobservedgap.Webelievethatourset-upgeneratingmultiple equilibriawithdifferentleversoftrustresonateswellwiththeirempiricalfindingsandanalysis.

Wealsoselectivelyrecallthecontributionsby

Knack and Keefer (1997)

,

La Porta et al. (1997)

,andmorerecently

Guiso et

al. (2004).Thesestudiesalldocumentinavarietyofwayshowthepresenceoftrustiscorrelatedwithdesirableeconomic outcomes.

Ourone-sidedset-upiscloselyrelatedtothestudyoftrustinexperimentswithasenderandareceiverthatgoesback to

Fehr et al. (1993)

and

Berg et al. (1995)

andismorerecentlyfoundin

Glaeser et al. (2000)

,

Irlenbusch (2006)

,

Sapienza

et al. (2013) andButler et al. (2016) amongothers.Onewaytoplaceourcontribution intheextantliterature isthatwe examinetheequilibriumpropertiesofacanonicalversionoftheseexperimentalgames.

Thesocialcomponentofour“costofcheating”isakintothepsychologicalcostwhich,inthespiritofthepsychological gamesfirstintroducedby

Genakoplos et al. (1989)

,wasassumedby

Huang and Wu (1994)

andby

Dufwenberg (1996, 2002)

toanalyzetheequilibriaofatrustgamebetweenatrustor(whatwecallinthispapertheofferer)andatrustee(thereceiver, inoursetting).Inthosepapersthetrusteewhocheatsincursapayoffpenaltyproportionaltohisexpectationofthetrustor’s expectation of the probability that the trustee doesnot cheat (in Dufwenberg, 1996, 2002) orof the proportion of the trusteeswhodonotcheat(in

Huang and Wu, 1994

).5Theideaisthatthetrusteeexperiencesabelief-based,psychological

cost ifhecheatswhen thetrustordoesnot expecthim todoso. Thismightsustainan equilibriuminwhich thetrustee doesnotcheat.

Since inequilibriumbeliefsarecorrect, theroleofhigherorderbeliefsinshapingthecost,postulated inthesepapers, boils down toincludingin thecost acomponentproportional to theequilibriumshareoftrusteewho donot cheat(the equilibrium trust, in our setting), whichis precisely what we do inthis paper. In fact, the equilibriumof our model is isomorphic totheequilibriumofatwo-stagepsychologicalBayesiangamewithheterogeneousreceivers,whoseutilityhas bothan idiosyncraticandapsychologicalcomponent, thelatterdependentontheoffererbeliefaboutthereceiver’saction (or, equivalently in equilibrium, on the receiver second-order belief about his own action).6 There is, however, a subtle differencebetweeninterpretingthecostasresultingfromnoncompliancewithasocialnormorfrombetrayalofthe expec-tationsofa wellidentifiedpartner.Webelievethattheformerismoreinlinewiththeone-shot,anonymousnatureofthe interactionwhichweemphasizeinthispaper.

Dufwenberg (2002) interpreted the psychological component of the cost as reflecting “guilt aversion.” In later work (Battigalli and Dufwenberg, 2007, 2009), guilt aversion has beengiven a moreprecise meaning, based onthe difference betweenthe outcome theofferer wasexpectingto obtain andtheactual outcome.7 To beinterpretable asreflectingthis more specificnotionofguiltaversion,ourmodelwouldneedto includeinthecost alsoatermdependingonthe sizeof thetransaction,sincetheexpectedoutcomefortheoffererisgivenbytheproductoftheprobabilitythatthereceiverdoes not cheatandthesizeofthetransaction.Indeed,makingthecostofcheatingsensitivetothesizeofthetransactiontobe cheated seemsa naturalandinteresting extensionofourbaseline assumption,whichmightbeconsistent alsowithother psychologicaljustifications(forexample,reciprocity,asin

Rabin, 1993

and

Dufwenberg and Kirchsteiger, 2004

).

We explorethisissueandtherobustnessofourresultstoalternativeformulationsofthecostofcheatinginSection5. There,we showthatourqualitativeresultscontinuetohold,aslongasthecostofcheatingincreaseswiththesizeofthe

4

Theliteraturedocumentingtheimpactof“socialcapital”or“trust”onincome,wealthandgrowthratesalsoincludessomenotableskeptics.Toour knowledge,themostprominentoneisSolow (1995),whointurncitesevidencefromKimandLau (1994)andYoung (1994).Forasurveyofmuchofthe literatureontrustandsocialcapitalwecitehere,includinganaccountofthedebatewehavejustmentioned,seeSobel (2002).

5 HuangandWu (1994)[p. 393]wrote:“themoreprevalentcorruptionis,thelessintenseistheremorsesufferedfromcorruptbehavior;andconversely, thelesscorruptionthereis,themoreregretfromviolatingasocialnormnottobecorrupt.”Thisispreciselyourinterpretationofthesocialcomponentof thecost.

6 Wearegratefultoananonymousrefereeforpointingthisout.

7 InHuangandWu (1994)andDufwenberg(1996, 2002)thedecisionbythetrusteeisbinary(whethertocheatornot),sothebeliefabouthisaction andthebeliefabouttheoutcomeresultingfromhisactionareproportional.Thereforethelooserandthemoreprecisenotionsofguiltaversionareinfact equivalent.

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transactioninalessthanproportionalway.This,webelieve,isfairlyintuitive.Thetensionweareexploringinthispaperis onebetweenthebenefitofincreasingthescaleofthetransactions,whichrequirestrust,andtheincentivetocheat,which increaseswiththat scale.Inourset-up,thebenefitisassumedto increaseproportionallywiththesizeofthetransaction. As long asthe cost ofcheating doesnot increase morethan proportionally with that size, thetension remains andour qualitativeresultsareconfirmed.Ifinsteadthecostisassumedtoincreasemorethanproportionally,thetensioneventually disappearsandthesetofequilibriaofthegamechangesqualitatively:whentheupperboundtothesizeofthetransaction islargeenough,theonlyequilibriumisoneoffulltrust(no cheating)andmaximaltransactionsize.Asimilaroutcomeis obtainedassumingthattheincentivetocheatiskeptincheckbyreciprocity.Thereagain,alargeenoughofferedtransaction isinterpretedasakindactbytheofferer,towhichthereceiverwantstorespondkindly.Therefore,alargeofferisnolonger a temptationto cheat,andthetension evaporates.Theseresults clarify thatthe scope ofouranalysisis limitedtothose casesinwhich aconflictbetweenthe gainsfromtrustinga transactionpartner andtherisk ofbeingtakenadvantageof remainsinplace.Ourset-upisnotsuitedtoanalyzethosecasesinwhichtheinterestsofthetwopartiesbecomealigned.

Bysimplyassumingacostcomponentproportionaltotheequilibriumtrust,insteadofexplicitlyadoptingapsychological gameapproach,wegainintractabilityandwecanexploreaspectsoftheproblemsofarneglected.Inparticular,ourofferer hasacontinuouschoicevariableasopposedtothebinaryonein

Huang and Wu (1994)

and

Dufwenberg (1996, 2002)

,and thisisessentialto studyboth extensiveandintensivemarginsofthetrustingbehavior. Moreover,ourset-up allowsfora simpletreatment ofheterogeneity amongthetrustees,another aspectwhichis clearly relevanttounderstandthe role of trust andwhich istechnicallymuch morechallenging inthecontext ofafull blownpsychologicalgame.8 Heterogeneous

cheatingcostsyield,inasimpleway,anequilibriuminpurestrategieswithaninteriorleveloftrust(neitherfullnorzero trust),whilepsychologicalgameswithnoheterogeneityonlyobtaininteriorlevelsoftrustinamixedstrategyequilibrium. Admittedly,ourapproachhasacost.Bycuttingthroughthechainofhigherorderbeliefswecannotexploretheforward inductionargumentswhich

Dufwenberg (1996, 2002)

and

Battigalli and Dufwenberg (2009)

haveusedtoselectamongthe possibleequilibria.Butwhiletheseargumentsareapowerfultoolinthecaseofasingleoffererandasingle receiver,we pursue amodelin whichofferers andreceiversare anonymouslymatched afterbeingdrawn fromlargepopulations. We believethataforwardinductionargumentlosessomeofitsappealinthiscase.9

Charness and Dufwenberg (2006)presentexperimentalevidenceshowingthattrusteeswhoguessthatthetrustors’ av-erage guessaboutthe proportionof trusteesnot cheating is higherare lesslikely to cheat.Although their experimental settingonlyexploresbest-responsebehavior,andcannotbe compareddirectlytoourequilibriumset-up,theirresults em-piricallyconfirmthepresenceofasocialfeedbackinthecostofcheating.10Fehr (2009)alsodiscussesatlengththepossible

multiplesources oftrustingbehavior, identifying socialfeedbacks (thoughhis analysisfocuseson feedbacks affectingthe offerer,whilewearehereconcernedwiththereceiver).

Feddersen and Sandroni (2006) explore the effects of “ethical” social feed-back mechanisms not unlike the one we considerhere,andtheirimpactontheequilibriaofvotingmodels.11Horst and Scheinkman (2006)areconcernedwiththe generaltheoretical problemsof models withsocial feedback variables, particularlywith the(far fromtrivial) issues that ariseinprovingtheexistenceofequilibriumingeneralinthisclassofmodels.

Dixit (2003)and

Tabellini (2008)

areboththeoreticalcontributionstotheliteratureontrust.12Theirmainfocusisonthe differentialeffectsofdistanceandsociety’ssizeonthesustainabilityoftrust.Inbothcases,trustismodeledasadynamic two-sidedphenomenonwithrepeatedinteractionbetweenplayers.Inourmodel,playonlytakesplaceonce,andtrusting andtrustworthy equilibriumbehavior can betraced directlyback to ourpreferences embodying thesocial feed-back we havedescribedabove.

Finally,thetheoreticalliteratureonrepeatedinteractions,fromwhichwepurposedlystayaway,isalsovast.Wesimply refer the interested readerto the monographby

Mailath

and Samuelson (2006), whichalso hasa comprehensive list of references.

1.3. Planofthepaper

The rest of the paperis structured asfollows. In Section 2 we describe the basic model indetail and make precise whatconstitutesan equilibriuminourset up.InSection 3wecharacterizethe setofpossibleequilibriaofthemodel.In Section4wehighlighthowhighandlowtrustequilibriamayarisefromeitherdifferencesinthefundamentalparameters ofthemodel,oraswitchacrossmultipleequilibriasupportedbythesamesetofparametervalues.Inthissectionwealso

8 ArecentworkingpaperbyAttanasietal. (2016),perhapsthefirsttoallowheterogeneityinapsychologicalgameoftrust,highlightsthetechnical difficultiesofdealingwithheterogeneoushigherorderbeliefs.

9 InHuangandWu (1994)thetrustgameconsideredis,asinourcase,oneinwhichtheoffererfacesacontinuumofreceivers,andforwardinduction isnotused.

10 CharnessandDufwenberg (2006)recognizesometensionbetweenthetheoreticalmodelpostulatingguiltaversion,whichreferstoone-on-onepairings betweentrustorandtrustee,andthesetupoftheirexperiment,whichelicitexpectationsaboutaveragesofpairings.Giventhatwepostulaterandom pairingsamonglargepopulationsofofferersandreceivers,andfocusonthefeedbackfromthe equilibriumshareof(non)cheatedtransactions,their experimentaldesignseemsrelevantforourapproach,notwithstandingthefactthattheystayclearfromequilibriumpredictions.

11 Inotherrelatedworks,Dufwenbergetal. (2011)investigate“other-regarding”preferencesingeneralequilibriummodels,whileBlume (2004)studies theeffectof“stigma”inadynamicmodel.

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spellouttheidentifiablecharacteristicsofhighandlowtrustequilibriainthesetwocases,andweproceedtocharacterize aggregate activitylevelsofthedifferentequilibria.InSection 5weconsiderseveralvariantsofourbaseline costfunction, andweassesstherobustnessofourresultstothesealternativeformulations.Section 6providesare-interpretationofthe socially generated component ofcheating costs as stemming froman enforcement technology that targets andpunishes cheaterswithlimitedresourcesavailable.Finally,Section8brieflyconcludes.

Foreaseofexposition,allproofshavebeenrelegatedto

Appendix A

.

2. Set-up

2.1. Themodel

Therearetwovarietiesofplayers,offeringplayers—orsimply

O

players—andreceivingplayers—orsimply

R

players. Thereisacontinuumof

O

playersofmass1,andsimilarlyacontinuumof

R

playersofmass1.

The

O

and

R

playersarethen randomlymatchedtoformaunit massofpairs.Theonlything thatisofconsequence hereisthatan

O

playershouldnotknowthe“costofcheating”(tobedefinedshortly)oftheplayersheismatchedwith.13 Each

O

player makesan offerx

∈ [

0

,

1

]

tothe

R

player inhermatch. Theoffergeneratesa totalsurplusof2x to be splitequallybetween

O

and

R

ifthetransactiongoesthroughwithout“cheating”onthepartof

R

.So,if

R

doesnotcheat, anofferofx generatesapayoffofx forboth

O

and

R

.14

Itisthe

R

playerinthematchwhodecideswhethertocheatornot.Afterreceivinganofferx fromthe

O

playerinthe match,

R

maydecidetocheatandgrabtheentiresurplus2x insteadofabidingbywhatthesplittingproceduresuggests. However,ifhecheats,

R

willalsosufferacostc.15Therefore,

R

willcheatif2x

c

>

x orequivalently x

>

c,andwillnot

cheatotherwise.16

Thetotalcostofcheatingc hastwocomponents.Onedependsontheexogenouslygiven“type”ofthe

R

playerandthe otherisdeterminedbythebehaviorofotherplayersinthemodel.

Forsimplicity,weassume thatthereare justtwo typesof

R

players, “high”(H )and“low”(L).17 Theexogenous

com-ponentofthecostofcheatingistL

∈ (

0

,

1

)

fortype L andtH

∈ (

0

,

1

)

fortype H ,withtL

<

tH.Theproportionoftype H is

denotedby p

∈ (

0

,

1

)

throughout.

Thecomponentofc thatis“sociallydetermined”isthesameforallplayers.18Lets betheproportionof

R

playerswho donot cheat.Wesimplysetthesocialcomponentofthecheatingcosttoequals andwetakethetwocomponentsofthe cheatingcosttocombineinalinearway.19Foran

R

playeroftype

τ

∈ {

L

,

H

}

,thetotalcostofcheatingifaproportions

oftransactionsgothroughwithoutcheatingisgivenby

c

(

τ

,

s

,

α

)

=

α

+ (

1

α

)

s (1)

with

α

∈ (

0

,

1

)

aparameter thatmeasures (inversely)the relativesocial sensitivityoftheplayers’ cost ofcheating.Note thatassumingthatthecoefficientsoftτ ands sumto1inequation

(1)

isrestrictive,sinceitfixestheoverallsensitivityto totalcost.However,thiscanbeviewedasanormalizationensuringthattheorderofmagnitudeofx andc isthesame.

As we have already remarked, the social component s of c is a critical ingredient of our model. We think of it as embodyingtheinfluenceofsocialnormsonindividualbehavior. Whenfewerpeopleinsocietycheat,thosewhodoarein somesensefurtherawayfromthesocialnorm,andthishasa“moralcost.”

The fact that therighthand side of(1)is linearinits two arguments simplifiesour analysisconsiderably. Theactual functionalformislargelyinessential,butoneofitsimplicationsisnot.Inparticular,ourpartialidentificationresult( Propo-sition 3)belowdoesdependonthefactthattheelasticity ofthetotalcostofcheatingtos shouldnotbe“toohigh”.Roughly speaking,ifwegofromanequilibriumwherethe L typescheattoanotherinwhichtheydonot,then,foragiven valueof

s theequilibriumlevelofx mustbe lowertomakecheatingunprofitablefortheL types.However, asthe L typeschange theirbehaviorfromcheatingtonotcheatingtheequilibriumlevelofs becomeshigher,andthisraisesthecheatingcostfor all types.Forourpartial identificationresulttoholdthe firsteffectmustdominateover thesecond. Asit turnsout, this

13 Sincewedonotconsiderrepeatedinteraction,theotherdetailsofthematchingprocessarecompletelyinessential. 14 Thefactthatthesurplusissplitequallysimplifiesourcalculations,butisinessential.

15 NotethatweareassumingthatRhasthechoiceofwhethertocheatornotevenwhenx=0.Ifhecheatsafteranofferequaltozero,herpayoffwill thereforebe−c.However,whenx=0,thereis,sotospeak,nothingtograb.Hence,analternativewouldbetoassumethatRdoesnothaveachoiceof whethertocheatornotwhenx=0.Proceedingaswedosimplifiestheanalysisbutdoesnotimpacttheresults.

16 OurimplicitassumptionthatwhenRisindifferenthewillnecessarilynotcheatsimplifiestheanalysisbutisinfactwithoutlossofgenerality.In equilibrium,thecheatingsetdefinedhereandformallyin(2)belowmustbeopenevenifitwereallowedinprincipletobeclosed.Thereasonisthatifit werenot,thentheoptimalofferofOplayerscouldnotbedefinedbecausetheacceptancesetwouldhavetobeopen,andhenceO’soptimaloffercould notbewelldefined.

17 Theoverallflavorofourresultseasilygeneralizestoanworldwithanyfinitenumberoftypes.Manyofourresultsalsohaveanaloguesinaworldwith acontinuumoftypes.Weproceedinthiswayforthesakeofsimplicity.

18 Again,thisistokeepthingssimple.Onecouldimagineheterogeneouscost“sensitivities”tothebehaviorofothers,andthiscouldeasilybe accommo-datedinoursetup.

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requiresthattheelasticityofc

(

τ

,

α

,

s

)

withrespecttos belessthanone,andthelinearitypostulatedin

(1)

issufficientto guaranteethatthisisthecase.20

Finally, as we have mentioned already, the effect of s on c can also be re-interpreted asa cost stemming from an enforcementtechnology.Forgivenresourcesdevotedtoenforcement,theprobabilitythatacheateris“caught”increasesas fewerpeoplecheat,thusincreasingtheexpectedcostofcheating.WepursuethisinterpretationmoreformallyinSection6

below.

2.2. Equilibriumdefinition

Astrategyprofile

σ

= (

σ

O

,

σ

L

,

σ

H

)

assignsanumber

σ

O

∈ [

0

,

1

]

toevery

O

player—theofferx thatshemakes—and

a cut-offvalue

σ

τ

∈ [

0

,

1

]

toeach

R

player oftype

τ

∈ {

L

,

H

}

,indicatingthat hewillcheatifandonly ifhereceives an

offerstrictly above

σ

τ .21Notice thatoncea profile

σ

isgiven,a valueof s isalsogivensincetheexpectedproportionof

transactionsthatwillnotinvolvecheatingisdetermineddirectlyby

σ

.22

Anequilibriuminourmodelis astrategyprofile

σ

= (

σ

O

,

σ

L

,

σ

H

)

such that the

O

playersmaximize their expected return,giventhestrategiesofthe

R

playersofeach type(andtheirrelative weightinthepopulation), andthe

R

players ofeach typedecide optimallywhetherto cheatorsplitwhateverofferthey received,giventhestrategies oftheother

R

playersofeachtype(andtheirrelativeweightinthepopulation).

Webegintheanalysisoftheplayers’maximizationproblemonthe

R

side.Fixa

σ

,andthereforeavalueofs.Consider an

R

playeroftype

τ

∈ {

L

,

H

}

.Hewillcheatifandonlyiftheofferx hereceivessatisfies

x

>

c

(

τ

,

s

,

α

)

=

α

+ (

1

α

)

s

.

(2)

For given

σ

, and hence s, using (2) we can compute the mass of players

R

who will not cheat on any givenoffer

x

∈ [

0

,

1

]

,denotedby P

(

x

,

s

)

: P

(

x

,

s

)

=

0 if x

∈ (

c

(

H

,

s

,

α

),

1

]

p if x

∈ (

c

(

L

,

s

,

α

),

c

(

H

,

s

,

α

)

]

1 if x

∈ [

0

,

c

(

L

,

s

,

α

)

]

(3)

Forgiven

σ

,wecanthereforewritetheexpectedpayoffofan

O

playeroffering x asxP

(

x

,

s

)

(recallthatofferingx,she getsapayoffofx wheneversheisnotcheated).Henceshewillchooseanx thatsolves

max

x∈[0,1]x P

(

x

,

s

)

(4)

The solutionto

(4)

isimmediate tocharacterize. Since c

(

L

,

s

,

α

)

<

c

(

H

,

s

,

α

) <

1, the solutionto (4) dependsonthe

comparisonbetween

c

(

L

,

s

,

α

)

and p c

(

H

,

s

,

α

)

(5)

Ifc

(

L

,

s

,

α

)

>

pc

(

H

,

s

,

α

)

thenitisuniquelyoptimaltosetx

=

c

(

L

,

s

,

α

)

—thelargestlevelthatensuresthatno

R

players cheat.Ifinsteadc

(

L

,

s

,

α

)

<

pc

(

H

,

s

,

α

)

theuniquesolutionistosetx

=

c

(

H

,

s

,

α

)

—thelargestlevelthatensuresthatonly

R

playersoftype L cheat.Finally ifc

(

L

,

s

,

α

)

=

pc

(

H

,

s

,

α

)

, then an

O

player is indifferentbetweenmaking an offer of

c

(

L

,

s

,

α

)

andanofferofc

(

H

,

s

,

α

)

,andhencebothvaluessolve

(4)

.

Intuitively,increasing x increases (injumps, becauseofthe discretenature ofthetypes) theprobability that theoffer will be cheatedon. The trade-off betweenincreasedpayoff conditional onnot being cheatedon andthe increase inthe probabilityofcheatingiswhatdeterminestheoptimalbehaviorof

O

players.

Beforeproceedingfurther,weintroduceasimplifyingassumptiononthebehaviorof

O

players.

Assumption1(Tiebreak).Wheneverc

(

L

,

s

,

α

)

=

pc

(

H

,

s

,

α

)

,all

O

playersmakean offerofc

(

L

,

s

,

α

)

whichisnotcheated onwithprobabilityone,ratherthananofferofc

(

H

,

s

,

α

)

whichischeatedonbyall

R

playersoftypeL.

Foreaseofexposition,fromnowon,whenwe saythat“x solves (4)” wewillmeanasolutionthat complieswiththe tie-breakingrulepositedhere.

20 ThedetailsofhowthiselasticityconditionguaranteesthatthefirsteffectdominatesthesecondarediscussedinSubsection4.2,followingthestatement ofProposition 5.

21 Forsimplicitywearefocusing onstrategieswhereallplayersofagiventype,thatisO,RoftypeL andRoftypeH ,actinthesameway.Moreover, weassumethatthestrategiesoftheRplayerscanbesummarizedbyacut-offvalue,albeitinprincipletheplayers’cheatingresponsescouldbebasedon morethanasimplecut-offvalue.However,itiseasytoshowthatwecanrestrictthestrategyspacesofRplayersinthiswaywithoutlossofgenerality, usingastandardweak-dominanceargument.Thecheatingsetcanalsobeshowntobeopenwithoutlossofgeneralityinequilibrium—seefootnote16 above.

22 Analternativeinterpretationofs,intheframeworkofpsychologicalgames,wouldseeitastheexpectationoftheRplayeroftheprobabilitythatthe

(7)

Assumption 1 makes the behavior ofall

O

playersuniquely determinedfor anyparameter valuesandanylevel of s,

substantiallysimplifyingtheanalysis.

Notethatthebehaviorthat

Assumption 1

postulates canbeinterpretedastheresultof“lexicographic”risk-aversionof the

O

players(addedtotheirbasicrisk-neutrality).Wheneverexpectedvaluesareequal(andonlythen),randomvariables withalowerriskarepreferred. Inparticular,whenc

(

L

,

s

,

α

)

=

pc

(

H

,

s

,

α

)

,thesurepayoffofc

(

L

,

s

,

α

)

willbepreferredto arandompayoffequalto0 withprobability1

p andtoc

(

H

,

s

,

α

)

withprobability p.

Assumption 1 simplifiesour analysissinceit rulesout, forparameterconfigurations supportingmultipleequilibria, an equilibriumthatis“intermediate”betweenthetwothatwewillfocuson.Thisequilibriumis“between”thetworemaining ones,anditspresencewouldnotaffectourqualitativeconclusionsinanyway.23

We can nowprovide aworkingdefinitionof whatconstitutesan equilibriuminour model.Notethat, using Assump-tion 1,theequilibriumbehaviorofall

O

playersissummarizedbyasinglenumberx

∈ [

0

,

1

]

—thesolutionto

(4)

,whichis theoffertheyallmaketothe

R

playertheyareeachmatchedwith.Ofcourse,inequilibriumitmustalsobethecasethat thevalueofs thatappearsin

(4)

isthecorrectone,asdeterminedbythebehaviorofthe

R

playersgivenx.Thisjustifies thefollowingdefinitionofequilibrium.

Definition1(Equilibrium).Anequilibriumisapair

(

x

,

s

)

suchthatx solves

(4)

givens andsuchthat

P

(

x

,

s

)

=

s (6)

Itisworthnotingthatourdefinitionofequilibriumimplicitlyinvokesanotionofsequentialrationality,asthe

R

players reactoptimally,andareknownbythe

O

playerstoreactoptimally,toany offertheymightreceive.24Suchanotionwould correspondtotherequirementofsubgameperfection,ifweweretomaketheassumptionthateach

R

playerobservesthe offerreceivedbyalltheother

R

players,sothatthestageofthegameinwhich

R

playersactwouldbeapropersubgame. Alternatively,ourequilibriumwouldcorrespondtoasequentialequilibriumofadiscretizedversion ofourgame(toavoid technicalmeasurabilityproblems).Fleshingoutthesealternativepossibilitiesformallywouldhoweverbetediousandwould notaddanythingofsubstancetoouranalysis.

Givenwhat weknowaboutthe behaviorof

O

players, itiseasy to checkthat onlytwo possibilitiesare open forthe value of s in anyequilibrium. Setting an x suchthat both types of

R

players cheat is clearly never optimal. Hence in equilibriumitmustbethat eithers

=

1 (andnocheatingatalltakesplace),ors

=

p (andall

R

playersoftype L cheat,

whileallthoseoftype H donot).

Itfollowseasilythat,inequilibrium,ifs

=

1 thenx

=

c

(

L

,

1

,

α

)

=

α

tL

+

1

α

,andsimilarlyifs

=

p thenx

=

c

(

H

,

p

,

α

)

=

α

tH

+ (

1

α

)

p.25

Atthispoint,itisusefultocrystallizesometerminologyforfutureuse.

Definition2(NCandLCequilibria).An equilibrium

(

x

,

s

)

withs

=

1 andx

=

c

(

L

,

1

,

α

)

—inwhichno

R

playerscheat—is calledaNoCheating(NC) Equilibrium.Anequilibrium

(

x

,

s

)

withs

=

p andx

=

c

(

H

,

p

,

α

)

—inwhich

R

playersoftype L

cheat—iscalledaLowCheating(LC) Equilibrium.

3. Equilibriumcharacterization

3.1. NCequilibrium

WecannowworkouttheconditionsunderwhichthemodelhasanNCequilibrium.

Bydefinitioninthiscasetheequilibriumvalueofs is1 —theprobabilityofcheatingis0.Therefore

c

(

L

,

1

,

α

)

=

α

tL

+

1

α

and c

(

H

,

1

,

α

)

=

α

tH

+

1

α

(7)

We alreadyknow thatin an NCequilibrium x

=

c

(

L

,

1

,

α

)

.Therefore,to confirm that

[

c

(

L

,

1

,

α

),

1

]

is an equilibrium,we justneedtocheckthattheparametersofthemodelaresuchthatno

O

playerhasanincentivetodeviateunilaterallyfrom offering x

=

c

(

L

,

1

,

α

)

(whichgivesherapayoffofpreciselyx

=

c

(

L

,

1

,

α

)

sincenocheatingtakesplace).

Deviatingtoanofferbelowc

(

L

,

1

,

α

)

isneverprofitablesinceityieldsalowerpayoffconditionalonnocheatingtaking place,butobviouslycannotdecreaseanyfurthertheprobabilitythatcheatingoccurs.Deviatingtoanofferabovec

(

L

,

1

,

α

)

, andhenceacceptingthat

R

playersoftype L will cheat,canyieldatmostapayoffof pc

(

H

,

1

,

α

)

.Infact thisiswhatan 23 Itisanasymmetricequilibriumofthegamewithacontinuumofplayerswhereplayersofthesametypebehavedifferently.Detailsareavailablefrom theauthorsonrequest.

24 Thus,astrategyprofileinwhichallRplayerscheatanyoffergreaterthanx<αtL+1αandallOplayerssendxwouldnotbeanequilibrium, accordingtoourdefinition.Howeveritwouldbeanon-subgame perfect,oranon-sequential,Nashequilibriumoftheappropriatelymodifiedversionof ourgame.Wearegratefultoananonymousrefereeforpointingthisout.

25 Notethatwhilethelinearityofc(·)allowedustoeasilycomputetheequilibriumoffers,asidefromguaranteeingmonotonicity,itplayednocritical roleintheargumentsofar.

(8)

O

playergetsifshemakes thelargestofferthat

R

playersoftype H willnotcheatupon,takingasgiventheequilibrium valueofs

=

1.Hence,using

(7)

,anecessaryandsufficientconditionforanNCequilibriumtoexististhat

α

tL

+

1

α

p

[

α

tH

+

1

α

]

(8)

3.2. LCequilibrium

The conditionsunder whichan LC equilibrium existscan be worked out ina parallel way.By definitionin thiscase,

s

=

p.Hence

c

(

L

,

p

,

α

)

=

α

tL

+ (

1

α

)

p and c

(

H

,

p

,

α

)

=

α

tH

+ (

1

α

)

p (9)

Aswenotedabove,inequilibriumwhens

=

p itmustbethat x

=

c

(

H

,

p

,

α

)

.Toensurethat

[

c

(

H

,

p

,

α

),

p

]

isan equi-libriumwe then need tocheck that the parameters ofthemodel are suchthat no

O

player hasan incentive to deviate unilaterallyfromoffering x

=

c

(

H

,

p

,

α

)

,whichyields her an expectedpayoffof pc

(

H

,

p

,

α

)

. Withalogic thatis bynow familiar,withoutlossofgeneralitywecanconsideronlythedeviationtoofferingx

=

c

(

L

,

p

,

α

)

—thelargestofferthatwill inducenocheatingfromeithertypeof

R

player,takingasgiventheequilibriumvalues

=

p.Thisdeviationyieldsapayoff ofc

(

L

,

p

,

α

)

.Hence,using

(9)

,anecessaryandsufficientconditionforthemodeltohaveanLCequilibriumis

p

[

α

tH

+ (

1

α

)

p

] ≥

α

tL

+ (

1

α

)

p (10)

3.3. Multipleanduniqueequilibria

It isusefultosumup andsharpen ourpicture ofthepossible equilibriaofthemodelasa functionof theparameter quadruple

(

α

,

p

,

tL

,

tH

)

.Forthesakeofsimplicity,fromnowonwerestrictattentiontoquadruplesawayfromtheboundary

of

[

0

,

1

]

4,satisfyingt

H

>

tL.Thisdispensesusfromhavingtoconsiderseparatelysomeoftheboundarycaseswhichwould

notaddanythingofinteresttoourresults.

Proposition1(Equilibriumset).Theequilibriumsetofthemodelisguaranteedtobenon-empty,andcanbeofthreetypes.Aunique NCequilibrium,denotedastheNCUregime;auniqueLCequilibrium,denotedastheLCUregime;twoequilibria,oneLCandoneNC equilibrium,denotedastheLCNC(ormultipleequilibria)regime.

If(8)issatisfiedand(10)isnot,thenweareintheNCUregime.If(10)issatisfiedand(8)isnot,thenweareintheLCUregime. Finally,if(8)and(10)arebothsatisfied,thenweareinthemultipleequilibriaregime.

Although aformal proof of

Proposition 1

doesnot requiremuch more than usingsome ofthe observations we have alreadymade,forthesakeofcompletenesswepresentonein

Appendix A

.Notethatinthemultipleequilibriaregimeonly two equilibriaare possiblebecauseof oursimplifying

Assumption 1

.As we mentionedabove,without itwe wouldgeta third“intermediate”equilibrium,inwhichtheproportionoftransactionsnotcheateduponisstrictlybetweenp and1.

Thethreeequilibriumregimesof

Proposition 1

arealsoallrobustinthestandardsense.

Proposition2(Parametricconditions).Thesetofparameterquadruples

(

α

,

p

,

tL

,

tH

)

thatyieldtheNCUregimecontainsanopenset.

ThesameistrueforthesetofquadruplesyieldingLCU,andforthoseyieldingLCNC.

Aformalproofof

Proposition 2

isin

Appendix A

.Thesimpleargumentbehindithingesonthefactthat

(8)

and

(10)

can bejointlyrewrittenas

(

1

α

)

p

(

1

p

)

α

(

ptH

tL

)

≤ (

1

α

)(

1

p

)

(11)

withthefirstinequalitycorrespondingto

(10)

andthesecondto

(8)

.Since

(

1

α

)

p

(

1

p

)

< (

1

α

)(

1

p

)

forall

α

and

p intheinterval

(

0

,

1

)

,weareguaranteedtofindrobustparameterconfigurationsthatsupporteachofthethreeregimes. Theparameterspace

P = {(

α

,

p

,

tL

,

tH

)

∈ (

0

,

1

)

4

,

tH

>

tL

}

can bepartitionedintoregions correspondingtothedifferent

equilibriumregimes.Inordertopresentan easytoreadpicture,wefixonevaluefor

α

andonevaluefor p,andwethen dividethesetofallrelevanttH andtL,i.e.theupperdiagonaloftheunitsquare,inthreeregionscorrespondingtothethree

regimes: NC U , LC NC andLC U .The figurebelowisdrawnfor

α

=

1

2 and p

=

3

5.Allpairs

(

tH

,

tL

)

abovethedashed line, representingthelocus tLp

+

1−αα

(

1

p

)

,generateLC U equilibria.Allpairs

(

tH

,

tL

)

belowthedot-dashedline,representing

the locus tLp

+

1−αα(1−pp), give rise to NC U equilibria.Clearly, all pairs

(

tH

,

tL

)

betweenthe dashed anddot-dashed lines

yield LC NC equilibria. Notethat the two lines are paralleland theintercept ofthe dot-dashed one is always above the interceptofthedashedone(foranychoiceof

α

andp in

(

0

,

1

)

).

(9)

The figure is drawn fora particular pair of

α

and p such that all three regions are present;a similar picture would obtain foranypairof

α

and p suchthat

α

>

1

p.Withpairsof

α

and p outsidethisrange,someoftheregionswould disappear.26

We concludethissectionwithan observation,whichwedonotfullyformalizepurelyforreasonsofspace.27Although

they embodya level oftrust s exactly equal to 1,the NCU regime and theNC equilibriumof theNCLC regime are less specialthanmightseematfirstsight.Thefactthattheyyields

=

1 ratherthanjusta“high”s isanartifactofourchoice toconsideronlytwotypesof

R

players,inordertokeepthemodelassimpleaspossible.

Suppose that a small proportion (say

ε

) of a third “very low” type were introduced, characterized by an exogenous componentofthecostofcheatingbetween0 andtL.Thenitwouldbeeasytocheckthatthesameparameterssupporting

an NCequilibriuminthetwo-typesmodelwouldyieldanequilibriumwiths

=

1

ε

,inwhichonlythe“verylow”type of

R

playerscheat.Thisisclearly possibleinastraightforwardwaywheneverthemodelyieldsan NCequilibriumwhich is“strict”inanappropriate sense.Withthisobservationinmind,wewillofteninterprettheNCequilibriaas“hightrust” equilibriaofageneralkind,ratherthanfocusingspecificallyonthefactthattheydisplay“fulltrust”.

4. Societieswithdifferentlevelsoftrust

4.1. Differencesinfundamentalsvs.equilibriumswitch

Differencesoftrustlevelsacrossdifferentsocietiesarewelldocumented.Moreover,theyareoftenseentobecorrelated with (and sometimes held responsible for) phenomena of primary economic importance like income levels and growth rates.

Oursimplemodelhighlightsthatsuchdifferencescanbetracedtotwoconceptuallydistinctsources.Twosocietiesmay exhibit differentlevels oftrustbecausethey differinsomeoftheir fundamentalparameters (thevector

(

α

,

p

,

tL

,

tH

)

inour

model).Alternatively,thetwosocietiesmay,infact,bethesame intermsoftheirfundamentalparameters,buthaveselected differentequilibria among those supported by their commonparameter vector. Forbrevity, we refer to thispossibility by sayingthattheirdifferentlevelsoftrustareattributabletoaswitch betweendifferentequilibria.

Inshort,wesaythatanequilibriumswitch isobtainedwhenthetwosocietiesarecharacterizedby thesame parameter

quadruple generatingtheNCLCregime,theLCequilibriumisrealizedinoneofthesocieties (thelowtrustone)whilethe NCequilibriumisrealizedintheother(thehightrustsociety).

We insteadrefer to differencesinfundamentals whendifferentlevels oftrust reflectthe fact thatthe two societiesare characterized by different parameterquadruples, leading to different levels ofequilibrium trust. Thisin turncan happen

26 Forvaluesofαandp suchthatα

<12−−pp,theinterceptofthedashedline(andofcourseofthedot-dashedlineaswell)wouldbeabove1.Hence,in thiscaseallpairs(tH,tL)wouldyieldanequilibriumoftypeNC U .Forvaluesofαandp suchthat12−−pp<α<1−p,theinterceptofthedot-dashedline wouldbeabove1,whilethatofthedashedlinewouldbebelow1.Hence,inthiscaseallpairs(tH,tL)thedashedlinewouldyieldanequilibriumoftype NC U ,allpairsabovewouldgiverisetoanequilibriumLC NC .

(10)

in severaldifferentways, asthe two quadruplescan be in differentregions of theparameter space supporting different equilibriumregimes,ortheymightsupportthesameequilibriumregimebutwithdifferentequilibriumtrustlevels.

Amongthemanywaysinwhichadifferenceinfundamentalscanarisewewilllatersingle outthecaseinwhichonly the probability mass of the two types of

R

players changes, with all other parameters constant. This we refer to as a difference in thedistributionoftypes.Specifically, in one societythe parameter quadruple is

(

α

,

p

,

tL

,

tH

)

,in theother is

(

α

,

p

,

tL

,

tH

)

, and in both the LC equilibrium obtains.28 In the first casethe equilibrium trust level is s

=

p and in the

seconditiss

=

p

<

s.

Whileitisclearthat notwosocietieswillinfact everbeidenticalintheirfundamentals,thetheoreticalpossibilityof an equilibriumswitch cautions againsthastilyinterpretingdifferentlevels oftrustasevidenceofdeep-rooteddifferences. Moreover,the“partialidentifiability”oftheequilibriumswitchregimethatwehavementionedabove(andtowhichwewill turnnext)providessomeguidancetotheempiricalinvestigationontheultimatesourcesoftheobserveddifferentlevelsof trust.

Clearly,thedistinctionbetweendifferencesinfundamentalsandequilibriumswitch hasinteresting policyimplications. Differentpolicy prescriptionswillbemoreappealingaccordingtowhether,inordertoraisetrust levels,achangein fun-damentalsoran equilibriumswitch isneeded.Consider aGovernmentseekingtointervenetorectify alowleveloftrust. Ifthelow trust levelistheresultofan equilibriumswitch, thentheGovernment canattempttoengendera switch toa higherleveloftrustbyre-focusing theexpectations oftheplayers.Convincingeveryoneinsocietythattheleveloftrustiss

insteadofs willdothejob,sincethisiscapableofbecomingaself-fulfillingprophecy.

Ifinsteadthelowleveloftrustisduetoadifferenceinfundamentals,thentheonlywaytoensureahighleveloftrust istoengineerachangeintheparametersofthemodel.Thisisconceptuallyandoperationallydifferentfromaself-fulfilling change in beliefs, and likely harder to achieve, although one might imagine a range of tools that vary from economic incentivestoeducationalprogramsbasedon“civicculture”thatcouldbebroughttobear.

Theconceptualdifferencebetweenthetwohypotheses,differenceinfundamentalsversusequilibriumswitch,mightalso beofinteresttoscholarsinotherdisciplines,suchaspoliticalscienceorsociology.Inessence,adifferenceinfundamentals points todifferentnorms ofbehaviorbeingrootedin“anthropological”factors,whilean equilibriumswitchpoints inthe directionofrandomeventstriggeringchangesthatbecomelong-lastingbecauseofself-reinforcementmechanismsatwork insociety. Ourstaticmodelis ofcourse silent abouttheprocess that mighthaveled to oneequilibriumor theother to prevail, but hasthe merit ofverifying that both possibilities are logically consistentand suggestsan empirical test that, undersomeconditions,canidentifywhichofthetwohypothesesistrue.

4.2. Partialidentification:trustandtransactionlevels

Supposethatweobservetwosocieties,onewithahighandtheotherwithalowleveloftrust.Canweidentifywhether thedifferingtrustlevelsareduetoanequilibriumswitchortoadifferenceinthefundamentals?Aswediscussedalready inSection1andSubsection

2.1

,undersomeconditionstheanswerisaqualifiedyes.

If the elasticity of the cost of cheating withrespect to s issmaller than 1, there are observations that allow us to rule out an equilibriumswitch, while other observations are inconclusive.In thisspecific sense we then say that partial identification29ispossible.Theconditionontheelasticityissatisfiedinthemodelasspecifiedabove,duetothelinearityof

thecostofcheatingins.

Full identificationcan be achievedifthe range ofpossible alternatives is further restricted inan appropriate way.In particular, if we know that the only differences in the fundamentals that need to be considered are variations in the distributionoftypes,thenfullidentificationbecomespossible.

Webeginwiththegeneralcase,inwhichallpossiblevariationsinparameterquadruplesareconsidered.

Thekeytoidentification—partialorfull,asthecasemaybe—istheequilibriumlevelofx,theoffermadeinequilibrium byallthe

O

players,whichisalsotheequilibriumindividualtransactionlevel.

Thefollowingtwopropositionsmakeourclaimpreciseforthegeneralcase.

Proposition3(Individualtransactionlevels—equilibriumswitch).LetaparameterquadruplesupportingtheNCLCregimebegiven. Thenthelevelofx intheassociatedNCequilibriumis lowerthanthelevelofx intheassociatedLCequilibrium.

Itfollowsthatifthedifferenceinequilibriumtrustbetweentwosocietiesisduetoan equilibriumswitchinthesenseof Subsec-tion4.1,thentheequilibriumlevelofx is negativelycorrelatedwiththeequilibriumtrustlevel.

Ifwethenobservethattheequilibriumlevelofx ishigherinthesocietywithhigherleveloftrustwecanruleoutthe possibilityofanequilibriumswitch.

Whenitcomestoadifferenceinfundamentals,inprincipleweshouldconsideranypairofequilibria,oftheNCand/or LC type. For completeness, the following proposition examines four cases. The bottom line is that when we allow for

28 So,clearly,bothparameterquadruplesmustgiverisetoeithertheLCUortheNCLCregimes. 29 Seefootnote3above.

(11)

unrestricteddifferencesinfundamentals,therelationshipbetweendifferencesintheequilibriumtrustlevelandthelevelof individualtransactioncannotbepinneddown.

Proposition4(Individualtransactionlevels—changeinfundamentals).FixarbitrarilyaparameterquadruplesupportinganNC equilibrium.ThenwecanfindparameterquadruplesthatsupportLCequilibriawithindividualtransactionlevelsthatcanbeboth higherandlowerthaninthegivenNCequilibrium.

FixagainarbitrarilyaparameterquadruplesupportinganNCequilibrium.ThenwecanfindparameterquadruplesthatsupportNC equilibriawithindividualtransactionlevelsthatcanbebothhigherandlowerthaninthegivenNCequilibrium.

FixarbitrarilyaparameterquadruplesupportinganLCequilibrium.ThenwecanfindparameterquadruplesthatsupportNC equi-libriawithindividualtransactionlevelsthatcanbebothhigherandlowerthaninthegivenLCequilibrium.

Finally,fixagainarbitrarilyaparameterquadruplesupportinganLCequilibrium.Thenwecanfindparameterquadruplesthat supportLCequilibriadisplayingahigherleveloftrustandalowerlevelofindividualtransactionsthanintheoriginalLCequilibrium, oralowerleveloftrustandahigherlevelofindividualtransactionsthanintheoriginalLCequilibrium.

The picture yieldedby

Propositions 3 and 4

— namely,partialidentification— changeswhen weconsider thecasein which parameterdifferencesare restrictedtobedifferencesinthedistributionoftypesinthesense mentionedabove. In thiscasethe ambiguityhighlighted by

Proposition 4

nolonger holdsanda definiteconclusion canbe reachedaboutthe correlationbetweentrustandthesizeofindividualtransactionsfollowingthedifferenceinparametervalues.

Proposition 5 (Transaction levels change in distribution of types).Consider two parameter quadruples

(

α

,

p

,

tL

,

tH

)

and

(

α

,

p

,

tL

,

tH

)

withp

>

p,eachgivingrisetoanLCequilibriumwithhigh(s

=

p)andlow(s

=

p)trustlevelsrespectively.

Thentheequilibriumlevelofx associatedwith

(

α

,

p

,

tL

,

tH

)

ishigherthantheequilibriumlevelofx associatedwith

(

α

,

p

,

tL

,

tH

)

.

Itfollowsthatifthedifferenceinequilibriumtrustbetweentwosocietiesisduetoadifferenceinfundamentalsinthenarrower senseofa differenceinthedistributionoftypes,thentheequilibriumlevelofx is positivelycorrelatedwiththeequilibriumtrust level.30

Formal proofsof

Propositions 3, 4 and 5

,which,again, consistoffairly simplealgebra,appearin

Appendix A

.Here,we elaborateontheintuitionbehindourresults,beginningwith

Proposition 4

.

RecallthatinanyNCequilibriumthelevelofindividualtransactionis

α

tL

+

1

α

whileinanyLCequilibriumitisgiven

by

α

tH

+(

1

α

)

p.

Proposition 4

isthentheresultofthefollowingobservations.Ifweareallowedtovaryalltheparameters

atwill,thenecessaryandsufficientconditions

(8)

foranNCequilibriumtoexistarecompatiblewithanyvalue of

α

tL

+

1

α

in

(

0

,

1

)

.Similarly, ifwe areallowedtovaryall theparametersatwill,thenecessaryandsufficientconditions

(10)

foranLCequilibriumtoexistarecompatiblewithanyvalueof

α

tH

+ (

1

α

)

p in

(

0

,

1

)

.

Thestatementof

Proposition 5

isanimmediateconsequenceofthefactthatinanyLCequilibriumthelevelofindividual transactionsisgivenby

α

tH

+ (

1

α

)

p.Hence,sinceallotherparametersarekeptconstant,itmustincreaseasp increases.

The intuition behind Proposition 3 requires some intermediate steps. Recallthat inthis casewe are concerned with a single parameter quadruple supporting the NCLCregime. Call thelevel ofindividual transactions in theLC equilibrium

xLC

=

α

tH

+ (

1

α

)

p.IntheLCequilibriuman

O

playergetsanexpectedpayoffofp xLC,sincewithprobability1

p she

ischeatedbythe

R

playershemeetsandendsup withapayoffofzero.Shecould howeverdeviateandmakethelargest offerthatwillinduceno cheatingfromanytypeof

R

players,

α

tL

+ (

1

α

)

p.Denotethisby xDLC.Bydeviating,shewould

getapayoffequaltoherofferforsure,henceherincentiveconstraintallowsustoconcludethat p xLC

xLCD mustbetrue.

Now considerthe NC equilibriumassociated withthe given parameterquadruple. The equilibrium level ofindividual transactions nowisxNC

=

α

tL

+

1

α

,since thisisthe largestofferthatwillkeep all

R

playersfromcheating.The only

difference betweenxDLC andxNC is infact given by the higherlevel of equilibriumtrust reflected in xNC. This makes it

immediatetocheckthatwehavexD

LC

>

pxNC.

Thetwostepswehaveoutlinedgivethat pxLC

xLCD andxDLC

>

pxNC respectively,whichtogetherimmediatelyyieldthe

claimof

Proposition 3

,namelyxLC

>

xNC.

Toseethecriticalroleoftheelasticityofthecostfunctionmentionedabove,takeageneralfunctionalformforthecostof cheatingc

(

t

,

s

)

.ItmuststillbetruethatinthelowtrustequilibriumtheofferisthelargestonethatinducestypeH players

not tocheat—namelyc

(

tH

,

p

)

—andinthehightrust(nocheating) equilibriumtheofferisthelargestonethatinduces

type L playersnottocheat—namelyc

(

tL

,

1

)

.31Theincentiveconstraintof

O

playersinthelowtrustequilibriumrequires

that what they expectto obtain— namely pc

(

tH

,

p

)

—be atleastaslargethan whatthey can getdeviating unilaterally

(i.e., not changing the equilibriumlevel oftrust) andloweringtheir offer, soas to inducealso the morecheating-prone players(type L)not tocheat,andatthesametime raisingthe probabilityofcompleting thetransaction(in fact,making surethatthetransactioniscarriedout).Formally,itmustthenbethecasethat pc

(

tH

,

p

)

c

(

tL

,

p

)

.Comparenowc

(

tL

,

p

)

30 Notethatoncewefixallparametersexceptfor p,allNCequilibriaarethesame. Hence,since theeffectofaswitchfromLCtoNCisalready characterizedbyProposition 3,theonlyrelevantcomparisonistheonetreatedhere—twoLCequilibria.AlltheothercaseswetreatedinProposition 4 canbeignored.

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