On the nature of reciprocity : evidence from the ultimatum reciprocity measure

On the nature of reciprocity: Evidence from the ultimatum reciprocity measure ^夽

Andreas Nicklisch

, Irenaeus Wolff

aUniversityofHamburg&MaxPlanckInstituteforResearchonCollectiveGoods,Bonn,Germany

bUniversityofKonstanz,Germany

cThurgauInstituteofEconomics(TWI),Kreuzlingen,Switzerland

a rt i c l e i nf o

JELclassification:

C91 D03 D63 Keywords:

Distributionalfairness Experiments

Intention-basedfairness Reciprocity

Ultimatumbargaining

a b s t ra c t

Weexperimentallyshowthatcurrentmodelsofreciprocityareincompleteinasystem- aticwayusinganewvariantoftheultimatumgamethatprovidessecond-moverswith amarginal-cost-freepunishmentoption.Forasubstantialproportionofthepopulation, thedegreeoffirst-moverunkindnessdeterminestheseverityofpunishmentactionseven whenmarginalcostsareabsent.Theproportionoftheseparticipantsstronglydependson atreatmentvariation:higherfixedcostsofpunishmentmorefrequentlyleadtoextreme responses.Thefractionsofpurelyselfishandinequity-averseparticipantsaresmalland stable.Amongthevarietyofreciprocitymodels,onlyoneaccommodates(ratherthanpre- dicts)partsofourfindings.Wediscusswaysofincorporatingourfindingsintotheexisting models.

1. Introduction

Despiteatraditionofresearchonreciprocalbehaviorthatspansalmostthreedecades,thedevelopmentoftheoriesof reciprocalbehaviorstillisfarfromcomplete.Oneindicationisthattherehasbeenaproliferationofreciprocitymodels(e.g., Rabin,1993;DufwenbergandKirchsteiger,2004;Sobel,2005;FalkandFischbacher,2006;Coxetal.,2007)thatallseemto fitspecificsituationsbetterthanothers,andyetthereisnoclearindicationofwhichmodeltochooseinwhatsituation.In his2005reviewarticle,Sobelcriticizestheexistingmodelsofreciprocalbehaviorforpresentingautilityfunctionofothers’

andownincomewithoutprovidinganexplanationforhowmuchweightplayersarelikelytoputonothers’incomerelative totheirown.Morespecifically,allofthemodelspositthattheharshnessofareactiontoanunkindactionisdeterminedby thetrade-offbetweenareductionintheotherplayer’spayoffandthecostsofpunishment.Forcostsofpunishmentthatare sufficientlylow,thesemodelsthereforepredicttheharshest-possiblereactiontoeventheslightestdegreeofunkindness.We

夽 WegratefullyacknowledgethemanyusefulcommentsprovidedbyChristophEngel,BerndIrlenbusch,Hans-TheoNormann,theTWIresearchgroup, anassociateeditor,twoanonymousreferees,andtheparticipantsoftheEconomicScienceAssociationmeeting2011inLuxemburg,the4thMaastricht BehavioralandExperimentalEconomicsSymposium2011,andtheEuropeanEconomicAssociationconference2011inOslo.Wearedeeplyindebtedto JörgOechsslerforpushingusintherightdirection,andtheMaxPlanckSocietyforfinancialsupport.

∗ Correspondingauthorat:SchoolofBusiness,EconomicsandSocialScience,UniversityofHamburg,von-Melle-Park5,20146Hamburg,Germany.

Tel.:+4940428388976;fax:+4940428386713.

E-mailaddresses:andreas.nicklisch@wiso.uni-hamburg.de,nicklisch@coll.mpg.de(A.Nicklisch),wolff@twi-kreuzlingen.ch(I.Wolff).

Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-0-320244

https://dx.doi.org/10.1016/j.jebo.2012.10.009

argue–andshowempirically–thatthisiswrong.However,aslongasthemarginalcostsofpunishmentarestrictlypositive, itisimpossibletofalsifytheabove-mentionedmodelsalongtheselines:itisalwayspossibletoadjustthereciprocation parameterssuchastoaccommodatethedata,giventhereciprocation-parameterdistributionisleftunspecifiedinthemodel expositions.ThissubstantiatesasecondcriticismSobel(2005,p.407)expresses,namelythattheabilityofintention-based modelsofreciprocitytoaccountforexperimentalresultsis“atributetotheirflexibilityratherthanactualsupportforthe formulation.”Tocorroboratetheargument,weintroducetheultimatumreciprocitymeasurewhicheliminatesthemarginal costsofpunishmentaltogether.Ourexperimentaldatashowthatasubstantialproportionofthepopulationdeviatesfrom themodels’extremepreditioninasystematicway,providingvaluableinsightsintohowexistingmodelsneedtobeamended.

Inarecentcontribution,Coxetal.(2008)abandonthedomainofexplicitfunctionalformsandmakeafirststepto addressSobel’s(2005)firstcriticism.Ourexperimentsuggeststhattheirmodelmaybeanimportantstepforward,being abletoaccommodate27–47%ofourobservationsinadditiontowhatcanbeexplainedusingthemoreconventionalmodels.

Nevertheless,themodelstillispronetoSobel’ssecondcriticismofalackofspecificity:aswediscussinSection3,themodel accommodatesratherthanpredictsourobservations.Thewaysinwhich itfailsonthespecificitydomainwillprovide guidancewithrespecttothedirectioninwhichtorefinethemodel.

Anotherquestionthathasattractedincreasedattentionintherecentscholarlydiscussionisthatofpreferencehetero- geneity.Inthecontextofourgame,thisparticularlyconcernstherelativeimportanceofintention-basedreciprocalmotives andinequityaversion(notablyproposedbyBoltonandOckenfels(2000)andFehrandSchmidt(1999)).Dependingonthe situation,oneortheotherseemstodominate.Infact,thereissomeindicationthatbothplayarole:theresultsofthemini- ultimatumgameexperimentsbyFalketal.(2003)andCoxandDeck(2005)demonstratetheimportanceofbothapproaches.

Whentheproposerhastheoptiontoofferanequaldistributionofearningsandanunequalonefavoringherself,therespon- derrejectsthelattersignificantlymoreoftenthanwhentheproposerhastochoosebetweentheunequalandanevenmore unequaldistributionofearnings(inFalketal.,44.4%versus8.9%).Obviously,thisresultpointstotheimportanceofreci- procity.However,whentheproposerhasnooptionbuttochoosetheunequaloffer,stillasubstantialnumberofresponders (18%)reject.Asthereisnointentiontofavorherselfonthepartoftheproposer,thisobservationsuggeststhatinequity aversionisasecondempiricallyrelevanttriggerforrejections.Otherexperimentshaveshownsimilarpatterns(e.g.,onthe convexultimatumgame,Andreonietal.(2003),onthree-personultimatumgames,Bereby-MeyerandNiederle(2005),and onathree-persongiftexchangegame,ThöniandGächter(2007)).

Theultimatumreciprocitymeasure(urmgame)hasthefollowingstructure:aproposermakesaproposalofhowtodivide anendowmentE.¹Therespondercaneitheracceptorreject.Inthefirstcase,theproposalisimplemented,inthesecond, theresponderobtainsafixedfraction,<1,oftheofferxandfreelychoosestheproposerpayofffromtheinterval[0, E−x].Theimportantfeatureoftheurmgameisthat(incontrasttomostothergameswithpunishmentintheliterature) punishmentisfreeofmarginalcosts,onlycomingatacostthatisfixedoncetheofferismade.²Thisfixedcostiseither equaltohalftheofferortothreequartersoftheoffer,dependingonthetreatment.Aswewillshowbelow,modelsof inequity-aversionandreciprocityleadtoverydifferentpredictionsforbehaviorintheurmgame:thefirstclassofmodels predictsthatresponders–iftheyrejectanoffer–leavetheproposerswithapayoffwhichequalstheirearnings.Incontrast, themajorityofreciprocitymodelspredictsthatrespondersleavetheproposerswithzeroearnings.

Theresultsweobtainarestriking.Lessthan10%oftheobservationscanbecharacterizedasstemmingfrompayoff- maximizers,modelsofinequityaversionaccountfor16–18%,conventionalmodelsofreciprocityfor17–38%.³Atthesame time,wefindasubstantialfractionofafourthtypethatdeviatesfromthesepredictionsinasystematicway,whichwecall gradualreciprocators.Theseplayersarecharacterizedbypunishmentpatternsthatleavetheirproposerswithpayoffsthat areincreasingintheoffermadebutgenerallyleadtounequalpayoffs.Moreover,thefractionoftheseplayersisdetermined bythetreatmentparameter.Inthetreatmentwithahighfixedcostofpunishment,20%ofthepopulationseemtoswitch frombeinggraduallyreciprocaltoconformingtoconventionalreciprocitymodels.Theseobservationscallforanextension ofexistingmodelsofreciprocityinthespiritofSobel’sfirstcriticism:acharacterizationofthesituationthatleadstothe predictionofthetypedistributioninducedbythesituation.

InSection5,wediscussanumberofapproaches ofhowtomodifytheexistingmodelsinlightofourobservations.

Inparticular,wecharacterizethegradual-reciprocatortypewithintheframeworkofCoxetal.(2008),havingdismissed theideaofmatchingtheother’sdegreeofkindnessduetoalackofobservationsofthecorrespondingresponse-pattern predictions.Withrespecttoourtreatmenteffect,wenotethatwhatappearsasanauxiliaryassumptionthatis“sometimes (...)useful”(Coxetal.,2008,p.34)seemstobeanessentialingredientofatheoryofreciprocalbehavior.Asanalternative, weproposethesituation’scoercivenessasapromisingexplanation,definedintermsofthegapbetweenthehighestpayoff theplayercanobtaininthegivensituationandthenext-lowerobtainablepayoff.Anevaluationoftheidea’spredictive power,however,isbeyondthescopeofthisarticleandisleftforfutureresearch.

Theremainderofthepaperisorganizedasfollows:Section2introducestheurmgameandpresentstheexperimental designandprocedure.Section3analyzesthegameaccordingtothepayoff-maximizationmodel,inequityaversion,and

1AsymbolstablecanbefoundinAppendixA.

2Forgamesthatallowforachangeintheotherplayer’spayofffreeofmarginalcosts,cf.,e.g.,EngelmannandStrobel(2004),orFismanetal.(2007), whoexaminethisquestioninthedictatorgame.

3Notethatwedonotconsidertheproposersinourgame;cf.Section3.

Proposer

0 E

x Responder

accept reject

0 E κx

y

{E x, x} {y, κx}

{ , }:π π

Fig.1. Gametreeoftheultimatumreciprocitymeasure.

severaltypesofreciprocitymodels,alwaysfocusingonresponderbehavior.Subsequently,weanalyzetheexperimentaldata withrespecttothesepredictionsandpointtotheexistenceofaplayertypethathasreceivedlittleattentionintheliterature sofarinSection4.InSection5,weexplorepossibledirectionsinwhichtoextendexistingmodelsofreciprocalbehaviorto enablethemtopredictthekindofbehaviorobserved.Finally,wesummarizeourfindingsandconcludeinSection6.

2. Thegame,experimentaldesign,andprocedure 2.1. Theultimatumreciprocitymeasure(urmgame)

Liketheclassicultimatumgame,theurmgamehastwoplayers,aproposerandaresponder.Theproposerisgivenan endowmentofEandoffersx,0≤x≤E,totheresponder.Iftheresponderacceptstheoffer,theproposerkeepsE−x,while theresponderearnsx.Iftheresponderrejectstheoffer,theresponderearnsx(theconflictpayoff^c_r)withacommonly knownparameter∈[0,1),whiletheproposer’sconflictpayoff^c_pisanyamounty,y∈[0,E−x],whereyisfreelychosen bytheresponder.Therefore,thepayofffunctionsfortheproposer,_p,andtheresponder,_r,respectively,are

_p =

y, otherwise, and

r =

Fig.1illustratesthegametreeoftheurmgame.Notethatrestrictingtheresponsesettoy=0andsetting=0yieldsthe standardtwo-personultimatumgame(Güthetal.,1982).Duetotheserestrictions,thestandardultimatumgameprovides littleinformationaboutnegativereciprocityasadriverforrejection(sinceitreducestheresponder’sdecisiontoachoice betweenonlytwoalternatives).Incontrast,byimposingnomarginalcostsonresponderstoalterproposers’payoffsafter arejection,the‘unrestricted’urmgameisabletoprovideaverydetailedpictureofparticipants’motivationsforrejections (aswillbecomeclearfromthediscussionoftheoreticalpredictionsinthenextsection).Inparticular,thelackofatrade-off betweenownmonetaryincomeandproposerpayoffprovidesnewinsightsintothenatureofother-regardingpreferences.

2.2. Experimentaldesignandprocedure

Eachparticipantplayedoneanonymousurmgameeitherintheroleoftheproposerorintheroleoftheresponder.Inthe instructions,wereferredtoproposersaspersonAandtorespondersaspersonB.ThepiesizewassettoE=12Euros.Offers couldonlybemadeinintegers.Inordertoanalyzeindividualheterogeneityofresponsescorrespondingtodifferentoffersin greaterdetail,weappliedthestrategy-vectormethodtoelicitresponders’choices(Selten,1967).Thismeansthatresponders hadtomakeadecisionforeachpossible(integer)offerbeforetheywereinformedabouttheactualoffer.Then,theofferand thecorrespondingresponderdecisiondeterminedthepayoffs.Thisprocedureimpliedthatrespondershadtomakeatotal of13acceptance/rejectiondecisions.Additionally,theyhadtodeterminethepayoffofproposersforanyoffersrejected.

Incontrasttothestandardprocedureofthestrategy-vectormethod,responderswerenotprovidedwithachoicemenu, thatis,adecisionsheetthatpresentsallpotentialoffersinanascendingordescendingorder.Rather,potentialofferswere presentedsequentiallywithoutapossibilityofreviewingearlierdecisions,andtheorderofpossibleoffersdifferedrandomly forallresponders.Weintroducedthisprocedureforseveralreasons.Theone-by-oneprocedurewaschosentomakeeach decisionassalientaspossible.Further,elicitingdecisionsonebyoneincombinationwitharandomorderwasintendedto keepanypotentialexperimenter-demandeffectsmallbyisolatingdecisionsasmuchaspossible:to‘smoothen’aresponse- patternoveralldecisionsoutofatasteforconsistencywouldinflicthighcognitivecostsonparticipants.Consequently,a smoothresponse-patternshouldonlybeobservedifparticipantsexhibitedunderlyingpreferencesgivingrisetoit.Finally, theorderwasrandomlydeterminedforeachparticipantindividually,inordertocontrolforpossibleordereffects.

Theexperimentstartedsuchthatcopiesoftheinstructionswerehandedouttoparticipantsandreadaloud.Subsequently, participants’questionsconcerningtheexperimentswereansweredprivatelybytheinstructors.Finally,allparticipantshad toansweranelectronicquestionnairetestingtheirunderstandingofthegameandthepayoffstructure.⁴Beforeparticipants answeredthequestionnaire,itwasmadeclearthattheonlypurposeofthequestionnairewastoimprovetheunderstanding oftherulesofthegame.Wronganswerswereprivatelyexplainedandcorrectedbeforetheexperimentstarted.

Aftertheyhadmadeallpayoff-relevantdecisions,responderswereaskedtostatewhichoffertheyconsideredasfair, andwhichoffertheyexpectedtoreceive.Subsequently,werandomlymatchedeachrespondertoaproposerandpayoffs wererealizedaccordingtothedecisionsmade.Participantswereinformedabouttheirpayoffsandaskedtoanswerashort socio-demographicquestionnaire,beforeprivatelybeingpaid.

Inordertolearnmoreaboutthenatureofreciprocalpreferences,weplayedthegameundertwotreatmentconditions.

Inthehigh-condition,thecommonlyknownparameterwassetto=0.5,whileinthelow-condition,weset=0.25.

Aswewillshowbelow,this(ratherinnocent)variationhaslittleimplicationforthepredictionsoftheconsideredsocial- preferencemodels,whilethereareimportantdifferencesinactualbehavior.Intotal,76pairsofproposersandresponders participatedinthehigh-treatment,whilewehad77pairsinthelow-condition.

ThelaboratoryexperimentswereconductedattheEconLabattheUniversityofBonn,Germany,inOctoberandNovember 2006.⁵Intotal,306participantsparticipated;50%oftheparticipantswerefemale,themedianagewas23years.Participants weremostlyundergraduatestudentsfromvariousfieldsofstudies.Approximatelyonethirdofthestudentswereeconomists ormathematicians.Furtherinformationconcerningthesocio-demographicbackgroundoftheparticipantsissummarized intheonlinesupplementarymaterial(availableonthejournal’swebsite).Averagepaymentwas5.15Euros(noshow-up fee)foranaveragedurationof30min,includingtheinstructiontimeandthetimeforpayingparticipants.

3. Theoreticalpredictions

Ourcentralresearchinterestliesintheempiricalanalysisofreciprocalbehavior.Forthisreason,wewillfocusonthe behaviorofrespondersthroughoutthepaper.Proposerbehaviorisunsuitableforourpurposes:proposalsreflect both proposers’other-regardingpreferencesaswellasproposers’strategicconsiderationsconcerningtheother-regardingpre- ferencesofresponders.

Wewillanalyzeresponders’best-responsefunctionsaccordingtoallmajormodelsthatarepotentialcandidatesfor theexplanationofreciprocalbehavior.Forbrevityandeaseofexposition,werefrainfrompresentingthecompletesetsof equilibriaastheydonotshedfurtherlightonourresearchquestion.Inthefollowing,wediscussthree(groupsof)models, the‘standard’game-theoreticprediction,modelsofinequity-aversion,andintention-basedmodelsofreciprocalbehavior.

Beforewedoso,letusclarifysomenotation.Ifamodelpredictsrejectionofanoffer,itwillhavetospecifyavalueforthe responseythatmaybedifferentdependingontheoffer.Toreflectthis,wewillwritey=y(x)todenotethe(offer)response function.Yet,thereisasecondwaytothinkaboutresponses,whichwillproveusefulparticularlyinthecontextoftreatment comparisons.Forthispurpose,weintroducetheconflict-payoffresponsefunction(definingtheresponseyintermsofthe conflictpayoff_r^c=x),whichwewilldenotebyy=(^c_r).

3.1. Purepayoff-maximizingpreferences

Thebestreplyofaresponderexclusivelydrivenbymaterialself-interestisobvious:given0<<1,wehavex>xfor anyx>0,andx=xforx=0.Consequently,payoff-maximizingresponders’bestreplyistoalwaysacceptanypositiveoffer x,andarbitrarilyacceptorrejectaproposalofx=0.Giventhisfeature,wewillnotobservevaluesofyfortheseplayers.Ifat all,weobserveavalueforyinresponsetox=0;however,thetheorydoesnotgiveanypredictionforthisvalue.Therefore, payoff-maximizingresponders’best-responsefunctionisgivenbybr_pm:x→(ı,y),whereı∈{0,1}representsrejection,ı=0, oracceptance,ı=1:

br_pm(x)=

(ı,y) ifx=0, (1)

where(ı,y)∈

(ı,y)|ı∈{0,1},y∈[0,E]

.Ofcourse,notreatmentdifferencesareexpected.

3.2. Inequity-aversepreferences

Inafirststep,notethatinequity-averseresponderswillalwayschoosetoequalizepayoffsafterarejection,sinceitis costlesstoaltertheproposer’spayoffoncethecostsofrejectingaresunk.Inotherwords,theirresponseywillbey(x)=xfor allrejectedoffersx.Whichofferswillberejected?BoththemodelbyFehrandSchmidt(1999)andbyBoltonandOckenfels (2000)predictacceptedoffersxtocomefromaconvexset[x;x],where0≤x<E/2<x≤E.Thespecificvaluesofxandx

4TranslationsoftheGermaninstructionsandthequestionnaireareprovidedintheonlinesupplementarymaterial(availableonthejournal’swebsite).

5ExperimentswerecomputerizedusingzTree(Fischbacher,2007).Fortherecruitmentofparticipants,weusedORSEE(Greiner,2004).

dependontheparametersofthemodel,notablyon(sinceitdeterminesthemonetaryearningsincaseofaconflict)and theimportancetheindividualresponderplacesonequityconcerns.Toindicatethedependencebetweenxand,andxand ,wewillwritexandx.Bothmodelswouldsuggesttheretobeheterogeneityinthecut-offvaluesforrejections,whileall modelsofinequityaversionmaketheuniquepredictiony(x)=x.Insummary,weobtainthefollowingbest-replyfunction br_ia:x→(ı,y):

br_ia(x)=

(1,.) ifx≤x≤x. (2)

Thepredictedtreatmenteffectsareevident:anincreaseinshiftsbothacceptancethresholds‘inwards’towardstheegali- tarianpayoffdistribution(E/2,E/2).Withrespecttoresponsesasafunction(^c_r)ofconflictpayoffs,notreatmentdifferences areexpected.

3.3. Intention-basedpreferences

Forourdiscussionofthesemodels,wesub-dividethisclassintofoursub-classes:(i)oneinwhichutilityfunctionsconsist ofalinearcombinationofownincomeandareciprocityterm(whichitselfisaproductofseveraltermsasdescribedbelow, cf.Rabin,1993;Levine,1998⁶;DufwenbergandKirchsteiger,2004;FalkandFischbacher,2006),(ii)thenon-linearmodel ofCoxetal.(2007),(iii)modelsmixingreciprocityconcernsandinequalityaversion,andfinally,(iv)themodelpresented byCoxetal.(2008).

‘Linear’reciprocitymodels.Inthemodelssubsumedunderthisclass,utilityisalinearcombinationofownincomeand reciprocity.Here,reciprocityisaproductofthreeterms.Thefirstweightstheimportanceofreciprocalbehaviortothe person.Thesecondtermcapturesthekindnessoftheotherplayer’spastbehavior.Offersarerankedfromunkind(i.e.,small) tokind(i.e,large)ones,suchthatthetermforanofferwhichisneitherkindnorunkindiszeroandincreases(decreases) monotonicallywitheachrankabove(below)thatoffer.Consequently,unkindoffershavenegativevaluesandkindoffers havepositivevalues.Thethirdtermmeasuresthedegreeofkindnessintheperson’sreaction.Again,responsesareranked suchthataresponsethatisneitherkindnorunkindcorrespondstoavalueofzero,andresponsesabove(below)thatleadto valuesincreasing(decreasing)monotonicallywitheachrank.Duetothemonotonicityofkindness,acceptedoffersforman interval:ifrejectinga(un)kindofferyieldslessutilitythanacceptance,rejectingaless(un)kindofferyieldsalsolessutility thanacceptance.

Atthetimeoftheresponder’sdecisionintheultimatumreciprocitymeasure,thereciprocityweightandthekindness termintheactingplayer’sutilityfunctionarefixed.Consequently,maximizationofutilityincombinationwiththepossibility ofchoosingtheproposer’spayofffreeofmarginalcostimpliesthefollowingforrejectedoffers:thebestreplytoanyunkind offerxmustbethemostunkindresponsepossible,thatis,y(x)=0,∀^x<x.Conversely,anyrejectedkindoffermustbe answeredwithy(x)=E−x,∀^x>x,thekindestresponsepossible.Inotherwords,therecannotbearejectionfollowedbya responsey(xr),sothat0<y(xr)<E−xr.Asfortheinequity-aversionmodelsabove,theswitchingpointbetweenacceptance andrejectionisplayer-specificandgenerallycannotbepredicted.Thepredictedbest-replyfunctionbr_rl:x→(ı,y)isgiven by:

br_rl(x)=

⎧ ⎪

⎨

⎪ ⎩

(0,E−x) ifx>x, (1,.) ifx≤x≤x

(0,0) ifx<x.

(3)

Notreatmentvariationsarepredictedwithrespecttoy(x)or(^c_r).Theloweracceptancethresholdxriseswith,asa highermakesrejectionlesscostly.Atthesametime,thereisnoclearpredictionwithrespecttox:whileahigherimplies ahigher‘conflict’payoffx,italsoleadstoalowerpotentialforrewardingactions:E−x<E−xfor>.Therefore,the signofthechangeintheupperacceptancethresholddependsontheweighttheresponderplacesonreciprocity.

Non-linearmodelsofreciprocity.EventhoughCoxetal.(2007)proposearemarkablemodelthatgeneralizestheabove reciprocity-modelsinanimportantway,ityieldsthesamepredictionsforresponderbehaviorintheultimatumreciprocity measureasthe‘linear’reciprocitymodels.Infact,utilityisagainalinearcombinationofownincomeandareciprocity term,wherethelattermultipliestheproposer-payoffwithan“emotional-state”function.isafunctionoftheproposer’s previousbehavior,andtherefore,afixedfactoratthetimeoftheresponder’sdecision.Consequently,theargumentsfrom ourdiscussionofthe‘linear’modelscarryoverandhence,thepredictedbest-replyfunctionhasthesameformasEq.(3) above.

6Strictlyspeaking,themodelofLevine(1998)isdifferentfromtheothermodelslistedinanumberofimportantaspects.However,thebestrepliesare verysimilar,givenLevine(1998)definesaplayer’sutilityfunctionasalinearcombinationofallplayers’monetarypayoffs.

my money

your money

6 5

4 3

2 1

0

086421012

payoff set for rejections n HIGHκ

offer payoff set for reject ons in LOWκ

Fig.2.Payoffspaceforresponders.

Mixedapproach.AspecialvariationofreciprocitymodelsistheapproachbyCharnessandRabin(2002)whichmixes reciprocityconcernsandinequalityaversion.⁷Inthismodel,aresponder’sutilityfunctionaddsownpayoffandtheproposer’s payoff,weightedbyatermthatintegratesinequalityaswellasreciprocityconcerns.Inparticular,thisweightislowered iftheproposerreceivesmoremoneythantheresponderandiftheproposermisbehaves.However,eveniftheweightfor reciprocitydependsonthedegreeofmisbehavior(asintheextendedmodelintheappendixofCharnessandRabin,2002), thesumofweightsiseitherpositiveornegative.⁸ Onceagain,thesameargumentsasforthe‘linear’reciprocitymodels apply,leadingtothesamepredictions.

Generalapproachtoreciprocity.Coxetal.(2008)presenttheirnovelapproachtoreciprocalbehaviorwithintheframework oftheproposer-payoff–responder-payoffspace.Inthisspace,thechoicesetSoftheresponderconsistsofonepointandaray paralleltotheproposer-payoffaxis.Thepointdescribestheoffer,whiletheraycharacterizespossiblepayoffcombinations incaseofarejection,asdepictedinFig.2.Noticethatourtreatmentvariationdoesnotchangethelocationofthepoint,but shiftstherayinlow-closertotheproposer-payoffaxiscomparedtothesituationinhigh-.

Responderpreferencesarerepresentedbyindifferencecurves∈,whereisaplayer’sindifference-curvesetfor agivensituation.Toillustrate,indifferencecurvesofpayoff-maximizingplayersarelinesparalleltotheproposer-payoff axis,thoseofinequity-averseplayersareeitherconvex(BoltonandOckenfels,2000)orpiece-wiselinearwithakinkat the45-degreeline(FehrandSchmidt,1999),indifferencecurvesinthe‘linear’reciprocitymodels(aswellasinthe‘mixed’

modelbyCharnessandRabin,2002)arestraightlinesthatareeitherupward-sloping(negativereciprocity)ordownward- sloping(positivereciprocity),whilethemodelofCoxetal.(2007)generalizesthe‘linear’reciprocitymodelsbyallowing theindifferencecurvestobenon-linear;however,theirslopecannotchangesigns.Irrespectiveoftheirshape,indifference curvesalwayscanberankedsuchthatissaidtobe‘higher’thanifpointsassociatedwitharepreferredtopoints associatedwith.Finally,wheneverwetalkaboutthe-definedpointinS,wemeanthepointassociatedwiththehighest indifferencecurveinwhichstillisinthechoicesetS.⁹

AtthecenteroftheapproachbyCoxetal.(2008)aretwobasicdefinitions,oneconcernsperceivedkindnessandthe otherkindnessin(re-)actions.First,letusdefineperceivedkindness,or“generosity”.Inthismodel,thenotionofgenerosity isattachedtoresponders’opportunitysets,or,moreprecisely,toopportunitysetsaftertheyhavebeenalteredbytheaction oftheproposer.¹⁰Particularly,considerthesetSxofpossiblepayoffcombinations(_p,_r)whichproposerandresponder cangainaftertheproposerhaschosenx.Letusdefine ˆ_i(x)=sup

_i Sxfori=p,r.AsetSxiscalled“moregenerousthan”a setSx if(i) ˆ_r(x)−ˆ_r(x)≥0and(ii) ˆ_r(x)−ˆ_r(x)≥ˆ_p(x)−ˆ_p(x).Inotherwords,theproposerismoregenerousby choosingxthanbychoosingxif(i)theproposer’schoiceofxoverxdoesnotleadtoadecreaseinthemaximumpayoff

7Infact,theapproachbyFalkandFischbacher(2006)alsorepresentsamixtureofreciprocityandinequalityconsiderations,asreciprocationbythe responderistriggeredbyproposerchoicesthatleadtounequalpayoffs.

8Thisassertionisnotcompletelycorrect.Underaspecificparametercombination,theextendedmodelintheappendixofCharnessandRabin(2002) allowsforrejectionsinconjunctionwithresponderutilityincreasinginproposerincomeifthelatteriscloseto0,anddecreasingifitisaboveathreshold ofx−b,wherebmeasureshowstronglyanundeservingpoorestsocietymemberisdisregarded.Inthatcase,responsesarepredictedtobey(x)=max{0, x−b},i.e.,theresponsefunctionisparalleltotheresponsefunctionofaninequity-averseplayer(neglectingthecornersolutionsforlowx).Aswefind onlytwooutof153participantsinourdatawhoseresponsepatternisinlinewiththisprediction(apartfromtheinequity-aversionequivalentb=0),we holdthatthisspecialcasecanbeneglectedforeaseofexposition.Thespecificparameterconstellationrequiresthatthecombinedweightplacedona Rawlsiansocialoptimum,ı,isclosetoone(butı<1),thespiteparameterwithrespecttoundeservingplayers,f,issufficientlysmall,f<ı/(1−ı),andthe undeservingarenotdisregardedinthetotal-surplus-maximisingpart,i.e.,k≈0fortheparameterkmeasuringthediscountingofundeservingproposers’

payoffsinthispart,norintheRawlsiansocial-welfarepart,i.e.,b<xforsomerejection-worthyoffers.

9Wedonotrefertothispointasthetangentialpoint,asincaseofacceptanceaswellasforsomeofthemodels,itwouldbeinadequatetospeakof tangents:therecannotbeatangenttoapoint,andinsomecases,the(highest)indifferencecurvewillhaveakinkatthe-definedpoint.

10Strictlyspeaking,thenotionofanopportunitysetasusedbyCoxetal.(2008)wouldruleoutapplicationoftheirmodeltoourgame,astheyrequire opportunitysetstobeconvex.However,wedonotseewhynon-convexityofopportunitysetswouldleadtoproblemsintheanalysis.Hence,wedropthe convexityassumption,asweareconvincedthattheirmodelisanimportanttooltounderstandbehaviorinourgame.

(a)

my money

your money

0 1 2 3 4 5 6

024681012 (b)

my money

your money

0 1 2 3 4 5 6

024681012

Fig.3.Indifferencecurves(a)and(b)(beingmorealtruisticthan).

therespondercanearn,and(ii)theincreaseofresponder’spayoffasaresultofdecisionxcomparedtoxisnotlessthanthe correspondingincreaseintheproposer’spayoff.Accordingtothisdefinition,anofferxintheurmgameismoregenerous thanxifandonlyifx≥x.

Second,letusdefinekindnessinaction,whichistermed“altruism”.Altruismreferstototheresponder’sutilityfunction ur(_r,_p)andthecorrespondingcurvatureoftheresponder’sconvexindifferencecurvesinthepayoff-space{(p,_r)}.

Forconvenience,itisdefinedintermsofaplayeri’swillingness-to-payforamarginalincreaseinthepayoffofplayerj, WTP_i=[∂u_i(_i,_j)/∂_j]/[∂u_i(_i,_j)/∂_i],ratherthani’smarginalrateofsubstitutionMRS_i=1/WTP_i.Theresponder’sutility functionu_r(_r,_p)issaidtobe“morealtruisticthan”u_r(_r,_p)ifWTP_r≥WTP_r,∀⁽r,_p).Equivalently,theutilityfunction associatedwithindifferencecurvesismorealtruisticthanthefunctionassociatedwithif,comparedto,thecurves inarerotatedcounter-clockwise(compareFig.3).Asaconsequence,theproposerpayoff_pinthe-definedpointin Sxmustnotbesmallerthan_pinthe-definedpointinthesameset.

Withinthisframework,reciprocityisdefinedasfollows:aproposer’sdecisionleadingtoSxratherthanSx(Sxbeing moregenerousthanSx)inducesindifferencecurvesratherthanonthepartoftheresponder(withu_r(_r,_p)more altruisticthanu_r(r,p)).Looselyspeaking,moregenerousoffersleadtomorealtruisticpreferences.

Havingoutlinedthemodel,wenowapplyittotheurmgame.Recallthattheconvexindifferencecurvesarerotated clockwiseforlessgenerousoffers.Thatis,thesmallertheoffer,thesteeper–orflatter,incaseofupward-slopingcurves –theindifferencecurves.Asaconsequence,theintersectionbetweenthehighestindifferencecurveandthechoiceset decreasesorremainsconstant,butneverincreasesintheproposer-payoffdimensionforoffersofdecreasinggenerosity.¹¹ Sincealteringtheproposer’spayoffiscostlessandinf

pSx=0,∀^x(i.e.,thelowerboundofthechoicesetdoesnotchangefor differentoffers)wecanconcludethatfortworejectedoffersxandxsuchthatx>x,y(x)≥y(x)musthold.

Whichofferswillberejected?Likeinanyoftheothermodelspresented,themodelproposedbyCoxetal.(2008)assumes thatutilityfromownincomeistradedoffagainstasecondutilitycomponentthatisinfluencedbyothers’income.Ifthe responderrejectsanoffer,theutilitygainsfromthissecondcomponentmustoutweighthedecreaseinone’sownincome.

Hence,therespondermusthaveapositiveWTPinresponsetoverygenerousoffers(e.g.,rejectingx≥xandrespondingby y>E−x)–althoughthisscenarioappearshardlyintuitiveatthefirstglance–orhaveanegativeWTPfor_pinresponse toveryungenerousoffers(e.g.,rejectingx≤xandrespondingbyy<E−x),sothatacceptedofferscomefromaconvexset [x;x].Thespecificvaluesofxandxagaindependontheimportancetheindividualresponderplacesonreciprocity.Thus, weobtainthefollowingbest-replyfunctionbr_re:x→(ı,y):

brre(x)=

(1,.) ifx≤x≤x. (4)

wherey(x)mustsatisfy∂y/∂x≥0.

Withrespecttotreatmenteffects,wefirstturntochangesintheoffer-responsefunctiony(x).Inresponsetoanincrease in,themodelallowsforbothmonotonicincreasesandinvarianceatanygivenlevel,merelyrulingoutreductionsinthe

11Strictlyspeaking,thisargumentrequirespreferencestohavetheincreasingbenevolenceproperty,whichCoxetal.(2008)defineasawillingnesstopay fortheotherplayer’sincomethatdoesnotdecreaseinownincome.

my money

6 5

4 3

2 1

0 your money 086421012

payoff set for rejections in LOWκ

payoff set for rejections in HIGHκ

offer

Fig.4. Examplefortheinfluenceofavariationonofferacceptance.

response.¹²Turningtotreatmenteffectsontheconflict-payoffresponsefunction,recallthatourtreatmentvariationdoes notalterthesupremumof_rinthesetSxforagivenofferx,sincethevalueofchangestheray,butnotthepoint(i.e.,the offer,whichdefinesthesupremum).Hence,thetreatmentvariationdoesnotchangethegenerosityofoffers.¹³Ontheother hand,thesameconflictpayoff^c_risassociatedwithdifferentoffersinthedifferenttreatments:underalow,ahigheroffer isassociatedwiththesameconflictpayoffthanunderahigh.Atthesametime,higheroffersareattachedtohigherlevels ofgenerosityandtherefore,metwithhigherdegreesofaltruism.Asaconsequence,theconflict-payoffresponsefunction (_r^c)maydifferbetweenthetreatments:for<,thesameconflictpayoff^c_r=x=ximplies(x)≥(x).

Withrespecttoacceptancethresholds,themodelpredictsadecreaseoftheacceptancethresholdxfordecreasing.To seethis,taketheofferxthatmakestheresponderindifferentbetweenacceptingandrejectingunder.Letusnowdecrease .Recallthatachangeinleavestheresponder-payoffsupremumunaffectedandhence,indifferencecurvesdonotchange astheresponder’sdegreeofaltruismremainsthesame.Butwiththeindifferencecurvesremainingthesameandtheray ofSxshiftingleft,therespondermustnowprefertoaccepttheoffer,asshowninFig.4.Bythesametoken,changing maychangetheupperthresholdx.Bytheconvexityofpreferencesandthelinearitywithrespecttovariationsofofthe maximum-possiblereward(E−x),itisimmediatelyobviousthatanincreaseincannotbeassociatedwithanincreasein theupperacceptancethreshold(andmoreoftenthannot,itwillleadtoadecreaseinx).

Intheprecedingparagraphs,wehavepresentedqualitativepredictionsthatcanbederivedfromthegeneralmodelby Coxetal.(2008).Virtuallyallofthesepredictionshavebeenweak,inthesenseexemplifiedbythestatementthat“foroffers ofdecreasinggenerosity,theresponsecannotincrease.”Byemployingweakinequalitiesinalloftheirdefinitions,Coxetal.

(2008)encompassalloftheexistingmodelpredictionsinoneframework.Themodelwouldbeabletoaccountevenfor offer-responsefunctionsequaltoastrictlypositiveconstant,incontrasttoanyoftheothermodels.Itdoesplaceanumber ofrestrictionsonbehaviorthatcanbeexpected,mostnotablyperhapstherequirementofacertaindegreeofconsistency.

However,itdoesnotmakeclearpredictionsliketheremainingmodelspresented.Inotherwords,whatthemodelgainsin generality,itloosesintermsofspecifity.Weconsiderthisanimportantshortcomingandbrieflyreviewpotentialdirections ofmodelrefinementtoeschewthisprobleminSection5ofthisarticle.Topreparethefloorfortheresults,wesummarize thepredictionsfromthedifferentmodelsinTable1.

4. Results

Westructurethepresentationofourresultsasfollows:first,wecharacterizerejectionbehaviorofresponders.Second,we analyzeresponsepatterns.Third,wesystematicallyrelateresponsepatternstorejectionbehavior.Werelegatepresentation ofdataonoffers,expectedoffers,aswellasofrole-contingentaveragepayoffstotheonlinesupplementarymaterial(available onthejournal’swebsite),asthesearenotinthefocusofthisstudy.

4.1. Rejections

84%ofactualoffersin thehigh- condition(81%inthelow-condition) wereaccepted.Followingourtheoretical discussionfromtheprevioussection,wedefineanupperandaloweracceptancethresholdforeachresponderi,x_iandx_i, asfollows:

12Onceagain,thisrequiresinvokingtheincreasingbenevolenceproperty,cf.footnote11.

13Strictlyspeaking,thisstatementisnotcorrect.Please,refertothediscussioninSection5forwhyweholdtheaboveassertiontobeinthespiritofthe model.

Table1

Predictionsofthemodelsdiscussed.

Responsey(x)tooffers 0<x<x x>x

Payoff-maximization Notapplicable:alloffersareaccepted

Inequityaversion x x

Reciprocity(linear/non-linear/mixed) 0 E−x

Coxetal.(2008) ∂y/∂x≥0 y>E−x,∂y/∂x≥0

Treatmenteffectony(x) 0<x<x x>x

Payoff-maximization Notapplicable:alloffersareaccepted

Inequityaversion + +

Reciprocity(linear/non-linear/mixed) 0 −

Coxetal.(2008) +or0 −

Acceptancethresholds ∂x/∂ ∂x/∂

Payoff-maximization Notapplicable:alloffersareaccepted

Inequityaversion + −

Reciprocity(linear/non-linear/mixed) + (0/−or0/0)

Coxetal.(2008) +or0 −or0

Note:Derivationsforpredictionsunderexcessivelyniceoffersareomitted.

Table2

Numbersofrespondersaccordingtoacceptancethresholds.

x_i xi

−1 0 1 2 3 4 5 6 9 10 11 12

high- 0 3 5 5 9 42 11 1 1 1 10 64

low- 1 8 6 9 19 27 6 1 0 1 5 71

x_i = max{x|ı_i(x)=1}and

x_i =

−1, otherwise,

(5)

whereı_i(x)denotestheacceptancedecisionofresponderiforacertainofferx.Notethatinequityaversionandreciprocity modelspredictregularitywithrespecttorejections,thatis,ı_i(x)=1,∀^x∈(x_i,x_i],andı_i(x)=0,otherwise.Intotal,32outof 153respondersexhibitrejectiondecisionsthatviolateregularity.Whilethisismorethantypicallyobserved(seeCamerer, 2003),only9ofthem(6%ofallresponders)makemorethanonedecisionthatwouldcontradictregularity.Weattribute theremaining23violationstothedifficultyarisingfromtherandom-orderone-by-onepresentationofpossibleoffers.To accountforthisfactanduseasmuchinformationaspossible,wechosetheabovedefinitionofx_i.¹⁴Thefurtheranalysis includesthedataofallresponders.Respondersareclassifiedaccordingtotheiracceptancethresholds;Table2reportsthe numberofrespondersineachlowerandupperacceptanceclass,||x_i||and||x_i||,respectively.

Thelowerthresholdsx_iaresignificantlyhigherinhigh-(withanaverageof3.57vs.2.97inlow-,p=0.003;also,ascan beeasilyseenfromTable2,x_ifromthehigh-treatmentfirst-orderstatisticallydominatesx_ifromthelow-condition).¹⁵ Atthesametime,thetreatmentdifferencebetweenupperacceptancethresholdsx_ifailstoreachsignificance(11.80vs.

11.91,p=0.114).However,thisisnotastrongindicationthatthereisnoeffect:whilemostrespondersneverrejectanoffer abovetheequalsplit,thenumberofthosewhodoinhigh-(12outof76)isdoublethecorrespondingnumberfromthe low-treatment(6outof77;again,statisticaldominanceholds).

4.2. Responses

Givenourmainresearchinterestliesinthestudyofreciprocalbehavior,responsesfollowingarejectionarethecentral elementinouranalysis.Inthefollowing,wewillidentifyrejectedoffersasxr,sothattheresponsetoarejectedofferisy(xr).

Inlightoftheexploratorynatureofourstudy,wewanttogetascloseaspossibletotherawdata.Therefore,weclassifythe responsefunctionsaccordingtomutuallyexclusivetypecategories.Wedefinethetypesbasedonthetheoreticalpredictions summarizedinTable1,focussingonthenegative-reciprocitypart,giventhereislittlevarianceinthedomainofpositive

14Ourqualitativeresultsandstatisticalinferencesdonotchangeifwedefinex_iusingthemorestraightforwarddefinitionx_i=min{x|ıi(x)=1},indicating therobustnessofourfindings.

15Unlessotherwiseindicated,allcomparisonsarebasedontwo-sidedWilcoxonrank-sumtests.Behaviorofeconomists/mathematiciansandother participantsdoesnotdiffersignificantlywithrespecttoanyvariablemeasured.

Table3

Frequencyofrespondertypes.

High- (in%) Low- (in%)

Null 3 3.9 6 7.8

Accepters 0 0.0 1 1.3

Symbolic 2 2.6 1 1.3

Gentlepunishers 3 3.9 3 3.9

Inequity-averse 14 18.4 12 15.6

Reciprocal:linear 13 17.1 29 37.7

Reciprocal:gradual 33 43.4 20 26.0

Between 3 3.9 3 3.9

Unclassified 5 6.6 2 2.6

Total 76 77

reciprocity.Participantswhoseresponsescouldnotbefittedintothecategoriesdefinedbythepresentedmodelswere groupedaccordingtothebroadcharacteristicsoftheirresponses.

Null.Subjectsfallingintothiscategoryacceptalloffersexceptforoffersx=0,inwhichcasetheyrespondbyy(0)=0.This behaviorcanbeclassifiedaseitherselfish,inequity-averseorreciprocal,andthereforedoesnotprovidemuchinformation aboutthenulls’motivations.

Accepters.Subjectsfallingintothiscategoryacceptalloffers.

Symbolic.Subjectsfallingintothiscategoryacceptmostoffersbutrejectatleastone.However,theirrejectionisonly symbolic–theyadministertheproposertheamountthelatteraskedfor:y(xr)=E−xr.

Gentlepunishers.Subjectsfallingintothiscategoryrejectsomeoffersbutleavetheproposerbetteroffthanwhatthey wereofferedforxr<6,i.e.,y(xr)>xr.

Inequity-averse.Inequity-averseplayersconformtothepredictionsofthecorrespondingmodels:whenevertheyreject anofferx_rwhichtheydoforatleasttwooffersx,theyrespondbychoosingy(x_r)=x_r.

Reciprocal:linear.Subjectsfallingintothiscategoryconformtothepredictionsofallmajorreciprocitymodels:whenever theyrejectanofferxrwhichtheydoforatleasttwooffersx,theyrespondbypunishingtheotherplayerasharshlyas possible,y(xr)=0.

Reciprocal:gradual.Subjectsfallingintothiscategoryexhibittwocharacteristics:(i)theyarenotinequity-averseplayers and(ii)theirofferresponsefunctionfulfillsy(x_j)≥y(x_i)foranypairofrejectedoffersx_iandx_jsuchthatx_i<x_jandy(x_j)>y(x_i) foratleastonesuchpair.

Between.Subjectsarecategorizedtofallinbetweentheothercategoriesiftheyrejectvariousoffersandchoosey(x_r)such thattheywouldbelongtodifferentcategoriesfordifferentxr.

Table3summarizestheresultsofourclassificationanalysis.¹⁶Wemakethefollowingobservations:First,takinginto accountallresponsepatternsthatcouldpotentiallyresultfrompreferencesofapayoff-maximizingplayer–null,accepters, onesymbolictypeineachtreatmentplusonegentle-punisher–wecountonly13participants(4inhigh-,9inlow-)out of153(8%).Hence,comparedtotypicalresultsfromothervariationsoftheultimatumgame(e.g.,seeAndreonietal.,2003), theurmgameyieldsmuchless‘selfish’behaviorbyresponders.¹⁷

Second,thesumofallplayerswhosebehaviorcanbedescribedbyoneofthetheoreticmodelsoutlinedinSection3 excludingthemodelbyCoxetal.(2008)makesupforonly31outof76inhigh-and50outof77inlow-.Inother words,conventionalmodelsaccountforonlyabout40%(65%)oftheobservedresponsepatternsinhigh-(low-).More specifically,weobserveastable16-18%inequity-averseplayers,whilethenumberbelongingtodifferentsubclassesof reciprocitydifferssubstantiallyacrosstreatmentconditions.Mostparticipantsnotexhibitingbehavioraspredictedbythe abovemodelscanbecategorizedasgraduallyreciprocal.ThemodelofCoxetal.(2008)canaccountfortheseobservations.

However,itaccommodatesratherthanpredictsthem.Below,weexploreanumberofwaysinwhichourunderstandingof graduallyreciprocalbehaviormaybecharacterizedonthebasisoftheirmodel.

Third,inhigh-,thefractionofplayerscategorizedasgradualreciprocatorsishigherthaninlow-byalmost20%.Atthe sametime,thehigh-fractionoflinear-reciprocalplayersislowerbythesame20%.Inotherwords,thedatalookasifachange infrom0.5to0.25changedtheresponsefunctionofabout20%ofthepopulationsuchthattheynolongerdifferentiatethe severityofpunishmentwithrespecttoanoffer’sunkindness.¹⁸Whetherthisisanactualtypeshiftorwhetheritismerely theslopeoftheresponsefunctionbeingshifteddownwardverystronglyissomethingwecannotanswer.Whatwedoknow isthatthefractionofgradualreciprocatorswitharesponse–functionslopelargerthan0.4changesfrom17outof33in

16Notethatweallowedforasingledeviationfromtherespectivepredictions;thiscouldbeanacceptancebelowx_ioraslightnon-monotonicity,e.g.,for gradualreciprocators.Notdoingsowouldleaveuswith24(12)gradualreciprocatorsinhigh-(low-),andwith20(13)unclassifiedresponders.

17Aplausiblereasonforthisobservationisthatpunishmentcostsarelowerinoursetup.Wearethankfultoananonymousrefereeforpointingthisout.

18A ²-testsuggeststhatthetypedistributionsofparticipantsclassifiableaspayoff-maximising,inequity-averse,linear-reciprocal,graduallyreciprocal, andothers,differbetweentreatments,(p=0.015).

Table4

Meanloweracceptancethresholdsbytreatmentandtype.

Respondertype HIGH LOW

Inequity-averse 4.08 3.50

Reciprocal:linear 3.71 3.31

Reciprocal:gradual 4.00 3.15

high-to5outof20inlow-.Inotherwords,responsesingeneraldogetharsherwithincreasingfixedcostsofpunishment alsowithinthegroupofgradualreciprocators.

Inourview,theseobservationsarecritical:furtherdevelopmentofanyreciprocitymodelshouldaccountforbothgrad- uallyreciprocalbehaviorandwhatlookslikeaparameter-inducedshiftinthetypedistribution,ifitistobeseenasastep forwardinourunderstandingofreciprocalbehavior.Forthisreason,wedevoteSection5tosomeideasonpossibledirections inwhichtoextendexistingmodelsofreciprocity,discussingtheminlightofourobservations.Beforewedoso,weshed somelightontheinteractionbetweenresponsepatternsandrejectionsinthefollowingpart.

4.3. Theinteractionbetweenresponsepatternsandrejections

Inthefollowingparagraph,webrieflyreporttheresultsofacomparisonofloweracceptancethresholdsbetweenthethree maintypes,reportedinTable4.Whilewedonotfindanysignificantdifferencesofloweracceptancethresholdsbetween inequity-averse,linear-reciprocal,andgraduallyreciprocalplayerswithineachtreatment(allpair-wisecomparisonsyield p>0.15),weobserveaverydifferentiatedpictureacrosstreatments.Bothforinequity-averseandlinear-reciprocalplayers, thetreatmentdifferencein loweracceptancethresholdsisinthepredicteddirectionbutclearlyfailstobesignificant (p=0.249andp=0.301,respectively).However,forparticipantsclassifiedasgradualreciprocators,thereisatreatment effect:inlow-,theyacceptsignificantlyloweroffersthaninhigh-(p<0.001).Thusforasubstantialfractionofthese players,fairnessconsiderationsaresubstantiallyinfluencedbyasituationalvariation.Onepossiblereadingofthisisthat playerswithhighacceptancethresholdsexhibitparticularsensitivitytothefixedcostsofpunishment:iftheirresponse functionshiftsenoughsothatinlow-,theyareclassifiedaslinearreciprocators,thiswouldexplainthe(non-)significance ofthetreatmentcomparisonsofbothlinearandgradualreciprocators.Onceagain,furtherresearchisneededtoassessthe plausibilityofthisinterpretation.

5. Generalized-reciprocalbehavior

Inthissection,wesetouttoexplorepossibledirectionsinwhichexistingmodelsmaybechangedtoaccountforour findings.Ourdatacallfortwothings.Thepresenceofasubstantialfractionofparticipantswhocanbecategorizedasgradual reciprocatorscallsforatheoreticcharacterizationofsuchplayers.Andtheshiftinthetypedistributioninresponsetoour treatmentvariationcallsforatheorythatisabletopredictthatshift.Ourdiscussionofwaystomeetthesechallengeswill bedividedintotwoparts.First,wereviewanddiscardasimpleextensionoflinearmodelsofreciprocity.Subsequently,we provideamoredetaileddiscussionwithinthemodelofCoxetal.(2008),payingtributetothefactthatitistheonlyavailable modelabletoaccommodateourfindings.

5.1. Linearreciprocitymodels

Inthereciprocitymodelsreviewedinthispaper,therearetwocomponentsofreciprocalbehavior:anassessmentof theotherplayer’skindness,orgenerosity,andthedegreeofreciprocation,oraltruisminaplayer’sresponse.Linearmodels likeCharnessandRabin(2002),DufwenbergandKirchsteiger(2004),orFalkandFischbacher(2006)aggregateanaction’s degreeofgenerosityintoa(relative)weightthatisputontheotherplayer’spayoff;thedegreeofkindnessof–oraltruismin –aresponsethenfollowsfromthemaximizationoftheweightedpayoffsum.Averysimpleideathatwouldbeabletomeet bothchallengesposedbyourdataistomodifythemodelsbyspecifyingplayers’utilityfunctionsuchthatitismaximizedif thedegreeofaltruismmeetsacertaintarget,namelythedegreeofgenerosityoftheotherplayer’saction.Thisisakintowhat mostlegalsystemsdo:matchingpunishmenttotheseverityofanoffence,ratherthanassigningthemaximumpenaltyto allinfringementsalike.

Inprinciple,therearethreewaystoapplythisideatoourgame;however,onlyoneofthemcanaddressbothchallenges posedbyourdata.Asintheearlierreciprocitymodels,aresponderwouldevaluatekindnessagainstthe‘fairness’orneutrality benchmark(inourgame,presumablycorrespondingtotheequalsplit),sothatthedegreeofgenerositytobematchedis givenbyx/6.Afterarejection,theresponderwouldassigntheproposertheabovefraction(i)oftheproposer’sfairshareof6, yieldingy(x)=6·x/6=x,(ii)oftheproposer’sclaim,sothaty(x)=(12−x)·x/6,or(iii)oftheproposer’sclaimaftershrinking