Contents lists available atScienceDirect
Games
and
Economic
Behavior
www.elsevier.com/locate/geb
Equilibrium
trust
✩
Luca Anderlini
a,
∗
,
Daniele Terlizzese
baGeorgetownUniversity,UnitedStates bEIEFandBankofItaly,Italy
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c
l
e
i
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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.
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 evidencepresentedinCharness 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.
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. MorerecentlyBigoni
et al. (2016)findthat the differencesincooperativebehavior between northernandsouthernItalycannotbeexplainedbyproxiesforeithersocialcapitaloramoralfamilism.Theyconcludethat persistentdifferencesinsocialnormsareresponsiblefortheobservedgap.Webelievethatourset-upgeneratingmultiple equilibriawithdifferentleversoftrustresonateswellwiththeirempiricalfindingsandanalysis.Wealsoselectivelyrecallthecontributionsby
Knack and Keefer (1997)
,La Porta et al. (1997)
,andmorerecentlyGuiso et
al. (2004).Thesestudiesalldocumentinavarietyofwayshowthepresenceoftrustiscorrelatedwithdesirableeconomic outcomes.Ourone-sidedset-upiscloselyrelatedtothestudyoftrustinexperimentswithasenderandareceiverthatgoesback to
Fehr et al. (1993)
andBerg et al. (1995)
andismorerecentlyfoundinGlaeser 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)
,wasassumedbyHuang and Wu (1994)
andbyDufwenberg (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,psychologicalcost 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
andDufwenberg 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.
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)
andDufwenberg (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 Heterogeneouscheatingcostsyield,inasimpleway,anequilibriuminpurestrategieswithaninteriorleveloftrust(neitherfullnorzero trust),whilepsychologicalgameswithnoheterogeneityonlyobtaininteriorlevelsoftrustinamixedstrategyequilibrium. Admittedly,ourapproachhasacost.Bycuttingthroughthechainofhigherorderbeliefswecannotexploretheforward inductionargumentswhich
Dufwenberg (1996, 2002)
andBattigalli and Dufwenberg (2009)
haveusedtoselectamongthe possibleequilibria.Butwhiletheseargumentsareapowerfultoolinthecaseofasingleoffererandasingle receiver,we pursue amodelin whichofferers andreceiversare anonymouslymatched afterbeingdrawn fromlargepopulations. We believethataforwardinductionargumentlosessomeofitsappealinthiscase.9Charness 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.
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—orsimplyR
players. ThereisacontinuumofO
playersofmass1,andsimilarlyacontinuumofR
playersofmass1.The
O
andR
playersarethen randomlymatchedtoformaunit massofpairs.Theonlything thatisofconsequence hereisthatanO
playershouldnotknowthe“costofcheating”(tobedefinedshortly)oftheplayersheismatchedwith.13 EachO
player makesan offerx∈ [
0,
1]
totheR
player inhermatch. Theoffergeneratesa totalsurplusof2x to be splitequallybetweenO
andR
ifthetransactiongoesthroughwithout“cheating”onthepartofR
.So,ifR
doesnotcheat, anofferofx generatesapayoffofx forbothO
andR
.14Itisthe
R
playerinthematchwhodecideswhethertocheatornot.Afterreceivinganofferx fromtheO
playerinthe match,R
maydecidetocheatandgrabtheentiresurplus2x insteadofabidingbywhatthesplittingproceduresuggests. However,ifhecheats,R
willalsosufferacostc.15Therefore,R
willcheatif2x−
c>
x orequivalently x>
c,andwillnotcheatotherwise.16
Thetotalcostofcheatingc hastwocomponents.Onedependsontheexogenouslygiven“type”ofthe
R
playerandthe otherisdeterminedbythebehaviorofotherplayersinthemodel.Forsimplicity,weassume thatthereare justtwo typesof
R
players, “high”(H )and“low”(L).17 Theexogenouscom-ponentofthecostofcheatingistL
∈ (
0,
1)
fortype L andtH∈ (
0,
1)
fortype H ,withtL<
tH.Theproportionoftype H isdenotedby p
∈ (
0,
1)
throughout.Thecomponentofc thatis“sociallydetermined”isthesameforallplayers.18Lets betheproportionof
R
playerswho donot cheat.Wesimplysetthesocialcomponentofthecheatingcosttoequals andwetakethetwocomponentsofthe cheatingcosttocombineinalinearway.19ForanR
playeroftypeτ
∈ {
L,
H}
,thetotalcostofcheatingifaproportionsoftransactionsgothroughwithoutcheatingisgivenby
c
(
τ
,
s,
α
)
=
α
tτ+ (
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.
requiresthattheelasticityofc
(
τ
,
α
,
s)
withrespecttos belessthanone,andthelinearitypostulatedin(1)
issufficientto guaranteethatthisisthecase.20Finally, 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)
assignsanumberσ
O∈ [
0,
1]
toeveryO
player—theofferx thatshemakes—anda cut-offvalue
σ
τ∈ [
0,
1]
toeachR
player oftypeτ
∈ {
L,
H}
,indicatingthat hewillcheatifandonly ifhereceives anofferstrictly above
σ
τ .21Notice thatoncea profileσ
isgiven,a valueof s isalsogivensincetheexpectedproportionoftransactionsthatwillnotinvolvecheatingisdetermineddirectlyby
σ
.22Anequilibriuminourmodelis astrategyprofile
σ
∗= (
σ
O∗,
σ
L∗,
σ
H∗)
such that theO
playersmaximize their expected return,giventhestrategiesoftheR
playersofeach type(andtheirrelative weightinthepopulation), andtheR
players ofeach typedecide optimallywhetherto cheatorsplitwhateverofferthey received,giventhestrategies oftheotherR
playersofeachtype(andtheirrelativeweightinthepopulation).Webegintheanalysisoftheplayers’maximizationproblemonthe
R
side.Fixaσ
,andthereforeavalueofs.Consider anR
playeroftypeτ
∈ {
L,
H}
.Hewillcheatifandonlyiftheofferx hereceivessatisfiesx
>
c(
τ
,
s,
α
)
=
α
tτ+ (
1−
α
)
s.
(2)For given
σ
, and hence s, using (2) we can compute the mass of playersR
who will not cheat on any givenofferx
∈ [
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
,wecanthereforewritetheexpectedpayoffofanO
playeroffering x asxP(
x,
s)
(recallthatofferingx,she getsapayoffofx wheneversheisnotcheated).Henceshewillchooseanx thatsolvesmax
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) dependsonthecomparisonbetween
c
(
L,
s,
α
)
and p c(
H,
s,
α
)
(5)Ifc
(
L,
s,
α
)
>
pc(
H,
s,
α
)
thenitisuniquelyoptimaltosetx=
c(
L,
s,
α
)
—thelargestlevelthatensuresthatnoR
players cheat.Ifinsteadc(
L,
s,
α
)
<
pc(
H,
s,
α
)
theuniquesolutionistosetx=
c(
H,
s,
α
)
—thelargestlevelthatensuresthatonlyR
playersoftype L cheat.Finally ifc(
L,
s,
α
)
=
pc(
H,
s,
α
)
, then anO
player is indifferentbetweenmaking an offer ofc
(
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
)
,allO
playersmakean offerofc(
L,
s,
α
)
whichisnotcheated onwithprobabilityone,ratherthananofferofc(
H,
s,
α
)
whichischeatedonbyallR
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
Assumption 1 makes the behavior ofall
O
playersuniquely determinedfor anyparameter valuesandanylevel of s,substantiallysimplifyingtheanalysis.
Notethatthebehaviorthat
Assumption 1
postulates canbeinterpretedastheresultof“lexicographic”risk-aversionof theO
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 theoffertheyallmaketotheR
playertheyareeachmatchedwith.Ofcourse,inequilibriumitmustalsobethecasethat thevalueofs thatappearsin(4)
isthecorrectone,asdeterminedbythebehavioroftheR
playersgivenx.Thisjustifies thefollowingdefinitionofequilibrium.Definition1(Equilibrium).Anequilibriumisapair
(
x,
s)
suchthatx solves(4)
givens andsuchthatP
(
x,
s)
=
s (6)Itisworthnotingthatourdefinitionofequilibriumimplicitlyinvokesanotionofsequentialrationality,asthe
R
players reactoptimally,andareknownbytheO
playerstoreactoptimally,toany offertheymightreceive.24Suchanotionwould correspondtotherequirementofsubgameperfection,ifweweretomaketheassumptionthateachR
playerobservesthe offerreceivedbyalltheotherR
players,sothatthestageofthegameinwhichR
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 ofR
players cheat is clearly never optimal. Hence in equilibriumitmustbethat eithers=
1 (andnocheatingatalltakesplace),ors=
p (andallR
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.25Atthispoint,itisusefultocrystallizesometerminologyforfutureuse.
Definition2(NCandLCequilibria).An equilibrium
(
x,
s)
withs=
1 andx=
c(
L,
1,
α
)
—inwhichnoR
playerscheat—is calledaNoCheating(NC) Equilibrium.Anequilibrium(
x,
s)
withs=
p andx=
c(
H,
p,
α
)
—inwhichR
playersoftype Lcheat—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 justneedtocheckthattheparametersofthemodelaresuchthatnoO
playerhasanincentivetodeviateunilaterallyfrom offering x=
c(
L,
1,
α
)
(whichgivesherapayoffofpreciselyx=
c(
L,
1,
α
)
sincenocheatingtakesplace).Deviatingtoanofferbelowc
(
L,
1,
α
)
isneverprofitablesinceityieldsalowerpayoffconditionalonnocheatingtaking place,butobviouslycannotdecreaseanyfurthertheprobabilitythatcheatingoccurs.Deviatingtoanofferabovec(
L,
1,
α
)
, andhenceacceptingthatR
playersoftype L will cheat,canyieldatmostapayoffof pc(
H,
1,
α
)
.Infact thisiswhatan 23 Itisanasymmetricequilibriumofthegamewithacontinuumofplayerswhereplayersofthesametypebehavedifferently.Detailsareavailablefrom theauthorsonrequest.24 Thus,astrategyprofileinwhichallRplayerscheatanyoffergreaterthanx∗<αtL+1−αandallOplayerssendx∗wouldnotbeanequilibrium, accordingtoourdefinition.Howeveritwouldbeanon-subgame perfect,oranon-sequential,Nashequilibriumoftheappropriatelymodifiedversionof ourgame.Wearegratefultoananonymousrefereeforpointingthisout.
25 Notethatwhilethelinearityofc(·)allowedustoeasilycomputetheequilibriumoffers,asidefromguaranteeingmonotonicity,itplayednocritical roleintheargumentsofar.
O
playergetsifshemakes thelargestofferthatR
playersoftype H willnotcheatupon,takingasgiventheequilibrium valueofs=
1.Hence,using(7)
,anecessaryandsufficientconditionforanNCequilibriumtoexististhatα
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.Hencec
(
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 noO
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 inducenocheatingfromeithertypeofR
player,takingasgiventheequilibriumvalues=
p.Thisdeviationyieldsapayoff ofc(
L,
p,
α
)
.Hence,using(9)
,anecessaryandsufficientconditionforthemodeltohaveanLCequilibriumisp
[
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,fromnowonwerestrictattentiontoquadruplesawayfromtheboundaryof
[
0,
1]
4,satisfyingtH
>
tL.Thisdispensesusfromhavingtoconsiderseparatelysomeoftheboundarycaseswhichwouldnotaddanythingofinteresttoourresults.
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,forthesakeofcompletenesswepresentoneinAppendix A
.Notethatinthemultipleequilibriaregimeonly two equilibriaare possiblebecauseof oursimplifyingAssumption 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
isinAppendix 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)
forallα
andp intheinterval
(
0,
1)
,weareguaranteedtofindrobustparameterconfigurationsthatsupporteachofthethreeregimes. TheparameterspaceP = {(
α
,
p,
tL,
tH)
∈ (
0,
1)
4,
tH>
tL}
can bepartitionedintoregions correspondingtothedifferentequilibriumregimes.Inordertopresentan easytoreadpicture,wefixonevaluefor
α
andonevaluefor p,andwethen dividethesetofallrelevanttH andtL,i.e.theupperdiagonaloftheunitsquare,inthreeregionscorrespondingtothethreeregimes: NC U , LC NC andLC U .The figurebelowisdrawnfor
α
=
12 and p
=
35.Allpairs
(
tH,
tL)
abovethedashed line, representingthelocus tLp+
1−αα(
1−
p)
,generateLC U equilibria.Allpairs(
tH,
tL)
belowthedot-dashedline,representingthe locus tLp
+
1−αα(1−pp), give rise to NC U equilibria.Clearly, all pairs(
tH,
tL)
betweenthe dashed anddot-dashed linesyield 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)
).The figure is drawn fora particular pair of
α
and p such that all three regions are present;a similar picture would obtain foranypairofα
and p suchthatα
>
1−
p.Withpairsofα
and p outsidethisrange,someoftheregionswould disappear.26We 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 toconsideronlytwotypesofR
players,inordertokeepthemodelassimpleaspossible.Suppose that a small proportion (say
ε
) of a third “very low” type were introduced, characterized by an exogenous componentofthecostofcheatingbetween0 andtL.Thenitwouldbeeasytocheckthatthesameparameterssupportingan NCequilibriuminthetwo-typesmodelwouldyieldanequilibriumwiths
=
1−
ε
,inwhichonlythe“verylow”type ofR
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)
inourmodel).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 .
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 theseconditiss
=
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.
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 byProposition 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,appearinAppendix A
.Here,we elaborateontheintuitionbehindourresults,beginningwithProposition 4
.RecallthatinanyNCequilibriumthelevelofindividualtransactionis
α
tL+
1−
α
whileinanyLCequilibriumitisgivenby
α
tH+(
1−
α
)
p.Proposition 4
isthentheresultofthefollowingobservations.Ifweareallowedtovaryalltheparametersatwill,thenecessaryandsufficientconditions
(8)
foranNCequilibriumtoexistarecompatiblewithanyvalue ofα
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 transactionsisgivenbyα
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.IntheLCequilibriumanO
playergetsanexpectedpayoffofp xLC,sincewithprobability1−
p sheischeatedbythe
R
playershemeetsandendsup withapayoffofzero.Shecould howeverdeviateandmakethelargest offerthatwillinduceno cheatingfromanytypeofR
players,α
tL+ (
1−
α
)
p.Denotethisby xDLC.Bydeviating,shewouldgetapayoffequaltoherofferforsure,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 allR
playersfromcheating.The onlydifference 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,whichtogetherimmediatelyyieldtheclaimof
Proposition 3
,namelyxLC>
xNC.Toseethecriticalroleoftheelasticityofthecostfunctionmentionedabove,takeageneralfunctionalformforthecostof cheatingc
(
t,
s)
.ItmuststillbetruethatinthelowtrustequilibriumtheofferisthelargestonethatinducestypeH playersnot tocheat—namelyc
(
tH,
p)
—andinthehightrust(nocheating) equilibriumtheofferisthelargestonethatinducestype L playersnottocheat—namelyc
(
tL,
1)
.31TheincentiveconstraintofO
playersinthelowtrustequilibriumrequiresthat 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.