A Theory of Fairness, Competition, and Cooperation

ATHEORY OFFAIRNESS, COMPETITION, ANDCOOPERATION*

ERNST FEHR AND KLAUS M. SCHMIDT

Thereis strong evidence that people exploit their bargaining power in competitivemarkets but not in bilateral bargaining situations. There is also strong evidencethat people exploit free-riding opportunities in voluntary cooperation games.Y et,when they are given the opportunity to punish free riders, stable cooperationis maintained, although punishment is costly for those who punish. Thispaper asks whether there is a simple commonprinciple thatcan explain this puzzlingevidence. We show that if somepeople care about equity the puzzles can beresolved.It turnsout that the economic environment determines whether the fairtypes or the sel sh typesdominate equilibrium behavior.

I. INTRODUCTION Almostall economicmodels assume that all people are exclusively pursuing theirmaterial self-interest and do notcare about‘ ‘social’’ goalsper se. This may be true for some (maybe many) people,but it is certainlynot true for everybody .By nowwe havesubstantial evidencesuggesting that fairnessmotives affect thebehavior of manypeople. The empirical results of Kahneman, Knetsch,and Thaler[1986], forexample, indicate that customers havestrong feelings about the fairness of rms’short-run pricing decisionswhich may explain why some rmsdo not fully exploit theirmonopoly power .Thereis alsoa lotof evidence suggesting that rms’wage setting is constrainedby workers’views about what constitutesa fair wage[Blinder and Choi1990; Agell and Lundborg1995; Bewley1995; Campbell and Kamlani 1997]. Accordingto these studies, a majorreason for rms’refusal to cut wagesin arecessionis thefear that workerswill perceivepay cuts as unfair whichin turnis expectedto affect workmorale ad- versely.Thereare also many well-controlled bilateral bargaining experimentswhich indicate that anonnegligiblefraction of the

*Wewould like to thank seminar participants at the Universities of Bonn andBerlin, Harvard, Princeton, and Oxford Universities,the European Summer Symposiumon Economic Theory 1997 at Gerzense ´e(Switzerland),and the ESA conferencein Mannheim for helpful comments and suggestions. We are particu- larlygrateful to three excellent referees and to Drew Fudenbergand John Kagel fortheir insightful comments. The rst authoralso gratefully acknowledges supportfrom the Swiss National Science Foundation (project number1214- 05100.97)and the Network ontheEvolution of Preferences and Social Norms of theMacArthur Foundation.The second author acknowledges nancialsupport by theGerman Science Foundation through grant SCHM119614-1. r 1999by thePresident and Fellowsof Harvard Collegeand theMassachusetts Institute of Technology. The Quarterly Journal ofEconomics, August 1999

817 818 QUARTERLY JOURNALOF ECONOMICS subjectsdo notcare solely aboutmaterial payoffs [Gu¨th and Tietz, 1990; Roth1995; Camererand Thaler1995]. However,there is alsoevidence that seemsto suggest that fairnessconsiderations arerather unimportant. For example, in competitiveexperimental marketswith completecontracts, in whicha well-dened homogeneousgood is traded, almost all subjectsbehave as if they areonly interested in theirmaterial payoff. Evenif thecompeti- tiveequilibrium implies an extremelyuneven distribution ofthe gains fromtrade, equilibrium is reachedwithin afewperiods [Smith and Williams1990; Roth,Prasnikar ,Okuno-Fujiwara,and Zamir1991; Kachelmeierand Shehata 1992; Gu¨th,Marchand, and Rulliere1997]. Thereis similarlycon icting evidencewith regardto cooperation.Reality providesmany examples indicating that peopleare morecooperative than is assumedin thestandard self-interest model.Well-known examples are that manypeople vote, pay their taxeshonestly ,participate in unionsand protestmovements, or workhard in teamseven when the pecuniary incentives go in the oppositedirection. 1 Thisis alsoshown in laboratoryexperiments [Dawes and Thaler1988; Ledyard 1995]. Undersome conditions it has evenbeen shown that subjectsachieve nearly full cooperation, althoughthe self-interest model predicts completedefection [Isaac and Walker1988, 1991; Ostromand Walker1991; Fehrand Ga¨chter1996]. 2 However,as wewill seein moredetail in Section IV,thereare also those conditions under which a vast majorityof subjectscompletely defect as predicted by theself-interest model. Thereis thusa bewilderingvariety of evidence.Some pieces ofevidence suggest that manypeople are driven by fairness considerations,other pieces indicate that virtually all people behaveas if completelysel sh, and still othertypes ofevidence suggestthat cooperationmotives are crucial. In this paper weask whetherthis conicting evidence can be explained by a single simple model.Our answer to this questionis affirmative if oneis willing toassume that, in addition topurely self-interested people,there are a fraction ofpeople who are also motivated by fairnessconsiderations. No other deviations from the standard

1.On votingsee Mueller [1989]. Skinner and Slemroad [1985] argue that the standardself-interest model substantially underpredicts the number of honest taxpayers.Successful team production in, e.g., Japanese-managed auto factories inNorth America is described in Rehder [1990]. Whyte [1955] discusses how workers establish‘ ‘productionnorms’ ’ underpiece-rate systems. 2.Isaac and Walker and Ostrom and Walker allow for cheap talk, while in Fehrand Ga ¨chter subjectscould punish each other at somecost. ATHEORY OFFAIRNESS,COMPETITION, ANDCOOPERATION 819 economicapproach arenecessary to account for the evidence. In particular, wedo not relax the rationality assumption. 3 Wemodel fairness as self-centeredinequity aversion. Ineq- uity aversionmeans that peopleresist inequitable outcomes; i.e., theyare willing togive up somematerial payoff tomove in the directionof more equitable outcomes. Inequity aversionis self- centeredif peopledo not care per se about inequity that exists amongother people but areonly interested in thefairness of their ownmaterial payoff relativeto the payoff ofothers.We show that in thepresence of someinequity-averse people ‘ ‘fair’’ and ‘‘cooperative’’ as wellas ‘‘competitive’’ and ‘‘noncooperative’’ behavioral patterns canbe explained in acoherentframework. A main insight ofourexamination is that theheterogeneity of preferences interactsin importantways with theeconomic environment. We show,in particular, that theeconomic environment determines thepreference type that is decisivefor the prevailing behaviorin equilibrium.This means, for example, that undercertain competi- tiveconditions a singlepurely sel sh player caninduce a large numberof extremely inequity-averse players tobehave in a completelysel sh manner ,too.Likewise, under certain conditions forthe provision of apublic good,a singlesel sh player is capable ofinducing all otherplayers tocontribute nothing to the public good,although the others may care a lotabout equity .Wealso show,however,that thereare circumstances in whichthe existenceof a fewinequity-averse players createsincentives for a majorityof purely sel sh types tocontribute to the public good. Moreover,theexistence of inequity-aversetypes mayalso induce selsh types topay wagesabove the competitive level. This revealsthat, in thepresence of heterogeneous preferences, the economicenvironment has awholenew dimension of effects. 4 Thereare a fewother papers that formalizethe notion of fairness.5 In particular, Rabin [1993] arguesthat peoplewant to benice to those who treat them fairly and want topunish those whohurt them. According to Rabin, an actionis perceivedas fair if

3.This differentiates our model from learning models (e.g., Roth and Erev [1995])that relax the rationality assumption but maintain the assumption that all playersare only interested in theirown material payoff. The issue of learning is further discussedin SectionVII below. 4.Our paperis, therefore, motivated by a concernsimilar to the papers by Haltiwangerand Waldman [1985] and Russell and Thaler [1985]. While these authorsexamine the conditions under which nonrational or quasi-rational types affectequilibrium outcomes, we analyze the conditions under which fair types affectthe equilibrium. 5.Section VIII deals with them in moredetail. 820 QUARTERLY JOURNALOF ECONOMICS the intention that is behind theaction is kind,and as unfair if the intentionis hostile.The kindness or thehostility of the intention, in turn,depends onthe equitability ofthe payoff distribution induced by theaction. Thus, Rabin’ s model,as ourmodel, is based onthe notion of an equitableoutcome. In contrastto our model, however,Rabin modelsthe role of intentions explicitly .We acknowledgethat intentionsdo play an importantrole and that it isdesirableto model them explicitly .However,the explicit model- ing ofintentions comes at acostbecause it requiresthe adoption of psychologicalgame theory that is muchmore difficult toapply than standard gametheory .In fact, Rabin’s modelis restrictedto two-personnormal form games, which means that veryimportant classesof games, like, e.g., market games and n-personpublic goodgames cannot be analyzed. Sincea majorfocus of this paper is therole of fairness in competitiveenvironments and the analysis of n-personcooperation games, we chose not to model intentionsexplicitly .Thishas theadvantage ofkeeping the model simpleand tractable.We would liketo stress, however, that— althoughwe do not model intentions explicitly— it is possibleto captureintentions implicitly by ourformulation of fairnessprefer- ences.We deal with this issuein SectionVIII. Therest of thepaper is organizedas followed.In SectionII we presentour model of inequity aversion. Section III applies this modelto bilateral bargaining and marketgames. In SectionIV cooperationgames with and withoutpunishments are considered. In SectionV weshow that, on thebasis ofplausible assumptions aboutpreference parameters, the majority of individual choicesin ultimatum and market and cooperationgames considered in the previoussections are consistent with thepredictions of ourmodel. SectionVI deals with thedictator game and with gift exchange games.In SectionVII wediscuss potentialextensions and objec- tionsto our model. Section VIII comparesour model with alterna- tiveapproaches in theliterature. Section IX concludes.

II. A SIMPLE MODEL OF INEQUITY AVERSION An individual is inequityaverse if hedislikesoutcomes that areperceived as inequitable.This de nition raises, of course,the difficult questionof how individuals measureor perceive the fairnessof outcomes. Fairness judgments are inevitably based on akind ofneutralreference outcome. The reference outcome that is usedto evaluate a givensituation is itself theproduct ofcompli- ATHEORY OFFAIRNESS,COMPETITION, ANDCOOPERATION 821 cated socialcomparison processes. In socialpsychology [Festinger 1954; Stouffer1949; Homans1961; Adams 1963] and sociology [Davis 1959; Pollis1968; Runciman1966] therelevance of social comparisonprocesses has beenemphasized fora longtime. One keyinsight ofthis literatureis that relative materialpayoffs affect people’s well-beingand behavior.As wewill seebelow ,withoutthe assumptionthat at leastfor some people relative payoffs matter,it is difficult, if notimpossible, to make sense of the empirical regularitiesobserved in manyexperiments. There is, moreover , directempirical evidence for the importance of relative payoffs. Agell and Lundborg[1995] and Bewley[1998], forexample, show that relativepayoff considerationsconstitute an importantcon- straint forthe internal wage structure of rms.In addition, Clark and Oswald [1996] showthat comparisonincomes have a signicant impact onoveralljob satisfaction. They construct a compari- sonincome level for a randomsample of roughly 10,000 British individuals by computinga standard earningsequation. This earningsequation determines the predicted orexpected wage of an individual with givensocioeconomic characteristics. Then they examinethe impact ofthis comparisonwage on overall job satisfaction.Their main result is that—holding otherthings constant—the comparison income has alargeand signicantly negativeimpact onoveralljob satisfaction. Strongevidence for the importance of relativepayoffs is also provided by Loewenstein,Thompson, and Bazerman[1989]. These authorsasked subjects to ordinally rankoutcomes that differ in thedistribution ofpayoffs betweenthe subject and acomparison person.On thebasis ofthese ordinal rankings,the authors estimatehow relative materialpayoffs enterthe person’ s utility function.The results show that subjectsexhibit a strongand robustaversion against disadvantageous inequality:for a given own income xi,subjectsrank outcomes in whicha comparison personearns more than xi substantially lowerthan an outcome with equalmaterial payoffs. Many subjectsalso exhibit an aversionto advantageous inequality althoughthis effectseems to besigni cantly weakerthan theaversion to disadvantageous inequality. Thedetermination of the relevant reference group and the relevantreference outcome for a givenclass ofindividuals is ultimatelyan empiricalquestion. The social context, the saliency ofparticular agents,and thesocial proximity among individuals areall likelyto in uence reference groups and outcomes.Because 822 QUARTERLY JOURNALOF ECONOMICS in thefollowing we restrict attention to individual behaviorin economicexperiments, we have to make assumptions about referencegroups and outcomesthat arelikely to prevail in this context.In thelaboratory it is usually muchsimpler to de ne what is perceivedas an equitableallocation by thesubjects. The subjectsenter the laboratory as equals,they do not know any- thing abouteach other ,and theyare allocated to different rolesin theexperiment at random.Thus, it isnatural toassumethat the referencegroup is simply theset of subjectsplaying against each otherand that thereference point, i.e., the equitable outcome, is givenby theegalitarian outcome. Moreprecisely ,weassumethe following. First, in addition to purelysel sh subjects,there are subjects who dislike inequitable outcomes.They experience inequity if theyare worse off in materialterms than theother players in theexperiment, and they alsofeel inequity if theyare better off. Second, however ,we assumethat, in general,subjects suffer more from inequity that is totheirmaterial disadvantage than frominequity that is totheir materialadvantage. Formally,considera setof n players indexed by i [ 1, . . . , n , and let x 5 x1, . . . , xn denotethe vector of monetary payoffs. Theutility functionof player i [ 1, . . . , n is given by 1 (1) Ui(x) 5 xi 2 a i max xj 2 xi,0 n 2 1 o jÞ i 1 2 b i max xi 2 xj,0 , n 2 1 o jÞ i wherewe assume that b i # a i and 0 # b i , 1.In thetwo-player case(1) simplies to

(2) Ui(x) 5 xi 2 a i max xj 2 xi,0 2 b i max xi 2 xj,0 , i Þ j. Thesecond term in (1) or(2) measuresthe utility lossfrom disadvantageous inequality,whilethe third termmeasures the lossfrom advantageous inequality.FigureI illustratesthe utility of player i as afunctionof xj fora givenincome xi.Givenhis own monetarypayoff xi, player i’sutility functionobtains amaximum at xj 5 xi.Theutility lossfrom disadvantageous inequality ( xj . xi) is largerthan theutility lossif player i is betteroff than player 6 j(xj , x i).

6.In allexperiments considered in this paper ,themonetary payoff functions ofall subjects were common knowledge. Note that for inequity aversion to be ATHEORY OFFAIRNESS,COMPETITION, ANDCOOPERATION 823

FIGURE I Preferenceswith Inequity A version

Toevaluatethe implications of this utility function,let us start with thetwo-player case.For simplicity ,weassume that the utility functionis linearin inequality aversionas wellas in xi. Thisimplies that themarginal rate of substitution between monetaryincome and inequality is constant.This may not be fully realistic,but wewill showthat surprisingly manyexperimental observationsthat seemto contradict each other can be explained onthebasis ofthis verysimple utility functionalready .However, wewill alsosee that someobservations in dictatorexperiments suggestthat thereare a nonnegligiblefraction of people who exhibitnonlinear inequality aversionin thedomain of advanta- geousinequality (seeSection VI below). Furthermore,the assumption a i $ b i capturesthe idea that a player suffersmore from inequality that is tohis disadvantage. Theabove-mentioned paper by Loewenstein,Thompson, and behaviorallyimportant it is not necessary for subjects to be informed about the nalmonetary payoffs of the other subjects. As long as subjects’ material payoff functionsare common knowledge, they can compute the distributional implica- tionsof any (expected) strategy prole; i.e., inequity aversion can affect their decisions. 824 QUARTERLY JOURNALOF ECONOMICS

Bazerman[1989] providesstrong evidence that this assumption is,in general,valid. Notethat a i $ b i essentiallymeans that a subjectis lossaverse in socialcomparisons: negative deviations fromthe reference outcome count more than positivedeviations. Thereis alargeliterature indicating therelevance of lossaversion inotherdomains (e.g., Tversky and Kahneman[1991]). Hence,it seemsnatural that lossaversion also affects socialcomparisons. Wealso assume that 0 # b i , 1. b i $ 0meansthat werule out theexistence of subjectswho like to be better off than others.We imposethis assumptionhere, although we believe that thereare 7 subjectswith b i , 0. Thereason is that in thecontext of the experimentswe consider individuals with b i , 0havevirtually no impact onequilibrium behavior .Thisis in itself an interesting insight that will bediscussed extensivelyin SectionVII. To interpretthe restriction b i , 1,suppose that player i has a higher monetarypayoff than player j.In this case b i 5 0.5 impliesthat player i is justindifferent betweenkeeping one dollar tohimself and giving this dollar toplayer j. If b i 5 1,then player i is prepared tothrow away onedollar in orderto reduce his advan- tagerelative to player j whichseems very implausible. This is why wedonot consider the case b i $ 1.On theother hand, thereis no justication to put anupper bound on a i.Toseethis, suppose that player i has alowermonetary payoff than player j.In this case player i is prepared togive up onedollar ofhis ownmonetary payoff if this reducesthe payoff ofhis opponentby (1 1 a i)/a i dollars.For example, if a i 5 4,then player i iswilling togive up onedollar if this reducesthe payoff ofhis opponentby 1.25 dollars. Wewill seethat observablebehavior in bargaining and public goodgames suggests that thereare at leastsome individuals with such high a ’s. If there are n . 2players,player i compareshis incomewith all other n 2 1players.In this casethe disutility frominequality has beennormalized by dividing thesecond and third termby n 2 1.Thisnormalization is necessaryto makesure that therelative impact ofinequality aversionon player i’stotalpayoff is independent ofthe number of players. Furthermore, we assume for simplicity that thedisutility frominequality is self-centeredin the sensethat player i compareshimself with eachof the other

7.For the role of status seeking and envy ,seeFrank [1985]and Banerjee [1990]. ATHEORY OFFAIRNESS,COMPETITION, ANDCOOPERATION 825 players,but hedoesnot care per se about inequalities within the groupof his opponents.

III. FAIRNESS, RETALIATION, AND COMPETITION: ULTIMATUM AND MARKET GAMES In this sectionwe apply ourmodel to a well-knownsimple bargaining game—the ultimatum game— and tosimple market gamesin whichone side ofthemarket competes for an indivisible good.As wewill seebelow ,aconsiderablebody ofexperimental evidenceindicates that in theultimatum game the gains from trade areshared relatively equally whilein marketgames very unequaldistributions arefrequently observed. Hence, any alterna- tiveto the standard self-interestmodel faces the challenge to explain both‘ ‘fair’’outcomesin theultimatum game and ‘‘competitive’’ and rather‘ ‘unfair’’ outcomesin marketgames.

A.TheUltimatum Game In an ultimatumgame a proposerand aresponderbargain aboutthe distribution ofa surplus of xed size.Without loss of generalitywe normalize the bargaining surplus toone. The responder’s shareis denotedby s and theproposer ’sshareby 1 2 s.Thebargaining rulesstipulate that theproposer offers a share s [ [0,1] totheresponder .Theresponder can accept or reject s. In caseof acceptancethe proposer receives a (normalized) monetary payoff x1 5 1 2 s,whilethe responder receives x2 5 s. In case of a rejectionboth players receivea monetaryreturn of zero. The self-interestmodel predicts that theresponder accepts any s [ (0,1] and is indifferent betweenaccepting and rejecting s 5 0. Therefore,there is auniquesubgame perfect equilibrium in which theproposer offers s 5 0,which is acceptedby theresponder. 8 Bynowthere are numerous experimental studies fromdiffer- entcountries, with different stakesizes and different experimental procedures,that clearlyrefute this prediction(for overviews

8.Given that the proposer can choose s continuously,anyoffer s . 0 cannot be anequilibriumoffer since there always exists an s8 with 0 , s8 , s whichis also acceptedby the responder and yields a strictly higherpayoff to the proposer . Furthermore,it cannot be an equilibriumthat the proposer offers s 5 0 which is rejectedby theresponder with positive probability .Inthis case the proposer would dobetter by slightlyraising his price— in which case the responder would accept withprobability 1. Hence, the only subgame perfect equilibrium is that the proposeroffers s 5 0whichis accepted by the responder. If there is a smallest money unit e ,thenthere exists a secondsubgame perfect equilibrium in which the responderaccepts any s [ [e ,1]and rejects, s 5 0whilethe proposer offers e . 826 QUARTERLY JOURNALOF ECONOMICS seeThaler [1988], Gu¨th and Tietz[1990], Camererand Thaler [1995], and Roth[1995]). Thefollowing regularities can beconsid- eredas robustfacts (seeTable I). (i) Thereare virtually nooffers above0.5. (ii) Thevast majorityof offersin almostany study is in theinterval [0.4, 0.5]. (iii) Thereare almost no offers below 0.2. (iv) Lowoffers are frequently rejected, and theprobability ofrejection tends todecrease with s.Regularities(i) to(iv) continueto hold for ratherhigh stake sizes, as indicated by theresults of Cameron [1995], Hoffman,McCabe, and Smith[1996], and Slonimand Roth [1997]. The200,000 rupiahs in thesecond experiment of Cameron (seeTable I) are,e.g., equivalent to three months’ income for the Indonesiansubjects. Overall, roughly 60– 80 percentof theoffers in TableI fall in theinterval [0.4, 0.5],while only 3 percentare belowa shareof 0.2. Towhat extentis ourmodel capable ofaccounting for the stylized facts ofthe ultimatum game? T oanswerthis question, supposethat theproposer ’spreferencesare represented by ( a 1,b 1), whilethe responder ’spreferencesare characterized by ( a 2,b 2). Thefollowing proposition characterizes the equilibrium outcome asafunctionof these parameters.

PROPOSITION 1.It is adominantstrategy for the responder to acceptany offer s $ 0.5,to reject s if

s , s8(a 2) ; a 2/(1 1 2a 2) , 0.5,

and toaccept s . s8(a 2).If theproposer knows the preferences ofthe responder, he will offer

5 0.5 if b 1 . 0.5

(3) s* [ [s8(a 2),0.5] if b 1 5 0.5

5 s8(a 2) if b 1 , 0.5 in equilibrium.If theproposer does not know the preferences ofthe responder but believesthat a 2 is distributed according tothe cumulative distribution function F(a 2), where F(a 2) has support [ a , a ] with 0 # a , a , ` ,thenthe probability (fromthe perspective of theproposer) that an offer s , 0.5 is goingto be accepted is givenby 1 if s $ s8(a ) (4) p 5 F(s/(1 2 2s)) [ (0,1) if s8(a ) , s , s8(a )) 0 if s # s8(a ). ATHEORY OFFAIRNESS,COMPETITION, ANDCOOPERATION 827

TABLE I PERCENTAGEOF OFFERS BELOW 0.2 AND BETWEEN 0.4 AND 0.5 IN THE ULTIMATUM GAME

Percentageof Percentageof Study Number of Stake size offers with offers with (Paymentmethod) observations (country) s , 0.2 0.4 # s # 0.5

Cameron[1995] 35 Rp 40.000 0 66 (AllSs Paid) (Indonesia) Cameron[1995] 37 Rp 200.000 5 57 (allSs paid) (Indonesia) FHSS [1994] 67 $5 and $10 0 82 (allSs paid) (USA) Gu¨th etal.[1982] 79 DM 4–10 8 61 (allSs paid) (Germany) Hoffman,McCabe, 24 $10 0 83 andSmith [1996] (USA) (AllSs paid) Hoffman,McCabe, 27 $100 4 74 andSmith [1996] (USA) (allSs paid) Kahneman, 115 $10 ? 75a Knetsch,and (USA) Thaler[1986] (20%of Ss paid) Rothet al. [1991] 116b approx. $10 3 70 (randompay- (USA, Slovenia, mentmethod) Israel,Japan) Slonimand Roth 240c SK 60 0.4d 75 [1997] (Slovakia) (randompay- mentmethod) Slonimand Roth 250c SK 1500 8d 69 [1997] (Slovakia) (randompay- mentmethod) Aggregate resultof 875 3.8 71 all studiese

a. percentage ofequal splits, b.only observations ofthe nal period, c. observations ofall ten periods, d. percentage ofoffers below 0.25,e. without Kahneman, Knetsch, and Thaler [1986].

Hence,the optimal offerof theproposer is givenby

5 0.5 if b 1 . 0.5

(5) s* [ [s8(a ), 0.5] if b 1 5 0.5

[ (s8(a ), s8(a )] if b 1 , 0.5. 828 QUARTERLY JOURNALOF ECONOMICS

Proof. If s $ 0.5,the utility ofaresponderfrom accepting s is

U2(s) 5 s 2 b 2(2s 2 1), whichis always positivefor b 2 , 1 and thus betterthan arejectionthat yields apayoff of0.The point isthat theresponder can achieve equality onlyby destroyingthe entire surplus whichis verycostly to himif s $ 0.5;i.e., if theinequality istohis advantage. For s , 0.5,a responderaccepts the offer only iftheutility fromacceptance, U2(s) 5 s 2 a 2(1 2 2s),is nonnega- tivewhich is thecase only if s exceedsthe acceptance threshold

s8(a 2) ; a 2/(1 1 2a 2) , 0.5.

At stage1 aproposernever offers s . 0.5.This would reducehis monetarypayoff as comparedwith an offerof s 5 0.5,which would alsobe accepted with certaintyand whichwould yield perfect equality.If b 1 . 0.5,his utility is strictlyincreasing in s for all s # 0.5.This is thecase where the proposer prefers to share his resourcesrather than tomaximizehis ownmonetary payoff, sohe will offer s 5 0.5. If b 1 5 0.5,he is justindifferent betweengiving onedollar tothe responder and keepingit tohimself; i.e., he is indifferent betweenall offers s [ [s’(a 2), 0.5]. If b 1 , 0.5, the proposerwould liketo increasehis monetarypayoff at theexpense ofthe responder .However,heis constrainedby theresponder ’s acceptancethreshold. If theproposer is perfectlyinformed about theresponder ’spreferences,he will simply offer s8(a 2). If the proposeris imperfectlyinformed about the responder ’stype,then theprobability ofacceptanceis F(s/(1 2 2s))whichis equal toone if s $ a (1 1 2a )and equalto zeroif s # a /(1 1 a ).Hence,in this casethere exists an optimal offer s [ (s8(a ), s8(a )]. QED Proposition1 accountsfor many of theabove-mentioned facts. It showsthat thereare no offers above 0.5, that offersof 0.5 are always accepted,and that verylow offers are very likely to be rejected.Furthermore, the probability ofacceptance, F(s/(1 2 2s)), isincreasingin s for s , s8(a ) , 0.5.Note also that theacceptance threshold s8(a 2) 5 a 2/(1 1 2a 2)is nonlinearand has someintui- tivelyappealing properties.It isincreasingand strictlyconcave in a 2,and it convergesto 0.5 if a 2 ` .Furthermore,relatively small values of a 2 already yield relativelylarge thresholds. For example, 1 a 2 5 3 impliesthat s8(a 2) 5 0.2 and a 2 5 0.75 impliesthat s8(a 2) 5 0.3. In SectionV wegobeyond the predictions implied by Proposi- tion1. There we ask whether there is adistribution ofpreferences ATHEORY OFFAIRNESS,COMPETITION, ANDCOOPERATION 829 that canexplain notjust the major facts oftheultimatum game but alsothe facts in marketand cooperationgames that will be discussed in thenext sections.

B.Market Gamewith Proposer Competition It is awell-established experimentalfact that in abroad class ofmarketgames prices converge to the competitive equilibrium. [Smith 1982; Davis and Holt1993]. Forourpurposes, the interesting fact is that convergenceto the competitive equilibrium can be observedeven if that equilibriumis very‘ ‘unfair’’ by virtually any conceivablede nition of fairness;i.e., if all ofthegains fromtrade arereaped by oneside ofthe market. This empirical feature of competitioncan be demonstrated in asimplemarket game in whichmany price-setting sellers (proposers) want tosell one unit ofagoodto asinglebuyer (responder) who demands onlyone unit of the good.9 Sucha gamehas beenimplemented in fourdifferent coun- triesby Roth,Prasnikar ,Okuno-Fujiwara,and Zamir[1991]: supposethat thereare n 2 1proposerswho simultaneously proposea share si [ [0,1], i [ 1, . . . , n 2 1 ,tothe responder. The responderhas theopportunity to accept or rejectthe highest offer s 5 maxi si .If thereare several proposers who offered s, one of themis randomlyselected with equalprobability .If theresponder rejects s,notrade takesplace, and all players receivea monetary payoff ofzero.If theresponder accepts s,hermonetary payoff is s, and thesuccessful proposer earns 1 2 s whileunsuccessful proposersearn zero. If players areonly concerned about their monetarypayoffs, this marketgame has astraightforward solution:the responder accepts any s . 0.Hence, for any si # s , 1, thereexists an e . 0suchthat proposer i canstrictly increase this monetarypayoff by offering s 1 e , 1.Therefore, any equilibrium candidate musthave s 5 1.Furthermore, in equilibriuma proposer i who offered si 5 1mustnot have an incentiveto lower his offer.Thus,there must be at leastone other player j who proposed sj 5 1,too. Hence, there is auniquesubgame perfect

9.We deliberately restrict ourattention to simple market games for two reasons:(i) the potential impact of inequityaversion can be seenmost clearly in suchsimple games; (ii) they allow for an explicit game-theoretic analysis. In particular,itiseasyto establish the identity between the competitive equilibrium andthe subgame perfect equilibrium outcome in these games. Notice that some experimentalmarket games, like, e.g., the continuous double auction as developed bySmith [1962], have such complicated strategy spacesthat no complete game-theoreticanalysis is yet available. For attempts in this direction see Friedmanand Rust [1993] and Sadrieh [1998]. 830 QUARTERLY JOURNALOF ECONOMICS equilibriumoutcome in whichat leasttwo proposers make an offer ofone,and theresponder reaps all gains fromtrade. 10 Rothet al.[1991] haveimplemented a marketgame in which nineplayers simultaneouslyproposed si whileone player accepted or rejected s.Experimentalsessions in fourdifferent countries havebeen conducted. The empirical results provide ample evidencein favorof theabove prediction. After approximately veto six periodsthe subgame perfect equilibrium outcome was reached ineachexperiment in eachof thefour countries. T owhat extent can ourmodel explain this observation?

PROPOSITION 2.Suppose that theutility functionsof the players aregiven by (1). Forany parameters( a i, b i), i [ 1, . . . , n , there is a unique subgameperfect equilibrium outcome in whichat leasttwo proposers offer s 5 1whichis acceptedby theresponder . Theformal proof of the proposition is relegatedto the Appendix, but theintuition is quitestraightforward. Note rst that,for similar reasons as in theultimatum game, the responder mustaccept any s $ 0.5.Suppose that herejects a ‘‘low’’ offer s , 0.5.This cannot happen onthe equilibrium path eithersince in this caseproposer i canimprove his payoff by offering si 5 0.5 whichis acceptedwith probability 1and giveshim a strictly higherpayoff. Hence,on theequilibrium path s mustbe accepted. Considernow any equilibriumcandidate with s , 1.If thereis one player i offering si , s,thenthis player should haveoffered slightly morethan s.Therewill beinequality anyway,but by winning thecompetition, player i canincrease his ownmonetary payoff, and hecan turn the inequality tohis advantage. Asimilar argumentapplies if all players offer si 5 s , 1.By slightly increasinghis offer,player i canincrease the probability of winning thecompetition from 1 /( n 2 1) to1. Again, this increases his expectedmonetary payoff, and it turnsthe inequality toward theother proposers to his advantage. Therefore, s , 1 cannot be part ofa subgameperfect equilibrium. Hence, the only equilibriumcandidate is that at leasttwo sellers offer s 5 1. This is a subgameperfect equilibrium since all sellersreceive a payoff of0, and noplayer canchange this outcomeby changinghis action. Theformal proof in theAppendix extendsthis argumentto the

10.Note that there are many subgame perfect equilibria in thisgame. As long astwo sellers propose s 5 1,any offer distribution of the remaining sellers is compatiblewith equilibrium. ATHEORY OFFAIRNESS,COMPETITION, ANDCOOPERATION 831 possibility ofmixed strategies. This extension also shows that the competitiveoutcome must be the unique equilibrium outcome in thegame with incompleteinformation where proposers do not knoweach others’ utility functions. Proposition2 providesan explanation forwhy marketsin all fourcountries in whichRoth et al. [1991] conductedthis experi- mentquickly converged to the competitive outcome even though theresults of the ultimatum game, that havealso been done in thesecountries, are consistent with theview that thedistribution ofpreferencesdiffers acrosscountries. 11

C.Market Gamewith Responder Competition In this sectionwe apply ourmodel of inequity aversion to a marketgame for which it is probably tooearly to speak of well-established stylized facts sinceonly one study with arela- tivelysmall number of independent observations [Gu ¨th,March- and, and Rulliere1997] has beenconducted so far. The game concernsa situationin whichthere is oneproposer but many responderscompeting against eachother .Therules of the game areas follows.The proposer ,whois denotedas player 1,proposes a share s [ [0,1] tothe responders. There are 2, . .., n responders who observe s and decidesimultaneously whether to accept or reject s.Thena randomdraw selectswith equalprobability oneof theaccepting responders. In caseall respondersreject s, all players receivea monetarypayoff ofzero.In caseof acceptanceof at leastone responder ,theproposer receives 1 2 s, and the randomlyselected responder gets paid s.All otherresponders receivezero. Note that in this gamethere is competitionin the secondstage of the game whereas in subsectionIII.B wehave competingplayers inthe rststage. Theprediction of the standard modelwith purelysel sh preferencesfor this gameis again straightforward. Responders acceptany positive s and areindifferent betweenaccepting and rejecting s 5 0.Therefore, there is auniquesubgame perfect equilibriumoutcome in whichthe proposer offers s 5 0 which is acceptedby at leastone responder . 12 Theresults of Gu¨th,March- and, and Rulliere[1997] showthat thestandard modelcaptures

11.Rejection rates in Slovenia and the United States were signi cantly higherthan rejection rates in Japan and Israel. 12.In the presence of a smallestmoney unit, e ,thereexists an additional, slightlydifferent equilibrium outcome: the proposer offers s 5 e whichis accepted byall the responders. T osupportthis equilibrium, all responders have to reject s 5 0.Weassume, however ,thatthere is nosmallest money unit. 832 QUARTERLY JOURNALOF ECONOMICS theregularities of this gamerather well. The acceptance thresholds ofrespondersquickly converged to very low levels. 13 Although thegame was repeatedonly vetimes, in the nal periodthe average acceptancethreshold is wellbelow 5 percentof the available surplus,with 71 percentof the responders stipulating a thresholdof exactly zero and 9percenta thresholdof s8 5 0.02. Likewise,in period5 theaverage offer declined to 15 percentof theavailable gains fromtrade. In viewof thefact that proposers had notbeen informed about responders’ previous acceptance thresholds,such low offers are remarkable. In the nal period all offerswere below 25 percent,while in theultimatum game such lowoffers are very rare. 14 Towhat extentis this apparent willingnessto make and toacceptextremely low offers compatible with theexistence of inequity-averse subjects? As thefollowing propositionshows, our model can account for the above regularities.

PROPOSITION 3.Suppose that b 1 , (n 2 1)/n.Thenthere exists a subgameperfect equilibrium in whichall respondersaccept any s $ 0,and theproposer offers s 5 0.The highest offer s that canbe sustained in asubgameperfect equilibrium is given by

a i 1 (8) s 5 mini[ 2,...,n , . (1 2 b i)(n 2 1) 1 2a i 1 b i 2 Proof. SeeAppendix. The rstpart ofProposition3 showsthat respondercompeti- tionalways ensuresthe existence of an equilibriumin whichall thegains fromtrade arereaped by theproposer irrespective of the prevailing amountof inequity aversion among the responders. Thisresult is notaffected if thereis incompleteinformation about thetypes ofplayers and is based onthe following intuition. Given that thereis at leastone other responder j whois goingto accept anofferof 0,thereis noway forresponder i toaffect theoutcome, and hemayjust as wellaccept this offer,too. However, note that theproposer will offer s 5 0 only if b 1 , (n 2 1)/n.If thereare n

13.The gains from trade were 50 Frenchfrancs. Before observing the offer s, eachresponder stated an acceptancethreshold. If s wasabove the threshold, the responderaccepted the offer; if itwasbelow ,sherejected s. 14.Due to the gap betweenacceptance thresholds and offers, we conjecture thatthe game had not yet reached a stableoutcome after veperiods. The strong andsteady downward trendin all previous periods also indicates that a steady statehad not yet been reached. Recall that the market game of Rothet al. [1991] wasplayed for ten periods. ATHEORY OFFAIRNESS,COMPETITION, ANDCOOPERATION 833 players altogether,than giving away onedollar toone of the respondersreduces inequality by 1 1 [1/(n 2 1)] 5 n/(n 2 1) dollars.Thus, if thenonpecuniary gain fromthis reductionin inequality, b 1[n/(n 2 1)], exceedsthe cost of 1,player 1prefersto givemoney away toone of the responders. Recall that in the bilateral ultimatumgame the proposer offered an equalsplit if b 1 . 0.5.An interestingaspect ofourmodel is that anincreasein thenumber of responders renders s 5 0.5 lesslikely because it increasesthe threshold b 1 has to pass. Thesecond part ofProposition 3, however, shows that there mayalso be other equilibria. Clearly ,apositiveshare s can be sustained in asubgameperfect equilibrium only if all responders can crediblythreaten to reject any s8 , s.Whenis it optimal to carryout this threat?Suppose that s , 0.5 has beenoffered and that this offeris beingrejected by all otherresponders j Þ i. In this caseresponder i canenforce an egalitarian outcomeby rejecting theoffer as well.Rejecting reduces not only the inequality toward theother responders but alsothe disadvantageous inequality toward theproposer .Therefore,responder i is willing toreject this offerif nobodyelse accepts it and if theoffer is sufficiently small, i.e.,if thedisadvantageous inequality toward theproposer is sufficiently large.More formally ,giventhat all otherresponders reject,responder i prefersto reject as wellif and onlyif theutility ofacceptanceobeys

a i n 2 2 (9) s 2 (1 2 2s) 2 b s # 0. n 2 1 n 2 1 i Thisis equivalentto

a i (10) s # s8i ; . (1 2 b i)(n 2 1) 1 2a i 1 b i Thus,an offer s . 0canbe sustained if and onlyif (10) holds for all responders.It is interestingto note that thehighest sustainable offerdoes not depend onall theparameters a i and b i but onlyon the inequity aversion of the responder with thelowest acceptancethreshold s8i.In particular,if thereis onlyone responderwith a i 5 0,Proposition 3 impliesthat thereis aunique equilibriumoutcome with s 5 0.Furthermore, the acceptance thresholdis decreasingwith n.Thus,the model makes the intuitivelyappealing predictionthat for n ` the highest 834 QUARTERLY JOURNALOF ECONOMICS sustainable equilibriumoffer converges to zero whatever the prevailing amountof inequityaversion. 15 D.Competitionand Fairness Propositions2 and 3suggestthat thereis amoregeneral principle at workthat is responsiblefor the very limited role of fairnessconsiderations in thecompetitive environments consid- eredabove. Both propositions show that theintroduction of inequityaversion hardly affects thesubgame perfect equilibrium outcomein marketgames with proposerand respondercompeti- tionrelative to the prediction of thestandard self-interestmodel. In particular,Proposition2 showsthat competitionbetween proposersrenders the distribution ofpreferences completely irrelevant.It doesnot matter for the outcome whether there are manyor only a fewsubjects who exhibit strong inequity aversion. By thesame token it alsodoes not matter whether the players knowor do not know the preference parameters of the other players.The crucial observation in this gameis that no single player can enforcean equitable outcome. Giventhat therewill be inequality anyway,eachproposer has astrongincentive to outbid his competitorsin orderto turn part ofthe inequality tohis advantage and toincrease his ownmonetary payoff. Asimilar forceis at workin themarket game with respondercompetition. As longas thereis at leastone responder who accepts everything, noother responder can prevent an inequitableoutcome. There- fore,even very inequity-averse responders try to turn part ofthe unavoidable inequality intoinequality totheir advantage by acceptinglow offers. It is,thus, the impossibility ofpreventing inequitableoutcomes by individual players that rendersinequity aversionunimportant in equilibrium. Therole of this factorcan be further highlighted by the followingslight modication of the market game with proposer competition:suppose that at stage2 theresponder may accept any ofthe offers made by theproposers; he is notforced to take the highestoffer .Furthermore,there is an additional stage3 at which theproposer who has beenchosen by theresponder at stage2 can decidewhether he wants tostick to his offeror whether he wants towithdraw—in whichcase all thegains fromtrade arelost for all

15.Note that the acceptance threshold is affected by the reference group. For example,if eachresponder compares his payoff only with that of the proposer but notwith those of theother responders, then the acceptance threshold increases for eachresponder ,anda higheroffer may be sustained in equilibrium. ATHEORY OFFAIRNESS,COMPETITION, ANDCOOPERATION 835 parties.This game would bean interestingtest for our theory of inequityaversion. Clearly ,in thestandard modelwith selsh preferences,these modi cations do not make any differencefor thesubgame perfect equilibrium outcome. Also, if someplayers havealtruistic preferencesin thesense that theyappreciate any increasein themonetary payoff ofother players, the result remainsunchanged because altruistic players donot withdraw theoffer at stage3. With inequity aversion the outcome will be radically different,however. A proposerwho is inequityaverse maywant todestroy the entire surplus at stage3 in orderto enforcean egalitarian outcome,in particular if hehas ahigh a i and if thesplit betweenhimself and theresponder is uneven.On theother hand, an evensplit will bewithdrawn by proposer i at stage3 onlyif b i . (n 2 1)/(n 2 2). Thus,the responder may prefer toacceptan offer si 5 0.5 ratherthan an offer sj . 0.5 becausethe ‘‘better’’ offerhas ahigherchance of beingwithdrawn. Thisin turn reducescompetition between proposers at stage1. Thus, while competitionnulli es the impact ofinequity aversion in the ordinaryproposer competition game, inequity aversion greatly diminishesthe role of competition in themodi ed proposer competitiongame. This change in therole of competitionis caused by thefact that in themodi ed gamea singleproposer can enforce an equitableoutcome. Weconclude that competitionrenders fairness considerations irrelevantif and onlyif noneof thecompeting players canpunish themonopolist by destroyingsome of thesurplus and enforcinga moreequitable outcome. This suggests that fairnessplays a smallerrole in mostmarkets for goods 16 than in labormarkets. Thisfollows from the fact that,in addition tothe rejection of low wageoffers, workers have some discretion over their work effort. By varying theireffort, they can exert a directimpact onthe relativematerial payoff oftheemployer .Consumers,in contrast, haveno similar option available. Therefore,a rmmay be reluctantto offer a lowwage to workers who are competing for a jobif theemployed worker has theopportunity to respond to a lowwage with loweffort. As aconsequence,fairness consider-

16.There are some markets for goods where fairness concerns play a role.For example,World Series or NBA playofftickets are often sold far belowthe market-clearingprice even though there is a greatdeal of competition among buyers.This may be explained by long-termpro t-maximizing considerations of themonopolist who interacts repeatedly withgroups of customerswho care for fair ticket prices.On thissee alsoKahnemann, Knetsch, and Thaler [1986]. 836 QUARTERLY JOURNALOF ECONOMICS ationsmay well give rise to wage rigidity and involuntary unemployment. 17

IV. COOPERATIONAND RETALIATION: COOPERATION GAMES In theprevious section we have shown that ourmodel can accountfor the relatively ‘ ‘fair’’outcomesin thebilateral ultimatumgame as wellas forthe rather ‘ ‘unfair’’or‘ ‘competitive’’ outcomesin gameswith proposeror responder competition. In this sectionwe investigate the conditions under which cooperationcan ourishin thepresence of inequity aversion. We show that inequityaversion improves the prospects for voluntary cooperationrelative to the predictions of the standard model.In particular, weshow that thereis an interestingclass ofconditions underwhich the sel sh model predicts completedefection, while inourmodel there exist equilibria in whicheverybody cooperates fully.But, thereare also other cases where the predictions of our modelcoincide with thepredictions of thestandard model. Westart with thefollowing public goodgame. There are n $ 2 players whodecide simultaneously on their contribution levels gi [ [0, y], i [ 1, . . . , n ,tothe public good.Each player has an endowmentof y.Themonetary payoff ofplayer i is given by

(11) xi( g1, . . . , gn) 5 y 2 gi 1 a gj, 1/n , a , 1, oj5 1 where a denotesthe constant marginal return to the public good n G ; S j5 1 gj. Since a , 1,a marginalinvestment into G causes a monetaryloss of (1 2 a);i.e.,the dominant strategy of a completelysel sh player is tochoose gi 5 0.Thus, the standard model predicts gi 5 0 for all i [ 1, . . . , n .However,since a . 1/n, the aggregatemonetary payoff is maximizedif eachplayer chooses gi 5 y. Considernow a slightly different public goodgame that consistsof two stages. At stage1 thegame is identical tothe previousgame. At stage2 eachplayer i is informedabout the contributionvector ( g1, . . . , gn)and cansimultaneously impose a punishmenton the other players; i.e., player i choosesa punishment vector pi 5 ( pi1, . . . , pin), where pij $ 0denotesthe punishmentplayer i imposeson player j.Thecost of this

17.Experimental evidence for this is provided by Fehr ,Kirchsteiger,and Riedl[1993] and Fehr and Falk [forthcoming]. We deal with these games in more detailin SectionVI. ATHEORY OFFAIRNESS,COMPETITION, ANDCOOPERATION 837

n punishmentto player i is given by c S j5 1 pij, 0 , c , 1. Player i, however,mayalso be punished by theother players, which n generatesan incomeloss to i of S j5 1 pji.Thus,the monetary payoff of player i is given by

n n n

(12) xi( g1, . . . , gn, p1, . . . , pn) 5 y 2 gi 1 a gj 2 pji 2 c pij. oj5 1 oj5 1 oj5 1 What doesthe standard modelpredict forthe two-stage game?Since punishments are costly ,players’dominant strategy at stage2 is tonot punish. Therefore,if selshness and rationality arecommon knowledge, each player knowsthat thesecond stage is completelyirrelevant. As aconsequence,players haveexactly thesame incentives at stage1 as theyhave in theone-stage game withoutpunishments, i.e., each player ’soptimal strategyis still given by gi 5 0.T owhat extentare these predictions of the standard modelconsistent with thedata frompublic goodexperi- ments?For the one-stage game there are, fortunately ,alarge numberof experimental studies (seeTable II). Theyinvestigate thecontribution behavior of subjects under a wide varietyof conditions.In TableII weconcentrate on thebehavior of subjects in the nal periodonly ,sincewe want toexclude the possibility of repeatedgames effects. Furthermore, in the nal periodwe have morecon dence that theplayers fully understand thegame that is beingplayed. 18 Thestriking fact revealedby TableII is that in the nal period of n-personcooperation games ( n . 3) withoutpunishment thevast majorityof subjects play theequilibrium strategy of completefree riding. If weaverage over all studies,73 percentof all subjectschoose gi 5 0in the nal period.It is alsoworth mentioningthat in addition tothose subjects who play exactly the equilibriumstrategy there are very often a nonnegligiblefraction ofsubjectswho play ‘‘close’’ tothe equilibrium. In viewof thefacts presentedin TableII, it seemsfair tosay that thestandard model ‘‘approximates’’ thechoices of a big majorityof subjects rather well.However ,if weturn to the public goodgame with punishment,there emerges a radically different picturealthough the standard modelpredicts thesame outcome as in theone-stage

18.This point is discussedin moredetail in SectionV .Notethat in some of the studiessummarized in Table II thegroup compositionwas the same for all T periods(partner condition). In others, the group compositionrandomly changed fromperiod to period (stranger condition). However ,inthelast period subjects in thepartner condition also play a trueone-shot public goods game. Therefore, Table IIpresentsthe behavior from stranger as well as from partner experiments. 838 QUARTERLY JOURNALOF ECONOMICS

TABLE II PERCENTAGEOF SUBJECTS WHO FREE RIDE COMPLETELY INTHE FINAL PERIOD OF A REPEATED PUBLIC GOOD GAME

Percentage Marginal Total of free Group pecuniary number riders Study Country size (n) return (a) of subjects (gi 5 0)

Isaacand W alker[1988] USA 4and100.3 42 83 Isaacand W alker[1988] USA 4and100.75 42 57 Andreoni[1988] USA 5 0.5 70 54 Andreoni[1995a] USA 5 0.5 80 55 Andreoni[1995b] USA 5 0.5 80 66 Croson[1995] USA 4 0.5 48 71 Croson[1996] USA 4 0.5 96 65 Keserand van Winden [1996] Holland 4 0.5 160 84 Ockenfelsand Weimann[1996] Germany 5 0.33 200 89 Burlandoand Hey [1997] UK,Italy 6 0.33 120 66 Falkinger,Fehr, Ga¨chter,and Winter-Ebmer [forthcoming] Switzerland8 0.2 72 75 Falkinger,Fehr, Ga¨chter,and Winter-Ebmer [forthcoming] Switzerland16 0.1 32 84 Totalnumber of subjects in all experiments and percentageof complete free riding 1042 73

game.Figure II showsthe distribution ofcontributionsin the nal periodof the two-stage game conducted by Fehrand Ga¨chter [1996]. Notethat the samesubjects generatedthe distribution in thegame without and in thegame with punishment.Whereas in thegame without punishment most subjects play closeto completedefection, a strikinglylarge fraction of roughly 80 percent cooperates fully in thegame with punishment. 19 Fehrand Ga¨chter

19.Subjects in theFehr and Ga ¨chter study participatedin both conditions, i.e.,in the game with punishment and inthe game without punishment. The parametervalues for a and n inthis experiment are a 5 0.4 and n 5 4. It is interestingto note that contributions are signi cantly higher in the two-stage gamealready in period 1. Moreover ,inthe one-stage game cooperation strongly decreasesover time, whereas in thetwo-stage game cooperation quickly converges tothe high levels observed in period10. ATHEORY OFFAIRNESS,COMPETITION, ANDCOOPERATION 839

FIGURE II Distributionof Contributionsin the Final Period of the Public Good Game with Punishment( Source: Fehrand Ga ¨chter [1996]) reportthat thevast majorityof punishments are imposed by cooperatorson thedefectors and that lowercontribution levels are associatedwith higherreceived punishments. Thus, defectors do notgain fromfree riding becausethey are being punished. Thebehavior in thegame with punishmentrepresents an unambiguousrejection of the standard model.This raises the questionwhether our model is capable ofexplaining boththe evidenceof theone-stage public goodgame and ofthepublic good gamewith punishment.Consider the one-stage public goodgame rst.The prediction of our model is summarizedin thefollowing proposition:

PROPOSITION 4. (a) If a 1 b i , 1 for player i,thenit isadominantstrategy for that player tochoose gi 5 0. (b) Let k denotethe number of players with a 1 b i , 1, 0 # k # n. If k/(n 2 1) . a/2,then there is auniqueequilibrium with gi 5 0 for all i [ 1, . . . , n . (c) If k/(n 2 1) , (a 1 b j 2 1)/(a j 1 b j)forall players j [ 1, . . . , n with a 1 b j . 1,then other equilibria with positivecontribution levels do exist. In theseequilibria all k players with a 1 b i , 1mustchoose gi 5 0, while all otherplayers contribute gj 5 g [ [0,y].Notefurther that (a 1 b j 2 1)/(a j 1 b j) , a/2. 840 QUARTERLY JOURNALOF ECONOMICS

Theformal proof of Proposition4 is relegatedto the Appendix. Toseethe basic intuitionfor the above results, consider a player with a 1 b i , 1.By spending onedollar onthe public good,he earns a dollars in monetaryterms. In addition, hemay get a nonpecuinarybene t ofat most b i dollars fromreducing inequality.Therefore,since a 1 b i , 1forthis player,it is adominant strategyfor him to contribute nothing. Part (b) oftheproposition says that if thefraction of subjects, for whom gi 5 0is adominant strategy,is sufficiently high,there is auniqueequilibrium in whichnobody contributes. The reason is that if thereare only a fewplayers with a 1 b i . 1,they would suffertoo much from the disadvantageous inequality causedby thefree riders. The proof of theproposition shows that if apotentialcontributor knows that thenumber of freeriders, k,is largerthan a(n 2 1)/2,then he will notcontributeeither .Thelast part oftheproposition shows that if thereare sufficiently manyplayers with a 1 b i . 1, they can sustain cooperationamong themselves even if theother players do notcontribute. However ,this requiresthat thecontributors are nottoo upset aboutthe disadvantageous inequality toward the freeriders. Note that thecondition k/(n 2 1) , (a 1 b j 2 1)/ (a j 1 b j)is lesslikely to be met as a j goesup. Toput it differently, thegreater the aversion against beingthe sucker ,themore difficult itis tosustain cooperationin theone-stage game. We will seebelow that theopposite holds truein thetwo-stage game. Notethat in almostall experimentsconsidered in TableII, a # 1/2.Thus, if thefraction of players with a 1 b i , 1 is larger 1 than 4,thenthere is noequilibrium with positivecontribution levels.This is consistentwith thevery low contribution levels that havebeen observed in theseexperiments. Finally ,it is worthwhile mentioningthat theprospects for cooperation are weakly increas- ing with themarginal return a. Considernow the public goodgame with punishment.T o what extentis ourmodel capable ofaccounting for the very high cooperationin thepublic goodgame with punishment?In the contextof our model the crucial point is that freeriding generates amaterialpayoff advantage relativeto those who cooperate. Since c , 1,cooperators can reduce this payoff disadvantage by punishing thefree riders. Therefore, if thosewho cooperate are suffi- cientlyupset by theinequality totheir disadvantage, i.e.,if they havesufficiently high a ’s,then they are willing topunish the defectorseven though this is costlyto themselves. Thus, the threatto punish freeriders may be credible, which may induce ATHEORY OFFAIRNESS,COMPETITION, ANDCOOPERATION 841 potentialdefectors to contribute at the rststage of the game. Thisis madeprecise in thefollowing proposition.

PROPOSITION 5.Suppose that thereis agroupof n’‘‘conditionally cooperativeenforcers,’ ’ 1 # n’ # n,with preferencesthat obey a 1 b i $ 1 and

a i (13) c , (n 2 1)(1 1 a i) 2 (n8 2 1)(a i 1 b i) for all i [ 1, . . . , n8 . whereasall otherplayers donot care about inequality; i.e., a i 5 b i 5 0 for i [ n’ 1 1, . . . , n .Thenthe following strategies,which describe the players’ behavior on and off the equilibriumpath, forma subgameperfect equilibrium. c In the rststage each player contributes gi 5 g [ [0, y]. c If eachplayer doesso, there are no punishments in the secondstage. If oneof the players i [ n’ 1 1, . . . , n deviatesand chooses gi , g,theneach enforcer j [ 1, . . . , n’ chooses pji 5 ( g 2 gi)/(n’ 2 c)whileall other players donot punish. If oneof the‘ ‘conditionallycoopera- tiveenforcers’ ’ chooses gi , g,orif any player chooses gi . g,orif morethan oneplayer deviated from g, then one Nash-equilibriumof thepunishment game is beingplayed. Proof. SeeAppendix. Proposition5 showsthat full cooperation,as observedin the experimentsby Fehrand Ga¨chter[1996], canbe sustained asan equilibriumoutcome if thereis agroupof n’‘‘conditionally cooperativeenforcers.’ ’ In fact, onesuch enforcer may be enough (n’ 5 1) if his preferencessatisfy c , a i/(n 2 1)(1 1 a i) and a 1 b i $ 1;i.e., if thereis oneperson who is sufficiently concerned aboutinequality .Toseehow the equilibrium works, consider such a‘‘conditionallycooperative enforcer.’ ’ Forhim a 1 b i $ 1, so he is happy tocooperateif all otherscooperate as well (this is why heis called ‘‘conditionallycooperative’ ’). In addition, condition(13) makessure that hecares sufficiently aboutinequality tohis disadvantage. Thus,he can credibly threaten to punish adefector (this is why heis called ‘‘enforcer’’). Notethat condition(13) is less demanding if n’ or a i increases.The punishment is constructed suchthat thedefector gets the same monetary payoff as the enforcers.Since this is lessthan what adefectorwould have receivedif hehad chosen gi 5 g,adeviationis notpro table. 842 QUARTERLY JOURNALOF ECONOMICS

If theconditions of Proposition5 aremet, then there exists a continuumof equilibriumoutcomes. This continuum includes the ‘‘goodequilibrium’ ’ with maximumcontributions but alsothe ‘ ‘bad equilibrium’’ wherenobody contributes to the public good.In our view,however,there is areasonablere nement argument that rulesout ‘ ‘bad’’ equilibria with lowcontributions. T oseethis, note that theequilibrium with thehighest possible contribution level, gi 5 g 5 y for all i [ 1, . . . , n ,is theunique symmetric and efficientoutcome. Since it is symmetric,it yields thesame payoff forall players.Hence, this equilibriumis anatural focalpoint that servesas acoordinationdevice even if thesubjects choose their strategiesindependently . ComparingPropositions 4 and 5,it is easyto see that the prospectsfor cooperation are greatly improved if thereis an opportunityto punish defectors.Without punishments all players with a 1 b i , 1will nevercontribute. Players with a 1 b i . 1 may contributeonly if theycare enough about inequality totheir advantage but not too much aboutdisadvantageous inequality.On theother hand, with punishment all players will contributeif thereis a(small) groupof ‘ ‘conditionallycooperative enforcers.’ ’ Themore these enforcers care about disadvantageous inequality, themore they are prepared topunish defectorswhich makes it easierto sustain cooperation.In fact, oneperson with asuffi- cientlyhigh a i is already enoughto enforce efficient contributions byall otherplayers. Beforewe turnto the next section, we would liketo point out animplicationof ourmodel for the Prisoner ’sDilemma(PD). Note that thesimultaneous PD is justa special caseof the public good gamewithout punishment for n 5 2 and gi [ 0, y , i 5 1,2. Therefore,Proposition 4 applies; i.e.,cooperation is an equilibrium if both players meetthe condition a 1 b i . 1.Y et,if onlyone player meetsthis condition,defection of bothplayers istheunique equilibrium.In contrast,in asequentiallyplayed PDapurely selsh rstmover has an incentiveto contribute if hefaces a secondmover who meets a 1 b i . 1.This is sobecause the second moverwill respondcooperatively to a cooperative rstmove while hedefects if the rstmover defects. Thus, due to the reciprocal behaviorof inequity-averse second movers, cooperation rates among rstmovers in sequentiallyplayed PDsare predicted tobe higherthan cooperationrates in simultaneousPDs. There is fairly strongevidence in favorof this prediction.W atabe,T erai,Haya- shi,and Yamagishi [1996] and Hayashi,Ostrom, Walker ,and ATHEORY OFFAIRNESS,COMPETITION, ANDCOOPERATION 843

Yamagishi [1998] showthat cooperationrates among rstmovers in sequentialPDs are indeed muchhigher and that reciprocal cooperationof secondmovers is veryfrequent.

V. PREDICTIONSACROSS GAMES In this sectionwe examine whether the distribution of parametersthat is consistentwith experimentalobservations in theultimatum game is consistentwith theexperimental evidence fromthe other games. It is notour aim here to show that our theoryis consistentwith 100 percentof the individual choices. Theobjective is ratherto offer a rsttest for whether there is a chancethat ourtheory is consistentwith the quantitative evidencefrom different games.Admittedly ,this testis rathercrude. However,at theend of this sectionwe make a numberof predictionsthat areimplied by ourmodel, and wesuggest how thesepredictions can be tested rigorously with somenew experiments. In manyof the experiments referred to in this section,the subjectshad toplay thesame game several times either with the sameor with varying opponents.Whenever available, wetake the data ofthe nal periodas thefacts tobeexplained. There are two reasonsfor this choice.First, it is well-knownin experimental economicsthat in interactivesituations one cannot expect the subjectsto play an equilibriumin the rstperiod already .Yet,if subjectshave the opportunity to repeat their choices and tobetter understand thestrategic interaction, then very often rather stable behavioralpatterns, that maydiffer substantially from rst-period- play,emerge.Second, if thereis repeatedinteraction between the sameopponents, then there may be repeated games effects that comeinto play .Theseeffects can be excluded if welook at thelast periodonly . TableIII suggestsa simplediscrete distribution of a i and b i. Wehave chosen this distribution becauseit is consistentwith the largeexperimental evidence we have on theultimatum game (see TableI aboveand Roth[1995]). Recall fromProposition 1 that for any given a i,thereexists an acceptancethreshold s8(a i) 5 a i/(1 1 2a i)suchthat player i accepts s if and onlyif s $ s8(a i). In all experimentsthere is afractionof subjects that rejectsoffers evenif theyare very close to an equalsplit. Thus,we (conserva- tively) assumethat 10 percentof the subjects have a 5 4 which impliesan acceptancethreshold of s8 5 4/9 5 0.444. Another, 844 QUARTERLY JOURNALOF ECONOMICS

TABLE III ASSUMPTIONS ABOUT THE DISTRIBUTION OF PREFERENCES

DISTRIBUTION OF a ’s AND DISTRIBUTION OF b ’s AND ASSOCIATED ACCEPTANCE ASSOCIATED OPTIMALOFFERS THRESHOLDS OFBUYERS OF SELLERS a 5 0 30 percent s8 5 0 b 5 0 30 percent s* 5 1/3 a 5 0.530 percent s8(0.5) 5 1/4 b 5 0.2530 percent s* 5 4/9 a 5 1 30 percent s8 (1) 5 1/3 b 5 0.640 percent s* 5 1/2 a 5 4 10 percent s8 (4) 5 4/9

typically muchlarger fraction of thepopulation insists ongetting atleastone-third of thesurplus, which implies a valueof a which is equalto one. These are at least30 percentof the population. Notethat theyare prepared togive up onedollar if this reduces thepayoff oftheir opponent by twodollars. Another ,say,30 percentof the subjects insist ongetting at leastone-quarter , whichimplies that a 5 0.5.Finally ,theremaining 30 percentof thesubjects do not care very much about inequality and arehappy toacceptany positiveoffer ( a 5 0). If aproposerdoes not know the parameter a ofhis opponent but believesthat theprobability distribution over a is given by TableIII, thenit is straightforward tocompute his optimaloffer as afunctionof his inequality parameter b .Theoptimal offer is given by

0.5 if b i . 0.5

(14) s*(b ) 5 0.4 if 0.235 , b i , 0.5

0.3 if b i , 0.235. Notethat it is neveroptimal tooffer less than one-thirdof the surplus,even if theproposer is completelysel sh. If welook at the actual offersmade in theultimatum game, there are roughly 40 percentof the subjects who suggest an equalsplit. Another30 percentoffer s [ [0.4,0.5), while 30 percentoffer less than 0.4. Thereare hardly any offersbelow 0.25. This gives us the distribution of b in thepopulation describedin TableIII. Letus now see whether this distribution ofpreferences is consistentwith theobserved behavior in othergames. Clearly ,we haveno problem in explaining theevidence on market games with proposercompetition. Any distribution of a and b yields the competitiveoutcome that is observedby Rothet al. [1991] in all ATHEORY OFFAIRNESS,COMPETITION, ANDCOOPERATION 845 theirexperiments. Similarly ,in themarket game with responder competition,we know from Proposition 3 that ifthereis at least oneresponder who does not care about disadvantageous inequality (i.e., a i 5 0), thenthere is auniqueequilibrium outcome with s 5 0.With veresponders in theexperiments by Gu¨th,March- and, and Rulliere[1997] and with thedistribution oftypes from TableIII, theprobability that thereis at leastone such player in eachgroup is givenby 1–0.7 5 5 83 percent.This is roughly consistentwith thefact that 71 percentof theplayers acceptedan offerof zero, and 9percenthad an acceptancethreshold of s8 5 0.02 in the nal period. Considernow the public goodgame. We know by Proposition 4that cooperationcan be sustained as an equilibriumoutcome onlyif thenumber k ofplayers with a 1 b i , 1 obeys k/(n 2 1) , a/2.Thus, our theory predicts that thereis lesscooperation the smaller a whichis consistentwith theempirical evidence of Isaac and Walker[1988] presentedin TableII. 20 In atypical treatment a 5 0.5, and n 5 4.Therefore, if all players believethat thereis at leastone player with a 1 b i , 1,then there is aunique equilibriumwith gi 5 0forall players.Given the distribution of preferencesof TableIII, theprobability that thereare four players with b . 0.5 is equalto 0.4 4 5 2.56 percent.Hence, we should observethat, on average, almost all individuals fully defect.A similarresult holds formost other experiments in TableII. Except forthe Isaac and Walkerexperiments with n 5 10 a single player with a 1 b i , 1is sufficient forthe violation of the necessary conditionfor cooperation, k/(n 2 1) , a/2.Thus, in all these experimentsour theory predicts that randomlychosen groups are almostnever capable ofsustaining cooperation.Table II indicates that this is notquite the case, although 73 percentof individuals indeed choose gi 5 0.Thus,it seemsfair tosay that ourmodel is consistentwith thebulk of individual choicesin this game. 21 Finally,themost interesting experiment from the perspective ofourtheory is thepublic goodgame with punishment.While in

20. For a 5 0.3,the rate of defectionis substantiallylarger than for a 5 0.75. TheIsaac and Walker experiments were explicitly designed to test for the effects of variationsin a. 21.When judging theaccuracy ofthemodel, one should also take into account thatthere is in general a signicant fractionof the subjects that play close to completefree riding in the nalround. A combinationof ourmodel with the view thathuman choice is characterized by a fundamentalrandomness [McKelvey and Palfrey1995; Anderson, Goeree, and Holt 1997] may explain much of the remaining25 percent of individualchoices. This task, however ,isleft for future research. 846 QUARTERLY JOURNALOF ECONOMICS thegame without punishment most subjects play closeto completedefection, a strikinglylarge fraction of roughly 80 percent cooperate fully in thegame with punishment.T owhat extentcan ourmodel explain this phenomenon?We know from Proposition 5 that cooperationcan be sustained if thereis agroupof n’ ‘‘conditionallycooperative enforcers’ ’ with preferencesthat satisfy (13) and a 1 b i $ 1.For example, if all fourplayers believethat thereis at leastone player with a i $ 1.5 and b i $ 0.6,there is an equilibriumin whichall fourplayers contributethe maximum amount.As discussed in SectionV ,this equilibriumis anatural focalpoint. Since the computation of the probability that the conditionsof Proposition5 aremet is abit morecumbersome, we haveput themin theAppendix. It turnsout that forthe preference distribution givenin TableIII theprobability that arandomly drawn groupof fourplayers meetsthe conditions is 61.1 percent. Thus,our model is roughlyconsistent with theexperimental evidenceof Fehr and Ga¨chter[1996]. 22 Clearly,theabove computations provide only rough evidence in favorof our model. T origorouslytest the model, additional experimentshave to be run. We would liketo suggest a few variants ofthe experiments discussed sofar that would be particularly interesting: 23 c Ourmodel predicts that underproposer competition two proposersare sufficient for s 5 1tobethe unique equilibriumoutcome irrespectiveof theplayers’ preferences. Thus, onecould conduct the proposer competition game with two proposersthat haveproved to be very inequity averse in othergames. This would constitutea particularly tough testof ourmodel. c Mostpublic goodgames that havebeen conducted had symmetricpayoffs. Ourtheory suggests that it will bemore difficult tosustain cooperationif thegame is asymmetric. Forexample, if thepublic goodis morevaluable tosome of theplayers, there will in generalbe a conict between efficiencyand equality.Ourprediction is that if thegame is sufficiently asymmetricit is impossibleto sustain cooperation even if a is verylarge or if players canuse punishments.

22.In this context one has to take into account that the total number of availableindividual observations in the game with punishment is muchsmaller thanfor the game without punishment or for the ultimatum game. Future experimentswill have to show whether the Fehr-Ga ¨chter resultsare the rule in thepunishment game or whetherthey exhibit unusually high cooperation rates. 23.We are grateful to a refereewho suggested some of these tests. ATHEORY OFFAIRNESS,COMPETITION, ANDCOOPERATION 847

c It would beinteresting to repeat the public goodexperi- mentwith punishmentsfor different valuesof a, c, and n. Proposition5 suggeststhat weshould observeless cooperation if a goesdown and if c goesup. Theeffectof an increase in thegroup size n is ambiguous,however .Forany given player it becomesmore difficult tosatisfy condition(13) as n goesup. On theother hand, thelarger the group, the higheris theprobability that thereis at leastone person with averyhigh a .Ourconjecture is that amoderate changein thesize of thegroup does not affect theamount of cooperation. c Oneof themost interesting tests of ourtheory would beto doseveral different experimentswith thesame group of subjects.Our model predicts across-situationcorrelation in behavior.Forexample,the observations from one experi- mentcould be used to estimate the parameters of the utility functionof each individual. It would thenbe possible totest whether this individual’s behaviorin othergames is consistentwith his estimatedutility function. c In asimilarfashion, one could screen subjects according to theirbehavior in oneexperiment before doing apublic good experimentwith punishments.If wegroupthe subjects in this secondexperiment according to theirobserved inequality aversion,the prediction is that thosegroups with high inequality aversionwill contributewhile those with low inequality aversionwill not.

VI. DICTATOR AND GIFT EXCHANGE GAMES Thepreceding sections have shown that ourvery simple modelof linear inequality aversionis consistent,with themost importantfacts in ultimatum,market, and cooperationgames. Oneproblem with ourapproach, however,is that it yields too extremepredictions in someother games, such as the‘ ‘dictator game.’’ Thedictator game is atwo-persongame in whichonly player 1,the ‘ ‘dictator,’’has tomake a decision.Player 1 has to decidewhat share s [ [0,1] ofagivenamount of moneyto pass on toplayer 2.For a givenshare s monetarypayoffs aregiven by x1 5 1 2 s and x2 5 s,respectively.Obviously,thestandard model predicts s 5 0.In contrast,in theexperimental study ofForsythe, Horowitz,Savin, and Sefton[1994] onlyabout 20 percentof subjectschose s 5 0,60 percentchose 0 , s , 0.5,and again 848 QUARTERLY JOURNALOF ECONOMICS roughly20 percentchose s 5 0.5.In thestudy by Andreoniand Miller[1995] thedistribution ofsharesis again bimodal but puts moreweight on the ‘ ‘extremes:’’ approximately40 percentof the subjectsgave s 5 0,20 percentgave 0 , s , 0.5,and roughly40 percentgave s 5 0.5.Shares above s 5 0.5 werepractically never observed.

Ourmodel predicts that player 1offers s 5 0.5 if b 1 . 0.5 and s 5 0 if b 1 , 0.5.Thus, we should observe only very‘ ‘fair’’ orvery ‘‘unfair’’outcomes,a predictionthat is clearlyrefuted by thedata. However,thereis astraightforward solutionto this problem.We assumedthat theinequity aversion is piecewise linear. The linearityassumption was imposedin orderto keep our model as simpleas possible.If weallow fora utility functionthat is concave intheamount of advantageous inequality,thereis noproblemin generatingoptimal offersthat arein theinterior of [0,0.5]. It is importantto note that nonlinearinequity aversion does notaffect thequalitative resultsin theother games we considered.This is straightforward in marketgames with proposeror respondercompetition. Recall that in thecontext of proposer competitionthere exists a uniqueequilibrium outcome in which theresponder receives the whole gains fromtrade irrespectiveof theprevailing amountof inequityaversion. Thus,it alsodoes not matterwhether linear or nonlinear inequity aversion prevails. Likewise,under responder competition there is auniqueequilib- riumoutcome in whichthe proposer receives the whole surplus if thereis at leastone responder who does not care about disadvan- tageousinequality .Obviously,this propositionholds irrespective ofwhetherthe inequity aversion of the other responders is linear ornot. Similar argumentshold forpublic goodgames with and withoutpunishment. Concerning the public goodgame with punishment,for example, the existence of nonlinear inequity aversionobviously does not invalidate theexistence of an equilibriumwith full cooperation.It onlyrenders the condition for the existenceof suchan equilibrium,i.e., condition (13), slightly more complicated. Anotherinteresting game is theso-called trust- orgift exchangegame [Fehr, Kirchsteiger, and Riedl 1993; Berg,Dick- haut,and McCabe 1995; Fehr,Ga¨chter,and Kirchsteiger1997]. Thecommon feature of trust- orgift exchangegames is that they resemblea sequentiallyplayed PDwith morethan twoactions for eachplayer .In someexperiments the gift exchangegame has been embeddedin acompetitiveexperimental market. For example, a ATHEORY OFFAIRNESS,COMPETITION, ANDCOOPERATION 849 slightly simplied versionof the experiment conducted by Fehr, Ga¨chter,and Kirchsteiger[1997] has thefollowing structure. Thereis oneexperimental rm,which we denote as player 1,and whichcan make a wageoffer w tothe experimental workers. Thereare 2, . .., n workerswho can simultaneously accept or reject w.Thena randomdraw selectswith equalprobability oneof theaccepting workers. Thereafter, the selected worker has to chooseeffort e fromthe interval [ e,e], 0 , e , e.In casethat all workersreject w,all players receivenothing. In caseof acceptance the rmreceives xf 5 ve 2 w, where v denotesthe marginal product ofeffort. The worker receives xw 5 w 2 c(e), where c(e) denotesthe effort costs and obeys c(e) 5 c8(e) 5 0 and c8 . 0, c9 . 0 for e . e. Moreover, v . c8(e) so that e 5 e is theefficient effort level. Thisgame is essentiallya marketgame with respondercompeti- tionin whichan acceptingresponder has tomake an effortchoice afterhe is selected. If all players arepure money maximizers, the prediction for this gameis straightforward. Sincethe selected worker always choosesthe minimum effort e,thegame collapses into a responder competitiongame with gains fromtrade equal to ve.In equilib- riumthe rmearns ve and w 5 0.Y et,since v . c8(e),thereexist many (w,e)-combinationsthat would makeboth the rmand the selectedworker better off. In sharp contrastto this prediction,and alsoin sharp contrastto what is observedunder responder competition without effortchoices, rmsoffersubstantial wagesto theworkers, and wagesdo not decrease over time. Moreover , workersprovide effort above e and thereis astrongpositive correlationbetween w and e. Towhat extentcan our model explain this outcome?Put differently,why is it thecase that underresponder competition without effortchoice the responders’ income converges toward the selsh solution, whereas under responder competition with effort choice,wages substantially abovethe sel sh solution can be maintained.From the viewpoint of our model the key fact is that—by varying theeffort choice— the randomly selected worker has theopportunity to affect thedifference xf 2 xw. If the rm offers‘ ‘low’’ wagessuch that xf . xw holds at any feasibleeffort level,the selected worker will always choosethe minimum effort. However,if the rmoffers a ‘‘high’’ wagesuch that at e the inequality xw . xf holds,inequity-averse workers with asuffi- cientlyhigh b i arewilling toraise e above e.Moreover,in the presenceof nonlinear inequity aversion, higher wages will be 850 QUARTERLY JOURNALOF ECONOMICS associatedwith highereffort levels. The reason is that by raising theeffort workers can move in thedirection of more equitable outcomes.Thus, our model is capable ofexplaining theapparent wagerigidity observedin gift exchangegames. Since the presence ofinequity-averse workers generates a positivecorrelation be- tweenwages and effort,the rmdoes not gain by exploitingthe competitionamong the workers. Instead, it has an incentiveto pay efficiencywages above the competitive level.

VII. EXTENSIONS AND POSSIBLE OBJECTIONS Sofar, we ruled out the existence of subjects who like to be betteroff than others.This is unsatisfactory becausesubjects with b i , 0clearlyexist. Fortunately ,however,such subjects have virtually noimpact onequilibrium behavior in thegames consid- eredin this paper.Toseethis, suppose that afractionof subjects with b i 5 0 exhibits b i , 0instead.This obviously does not change responders’behavior in theultimatum game because for them only a i matters.It alsodoes not change the proposer behavior in thecomplete information case because both proposers with b i 5 0 and thosewith b i , 0will makean offerthat exactlymatches the responder’s acceptancethreshold. 24 In themarket game with proposercompetition, proposers with b i , 0areeven more willing tooverbida goingshare below s 5 1,compared with subjectswith b i 5 0,because by overbidding theygain apayoff advantage relativeto the other proposers. Thus, Proposition 2 remains unchanged.Similar argumentsapply tothe case of responder competition(without effortchoices) because a responderwith b i , 0isevenmore willing tounderbid apositiveshare compared with aresponderwith b i 5 0.In thepublic goodgame without punishmentall players with a 1 b i , 1havea dominantstrategy tocontribute nothing. It doesnot matter whether these players exhibita positiveor a negative b i.Finally,theexistence of types 25 with b i , 0alsoleaves Proposition 5 unchanged. If there are sufficiently manyconditionally cooperative enforcers, it doesnot

24.It may affect proposer behavior in the incomplete information case althoughthe effect of a changein b i isambiguous. This ambiguity stems from the fact thatthe proposer’ s marginalexpected utility of s mayrise or fallif b i falls. 25.This holds true if, for those with a negative b i,theabsolute value of b i is nottoo large. Otherwise, defectors would have an incentive to punish the cooperators.A defectorwho imposes a punishmentof one on a cooperatorgains [2 b i/(n 2 1)](1 2 c) . 0innonpecuniary terms and has material costs of c. Thus, heis willing to punish if b i $ [c/(1 2 c)](n 2 1)holds. This means that only defectorswith implausibly high absolute values of b i arewilling to punish. For ATHEORY OFFAIRNESS,COMPETITION, ANDCOOPERATION 851 matterwhether the remaining players have b i , 0ornot.Recall that—according to Proposition 5— strategies that discipline potential defectorsmake the enforcers and thedefectors equally welloff in materialterms. Hence, a defectorcannot gain apayoff advan- tagebut is evenworse off relativeto a cooperatingnonenforcer. Thesepunishment strategies, therefore, are sufficient todisci- pline potentialdefectors irrespective of their b i-values. Anotherset of questions concerns the choice of thereference group.As arguedin SectionII, formany laboratory experiments ourassumption that subjectscompare themselves with all other subjectsin the(usually relativelysmall) groupis anatural starting point.However, we areaware of thepossibility that this maynot always bean appropriate assumption. 26 Theremay well beinteractive structures in whichsome agents have a salient positionthat makesthem natural referenceagents. Moreover ,the socialcontext and theinstitutional environmentin whichinterac- tionstake place is likelyto be important. 27 Bewley[1998], for example,reports that in nonunionized rmsworkers compare themselvesexclusively with their rmand with otherworkers in their rm.This suggests that onlywithin- rm social comparisons but notacross- rm comparisons affect thewage-setting process. Thisis likelyto be different in unionizedsectors because unions makeacross- rm and evenacross-sector comparisons. Babcock, Wang,and Loewenstein[1996], forexample, provide evidence that wagebargaining betweenteachers’ unions and schoolboards is stronglyaffected by referencewages in otherschool districts. An obviouslimitation of our model is that it cannotexplain theevolution of play overtime in theexperiments discussed. Instead, ourexamination aims at theexplanation ofthe stable behavioralpatterns that emergein theseexperiments after severalperiods. It is clear,that amodelthat solelyfocuses on equilibriumbehavior cannot explain thetime path ofplay .This limitationof our model also precludes a rigorousanalysis ofthe

example,for c 5 0.5 and n 5 4, b i $ 3isrequired. For c 5 0.2 and n 5 4, b i still hasto exceed 0.75. 26.Bolton and Ockenfels [1997] develop a modelsimilar to ours that differs in thechoice of the reference payoff. In their model subjects compare themselves only withthe average payoff of thegroup. 27.A relatedissue is the impact of social context on a person’s degreeof inequityaversion. It seemslikely that a personhas a differentdegree of inequity aversionwhen interacting with a friendin personal matters than in a business transactionwith a stranger.Infact, evidencefor this is provided by Loewenstein, Thompson,and Bazerman [1989]. However, note that in all experiments consid- eredabove interaction took place among anonymous strangers in a neutrally framedcontext. 852 QUARTERLY JOURNALOF ECONOMICS short-run impact ofequity considerations. 28 Theempirical evi- dencesuggests that equityconsiderations also have important short-runeffects. This is obviousin ultimatumgames, public good gameswith punishment,and gift exchangegames, where equity considerationslead tosubstantial deviationsfrom the sel sh solutionin theshort and inthelong run. However, they also seem toplay ashort-runrole in marketgames with proposeror respondercompetition or public goodgames without punishment; that is,in gamesin whichthe sel sh solution prevails in thelong run.In thesegames the short-run deviation from equilibrium is typically in thedirection of more equitable outcomes. 29

VIII. RELATED APPROACHESIN THE LITERATURE Thereare several alternative approaches that tryto account forpersistent deviations from the predictions of the self-interest modelby assuminga different motivationalstructure. The ap- proachpioneered by Rabin [1993] emphasizesthe role of inten- tionsas asourceof reciprocal behavior. Rabin’ s approach has recentlybeen extended in interestingways by Falkand Fisch- bacher[1998] and Dufwenbergand Kirchsteiger[1998]. Andreoni and Miller[1995] is based ontheassumption of altruistic motives. Anotherinteresting approach is Levine[1997] whoassumes that people are either spiteful or altruistic tovariousdegrees. Finally , thereis theapproach by Boltonand Ockenfels[1997] that is,like ourmodel, based onakind ofinequity aversion. Thetheory of reciprocityas developedby Rabin [1993] rests ontheidea that peopleare willing toreward fair intentionsand to punish unfair intentions.Like our approach, Rabin’s modelis also based onthe notion of equity: player j perceivesplayer i’s intentionas unfair if player i choosesan actionthat gives j less

28.In the short-run, minor changes in the(experimental) context can affect behavior.For example, there is evidence that subjects contribute more in a one-shotPD ifit is called‘ ‘communitygame’ ’ thanif itiscalled ‘ ‘WallStreet Game.’ ’ Underthe plausible assumption that the community frame triggers moreoptimis- tic beliefsabout other subjects inequity aversion our model is consistentwith this observation. 29.Such short-runeffects also are suggested by the results of Kahneman, Knetsch,and Thaler [1986] and Franciosi et al. [1995]. Franciosi et al. show that—in a competitiveexperimental market (without effort choices)— equity considerationssigni cantly retard theadjustment to the (sel sh) equilibrium. Ultimately,however,theydo not prevent full adjustment to the equilibrium. Note thatthe retardation effect suggests that temporary demandshocks (e.g., after a naturaldisaster) may have no impact on prices at all because the shock vanishes beforecompetitive forces can overcome the fairness-induced resistance to price changes. ATHEORY OFFAIRNESS,COMPETITION, ANDCOOPERATION 853 than theequitable material payoff. Theadvantage ofhis modelis that thedisutility ofan unfair offercan be explicitly interpretedas arising from j’sjudgmentabout i’sunfair intention.As aconsequence,player j’sresponseto i’sactioncan be explicitly interpretedas arising from j’sdesireto punish an unfair intention whileour model does not explicitly suggestthis interpretationof j’sresponse.On theother hand, disadvantages ofRabin’ s model arethat it isrestrictedto two-personnormal form games and that it givespredictions if it is applied tothe normal form of important sequentialmove games. 30 Thelack of explicit modelingof intentions in ourmodel does, however,notimply that themodel is incompatiblewith an intentions-basedinterpretation of reciprocal behavior .In our modelreciprocal behavior is drivenby thepreference parameters a i and b i.Themodel is silentas towhy a i and b i arepositive. Whetherthese parameters are positive because individuals care directlyfor inequality orwhether they infer intentions from actionsthat causeunequal outcomes is notmodeled. Y et,this meansthat positive a i’s and b i’scanbe interpreted as adirect concernfor equality as wellas areduced-formconcern for intentions.An intentions-basedinterpretation of our preference parametersis possiblebecause bad orgoodintentions behind an actionare, in general,inferred from the equity implications of the action.Therefore, people who have a desireto punish abad intentionbehave as if theydislike being worse off relativeto an equitablereference point and peoplewho reward good intentions behaveas if theydislike being better off relativeto an equitable referencepoint. As aconsequence,our preference parameters are compatiblewith theinterpretation of intentions-drivenreciprocity . Toillustratethis point furtherconsider, e.g., an ultimatum gamethat is played undertwo different conditions[Blount 1995]. c In the‘ ‘random’’ conditionthe rstmover ’sofferis deter- minedby arandomdevice. The responder knows how the

30.In the sequentially played Prisoner’ s Dilemma,Rabin’ s modelpredicts thatunconditional cooperation by the second mover is part ofanequilibrium; i.e., thesecond mover cooperates even if the rst moverdefects. Moreover ,conditional cooperationby the second mover is not part ofanequilibrium. The data in Watabe etal. [1996] and Hayashi et al. [1998], however, show that unconditional cooperationis virtually nonexistent while conditional cooperation is the rule. Likewise,in the gift exchangegame workers behaveconditionally cooperative whileunconditional cooperation is nonexistent.The reciprocity approachesof Falk andFischbacher [1998] and of Dufwenberg and Kirchsteiger [1998] do not share thisdisadvantage of Rabin’s model. 854 QUARTERLY JOURNALOF ECONOMICS

offeris generatedand that theproposer cannot be held responsiblefor it. c In the‘ ‘intention’’ conditionthe proposer makes the offer himselfand theresponder knows that this is theproposer ’s deliberatechoice. In theintention condition the responder may not only be directlyconcerned about inequity .Hemay also react to the fairnessof theperceived intentions of theproposer .In contrast,in therandom condition it is onlythe concern for pure equity that mayaffect theresponder’ s behavior.In fact, Blount [1995] reports that thereare responders who reject positive but unequaloffers in bothconditions. However, the acceptance threshold is signicantly higherin theintention condition. 31 Recall fromProposition 1that thereis amonotonicrelationship between the acceptance thresholdand theparameter a i.Thus,this resultsuggests that thepreference parameters do not remain constant across random and intentioncondition. Y et,for all gamesplayed in theintention conditionand, hence,for all gamesconsidered in theprevious sections,the preference parameters should beconstant across games. Altruismis consistentwith voluntarygiving in dictator and public goodgames. It is,however ,inconsistentwith therejection ofoffers in theultimatum game, and it cannotexplain thehuge behavioraldifferences between public goodgames with and withoutpunishment. It alsoseems difficult toreconcile the extremeoutcomes in marketgames with altruism.Levine’ s approach canexplain extremeoutcomes in marketgames as well as theevidence in thecentipede game, but it cannotexplain positivegiving in thedictator game. It also seems that Levine’s approach has difficulties in explaining that the same subjects behavevery noncooperatively in thepublic goodgame without punishment,while they behave very cooperatively in thegame with punishment. Theapproach byBoltonand Ockenfels[1997] is similarto our model,although there are some differences in thedetails. For example,in theirmodel people compare their material payoff with thematerial average payoff ofthegroup. In ourview the appropri- atechoice of the reference payoff is ultimatelyan empirical

31.Similar evidence is givenby Charness[forthcoming] for a gift exchange game.For further evidencein favor of intentions-drivenreciprocity ,seeBolle and Kritikos[1998]. Surprisingly ,andin contrast to these studies, Bolton, Brandts, andKatok [1997]and Bolton, Brandts, and Ockenfels [1997] ndnoevidence for intentions-drivenreciprocity . ATHEORY OFFAIRNESS,COMPETITION, ANDCOOPERATION 855 questionthat cannotbe solved on the basis ofthe presently available evidence.There may well be situations in whichthe averagepayoff istheappropriate choice.However ,in thecontext ofthepublic goodgame with punishment,it seemsto be inappro- priate becauseit cannotexplain why cooperatorswant topunish a defector.If thereare, say , n 2 1fully cooperatingsubjects and one fully defectingsubject, the payoff ofeachcooperator is belowthe group’s averagepayoff. Cooperatorscan reduce this difference betweenown payoff and thegroup’ s averagepayoff by punishing oneof theother players, i.e., they are indifferent betweenpunish- ing othercooperators and thedefector . Boltonand Ockenfels[1997] assumethat themarginal disutility ofsmall deviations from equality is zero.Therefore, if subjectsare nonsatiated in theirown material payoff theywill never propose an equalsplit in thedictator game. Likewise, they will—in caseof nonsatiation in materialpayoffs— never propose an equalsplit in theultimatum game unless a 2 5 ` forsufficiently manyresponders. Typically ,themodal offer in mostultimatum gameexperiments is, however ,theequal split. In addition, the assumptionimplies that completefree riding is theunique equilibriumin thepublic goodgame without punishment for all a , 1 and all n $ 2.Their approach thusrules out equilibria whereonly a fractionof all subjectscooperate. 32

IX. SUMMARY Thereare situations in whichthe standard self-interest modelis unambiguouslyrefuted. However ,inothersituations the predictionsof this modelseem to be very accurate. For example, in simpleexperiments like the ultimatum game, the public good gamewith punishments,or the gift exchangegame, the vast majorityof the subjects behave in a‘‘fair’’ and ‘‘cooperative’’ manneralthough the self-interest model predicts very‘ ‘unfair’’ and ‘‘noncooperative’’ behavior.Yet,there are also experiments like,e.g., market games or public goodgames without punishment,in whichthe vast majorityof the subjects behaves in a rather‘ ‘unfair’’ and ‘‘noncooperative’’ way—as predicted by the self-interestmodel. We show that this puzzling evidencecan be explained in acoherentframework if— in addition topurely selsh people— there is afractionof thepopulation that caresfor

32.Persistent asymmetric contributions are observed in Isaac, Walker ,and Williams[1994]. 856 QUARTERLY JOURNALOF ECONOMICS equitableoutcomes. Our theory is motivatedby thepsychological evidenceon socialcomparison and lossaversion. It is verysimple and canbe applied toany game.The predictions of ourmodel are consistentwith theempirical evidence on all ofthe above- mentionedgames. Our theory also has strongempirical implica- tionsfor many other games. Therefore, it is an importanttask for futureresearch to test the theory more rigorously against competing hypotheses.In addition, webelieve that futureresearch should aimat formalizingthe role of intentionsexplicitly forthe n-personcase. Amaininsight ofour analysis is that thereis an important interactionbetween the distribution ofpreferences in agiven population and thestrategic environment. We have shown that thereare environments in whichthe behavior of a minorityof purelysel sh people forces the majority of fair-minded peopleto behavein acompletelysel sh manner ,too.For example, in a marketgame with proposeror responder competition, it is very difficult, if notimpossible, for fair players toachieve a ‘‘fair’’ outcome.Likewise, in asimultaneouspublic goodgame with punishment,even a smallminority of selsh players cantrigger theunraveling of cooperation. Y et,we have also shown that a minorityof fair-minded players canforce a big majorityof selsh players tocooperate fully in thepublic goodgame with punishment.Similarly ,ourexamination of the gift exchangegame indicates that fairnessconsiderations may give rise to stable wage rigidity despite thepresence of strong competition among the workers.Thus, competition may or maynot nullify theimpact of equityconsiderations. If, despite thepresence of competition, singleindividuals haveopportunities to affect therelative material payoffs, equityconsiderations will affect marketoutcomes evenin verycompetitive environments. In ourview these results suggestthat theinteraction between the distribution ofprefer- encesand theeconomic environment deserves more attention in futureresearch.

APPENDIX

Proofof Proposition2 We rstshow that it is indeed asubgameperfect equilibrium if at leasttwo proposers offer s 5 1whichis acceptedby the responder.Note rstthat theresponder will acceptany offer s $ ATHEORY OFFAIRNESS,COMPETITION, ANDCOOPERATION 857

0.5,because 1 n 2 2 (A1) s 2 b (s 2 1 1 s) 2 b (s 2 0) $ 0. n 2 1 i n 2 1 i Toseethis, note that (A1) is equivalentto

(A2) (n 2 1) s $ b i(ns 2 1).

Since b i # 1,this inequality clearlyholds if (A3) (n 2 1)s $ ns 2 1, whichmust be the case since s # 1.Hence, the buyer will accept s 5 1.Giventhat thereis atleastone other proposer who offers s 5 1and giventhat this offerwill beaccepted, each proposer gets a monetarypayoff of0 anyway,and noproposer can affect this outcome.Hence, it is indeed optimal forat leastone other proposerto offer s 5 1, too. Next,we show that this is theunique equilibrium outcome. Suppose that thereis anotherequilibrium in which s , 1 with positiveprobability .Thisis onlypossible if eachproposer offers s , 1with positiveprobability .Let si bethe lowest offer of proposer i that has positiveprobability .It cannotbe the case that player i puts strictlypositive probability onoffers si [ [si, sj)becausethe probability that hewins with suchan offeris zero.T oseethis, note that in this caseplayer i would get

a i a i a i (A4) U (s ) 5 2 s 2 (1 2 s) 5 2 . i i n 2 1 n 2 1 n 2 1

Ontheother hand, if proposer i chooses si [ (maxjÞ i sj,0.5 ,1), then thereis apositiveprobability that hewill win—in whichcase he gets

a i n 2 2 (A5) 1 2 s 2 (2s 2 1) 2 b (1 2 s ) i n 2 1 i n 2 1 i i

n 2 2 a i a i . (1 2 s )[1 2 b ] 2 . 2 . i n 2 1 i n 2 1 n 2 1 Ofcourse,there may also be apositiveprobability that proposer i doesnot win, but in this casehe again gets 2 a i/(n 2 1). Thus, proposer i would deviate.It followsthat it mustbe the case that si 5 s for all i. Suppose that proposer i changeshis strategyand offers s 1 e , 1in all stateswhen his strategywould haverequired him to 858 QUARTERLY JOURNALOF ECONOMICS choose s.Thecost of this changeis that wheneverproposer i would havewon with theoffer s henow receives only 1 2 s 2 e . However, by making e arbitrarily small,this costbecomes arbitrarily small. Thebene t is that thereare now some states of theworld which havestrictly positive probability in whichproposer i doeswin with the offer s 1 e but in whichhe would nothave won with theoffer s. Thisbene tis strictlypositive and doesnot go to zero as e becomes small.Hence, s , 1cannotbe part ofan equilibriumoutcome. QED

Proofof Proposition3 We rstshow that s 5 1,which is acceptedby all responders, isindeed asubgameperfect equilibrium. Note that any offer s $ 0.5 will beaccepted by all responders.The argument is exactlythe sameas theone in thebeginning of theproof of Proposition1. The followingLemma will beuseful.

LEMMA 1. For any s , 0.5 thereexists a continuationequilibrium in whicheverybody accepts s. Giventhat all otherplayers accept s player i prefersto accept as wellif and onlyif 1 n 2 2 (A6) s 2 a (1 2 s 2 s) 2 b (s 2 0) n 2 1 i n 2 1 i 1 1 $ 0 2 a (1 2 s) 2 a s, n 2 1 i n 2 1 i whichis equivalentto

(A7) (1 2 b i)(n 2 1) 1 2a i 1 b i $ 0.

Sincewe assume that b i , 1,this inequality musthold. h Considernow the proposer .Clearly,it is neveroptimal to offer s . 0.5.Such an offeris always dominatedby s 5 0.5 whichyields ahighermonetary payoff and lessinequality .On theother hand, weknow by Lemma1 that forany s # 0.5 thereexists a continuationequilibrium in whichthis offeris acceptedby everybody.Thus,we only have to look for the optimal s fromthe point of viewof the proposer given that s will beaccepted. His payoff function is 1 n 2 2 (A8) U (s) 5 1 2 s 2 b (1 2 s 2 s) 2 b (1 2 s). 1 n 2 1 1 n 2 1 1 ATHEORY OFFAIRNESS,COMPETITION, ANDCOOPERATION 859

Differentiating with respectto s yields

dU1 2 n 2 2 (A9) 5 2 1 1 b 1 b , ds n 2 1 1 n 2 1 1 whichis independentof s and is smallerthan 0ifand onlyif

(A10) b 1 # (n 2 1)/n. Hence,if this conditionholds, it is an equilibriumthat the proposeroffers s 5 0whichis acceptedby all responders.We now showthat thehighest offer that can besustained in asubgame perfectequilibrium is givenby (8).

LEMMA 2.Suppose that s , 0.5 has beenoffered. There exists a continuationequilibrium in whichthis offeris rejectedby all respondersif and onlyif

a i (A11) s # ; i [ 2, . . . , n . (1 2 b i)(n 2 1) 1 2a i 1 b i

Giventhat all otherresponders reject s, responder i will reject s as wellif and onlyif

a i n 2 2 (A12) 0 $ s 2 (1 2 2s) 2 b s, n 2 1 n 2 1 i whichis equivalentto (A11). Thus,(A11) is asufficient condition fora continuationequilibrium in which s is rejectedby everybody. Suppose nowthat (A11) is violatedfor at leastone i [ 2, . . . , n .Wewant toshow that in this casethere is nocontinua- tionequilibrium in which s is rejectedby everybody.Note rstthat in this caseresponder i prefersto accept s if all otherresponders rejectit. Suppose nowthat at leastone other responder accepts s. In this caseresponder i prefersto accept s as wellif and onlyif (A13)

a i n 2 2 a i a i s 2 (1 2 2s) 2 b s $ 0 2 (1 2 s) 2 s. n 2 1 n 2 1 i n 2 1 n 2 1 Theright-hand side ofthis inequality issmallerthan 0.We know already that theleft-hand side is greaterthan 0since(A11) is violated.Therefore, responder i prefersto accept s as well. We concludethat if (A11) doesnot hold forat leastone i,thenat least oneresponder will accept s.Hence,(A11) is alsonecessary . h

If b 1 , (n 2 1)/n,an equilibriumoffer must be sustained by 860 QUARTERLY JOURNALOF ECONOMICS thethreat that any smalleroffer s˜ will berejected by everybody. But weknow from Lemma 2 that an offer s˜ maybe rejected only if (A11) holds forall i.Thus,the highest offer s that canbe sustained inequilibriumis givenby (8). QED

Proofof Proposition4

(a) Suppose that 1 2 a . b i for player i.Consideran arbitrary contributionvector ( g1, . . . , gi2 1, gi1 1, . . . , gn)ofthe other players.Without loss of generalitywe relabel the players suchthat i 5

1 and 0 # g2 # g3 # ? ? ? # gn.If player 1chooses g1 5 0,his payoff is given by n b n (A14) U1( g1 5 0) 5 y 1 a gj 2 gj. oj5 2 n 2 1 oj5 2

Note rstthat ifall otherplayers choose gj 5 0, too, then g1 5 0 is clearlyoptimal. Furthermore, player 1will neverchoose g1 . max gj .Suppose that thereis at leastone player whochooses gj . 0.If player 1chooses g1 . 0, g1 [ [ gk, gk1 1], k [ 2, . . . , n , then his payoff is givenby

U1(g1 . 0)

n n k b 1 a 1 5 y 2 g1 1 ag1 1 a gj 2 (gj 2 g1) 2 (g1 2 gj) oj5 2 n 2 1 j5 o k1 1 n 2 1 oj5 2

n n k b 1 b 1 , y 2 g1 1 ag1 1 a gj 2 (gj 2 g1) 1 (g1 2 gj) oj5 2 n 2 1 j5 o k1 1 n 2 1 oj5 2

n n b 1 5 y 2 g1 1 ag1 1 a gj 2 gj oj5 2 n 2 1 oj5 2

b 1 1 (n 2 1)g n 2 1 1

n n b 1 5 y 2 (1 2 a 2 b 1)g1 1 a gj 2 gj oj5 2 n 2 1 oj5 2

n n b 1 , y 1 a gj 2 gj 5 U1(g1 5 0). oj5 2 n 2 1 oj5 2

Hence, gi 5 0,is indeed adominantstrategy for player i. ATHEORY OFFAIRNESS,COMPETITION, ANDCOOPERATION 861

(b) It is clearlyan equilibriumif all players contribute nothingbecause to unilaterally contributemore than zeroreduces themonetary payoff and causesdisadvantageous inequality. Suppose that thereexists another equilibrium with positive contributionlevels. Relabel players suchthat 0 # g1 # g2 # ? ? ? # gn.By part (a) weknow that all k players with 1 2 a . b i must choose gi 5 0.Therefore, 0 5 g1 5 . . . gk.Considerplayer l . k who has thesmallest positive contribution level; i.e., 0 5 gl2 1 , gl # gl1 1 # ? ? ? # gn.Player1’ s utility is givenby

n n b l (A15) Ul( gl) 5 y 2 gl 1 agl 1 a gj 2 ( gj 2 gl) j5 o l1 1 n 2 1 j5 o l1 1

l2 1 n n a l b l 2 gl 5 y 1 a gj 2 gj n 2 1 oj5 1 j5 o l1 1 n 2 1 j5 o l1 1

n 2 l l 2 1 2 (1 2 a) g 1 b g 2 a g l l n 2 1 l l n 2 1 l

n 2 l l 2 1 5 U (0) 2 (1 2 a)g 1 b g 2 a g , l l l n 2 1 l l n 2 1 l where Ul(0) is theutility player 1getsif hedeviatesand chooses gl 5 0. Since a l $ b l, l $ k 1 1, and b l , 1, we have

n 2 l l 2 1 (A16) U ( g ) # U (0) 2 (1 2 a) g 1 b g 2 b g l l l l l n 2 1 l l n 2 1 l

n 2 2(k 1 1) 1 1 # U (0) 2 (1 2 a)g 1 b g l l l n 2 1 l

n 2 2k 2 1 , U (0) 2 (1 2 a)g 1 g l l n 2 1 l

(1 2 a)(n 2 1) 2 (n 2 2k 2 1) 5 U (0) 2 g . l n 2 1 l

Thus if

(1 2 a)(n 2 1) 2 (n 2 2k 2 1) (A17) $ 0, n 2 1 player l prefersto deviate from the equilibrium candidate and to 862 QUARTERLY JOURNALOF ECONOMICS choose gl 5 0.But this inequality isequivalentto (A18) (1 2 a)(n 2 1) $ n 2 2k 2 1 n 2 2k 2 1 Û a # 1 2 n 2 1 n 2 1 2 n 1 2k 1 1 2k Û a # 5 n 2 1 n 2 1 k a Û $ , n 2 1 2 whichis thecondition given in theproposition. (c) Suppose that theconditions of the proposition are satis- ed.We want toconstruct an equilibriumin whichall k players with 1 2 a . b i contributenothing, while all other n 2 k players contribute g [ [0, y].Weonly have to check that contributing g is indeed optimalfor the contributing players. Consider some player j with 1 2 a , b j.If hecontributes g,his payoff is givenby

(A19) Uj( g) 5 y 2 g 1 (n 2 k) ag 2 [a j/(n 2 1)] kg. It clearlydoes not pay tocontribute more than g.Sosuppose that player j reduceshis contributionlevel by D . 0.Then his payoff is

Uj( g 2 D ) 5 y 2 g 1 D 1 (n 2 k) ag 2 D a

a j b j 2 k( g 2 D ) 2 (n 2 k 2 1)D n 2 1 n 2 1

a j 5 y 2 g 2 (n 2 k)ag 2 kg n 2 1

a j b j 1 D 1 2 a 1 k 2 (n 2 k 2 1) n 2 1 n 2 1

a j b j 5 U ( g) 1 D 1 2 a 1 k 2 (n 2 k 2 1) . j n 2 1 n 2 1 Thus,a deviationdoes not pay if and onlyif

a j b j 1 2 a 1 k 2 (n 2 k 2 1) # 0, n 2 1 n 2 1 whichis equivalentto

(A20) k/(n 2 1) # (a 1 b j 2 1)/(a j 1 b j). ATHEORY OFFAIRNESS,COMPETITION, ANDCOOPERATION 863

Thus,if this conditionholds for all (n 2 k) players j with 1 2 a , b j,thenthis is indeed anequilibrium.It remainsto be shown that (a 1 b j 2 1)/(a j 1 b j) # a/2.Note that a j $ b j impliesthat (a 1 b j 2 1)/(a j 1 b j) # (a 1 b j 2 1)/(2b j).Furthermore,

a 1 b j 2 1 a # Û a1 b j2 1 # b j a Û a(1 2 b j) # 1 2 b j Û a # 1, 2b j 2 whichproves our claim. QED

Proofof Proposition5 Suppose that oneof the players i [ n8 1 1, . . . , n chooses gi , g.If all players stickto the punishment strategies in stage2, thendeviator i getsthe same monetary payoff as eachenforcer j [ 1, . . . , n8 .In this casemonetary payoffs of i and j are given by

g 2 gi (A21) x 5 y 2 g 1 a[(n 2 1)g 1 g ] 2 n8 i i i n8 2 c

g 2 gi n8 2 c (A22) x 5 y 2 g 1 a[(n 2 1)g 1 g ] 2 c 2 ( g 2 g ) j i n8 2 c n8 2 c i i

g 2 gi 5 y 2 g 1 a[(n 2 1)g 1 g ] 2 (n8 2 c 1 c) 5 x . j i n8 2 c i

Thus,given the punishment strategy of theenforcers, devia- torscannot get a payoff higherthan what theenforcers get. However,theyget a strictlylower payoff than thenonenforcers whodid notdeviate. We now have to check that thepunishment strategiesare credible; i.e., that an enforcercannot gain from reducinghis pij.If an enforcerreduces pij by e , he saves ce and experiencesless disadvantageous inequality relativeto those (n 2 n8 2 1) players whochose g but do notpunish. Thiscreates a nonpecuniaryutility gain of[ a i(n 2 n8 2 1) ce ]/(n 2 1). On the otherhand, theenforcer also has nonpecuniarycosts because he experiencesnow disadvantageous inequality relativeto the defec- torand adistributional advantage relativeto the other ( n8 2 1) enforcerswho punish fully.Thelatter generates a utility lossof b i (n8 2 1) ce /(n 2 1), whereasthe former reduces utility by a i(1 2 c)e /(n 2 1). Thus,the loss from a reductionin pij is greater 864 QUARTERLY JOURNALOF ECONOMICS than thegain if (A23) 1 ce [a (1 2 c)e 1 b (n8 2 1)ce ] . ce 1 a (n 2 n8 2 1) n 2 1 i i i n 2 1 holds.Some simple algebraic manipulations showthat condition (A23) is equivalentto condition (13). Hence,the punishment is credible. Considernow the incentives of oneof theenforcers to deviate inthe rststage. Suppose that hereduceshis contributionby e . 0.Ignoring possible punishments in thesecond stage for a moment,player i gains (1 2 a)e in monetaryterms but incursa nonpecuniaryloss of b 1e by creatinginequality toall other players.Since 1 2 a , b i by assumption,this deviationdoes not pay.If his defectiontriggers punishments in thesecond stage, thenthis reduceshis monetarypayoff whichcannot make him betteroff than hewould havebeen if hehad chosen gi 5 g. Hence, theenforcers are not going to deviateat stage1 either.It is easyto seethat choosing gi . g cannotbe pro table forany player either, sinceit reducesthe monetary payoff and increasesinequality . QED

Computationof theProbability That There Are Conditionally CooperativeEnforcers Tocomputethe probability that,in arandomlydrawn group offour,there are subjects who obey condition (13) and a 1 b i $ 1, wehaveto make an assumptionabout the correlation between a i and b i.Wementioned already that theempirical evidence suggeststhat theseparameters are positively correlated. For concrete- nesswe assume that thecorrelation is perfect.Thus, in termsof TableIII all players with a 5 1 or a 5 4areassumed to have b 5 0.6.This is clearlynot fully realistic,but it simplies theanalysis dramatically. In theFehr-Ga ¨chter[1996] experimentthe relevant pa- rametersare a 5 0.4, n 5 4,and (roughly 33)) c 5 0.2.The following summarystates the conditions on a i and b i implied by Proposition 5fora groupof n8 [ 1, . . . , 4 conditionallycooperative enforcers.

33.The cost function in Fehrand Ga ¨chter isactuallyconvex, so thatwe have toslightly simplify their model. Y et,the vast majority of actual punishments occurred where c 5 0.2. ATHEORY OFFAIRNESS,COMPETITION, ANDCOOPERATION 865

If oneof these conditions holds, cooperation can be sustained in equilibrium: (i) n8 5 1, a i $ 1.5, and b i $ 0.6; (ii) n8 5 2, a i $ 1 2 0.3b i, and b i $ 0.6; (iii) n8 5 3, a i $ 0.75 2 0.5 b i, and b i $ 0.6; (iv) n8 5 4, a i $ 0.6 2 0.6b i, and b i $ 0.6. Notethat foreach group n8 ofconditionallycooperative enforcers theconditions on a i and b i haveto hold simultaneously.Giventhe discretedistribution of a and b ofTableIII, this canonly be the case if c thereis at leastone player with a i 5 4 and b i 5 0.6, or c thereare at leasttwo players with a i 5 1 and b i 5 0.6, or c both. Giventhe numbers of Table III, itis notdifficult toshow that the probability that oneof these cases applies is equal to61.12 percent.

UNIVERSITYOF ZURICH UNIVERSITYOF MUNICH

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