Inequity Aversion Preference in the DynamicPublic Goods Game

Wei Hu ∗

Toulouse School of Economics (LERNA)

September 11, 2010

Abstract

This paper shows whether and how Fehr and Schmidt’s (1999)inequity aversion model can be used to explain cooperations in a dy-namic public goods game. We first extend the originally static modelinto a dynamic one by assuming agents’ non-material concern coversexpectations across all game rounds. This also allows us to redefine arepeated public goods game as dynamic under this model. Inequity-averse agents exhibit a conditional cooperation pattern of behavior inthe game, causing a severe multiple equilibria problem. This is fixedwhen we impose the Limited Punishment for Deviations principle,which allows us to predict nearly full cooperation over contributionsthroughout the game. Selfish agents can also be found to cooperateat high levels, provided that inequity-averse members (supposedly)exist. This result does not depend on whether there is uncertaintyabout players’ types. Other regularities like end-game effect, restarteffect and pulsing pattern are also captured. As a complement, Char-ness and Rabin’s (2003) quasi-maximin preference is also examined tocomply with most of these findings.

∗Contact: wei.hu@tse-fr.eu.

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1 Introduction

How did cooperative behavior evolve? This question is on the Science mag-azine among the top 25 “scientific puzzles that are driving basic scientificresearch”.1 To economists, cooperations do not seem to grow naturally outof the Homo economicus assumption. Resorting to people’s non-selfish be-havior certainly provides an explanation, but it is then a question about howsuch non-selfish behavior emerges. Equally important, we would also be in-terested in whether such behavior is sustainable over time, especially whenselfish and non-selfish individuals interact with each other. The objective ofthis paper is to shed some light over these later questions by using the gametheoretical tools.

The cooperative behavior that we look at here is contributions in a “re-peated” public goods game. This game is formed by finite repetitions of thestatic public goods game, as is usually practiced in the lab experiments. Forthe non-selfish preference, we refer to Fehr and Schmidt’s (1999) inequityaversion model. An inequity-averse agent bears extra psychological costs forany inequity between herself and others. Their behavior and impact haveso far been studied in various environments. For instance, how the opti-mal incentive scheme shall be decided in the workplace (e.g., Demougin andFluet, 2003, Itoh, 2004, Fehr, Klein and Schmidt, 2007, Torgler et al, 2008);how competition’s influence on bargaining is changed (Fischbacher, Fongand Fehr, 2009); formation of long-term trading relationships (Brown, Falkand Fehr, 2004); quantity collusions (Santos-Pionto, 2006, Muller, 2006) andprice rigidities of consumption goods (Rotemberg, 2004). The current paperthen contributes to this literature by extending the originally static-gamemodel into a dynamic framework, to our knowledge, for the first time.

To achieve this, we assume that what inequity-averse individuals careabout is the differences between expected payoffs from the entire game. Un-der this assumption, agents’ strategies include taking actions to correct pay-off differences generated earlier in the game, provided that they care enoughabout inequity. In the public goods game, that means if one has been con-tributing less than the others, she will later catch up with correspondinglyhigher level of contributions, as a difference in contributions translates di-

1Science 1 July 2005: Vol. 309. no. 5731, pp. 78 - 102.

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rectly into one in payoffs. The inequity-averse agents are therefore potentialconditional cooperators who adjust contributions with others. Their behav-ior would be hardly distinguishable from that of reciprocal agents who areborn conditional cooperators. We tend to also regard our inequity aversionmodel as a reduced reciprocity model in the current simple game.

For inequity-averse players who take with them the payoff differencesacross rounds, a later round in the game is not identical to an earlier one.The conventionally so-called repeated public goods game is hence redefinedas a dynamic one in our study.2

With the linear inequity aversion model, a multiple equilibria problemensues from the conditionally cooperative behavior. Like in a static game,any strategy that leads to symmetric aggregate payoffs can be taken in aNash equilibrium. The theory therefore has yet little help for making a clearprediction in this game.3 In an attempt to solve this problem, we introducea principle called Limited Punishment for Deviations to players’ actions. Byrestricting the harshness of punishment for equilibrium deviations, it helpseliminate a large amount of equilibrium that are only supported by harshenough punishments. The model then predicts nearly full contribution tothe public goods throughout the game.

This prediction stays valid even when selfish agents join the group. Imag-ine a group consisting of one selfish and one inequity-averse member. Startingfrom a zero contribution, suppose the selfish member alone contributes a pos-itive amount in an early round. The inequity-averse member will then wantto reduce their payoff difference by matching that contribution in the follow-ing rounds. As a result, the selfish one gets paid back twice by the publicgoods return for his contribution, which makes a net profit under the designof the public goods game.4 The inequity-averse agent’s willingness to match

2This sharply contrasts with the case under an alternative assumption that agents focuson single round inequities. If that was so, the “repeated” nature of the game would persist.

3Fehr and Schmidt (1999) focus on the static game. Claiming the final round of arepeated game to be same as a static game, they predict zero contributions based on acalibrated model using data from various experiments.

4His net payoff would be −g+ag+ag, where a represents public goods’ rate of return.g indicates his contribution; the first ag is the direct public goods return from his owncontribution, and the second is obtained from the inequity-averse member’s matching.

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contributions therefore provides a commitment device for a selfish memberto also cooperate. This device will be functioning till very end of the game,as long as the selfish member holds a contributing advantage, so that hisinequity-averse partner needs to perform enough catch-up even in the finalround. This result is further extendable to the situation where types areplayers’ private information, by employing Kreps et al.’s (1982) reputationgame method.

Our model indeed does not capture the stylized declining contributionsduring the dynamic public goods game. However, it does predict several otherregularities that are consistent with experimental evidences. They include:(i) an end-game effect as a sudden reduction of contribution in the finalround; (ii) a restart effect as a possible recovery of contributions when thegame has a surprise restart, after which nearly full cooperations follow, and(iii) a pulsing effect on the individual level, as one’s level of contribution maydrop and rebound for once during the game. We also consider an alternativeother-regarding preference model, known as the quasi-maximin preference(Charness and Rabin, 2003), for the same public goods game. We find thatalthough this model takes a different angle to depict one’s concern for others,it makes qualitatively similar predictions as the inequity aversion model.A distinction arises only for the private information case, where a quasi-maximin agent behaves as a more proactive contributor. This last study isrecorded in the appendix part for interested readers.

Related Literature

The current paper relates to several strands of literature in economics. Inbelow we will refer to the relevant works in game theory, in behavior eco-nomics about other-regarding preferences, and in experimental studies.

In the game theoretical literature, the folk theorem provides an explana-tion for cooperations in infinitely repeated relationships. As an equilibrium,cooperations can be sustained when players commit to strategies that pun-ish any deviation from equilibrium harsh enough (for a long enough period),so that no profit can be earned in any potential deviation. Theoretically,any outcome that gives each player more than his/her minimax payoff canbe supported by the folk theorem as an equilibrium. The cooperative equi-librium therefore is just one equilibrium out of many, and there is no prior

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reason for players to just pick this one. The folk theorem also fully appliesto finite but still long enough relationships, where some hidden informationabout a player’s commitment type is needed (see Fudenberg and Maskin,1986). The technique then relates to the reputation game demonstrated byKreps and Wilson (1982) and Kreps et al. (1982). A rational agent mimicsthe behavior of a committed type in each round to benefit from building areputation of irrationality. A small prior suspicion about the existence ofirrationality would do in this case, provided that the game is long enough.For our current study, we also use this technique.

The behavioral literature on other-regarding preferences has two classicalmodels, namely altruism and reciprocity. Pure altruism is featured with afixed concern about others as a characteristic of an individual. Andreoni andMiller (2002) show that their game attendants’ choices are consistent withsome nonlinear altruism models, which can be used to predict contributionsin the beginning of public goods games, as well as for the all-time average.However, the models keep silent about any interactive feature of the game.Contrastingly, reciprocity models highlight interactions by letting one’s con-cern for others dependent on her conjecture about their kindness (e.g., Rabin,1993, Bolton and Okenfels, 2000, Charness and Rabin, 2003, Dufwenberg andKirchsteiger, 2004). As an example, for a sequential public goods game Falkand Fischbacher (2006) suggest that individuals tend to reciprocate less toothers, hence slowly take down contributions. Our current inequity aversionstudy, if one wants to, can be regarded as using a reduced reciprocity modelto expand this literature into repeatedly played simultaneous move game.

For more recent behavioral studies, Ambrus and Pathak (2009) and Figu-ieres et al. (2009) propose their own models to understand regularities in thepublic goods game. The former supposes reciprocal agents who only careabout matching actions with a group average calculated over time. Theseagents if coexist with selfish ones will bring full cooperations in early roundsand declining average contributions in late rounds of the public goods game.The latter introduces “weakly morally motivated” agents who suffer frombehaving differently from their moral targets. This target is determined foreach round based on one’s prior ethical level and the previous round groupaction. It tunes lower along the game as one is always lured away from it,which in turn creates the declining contribution path from the start of game.

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The experimental literature of public goods game has flourished duringthe past decades. Among various designs, the voluntary contributions mech-anism that we use in this paper has been extensively examined (see chapter3 of Ledyard (1995) for a survey). Its primary regularities include: (i) pub-lic goods contributions in both one-shot and finitely repeated games; (ii)levels of contribution start high but decline with repetitions; (iii) contribu-tions plummet in the final rounds, but pick up in a surprise restart. Forthese regularities, Andreoni (1988, 1995) proposes altruism as the individualmotivation, and suggests dissolved confusion as the cause of declining coop-erations. Palfrey and Prisbrey (1997), however, find warm-glow to be moresignificant than pure altruism. Croson (1998, 2007) but comes up with condi-tional cooperation or reciprocty, after finding a positive link between players’contributions (also between one’s action and belief about other’s). Severalwork has since extended this finding. After categorizing players as condi-tionally cooperative or selfish, they find homogeneous groups of cooperatorsalways achieve high level of public goods, meanwhile heterogeneous groupsshould see declining cooperations along the game. (Gachter and Thoni, 2004,Burlando and Guala, 2005, Fischbacher and Gachter, 2008). Fischbacher etal. (2001) also notice that a majority of their conditional cooperative agentstend to contribute less than the group average. This selfish-biased conditionalcooperation is then suggested as another cause for the patterns (Neugebaueret al., 2009). Finally, Fehr and Gachter (2000) (also Masclet et al. 2003,Sefton et al. 2007) show that if given chances, players will punish free-ridersin the group even at their own costs. They relates this to negative emotionsagainst non-cooperative individuals, and one may as well consider it as asupport for the reciprocity motivation.

The remaining parts of this paper are organized as following. Chapter 2presents the model. In Chapter 3, the model’s performance is examined in astatic public goods game. Chapter 4 presents the complete dynamic analyses,including both the complete information and incomplete information cases.Chapter 5 then concludes. The quasi-maximin preference is examined in thesame manner and included in the appendix. The appendix also includes allrelevant proofs.

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2 The Inequity Aversion Model

The inequity aversion preference proposed by Fehr and Schmidt (1999) is astatic game model that captures an individual’s concern beyond own materialpayoff in the following way:

Ui(x) = xi − αi1

n− 1

∑j 6=i

max{xj − xi, 0} − βi1

n− 1

∑j 6=i

max{xi − xj, 0}

In a group of n members, individual i cares about not only her materialpayoff xi, but also the payoff differences between her own and the others. Thisextra concern is measured by αi and βi. Fehr and Schmidt suggest 0 ≤ βi < 1and αi ≥ βi, indicating that having either more or less payoff than othersgenerates a psychological cost of inequity, with a disadvantageous inequitybeing more costly than an advantageous inequity.5 The inequity among oth-ers is not a direct concern.

We will only look at the n = 2 case in this paper for analytical simplicity,and denote the two players by player i and j, respectively. The linearityof the model and the game also allows us to focus on uniform α and β forinequity-averse individuals, without missing any essential feature.6 Since wecare about how selfish and non-selfish agents interact for cooperations, a spe-cial attention will be paid to the case where inequity-averse and purely selfishmembers coexist in a group. Hereinafter we use “he” for a selfish people and“she” for a non-selfish people to avoid any confusions.

The inequity aversion model has two scenarios with n = 2:{Ui = (1 + α)xi − αxj, when xi ≤ xj;

Ui = (1− β)xi + βxj, when xi > xj.(1)

5This proposed value for β rules out status seeking (one enjoys being better off thanothers, i.e., βi < 0) or a willingness to sacrifice own payoff without improving the others’(βi ≥ 1) in order to reduce an advantageous inequity.

6As both the VCM game and individual preferences are linear, in all situations we willwork with threshold parameter values. If there are heterogeneity in parameter values,there are again only two situations: either one’s α β values are beyond this threshold andshe behaves non-selfishly, or in the contrary so she acts like a selfish one. Allowing thecoexistence of selfish and inequity-averse individuals equally capture these two cases.

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3 The Static Public Goods Game

We start with applying the model to a static public goods game. Fehr andSchmidt (1999) have a complete-information setup with k inequity-averseplayers in a n-member group. In stead, players in our setup do not haveexact knowledge about each other’s type, but hold a prior belief p as thepossibility that he or she is playing against a selfish one. This reproducesFehr and Schmidt’s original results.

In the static public goods game, each player in the group receives an en-dowment equal to ω. Everyone decides how much investment to make for thepublic goods, denoted by gi and gj. The rate of return of the public good isa ∈ (1/2, 1), and that of the private good is 1. Player i’s material payoff xiexpands to xi = ω − gi + a(gi + gj).

The strategy set of a player is a set {ω} that includes all integers between0 and ω. {ω}× {ω} forms a set that covers all possible public goods realiza-tions by the group, with each element being a pair (gi, gj). An equilibrium isan element of this set with which players’ utilities are maximized given theother’s action. Denote it by (g∗i , g

∗j ), then:

if i is selfish:g∗i = arg max

gi[ω + (a− 1)gi + ag∗j ]; (2)

if i is inequity-averse:

g∗i =

{arg maxgi [ω + (a− 1)gi + ag∗j − α(gi − g∗j )], if gi ≥ g∗j ;

arg maxgi [ω + (a− 1)gi + ag∗j − β(g∗j − gi)], if gi < g∗j .(3)

For a < 1, selfish players always choose the minimum level of contribu-tion. Hence based on the selfish assumption, the traditional theory predicts(0, 0) as the unique equilibrium, leaving either player with a payoff of ω.However, if players could contribute ω each, both would have received 2aωinstead, which is more profitable given a ∈ (1/2, 1). This is the well-knownpublic goods problem to arise in the current game.

The first row in (3) demonstrates an even stronger disincentive to con-tribute, if a player is inverse to inequity and expects to contribute more than

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the other. In that case, she will cut gi to reduce both the material and psycho-logical cost. But in the contrary, when i is expected to contribute less thanthe other, increasing gi by one unit costs materially 1 − a, but also reducesthe psychological by β. A high enough aversion to advantageous inequitywill then convince i to increase gi until it equals the other’s. Specifically, ifa+β− 1 > 0, the second row in (3) predicts public goods contributions . Tofocus on this more interesting case with contributions, we assume throughoutthe paper that:

Assumption 1. a+ β − 1 > 0.

Inequity-averse players will then coordinate contributions in an equilib-rium, even when p > 0 — otherwise, one who is supposed to contributemore could deviate with profit to a lower g as mentioned in above. Herep will only add to the chance that a contributor ends up over-contributingwith a psychological cost. This is clearly seen if one varies p in the followingequilibrium condition for an inequity-averse player i:

arg maxgi

p[ω+ (a− 1)gi−αgi] + (1− p)[ω+ (a− 1)gi + ag∗− β(g∗− gi)] = g∗

Proposition 1 concludes for the above equilibrium play in a static gamewith incomplete player type information. It follows closely Proposition 4 inFehr and Schmidt’s (1999) paper.

Proposition 1.

(a) It is always a dominant strategy for a selfish player to contribute 0, andweakly for an inequity-averse player to do so if a+ β − 1 ≤ 0;

(b) If a+ β− 1 > 0, and 1− p < 1−a+αα+β

, there is a unique equilibrium where

(0, 0) is realized as public goods contribution;

(c) If a+ β− 1 > 0, and 1− p ≥ 1−a+αα+β

, there is an infinity of equilibria. In

each of these equilibria, inequity-averse players choose gi = gj ∈ [0, ω].

Part (c) shows a practical impossibility to make predictions for the game,thanks to inequity-averse players’ willingness to coordination with each otherat any level of public goods contribution. One solution for this multiple-equilibria problem is to use refinements, for example, the payoff dominance

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and risk dominance criteria defined by Harsanyi and Selten (1988). In thefirst case, players shall coordinate on the equilibrium that is Pareto superiorin the equilibrium set; in the second, players expect each other to choose anyequilibrium action with equal probability, hence each picks up the one thatgives herself the highest expected payoff. Different refinement surely leadsto different equilibrium predictions. As there is no prior reason for playersto collude on any specific criterion, we put only two refinement examples tothe appendix for interested readers to read.

We are now ready to take the inequity aversion model into the dynamicgame.

4 Dynamic Public Goods Game

When the static game is repeatedly played for T times, it is convention-ally labeled as a “repeated public goods game”, with each repetition beingcalled a round. But for the inequity-averse agents, the payoff differences thatthey care about are not necessarily independent variables across rounds. Forexample, we assume:

Assumption 2. Concern for Expected Payoffs: An inequity-averse playerevaluates and compares expected total payoffs from the game at each round.

The expected differences between each other’s total game payoff update asplayers take actions through the game. Individual rounds hence distinguishfrom each other with difference characteristic variables of expected payoffdifference. Based on this, this game played by inequity-averse agents cannotbe considered as formed by simple repetitions. We’d rather reframe it as adynamic game.

An inequity-averse player will then not restrict her concern on any advan-tageous or disadvantageous payoff differences generated in a specific round.She can accept them as long as they are expected to be offset in the fu-ture. This player obviously distinguishes herself from a myopic individual:by myopia, we mean either one can only focus on the current-round expectedreturns, or somewhat better, focuses on the accumulated payoff differencesfrom the beginning till the current round. For either of these latter cases,the game would not be played the same as under our assumption in above.

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To characterize the dynamic game, number rounds in the ascending orderso that the game starts in round-1 and ends in round-T . Use set {t} toinclude all integers between 1 and T , and set {ω}for all integers between0 and ω. A = Ai × Aj is the action set for the group of players, withAi = Aj = {t} × {ω}. Each element of A is a vector gt = (gti , g

tj), indicating

a possible pair of public goods contributions of the group at round-t. Thecomplete realization of public goods contribution from round-1 till the startof round-t makes the history of this round, denoted by ht = (g1, g2, ..., gt−1).All the possible histories constitute the set of round-t history H t ∈ H, whereH = {t}×H t. Let Si be the set of pure strategy of player i and Sj of playerj. Each element in Si is a mapping from H to Ai, denoted by si. Finally, theintertemporal expected utility of player i at the beginning of any round-t isdenoted by ui(si, sj|ht) : H t × A 7→ R:

uti(si, sj|ht) = (t− 1) · ω −t−1∑k=1

gki + a( t−1∑k=1

gki +t−1∑k=1

gkj

)+ (4)

+ (T − t+ 1) · ω −T∑l=t

gli + a( T∑

l=t

gli +T∑l=t

glj

)−

− αmax{ T∑m=1

gmi −T∑

m=1

gmj , 0}− βmax

{ T∑m=1

gmj −T∑

m=1

gmi , 0}

The first row in (4) is i’s already realized material payoff until the startof round t; the second row is her expected payoff from round t on till the endof game, with glj indicating expected contribution of the other member. Thethird row records the expected overall payoff difference from the completegame, with upper bars being skipped for gmj ,m ≥ t.7 We notice that thepsychological cost of inequity arises if and only if the two members finish thegame with different total contributions to the public goods.

7(4) is obviously obtained by replacing instantaneous material payoffs xti and xtj in:

E[U ti ] =

t−1∑k=1

xki +

T∑l=t

xli − αmax{ T∑

m=1

xmj −T∑

m=1

xmi , 0}− βmax

{ T∑m=1

xmi −T∑

m=1

xmj , 0}

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4.1 Complete Information Game

Before we examine a group with mixed types of player and hidden infor-mation, let us first look at the complete information scenarios. We canisolate the influence of reputations and obtain a clear view of the influenceof inequity-averse agents.

Use backward induction, we draw a quick conclusion for the scenariowhere both players are selfish: since neither player would contribute in round-T , no matter what history, this round will provide no credible threat to sup-port any cooperations in the penultimate round, which in turn has to alsoend up in gT−1 = (0, 0). Roll this back to the first round, there is the uniqueequilibrium: (gti , g

tj) = (0, 0),∀t.

The other extreme scenario has both players as averse to inequity. Basedon (4), one can think about the whole game (from round-1 to round-T ) as asingle p = 0 case in part (c) of Proposition 1: given a + β − 1 > 1, the fearfor incurring any psychological cost of inequity drives both players to matchtheir total public goods contributions.

Then what could one expect for any individual round? The answer is,unfortunately, we do not know. Without further disciplines, an equilibriumfulfills only two requirements. The first is the equivalence of overall contri-butions, and the second relates to the final rounds’ play. Specifically:

Proposition 2. For the dynamic public goods game played between twoinequity-averse players, if a + β − 1 > 0, a pure strategy equilibrium thatis subgame perfect, (gti , g

tj), is fully characterized as following:

(a)∑T

t=1 gti =

∑Tt=1 g

tj,

(b) gT−1i + gTi > ω, gT−1j + gTj > ω.

Proof for this proposition as well as for all following ones are relegated tothe appendix. We look at the stories behind the veil here. Part (a) is nothingbut the Concern for Expected Payoffs assumption. As we already discussed,as long as (a) is satisfied, pairs of contribution like (0, ω) in some roundcan arise in an equilibrium. Part (b) is actually the minimum amount ofcontributions that are needed to support any equilibrium. They are supposed

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to be forfeited if deviations happen in any earlier rounds. Part (b) defines theminimum amount of public goods to be achieved in an equilibrium, whichis the sufficient threat if harshest punishments are always imposed againstdeviations. By punishment, we mean:

Definition 1. Following a player’s deviation from the equilibrium path, theother member can force future coordinations to happen on a lower level ofcontributions. The difference between this level and the original equilibriumlevel, excluding the deviation itself, forms a Punishment.

To illustrate it, suppose player i commits that, if j deviates from a{(gT−1i , gT−1j ), (gTi , g

Tj )} game profile by playing gT−1j + ∆ in round T − 1,

she will contribute gTi − P + ∆ in the next round, where P > 0. Since jwants to equalize overall contributions, she will then respond with gTj − Pin the final round. The two players then receive for the final two roundsgT−1i + gTi −P + ∆ each, as compared to gT−1i + gTi on the equilibrium path.This extra public goods loss P hence forms a punishment for the deviatingbehavior. The harshest punishment would be made when at least one player’sround-T contribution is suppressed to 0, e.g., P = gTj . At the same time, j’s

potential deviation gain is bounded by ∆ = ω − gT−1j . Part (b) then showsthe smallest gT−1 that can be supported by the highest P in an equilibrium,i.e., ω − gT−1j < gTj .

Once the final two rounds follow Proposition 2, they can jointly form apunishment period for deviations in any earlier round. The harshest pun-ishment in this period (e.g., P = gT−1j + gTj ) will be enough to prevent anysingle-round deviation (which is bounded below ω) in an equilibrium. Thisis exactly why individual round contributions are totally undefined for earlystages of the game.

Proposition 2 presents to us an even more severe multiple equilibria prob-lem, if compared to that in the static game. In an attempt to improve themodel’s predictability, we choose to put restrictions over deviation punish-ments. Imagining between players they hold such a mutual agreement: “de-viators can be punished, but not to infinity. Letting him pay up to whathe has deviated would be enough.” We define this “agreement” into thefollowing principle of Limited Punishment for Deviations (LPD):

Definition 2. Limited Punishment for Deviations (LPD): A devia-tion from equilibrium is punished in the remaining game, with the size of

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punishment not exceeding that of this deviation.

This LPD principle can be illustrated in the same manner that we haveused to define Punishment just now. Take an equilibrium path gt = (gti , g

tj).

The LPD rule says:

∀t < T , if gt = (gti + ∆, gtj),∀∆, then gk = (gki , gkj + ∆), and

gl = (gli, glj),∀l /∈ {t, k},

where: gki = gki −P, gkj = gkj −P . P for punishment and P ∈ [0, |∆|], with |∆|indicating ∆’s absolute value. Round-k is the first round following round-tthat could accommodate above actions.

If players do implement this principle, deviation from an equilibriumwould become less costly, and the equilibrium set will shrink. This set actu-ally downsizes into:

Proposition 3. For the dynamic public goods game played by two inequity-averse players, if a+β−1 > 0, there is a unique type of pure strategy equilibriaunder LPD. Each equilibrium features the following path of contribution tothe public goods:

(a) gti = gtj, ∀t;

(b) gti = gtj = ω,∀t with at most an exceptional k, when gki = gkj ∈ [0, ω].

Now an equilibrium profile no longer accommodates more than two roundswhere players contribute less than maximum. If that ever happened, LPD en-sures that either player could find deviation opportunities with non-negativeprofits, usually by over-contributing before the game ends. As there is nofurther room for one to raise contributions and benefit from the other’s cor-responding raise in a nearly full-contribution equilibrium in Proposition (3),such deviations are prevented.

Proposition 3 gives a clear and strong prediction for the game playedunder complete information. We shall expect players to coordinate at fullcontributions nearly through the game, if they ever plan to contribute at all.The chance to play the game for more than one-shot indeed functions as apowerful coordination device between inequity-averse players.

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The last scenario under complete information is when a group is com-posed of one inequity-averse player and one selfish player. Such a group canhave only zero contributions in a static game, so would it perform in a real“repeated” game. However, the dynamic game feature that we have empha-sized here will allow us to tell a complete different story. Imagine a situationwhere no one contributes to the public goods through the game. If the selfishplayer raises his contribution to δ in round T − 1, the inequity-averse playerwill certainly match by contributing δ in round-T . That improves the selfishplayer’s payoff by (2a−1)δ, and he surely wants to do so. A quick conjecturewould be, if the selfish player can keep a larger total contribution than theinequity-averse member until round T −1, the latter will make contributionseven in the final round.

An important equilibrium feature of the current scenario now emerges:if the inequity-averse player commits to make high enough contribution inthe final round, it forms a credible threat for the selfish player to contributein early rounds. The existence of inequity-averse players provides a kind ofcommitment device for the selfish ones to contribute. This selfish agent willbe glad to take it: by performing this opportunistic generousness, he receivesa higher overall payoff that cannot be achieved if playing against anotherselfish member.

As a result, a mixed group will coordinate in a way close to that of agroup with only inequity-averse players. The equilibria without LPD hasequity in overall contributions as well as a high end-game play. When LDPis employed, contributions will rise up to nearly all-time maximum by bothplayers. Denote the inequity-averse player’s contribution by gIA and theselfish player’s by gS, the public goods are realized in an equilibrium as:

Proposition 4.∑T

t=1 gtIA =

∑Tt=1 g

tS, gTS = 0,and

When the LPD rule is not imposed:

• either gTIA = ω, gT−2IA + gT−1IA > ω, gT−2S + gT−1S > ω;

• or gT−1S = ω, gTIA ∈ [1−aaω, ω], gT−2IA + gT−1IA > ω.

When the LPD rule is employed:

• gtS = ω,∀t < T , and gTIA ∈ [1−aaω, ω].

15

One can see that the final rounds are characterized with gTIA ≥ 1−aaω, and

max{gT−1S , gTIA} = ω. The former is the minimum final-round contributionthat has to be committed by the inequity-averse player, so that the selfishplayer stays on the contribution path in early rounds. The latter conditionensures that the selfish player has either no possibility to deviate to highercontributions in round T −1, or will receive no benefits after doing so. Mech-anism in the second condition also helps the selfish player to stick to the fullcontribution equilibrium under LDP. The remaining conditions more or lesscorrespond to those of the two inequity-averse players’ group.

Till now we have seen that, the existence of at least one inequity-aversemember will ensure public goods contributions of a group. The contribu-tions will be made equally between the members. With a restriction on thepunishment that one may give to a deviator, the model can be cured froma multiple equilibria problem. Our predictions would go for nearly full con-tributions in the dynamic public goods game. And noticeably, even a selfishplayer can commit to full contributions in any non-final rounds.

We tend to use this as a potential explanation for the regularity of restarteffect. According to Proposition 3 and 4, after players’ types are revealed atthe end of a game, a restart will be played under complete information, andwill still have contributions as long as not both players are selfish.

4.2 Incomplete Information Dynamic Game

Let now the type information be kept with each player herself, and one holda belief at the start of round-t that the other member is selfish by probabilityp(ht), or pt for simplicity. Player i’s expected utility at this round is thenuti(si, sj|ht, pt) in a pure strategy equilibrium, and uti(σi, σj|ht, pt) in a mixedstrategy equilibrium, where σi denotes i’s mixed strategy as a distributionover si. We still use p for the common prior belief before the game starts.Furthermore, LPD will be employed whenever possible.

We have shown that the existence of selfish players jeopardizes publicgoods contributions only when two such people know that they are meet-ing each other. In the asymmetric information game, when and how thisinformation should be revealed decides the way the game is played. Playerscan choose to reveal types either in one-time or slowly during the game, so

16

we also look separately at the game played under pure strategy and mixedstrategy.

In a pure strategy equilibrium, neither player may ever contribute if thechance to meet a selfish player, p, is too high. In this case, initiating con-tributions has a low chance to be paid back in the future, so a selfish playerlacks the incentive to deviate from a zero contribution equilibrium in the waysuggested by Proposition 4. An inequity-averse player would not want to dothat either, for she risks even an extra psychological cost of inequity whendeviates to contributions.

In the contrast, when p is small enough, the other extreme profile withboth contributing maximally in nearly all rounds becomes possible. Evenwithout new information about types, a small p still allows inequity-averseplayers to coordinate over any x ∈ [0, ω] in the final round, as Proposition1 shows. A high enough x will then be a credible threat to prevent eithercontributor from shirking in an early round, which does prevent types frombeing revealed until the final round. Meanwhile, any gti < ω in a non-finalround-t is prone to deviations: an inequity-averse player can at least save po-tential psychological costs by shifting contributions from the final round tothis round, meanwhile not necessarily changing the aggregate contributions.

The revelation of types may as well happen before the final round. How-ever, if that does happen, it should not be far from the end of game. Beingrevealed as selfish in the early game is costly, since the public goods willthen be contributed at a reduced level, although there are still cooperationsbetween revealed selfish and inequity-averse players, as in Proposition 4.

The above discussions are formalized in the following proposition:

Proposition 5. For a T round game with private type information, therecan arise the following equilibrium paths under pure strategies. The first twoare also unique under LPD:

(a) for 1− p < min{

1−aa, 1−a+α

α+β

}: gt = (0, 0),∀t;

(b) for 1− p ≥ max{

1−a+αα+β

, 1−a+αa+α

}:

- either gt = (ω, ω),∀t < T , gTIA = x ∈ [1−aa

ω1−p , ω], and gTS = 0;

17

- or gtS = gtQM = ω,∀t < T and t 6= k, with gkS = gkQM ∈ [0, ω];gTS = 0, gTQM = ω.

(c) for 1 − p ≥ (1−a)2+aαa(1−a)+aα : gt = (ω, ω),∀t < T − 1. gT−1IA = ω and gT−1S =

gTS = 0. Then gTIA = y ∈[

(1−a+α)p(2a−1)(1−p)ω,

(1−a)a

ω1−p

]after gT−1 = (ω, ω).

Otherwise, gTIA = 0.

(a) and (b) depict all the pooling equilibrium and (c) depicts all the sep-arating equilibrium. We shall talk a bit more about (c). In a separatingequilibria, an inequity-averse player risks contributing more than the otherwhen following the equilibrium to reveal herself through contributions. Shecould however avoid this psychological cost if instead waits for the other tocontribute first and then catches up. Comparing to a selfish player, this psy-chological cost makes an inequity-averse player a less-willing-to-contributemember in the revelation round, whenever the catch-up cost (a − 1)gT−1IA isnot so high comparing to the post-revelation coordination cost (a−1)gTIA. Inparticular, gTIA = gT−1IA = ω will not be feasible for a separating equilibrium.For this reason, gTIA need be manipulated downwardly so that the catch-upcost becomes significant enough to prevent an inequity-averse player fromchoosing to wait-and-see in the revelation round. This is how y is chosen inabove.8

In Proposition 5, the final-round contributions, x and y appear as abruptcontribution drops from the contribution levels in previous rounds. We tend

8This argument is seen more straightforwardly in math below. For an inequity-averseplayer, to contribute in the T −1 round instead of playing wait-and-see needs the followingto be positive:[

ω + (a− 1)gT−1IA + a(1− p)gT−1

IA + ω + (2a− 1)(1− p)gTIA − αpgT−1IA

]−

−[ω + a(1− p)gT−1

IA + ω + (a− 1)(1− p)gT−1IA

]= (a− 1)gT−1

IA + (2a− 1)(1− p)gTIA − αpgT−1IA − (a− 1)(1− p)gT−1

IA .

In contrast, for a selfish player to do so just needs the following to be positive:[ω + (a− 1)gT−1

IA + a(1− p)gT−1IA + ω + a(1− p)gTIA

]−

−[ω + a(1− p)gT−1

IA + ω]

= (a− 1)gT−1IA + a(1− p)gTIA,

which, when gTIA = gT−1IA , is easier to satisfy than the inequity-averse player’s.

18

to refer to them as a support for the end-game effect.

The pure strategy equilibria that we see in above predict no contributionswhen the prior belief 1− p is too low to attract one to initiate contributions.Alternatively, if players adopt mixed strategies and let beliefs updated grad-ually along the game, there could be a time when one’s belief 1 − pt finallyis updated over the threshold level, so that he/she makes contributions fromthen on, which wills turn back become a support for contributions in earlyrounds. This is the idea of the reputation game that is demonstrated byKreps and Wilson (1982) as a solution to the chain store paradox.

In their game that reproduces the chain store paradox, there is a monopolyincumbent in a market which lasts for finitely many rounds. In each round, apotential entrant decides whether to enter the market, whilst the incumbentdecides whether to fight if entry happens. For the entrant, entering the mar-ket without being fought against is payoff superior to staying out, which isstill payoff superior to entering the market and being fought. For the incum-bent, fighting is costly in a single round but deterring entry is beneficial inthe long run. The finite feature of the game determines the entry to happenfrom the beginning and no fighting whenever: there is no credible fighting inthe final round that can support fighting threats in all early rounds, and anentrant figures this out using backward induction.

Kreps and Wilson (1982) introduce the possibility that a player is pos-sessed by fighting and never succumbs. Though this probability might besmall, a normal incumbent can slowly build a reputation of being possessedby randomizing between fighting and succumbing in each round facing anentry. In this way, if fighting has been always observed along the game, thebelief about this incumbent being actually possessed is enhanced, and anentrant may not enter for certain in the final round of the game.

In our public goods game, players can employ a similar strategy in con-tributions. For example, if the inequity-averse player sticks to an actionsuch as contributing, a selfish player’s strategy will be randomizing betweencontributing and not in the same round. If one always observes the othercontributing, then the belief about that member being inequity-averse is im-proved until it allows contributions to happen even in the final round.

19

Let us denote rt as the probability that a selfish player contributes inround-t. Based on Proposition 5, we only focus on ω and 0 as potential lev-els of contribution before the final round. Use qt for the probability that ωis contributed by a player in round-t. So if inequity-averse players are sureto contribute ω in this round, there is qt = (1 − pt) + ptrt. Belief is thenupdated according to Bayes’ rule: 1− pt+1 = (1− pt)/qt. Whenever it is notfor certain that one will contribute, i.e., qt < 1, the other player will hold ahigher 1− pt+1 as compared to 1− pt after observing ω in round t.

The belief can be updated in the following way. In some round-k, selfishplayers randomize between contributing ω and 0, meanwhile inequity-averseplayers contribute 0 for certain. If gk = (ω, 0) is realized, the player who hascontributed ω is revealed as selfish, whereas the other can be believed to bemore likely inequity-averse, comparing to the prior belief. Call the revealedselfish player SR, and a selfish player who is still hiding SUR.

An inequity-averse player then matches the other member’s previousround contribution. She does not need to solve an explicit decision mak-ing problem, since this behavior ensures equal payoffs between the two. SUR

instead, randomizes between contributing ω and 0 in each round that is pre-ceded by ω contribution from SR, except for the penultimate round whenSUR contributes 0. In this way, {1 − pt} ascends with t after each ω be-ing contributed by the player whose type is yet unrevealed. As for SR, aslong as his ω contribution is matched by the other, he will alternate betweencontributing ω for sure and randomizing between ω and 0 in the followinground, until the penultimate round. But once the other fails to match his ωcontribution, SR will contribute 0 from then on.

Before round-k, both players will all the time contribute ω for sure.Any contribution pair like gk = (0, 0), or gk = (ω, ω), or gk = (ω, 0) butgti = 0,∀t > k, will lead to zero contributions from that round on till the endof the game.

It can be shown that:

Proposition 6. The above describes an equilibrium profile that is subgameperfect.

To elaborate details of the equilibrium is tedious and interested read-

20

ers are suggested to read it in the appendix. The tricks are, since aninequity-averse player is not naturally more willing to contribute than a self-ish member, it would be convenient to create a leading contributor, so that aninequity-averse player will only “mindlessly” match with contribution, whichis easy for a selfish player to mimic. At the same time, the leading contributormust not always reward matching behavior with new contributions, for thatwould fail to convince a selfish follower to randomize and let the belief update.

Like in the case of Kreps and Wilson (1982), the last equilibrium profileallows public goods to be contributed even with very low prior belief aboutinequity-averse players. The value of 1− p only matters for the length of thesecond stage in the above reputation game: the smaller is 1 − p, the morerounds will be needed to get it updated over the critical level. As long as agame has a length to accommodate this, public goods are contributed withcertainty in the early stage and at a probability in the late period, at eventhe maximal level. 9

5 Conclusions

In this paper we have examined the inequity aversion model in the publicgoods game. The model captures individuals’ dislike to distributional in-equities at the same time of enjoying their normal material payoffs. Thischaracteristic determines that an inequity-averse agent may tend to matchother’s action in order to achieve equal incomes. Their behavior will presenta conditional cooperation pattern in the public goods game.

To take the originally static inequity aversion model into the dynamicgame, we have assumed that non-selfish agents are not myopic but focus on

9This equilibrium does not have to be the unique one with contributions. Dependingon how off-equilibrium beliefs are defined when Bayes’ rule does not apply, other equi-libria might as well be specified. We however can be sure about one thing, that is, hadthere being no leading contributor, it would not happen that an inequity-averse individ-ual contributes with certainty in each round and to get mimicked by a selfish one. Thepotential psychological loss in contributions makes an inequity-averse player all the waymore demanding than a selfish member, whenever being called upon to make contributiondecisions. Analytically, expecting her to contribute with more certainty than a selfishmember will quickly run into a conflict between the two types’ decision making problems.

21

expected total payoffs from the game. With this assumption, each roundof the repeated public goods game is featured with a kind of state variablethat records expected payoff differences at that round. We hence suggest totreat this game as a dynamic game rather than a genuine repeated one. Ofcourse, this suggestions will be prone to challenges if inequity-averse agentsare assumed to be myopic, instead.

In the dynamic public goods game, inequity-averse agents will coordinatetheir contributions if they care enough about advantageous inequities, i.e.,earning more than others is uncomfortable enough so that one will not free-ride on other’s contributions. However, cooperation towards public goodsrealizations turns out to be only one result out of an infinity in the equi-librium. This multiple equilibria problem is a result of agents’ conditionalcooperation behavior. We introduce a special behavioral rule called the prin-ciple of Limited Punishment for Deviations. It effectively helps to downsizethe equilibrium set, and allows us to find nearly full cooperation betweennon-selfish agents on public goods contributions.

The cooperation is maintained even when selfish individuals join thegroup. As long as an inequity-averse agent exists in the group, her willingnessto match contributions can provide a commitment device for a selfish one tostay on the contribution equilibrium. Such a cooperation can be easily sus-tained until the game approaches the end, as long as the selfish agent keepsa strong enough leading position in overall contributions. The same LimitedPunishment for Deviations principle will again allow players to choose thenearly full cooperation towards achieving maximum amount of public goods.

Introducing hidden information about players’ types turns out to be notqualitatively changing the above results. However, it does bring more pre-dictions about other regularities of the public goods game. These includean end-game effect, which corresponds to selfish players revealing themselvesas the game ends, or inequity-averse players coordinating over lower level ofcontributions to encourage types being revealed earlier. The restart effectthen naturally emerges, as revealed selfish players can still cooperate withrevealed inequity-averse players in a surprise restart. An individual puls-ing pattern is also weakly captured, typically when an inequity-averse playerchooses the strategy of wait-and-see in a case when prior belief about theother being selfish is high.

22

There are still other regularities of the public goods game that are beyondthe explanation of the inequity aversion model, notably the declining contri-bution trend in the game. The Limited Punishment for Deviations principlealso excludes any less-than-maximum contributions in the equilibrium untilthe final rounds. No wonder, many more equilibrium patterns will appearif we drop this principle. However, we still prefer our choice in this tradeoffbetween predictability and fitting the data.

For a closing remark, we realize that it is not yet clear how one can im-prove the predicability of current inequity aversion model, or related other-regarding preference models (see the complementary study of the quasi-maximin preference in the appendix). This study does not show any helpby including hidden information, whereas experimental literature tends tosuggest using coexistence of heterogeneous types. Nonlinear utility functionscan of course easily bring up non-maximum level of contributions, whichare also sensitive to the rates of return to the public goods. However, itmay not directly help with forming the declining trending in contributions.Let us expect more theoretical work to join this literature in the short future.

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Appendices

A The Quasi-Maximin Model

In this section we use Charness and Rabin’s (2003) quasi-maximin model togo through all that we have done in the paper with the inequity aversionmodel. The two models differ in their angles used to look at interactions.And although the inequity aversion preference can capture both negativeinterpersonal mood like envy and prosocial actions such as donation, it ischallenged as being not helpful for our understanding of some helping be-havior that even enlarges payoff differences. Engelmann and Strobel (2004)point out that adding a concern for efficiency like Charness and Rabin (2003)do can give a hope.

The quasi-maximin model can be presented in the following weightedaverage form:

Ui = (1− ρ− σ)xi + ρmin{x1, x2, ...xn}+ σn∑j=1

xj. (5)

xi is again individual i’s material payoff. ρ ≥ 0 comes from the normal max-imin preference, and σ ≥ 0 captures how one cares about efficiency, in termsof aggregate group income. This is how the model gets its name. Besides,

27

1− ρ− σ ≥ 0. 10

A person with such a preference is willing to see other’s payoff rises, eventhough their payoff may already be superior to hers. And it is never her wishto destroy others’ wealth only to reduce some payoff differences. This clearlydistinguishes from the inequity aversion preference.

The payoff difference is, however, again a key factor that influences one’schoices. In a two-player game, who is receiving a smaller payoff determinesthe weight that a quasi-maximin player puts between her own and the other’spayoff. To see it more explicitly, restate the preference like following:{

Ui = xi + σxj, when xi ≤ xj;

Ui = (1− ρ)xi + (ρ+ σ)xj, when xi > xj.(6)

Comparing (6) to (1), one can find that the maximin parameter ρ hasthe same function as α + β in the inequity aversion preference: it shifts theutility weights between oneself and the other when the payoff distributionchanges, hence decides how one reacts to other’s actions. The efficiency-concern parameter σ at the same time keeps one’s concern for the otherpositive. 11

A.1 Static Game Analyses

Looking first at the two-player static game. Imagine first that both playersare quasi-maximin and have public goods contributions (gi, gj), with gi ≥ gj.

10Charness and Rabin (2003) proposed their model without the reciprocal item like this:Ui(x1, x2, ..., xn) = (1−λ) ·xi +λ [δmin{x1, x2, ..., xn}+ (1− δ)(x1 + x2 + ...+ xn)]. Thesimplified model that we have here suppresses some parameter to keep the model linear inparameters. Our simplification shall not change the model essentially but help to highlightthe weight that one puts on each part of social concerns.

11In the appendix of their paper, Charness and Rabin extend their basic model to includea negative reciprocity item. When this item is activated, it will allow one’s concern forothers to turn negative and may lead to destructive punishments. Their complete quasi-maximin model is then able to cover the complete parameter range of the inequity-aversemodel.

28

Their respective utilities are:

Ui = xi + σxj

= (1 + σ)ω + (a− 1 + σa)gi + (a+ σa− σ)gj

Uj = (1− ρ)xj + (ρ+ σ)xi

= (1 + σ)ω + (a− 1 + σa+ ρ)gj + (a+ σa− σ − ρ)gi

(7)

Each player decides independently whether to change her action, accord-ing to:

• if a− 1 + σa > 0 (so a− 1 + σa+ ρ > 0), contributions will increase tomaximum by both players;

• if a − 1 + σa + ρ ≤ 0 (so a − 1 + σa < 0), contributions will decreaseto zero by both players;

• if a − 1 + σa ≤ 0 < a − 1 + σa + ρ, the leading contributor reduceswhereas the other increases her contribution, until the two equalize.

A Nash equilibrium will be symmetric:gi = gj = ω, if a > 1

1+σ;

gi = gj ∈ [0, ω], if 1−ρ1+σ

< a ≤ 11+σ

;

gi = gj = 0, otherwise.

A quasi-maximin player has an obvious internal incentive for public goodscontributions. She can internalize the externality up to her scale of efficiencyconcern: a + σa, and even becomes an unconditional contributor if it over-whelms the material cost of contribution, i.e., a+σa > 1. The maximin con-cern ρ provides yet another incentive for contribution: if the other memberends up the worst-off by contributing more, concern for her/him immediatelyjumps up, leading to an even higher level of efficiency concern: a + σa + ρ.This extra incentive for contribution disappears until the current player fi-nally contributes more than the other member. As a result, ρ will completelydecide how one interacts with the other.

Following this characteristic, a group with one quasi-maximin and oneselfish player will have the following contributions in the equilibrium:

gS = 0, and

gQM = ω, if a > 1

1+σ;

gQM ∈ [0, ω], if a = 11+σ

;

gQM = 0, otherwise,

29

where gQM and gS represents the equilibrium contribution of a quasi-maximinplayer and of a selfish player, respectively.

When type information is not publicly known but a prior belief p aboutone being selfish is held within a group, a quasi-maximin player will maximizeher expected utility. Suppose a quasi-maximin player i chooses gi ≤ gQM ,she has:

E[Ui] = p[(1 + σ)ω + (a− 1 + σa)gi + (a+ σa− σ) · gS] +

+(1− p)[(1 + σ)ω + (a− 1 + σa+ ρ)gi + (a+ σa− σ − ρ)gQM ]

= (1 + σ)ω + [a− 1 + σa+ (1− p)ρ]gi + (1− p)(a+ σa− σ − ρ)gQM

The equilibrium is still symmetric between the same type of playes:

gS = 0, and

gQM = ω, if a > 1

1+σ;

gQM ∈ [0, ω], if 1−(1−p)ρ1+σ

< a ≤ 11+σ

;

gQM = 0, otherwise.

Notice that the existence of selfish players affects a quasi-maximin player’sdecision making by reducing her real value of ρ (think, e.g., ρ = (1 − p)ρ),since now a quasi-maximin player can put the other member on the worst-offposition only at a probability of 1− p, when ρ starts to work. A decreasing1 − p hence reduces the interactiveness between the players. Specifically,when 1 − p < 1−a−aσ

ρ, no more interactive concern will remain between the

players, and public goods realizations in a static game have to rely totallyon quasi-maximin players’ efficiency concerns.

The possible existence of unconditional or committed contributors makesa first difference between the quasi-maximin model and the inequity aver-sion model. The motivations that drive symmetric equilibrium contributionsmake a second. A quasi-maximin player catches up with her group member’scontribution in the hope of improving both efficiency (at a rate σa) and theworst-off’s payoff (at a rate ρ). In contrast, an inequity-averse player does sobecause it is the only way to avoid her psychological cost from advantageousinequities.

30

A.2 Dynamic Public Goods Game

For the dynamic game, the same Concern for Expected Payoffs assumptionapplies. The intertemporal expected utility of a quasi-maximin player is:

uti = (1−ρ−σ)

[t−1∑l=1

xli +T∑m=t

xmi

]+ρmin

{T∑n=1

xni ,

T∑n=1

xnj

}+σ

T∑n=1

(xni +xnj ),

in a T -round dynamic game, where up-bars again indicate the expected payofffrom a future round, and are skipped in the second and third part. The utilitycan also be expressed in gti ’s as:

uti = T · (1 + σ)ω + (a− 1 + aσ)T∑n=1

gni + (a+ aσ − σ)T∑n=1

gnj ,

ifT∑n=1

gni ≥T∑n=1

gnj , or otherwise:

uti = T · (1 + σ)ω + (a− 1 + σa+ ρ)T∑n=1

gni + (a+ σa− σ − ρ)T∑n=1

gnj .

It has been shown that with moderate concerns for others, quasi-maximinplayers want to match contributions between members. So like in the inequityaversion model, we will assume:

Assumption 3. 1−ρ1+σ

< a ≤ 11+σ

So we only focus on the situation where people are interactive enoughwith each other. A quasi-maximin player then also has a behavior pattern ofconditional cooperation, just like an inequity-averse player. So without sur-prises, we will see the quasi-maximin model making very similar predictionsas the inequity aversion model in what remains in this section. The LPDprinciple is automatically applied, and proofs are similar as for the othermodel, which can be referred to in Appendix B.

For a complete-information dynamic game with two quasi-maximin play-ers, an equilibrium has:

• if a > 11+σ

, then (gti , gtj) = (ω, ω), ∀t;

31

• if a ≤ 1−ρ1+σ

, then (gti , gtj) = (0, 0),∀t;

• if 1−ρ1+σ

< a ≤ 11+σ

, then (gti , gtj) = (ω, ω),∀t 6= k, and gki = gkj ∈ [0, ω].

For a complete-information dynamic game with heterogeneous type ofplayers, an equilibrium has:

• if a > 11+σ

, then (gtQM , gtS) = (ω, 0), ∀t;

• if a ≤ 1−ρ1+σ

, then (gtQM , gtS) = (0, 0),∀t;

• if 1−ρ1+σ

< a ≤ 11+σ

, then gtS = ω,∀t < T ; gTS = 0, and∑T

n=1 gnQM =∑T

n=1 gnS = (T − 1)ω.

For the dynamic game with hidden type information, the pure strategyequilibria are featured with the following contribution paths:

• if a > 11+σ

, then gtQM = ω, gtS = 0,∀t;

• if a ≤ 1−ρ1+σ

, then gtQM = gtS = 0,∀t;

• if 1−ρ1+σ

< a ≤ 11+σ

:

– for 1− p ≤ min{

1−a−aσρ

, 1−a−aσa+σa−σ

}, gtQM = gtS = 0,∀t;

– for 1 − p ≥ max{

1−a−aσρ

, 1−aa

}, there are two kinds of pooling

equilibrium:

(i) gtS = gtQM = ω,∀t < T , and gTS = 0, gTQM ∈ [1−aa

11−pω, ω];

(ii) gtS = gtQM = ω,∀t < T and t 6= k, and gkS = gkQM ∈ [0, ω];gTS = 0, gTQM = ω.

– for 1 − p ≥ 1−a−σaa+σa−σ , there are a series of separating equilibria

that distinguish types in the penultimate round. In each of them,gtS = gtQM = ω,∀t < T − 1; gT−1S = gTS = 0, gT−1QM = ω. Then

gTQM ∈[

(1−a−σa)p(2a−1)(1+σ)(1−p)ω,min

{ω, 1−a

a1

1−pω}]

if gT−1 = (ω, ω), and

gTQM = 0 otherwise.

32

For a separating equilibrium, ω is now a possible choice of a quasi-maximin player in the final round. Remember that it is not the case forthe inequity aversion model. Although inequity-averse players and quasi-maximin players both pursue equal contributions, a quasi-maximin playerbears no psychological costs when contributes more than the other. Instead,the ability to enjoy the efficiency gain from public goods makes her alwaysmore willing to contribute than a selfish player under the same situation. Thefinal round coordination hence does not need to be manipulated to supporta separating equilibrium. 12

In the end, we also refer to the reputation game to discover whether andhow the public goods is contributed when the prior belief about a memberbeing selfish is high. It corresponds to the parameter range in which purestrategy equilibria predict zero contribution. The reputation game is playedin the same logic as for the inequity aversion model.

The prior belief here is for 1 − p < 1−a−σaρ

, with 1−ρ1+σ

< a ≤ 11+σ

. Wepropose the following way to play the game.

12Like what we did for the separating equilibrium in the inequity aversion model, thisargument is shown more explicitly in math. For a quasi-maximin player to contribute inthe T − 1 round instead of playing wait-and-see, the following needs to be positive:[

ω + (a− 1)gT−1QM + a(1− p)gT−1

QM + ω + (2a− 1)(1− p)gTQM +

+σ{ω + agT−1QM + (a− 1)(1− p)gT−1

QM + ω + (2a− 1)(1− p)gTQM}]−

−[ω + a(1− p)gT−1

QM + ω + (a− 1)(1− p)gT−1QM +

+σ{ω + (a− 1)(1− p)gT−1QM + ω + a(1− p)gT−1

QM }]

= (a− 1)pgT−1QM + (2a− 1)(1− p)gTQM + σ{apgT−1

QM + (2a− 1)(1− p)gTQM}

In contrast, for a selfish player to do so will need the following to be positive:[ω + (a− 1)gT−1

QM + a(1− p)gT−1QM + ω + a(1− p)gTQM

]−

−[ω + a(1− p)gT−1

QM + ω]

= (a− 1)gT−1QM + a(1− p)gTQM ,

which, when gTQM = gT−1QM , is always more difficult to satisfy than the quasi-maximin

player’s.

33

In each round from the beginning till a certain round k− 1, both playerscontribute ω for sure. Deviations in any round will be exactly matched in thenext round, plus a punishment up to the size of this deviation. Starting fromround-k, a quasi-maximin player will contribute ω in each round, as long asnobody has contributed less than ω in the history. At the same time, a selfishplayer will randomize between contributing ω and 0. Whenever one makesa less-than-ω contribution, the other will match it in the next round plus arestricted punishment as usual. But after that, both players will contribute 0till the end. This continues till round T −1. Then in the final round, if (ω, ω)has been realized in each past round, quasi-maximin players contribute at alevel x for sure, and selfish players contribute 0.

We again focus our attention on contributions levels either at ω or 0,whenever possible. pt, rt and qt follow definitions in the paper.

The selfish players’ randomization phase, with a length of T − k rounds,will be long enough to allow the belief being updated for round-T contri-butions, meanwhile not yet pushing 1 − pT to beyond 1. For this purpose,we define K as the smallest even number that satisfies (1 − p)(1−a

a)−

K2 ≥

max{

1−a−aσρ

, 1−aa

}, and show the belief system below:

- k = T − K, if (1 − p)(1−aa

)−K2 < 1. In this scenario, for t < k:

qt = 1. For t ∈ [k, T − 1) and gt−1 = (ω, ω): qtqt+1 = 1−aa

withqt = (1−pt)+ptrt; otherwise, qt = 0. Finally, with gt = (ω, ω),∀t < T :

qT = 1− pT = (1− p)(1−aa

)−K2 , qT−1qT = 1−a

aωx.

- k = T − K + 1, if (1 − p)(1−aa

)−K2 > 1. Here, for t < k: qt = 1. For

t ∈ [k, T − 1) and gt−1 = (ω, ω): qtqt+1 = 1−aa

, with qt = (1− pt) + ptrt;otherwise, qt = 0. Finally, with gt = (ω, ω),∀t < T : qT = 1 − pT =

(1−p)(1−aa

)1−K2 /qT−1, where qT−1 is such that 1−pT > 1−a−aσ

ρ. Besides,

qT−1qT = 1−aa

ωx, with x = ω

1−p

(1−aa

)K2 . 13

13x should be carefully chosen to ensure the belief updating process. One can read thisin the proof. There is another way to play the final round in an equilibrium though. Withselfish players sticking to 0, quasi-maximin players can randomize between contributing ωwith a probability s = ( 1−a

a )K2 /(1 − p) and contributing 0 at the complementary proba-

bility. Everything else stays unchanged. Interested readers are encouraged to test it.

34

Proposition 7. For a T -round dynamic public goods game, when there arechances that players are the quasi-maximin type, the strategies and the beliefdescribed above form an equilibrium that is sebgame perfect.

With mixed strategies, public goods contributions are again realized formost time of the game, making the prior belief no long essential.

This reputation equilibrium profile however noticeably differs from theone for the inequity aversion model. In the rounds that a selfish playerrandomizes, a quasi-maximin player is always making contributions for cer-tain. In contrast, in order to have an inequity-averse player do so, it hasto be a situation where she has an inferior contribution position and reallyneeds to catch-up. As already said, an inequity-averse player’s fear for over-contribution decides that a selfish player will be the initiator of contributionsat some time in a reputation game, which is of course not the case for thequasi-maximin model.

B Proof for the Inequity Aversion Model

B.1 Selecting Equilibrium in a Static Game

We try two equilibrium refinements: Payoff Dominance (PD), and Risk Dom-inance (RD). Both are defined by Harsanyi and Selten (1988). These criteriahelp players to coordinate on one of the many equilibria.

Start with payoff dominance. A payoff dominant equilibrium is one thatis Pareto superior to any other equilibria in the game. Based on Proposition1, for a + β − 1 > 0, an inequity-averse player i’s payoff in an equilibriumwhere inequity-averse players contribute g∗ and selfish players contribute 0is:

ui(g∗, 0) = p[ω − (1− a)g∗ − αg∗] + (1− p)[ω − (1− a)g∗ + ag∗]

= ω + [(2a− 1)− (a+ α)p]g∗

The payoff dominant equilibrium is hence:{g∗ = ω, if p < 2a−1

a+α

g∗ = 0, otherwise.

35

Notice the condition for there being positive contributions is not the sameas part (b) of Proposition 1.

Now with Risk Dominance. We refer to a risk dominant equilibrium asone that gives the highest expected utility when a player thinks the otherchooses any equilibrium according to a uniform distribution between 0 andω. a + β − 1 > 0 is satisfied as a prerequisite. An inequity-averse player ichooses gi to maximize:

ui = p[ω − (1− a)gi − αgi] +

+ (1− p){Prob(gi ≥ gj)[ω − (1− a)gi + agj − α(gi − gj)] +

+ Prob(gi < gj)[ω − (1− a)gi + agj − β(gj − gi)]}

To simplify calculations, we treat the uniform distribution as continuousand the objective function is presented as:

ui = p[ω − (1− a)gi − αgi] +

+ (1− p){∫ gi

0

1

ω[ω − (1− a)gi + agj − α(gi − gj)]dgj +

+

∫ ω

gi

1

ω[ω − (1− a)gi + agj − β(gj − gi)]dgj

},

which reduces to:

ui = −1− pω

1

2(α+β)g2i +[a−1−α+(1−p)(α+β)]gi+ω+

1

2(1−p)(a−β)ω

The first order condition then gives the chosen level of contribution:{g∗i = (1−p)(α+β)−(1−a+α)

(1−p)(α+β) ω, if p < a+β−1α+β

,

g∗i = 0, otherwise.

B.2 Dynamic Game with Complete Information

B.2.1 Prove Proposition 2 and Proposition 3

Proposition 2 and Proposition 3 describe equilibrium paths when homoge-neous inequity-averse players play the dynamic game under complete infor-mation, with and without Limited Punishment for Deviations. They areproved side by side. Denote the two inequity-averse players by i and j re-spectively.

36

Lemma 1. In any pure strategy equilibrium,∑T

t=1 gti

∗=∑T

t=1 gtj

∗.

Proof. Either player chooses her public goods contribution to maximize ob-jective function (4) in the game:

T∑t=1

gt∗

i = arg max∑Tt=1 g

ti

[T · ω −

T∑t=1

gti + a( T∑t=1

gti +T∑t=1

gt∗

j

)−

− αmax{ T∑

t=1

gti −T∑t=1

gt∗

j , 0}− βmax

{ T∑t=1

gt∗

j −T∑t=1

gti , 0}],

The problem can be solved by splitting the objective function into threescenarios:

T∑t=1

gt∗

i = arg max∑Tt=1 g

ti

(8)Tω + (a+ α)

∑Tt=1 g

t∗j + (a− α− 1)

∑Tt=1 g

ti , if

∑Tt=1 g

ti >

∑Tt=1 g

t∗j ;

Tω + a∑T

t=1 gt∗j + (a− 1)

∑Tt=1 g

ti , if

∑Tt=1 g

ti =

∑Tt=1 g

t∗j ;

Tω + (a− β)∑T

t=1 gt∗j + (a+ β − 1)

∑Tt=1 g

ti , if

∑Tt=1 g

ti <

∑Tt=1 g

t∗j .

Given∑T

t=1 gtj

∗and a + β − 1 > 0 (as assumed), (8) clearly shows that∑T

t=1 gti

∗=∑T

t=1 gtj

∗is the dominant strategy and must be the outcome. As a

result, neither player incurs psychological costs of inequity in the equilibrium.

Lemma 2. All subgame perfect equilibrium profile must have gT−1i + gTi ≥ω, gT−1j + gTj ≥ ω. Same for T = 2.

Proof. Use contradiction, without loss of generality, we show that any equilib-rium with gT−1i +gTi < ω is prone to deviations. Try the following equilibriumstrategies:

(i) gT−1 = (gT−1i , gT−1j ), gT = (gTi , gTj ), with gT−1i + gTi < ω;

(ii) If, however, gT−1 = (gT−1i + ∆, gT−1j ), ∀∆, then:

• gT = (0, ω) if gTj − gTi + ∆ ≥ ω;

• gT = (0, gTj − gTi + ∆) if gTj − gTi + ∆ ∈ [0, ω);

37

• gT = (gTi − gTj −∆, 0) if gTj − gTi + ∆ < 0;

(ii) makes the harshest subgame perfect punishment for a deviation.14 Itprovides the most powerful protection for an equilibrium.

A player that makes a one-shot ∆-deviation in the penultimate roundthen has the following deviation benefit (her accumulated payoff from earlyrounds is unchanged and neglected for the calculation):

D = ω + (a− 1)(gT−1i + ∆) + agT−1j + ω + aω − α[∆− ω − gTi + gTj ](9)

−[ω + (a− 1)gT−1i + agT−1j + ω + (a− 1)gTi + agTj

]= (a− 1)[∆− gTi ] + a[ω − gTj ]− α[∆− ω − gTi + gTj ],

if gTj − gTi + ∆ ≥ ω, or:

D = ω + (a− 1)(gT−1i + ∆) + agT−1j + ω + a(gTj − gTi + ∆) (10)

−[ω + (a− 1)gT−1i + agT−1j + ω + (a− 1)gTi + agTj

]= (2a− 1)(∆− gTi ),

if gTj − gTi + ∆ ∈ [0, ω), or:

D = ω + (a− 1)(gT−1i + ∆) + agT−1j + ω + (a− 1)(gTi − gTj −∆) (11)

−[ω + (a− 1)gT−1i + agT−1j + ω + (a− 1)gTi + agTj

]= −(2a− 1)gTj ,

if gTj − gTi + ∆ < 0.

(9) involves a psychological cost, because i would love to choose a neg-ative contribution in round-T to keep overall aggregate payoff matched.Since D decreases in ∆, i should choose the smallest ∆ in this value range:∆ = ω + gTi − gTj . Then D = (2a − 1)(ω − gTj ) ≥ 0 makes a non-negativebenefit of deviation.

14It is the harshest since following a deviation, the next round public goods realizationis reduced to as low as possible, meanwhile complies with Lemma 1 to ensures that thepunishment is subgame perfect.

38

No psychological cost of inequity arises in (10) and (11) since Lemma 1is respected. Here D is non-decreasing in ∆, so the largest deviation benefitis obtained when ∆ = ω − gT−1i . Player i has then a profit of deviation:

D = (2a− 1)[ω − (gT−1i + gTi )],

according to (10).15 It is also a net gain under the assumption of (i).

The proposed equilibrium profile is hence prone to these deviations16,even though being protected by the harshest punishment. This forms thecontradiction to any equilibrium profile with gT−1i + gTi < ω, and completesthe current proof.

Lemma 1 and Lemma 2 make Proposition 2, which defines the largest setof subgame perfect equilibrium since the harshest deviation punishment isimposed.

Proposition 2: for the dynamic public goods game played by two inequity-averse players, Lemma 1 and 2 fully characterize the equilibrium path of allsubgame perfect equilibria.

Proof. First, under Lemma 1, no one deviates from an equilibrium in thefinal round.

Second, Lemma 2 shows that the harshest punishment dissuades the mostprofitable ∆ > 0 deviation in the penultimate round. So it also prevents anysmaller ∆ > 0 deviations.

Third, no ∆ < 0 deviations in the penultimate round is profitable, either.This can be directly observed in (10).

It remains to show that in the equilibrium, any combination of publicgoods contributions can appear in an earlier round, provided that Lemma 1

15(11) is not relevant here, because (i) has assumed gT−1i + gTi < ω, which makes

gTj + (ω − gT−1i ) ≥ gTi .

16In the case of (9), we assume that one still chooses to deviate from the equilibriumpath if she is indifferent to do so. This corresponds to the situation gTj = ω.

39

and 2 are complied with.

Suppose gt = (gti , gtj) is an equilibrium path that agrees with the two

Lemmas. We try the following equilibrium strategy:

(i)’ For deviations in the round T − 1, follow (ii) in Lemma 2;

(ii)’ For any t < T − 1, if gt = (gti + ∆, gtj), ∀∆, then gl = (gli, glj),∀l ∈

[t+ 1, T − 2], and:

• gT−1 = (0, gT−1j ), gT = (0, gTj ), where gT−1j + gTj =(gT−1j + gTj

)−(

gT−1i + gTi)

+ ∆, if(gT−1j + gTj

)−(gT−1i + gTi

)+ ∆ ∈ [0, 2ω);

• gT−1 = (gT−1i , 0), gT = (gTi , 0), where gT−1i + gTi =(gT−1i + gTi

)−(

gT−1j + gTj)−∆, if

(gT−1j + gTj

)−(gT−1i + gTi

)+ ∆ < 0.

(ii)’ is nothing more than an extended version of (ii): it uses the last tworounds instead of only the final one to punish a deviation. The deviationpunishment is hence harsher.17 Notice that there does not exist a version of(9), because that corresponds to:

gT−1 = (0, ω), gT = (0, ω), if(gT−1j + gTj

)−(gT−1i + gTi

)+ ∆ ≥ 2ω.

Given gT−1j + gTj > ω, gT−1i + gTi > ω and ∆ < ω, above situation wouldnever appear.

The ∆-deviation benefit D is calculated in the same manner. It is either:

D = (2a− 1)[∆− gT−1i − gTi

], or: D = −(2a− 1)[gT−1j + gTj ].

Either way, it is negative given gT−1i + gTi ≥ ω ≥ ∆. The final tworounds’ punishment threat hence can protect any equilibrium path againstdeviations.

Proposition 2 defines the largest set of equilibrium, and leaves no predic-tion for equilibrium actions until the very end of the game. The situationwill be improved if equilibrium protections are weaker, i.e., with less harsh

17One can also use, for example, a trigger strategy that pulls all remaining rounds’contributions towards 0 after a deviation. That is the harshest punishment for deviations.We will show that the current two-round-punishment strategy is already harsh enough forour purpose.

40

punishment. That is the Limited Punishment for Deviations rule and Propo-sition 3.

Limited Punishment for Deviations requires a deviation punishment tobe no larger than the size of that deviation. Take an equilibrium path gt =(gti , g

tj), it can be specified in the manner of (ii):

Limited Punishment for Deviations (LPD): ∀t < T , if gt = (gti +∆, gtj),∀∆, then gk = (gki , g

kj + ∆), and gl = (gli, g

lj),∀l /∈ {t, k},

where: gki = gki − P, gkj = gkj − P . P for punishment and P ∈ [0, |∆|], with|∆| indicating ∆’s absolute value. Round-k is the first round following t thatcould accommodate above values. 18

LPD requires the post-deviation contribution to be reduced to the vicinityof gk rather than 0, and to have the reduction in only one round if possi-ble. Proposition 3 tells us that the equilibrium set is downsized significantlythanks to this rule. It is proved in below.

Proof. Start with part (b) of Proposition 3. Again use contradiction. Sup-pose an equilibrium has at least round-t and round-k in which not bothplayers contribute ω. Without loss of generality, let k > t. The equilibriumstrategy following a deviation is LPD. If player i deviates from gti by a small

18In case no single round k can be located in the remaining game after round-t, one canuse two or even more rounds for punishment. For example, take the penultimate and finalround:

• For any t < T , if gt = (gti + ∆, gtj),∀∆, then gl = (gli, glj),∀l ∈ [t + 1, T − 2], and

gT−1 = (gT−1i , gT−1

j + δT−1), gT = (gTi , gTj + δT ),

where: {gT−1i + gTi = gT−1

i + gTi − PgT−1j + gTj = gT−1

j + gTj − P,

δT−1 + δT = ∆, and P ∈ [0, |∆|], with |∆| indicating the absolute value of ∆.

Given Lemma 2, the final two rounds are able to accommodate the proposed punish-ment for any ∆-deviations, as long as one chooses P properly. Furthermore, this two-punishment-rounds scenario can be shown to have no influence on the results that we willshortly prove.

41

δ, her deviation profit will be:

D = [ω + (a− 1)(gti + δ) + agtj] + [ω + (a− 1)gki + a(gkj + δ)]

− [ω + (a− 1)gti + agtj] + [ω + (a− 1)gki + agkj ]

= (2a− 1)(δ − P )

To get to this result, we have replaced gki with gki − P , and gkj accordingly.

LPD ensures that D is non-negative with δ > 0, meaning a positive δ-deviation is always happening. So there can be at most one round in anyequilibrium with players contributing less than maximum.

At the same time, since δ < 0 always leads to negative D, one need notworry if players would unilaterally step down from a full contribution profile.

Denote the only non-maximal contribution round by round-k. We haveobtained:

gti = gtj = ω,∀t 6= k ⇒∑t6=k

gti =∑t6=k

gtj = (T − 1) · ω.

As Lemma 1 requires∑T

t=1 gti =

∑Tt=1 g

tj, there should be:

gki = gkj ∈ [0, ω].

This completes our proof for Proposition 3.

B.2.2 Prove Proposition 4

Proposition 4 depicts how heterogeneous players coordinate over aggregatepublic goods contributions in the dynamic game. Use IA to represent aninequity-averse player and S for a selfish player, we prove their equilibriumactions in below.

Proof. First,∑T

t=1 gtIA =

∑Tt=1 g

tS. Given a+ β − 1 > 0, an inequity-averse

player always matches the other member’s action.

Second, in any equilibrium, gTS = 0, and:

42

- either gTIA = ω,

- or gT−1S = ω, gTIA ∈ [1−aaω, ω].

Suppose not. If max{gTIA, gT−1S } < ω, let player S deviate by contribut-ing gT−1S + δ in round T − 1, with a small δ > 0. According to theFirst argument, player IA will rebalance the aggregate contributionsby choosing gTIA + δ in round-T . The deviation hence brings player Sa net gain of:

D =[ω − (1− a)(gT−1S + δ) + agT−1IA + ω + a(gTIA + δ)]−

−[ω − (1− a)gT−1S + agT−1IA + ω + agTIA

]= (2a− 1)δ.

In the other case, if gT−1Sj = ω but gTIA <1−aaω, let player S deviate to

0 in round T − 1. Player IA then reacts with gTIA = 0 to reduce thepayoff inequity between the two. The deviation is then profitable forS, as for gTIA <

1−aaω:

D =[ω + agT−1IA + ω

]−[ω − (1− a)ω + agT−1IA + ω + agTIA

]= (1− a)ω − agTIA > 0.

Third, without imposing LPD :

- either gTIA = ω, gT−2IA + gT−1i > ω, gT−2S + gT−1j > ω,

- or gT−1S = ω, gTIA ∈[1−aaω, ω

], gT−2IA + gT−1IA > ω.

Using contradictions, the proof is nearly the same as for proposition 2.Readers can repeat the exercise here, but are suggested to use gTIA = ωfollowing any deviation in a subgame perfect equilibrium. The othersubgame perfect ending: gT−1S = ω is not able to accommodate punish-ments for deviations in round T − 2.

Finally, with LPD, player S contributes all the time ω till round T −1, andplayer IA fulfills gTIA ≥ 1−a

aω in addition to equal aggregate contribu-

tions. To show this, use contradictions and suppose a profile (gtIA, gtS)

with at least one round k < T in which gkS < ω.

43

- if glIA = ω,∀l > k, there must be at least one round k′ < kwith gk

′IA < ω to ensure equivalent aggregate contributions. Then

player IA can deviate in round-k′ with gk′IA = gk

′IA + δ, δ > 0.

The equilibrium path following that can be gtS = gtS,∀t 6= k, andgkS = gkS+δ−P ; meanwhile gtIA = gtIA,∀t 6= k′′, and gk

′′IA = gk

′′IA−P .

k′′ is a random round after k′. With P ≤ δ, that makes a non-negative deviation benefit for player IA.

- if instead ∃l > k where glIA < ω, player S can play the δ-deviatorand gain for sure from player IA’s matching contribution in thisround-l.

B.3 Dynamic Game with Incomplete Information

B.3.1 Prove Proposition 5

Proof. Basically, an IA player can choose any round k > t to match the othermember’s round-t deviation behavior. We fix the idea by assuming that shedoes so in the nearest future found that can accommodate this matchingaction.

(a) When 1−p < min[1−aa, 1−a+α

α+β

], there is a unique equilibrium with players

pooling on gt = (0, 0),∀t.

- gt = (0, 0) is the only pooling equilibrium, because:

There does not exist pooling equilibrium with positive contributions:Suppose there is one. Since in a pooling equilibrium round-T islike a static game, (b) of Proposition 1 requires gT = (0, 0). Letround-k be the last round with positive contributions. An S playerwill always choose 0 in this round as there is nothing to gain inthe future, which contradicts to the pooling feature of currentequilibrium.

gt = (0, 0),∀t < T is not subject to deviations, for deviations only givenegative benefits. Take an S player as an example, even thoughwith probability 1− p there is an IA player who catches up with

44

his deviation, the gain is:

DS = T · ω + (a− 1)ω + a(1− p)ω − T · ω

< 0 for (1− p) ≤ 1− aa

- There exists no separating equilibrium. For if there was one, there mustbe gTIA ≤ 1−a

a1

1−pgT−1IA for the revelation to happen in round T − 1, see

later part of this proof. An IA player would want to deviate in roundT − 1 by mimicking an S player, to get the benefit:

DIA =[ω + a(1− p)gT−1IA + ω + (1− p)(a− 1)gT−1IA

]−

−[ω + (a− 1)gT−1IA + a(1− p)gT−1IA + ω + (2a− 1)(1− p)gTIA − αpgT−1IA

]= (1− a+ α)pgT−1IA − (2a− 1)(1− p)gTIA≥ (1− a+ α)pgT−1IA − (2a− 1)

1− aa

gT−1IA > 0, for 1− p < 1− aa

Where the last line has used gTIA ≤ 1−aa

11−pg

T−1IA .

(b) When 1 − p ≥ max[1−a+αα+β

, 1−a+αa+α

], there are only two types of pooling

equilibria. The one with (ω, ω) till the penultimate round then gTIA = x ≥1−aa

ω1−p is proved directly in below. The other that has at most one round

gk < ω will be shown just as a by-product.

- It is an equilibrium because:

• An IA player does not deviate in round-T . Any gTIA = x−δ, ∀δ > 0makes no profit:

DIA =[ω + (a− 1)gTIA + a(1− p)x− β(1− p)(x− gTIA)− αpgTIA

]−

−[ω + (a− 1)x+ a(1− p)x− αpx

]= (1− a+ α)δ − (α + β)(1− p)δ

≤ 0, for 1− p ≥ 1− a+ α

α + β

• Let a deviation of gti = ω − δ,∀t < T lead to no belief changesconcerning deviator’s type but a punishment P ∈ [0, δ] in roundt + 1. It is easy to show that the deviation profit is negative for

45

both types if t + 1 < T . For t + 1 = T , the deviation profits arealso negative:

DIA =[ω + (a− 1)(ω − δ) + aω + ω + (a− 1)(x− P ) +

+a(1− p)(x− P − δ)− αp(x− P )]−

−[ω + (a− 1)ω + aω + ω + (a− 1)x+ a(1− p)x− αpx

]=

[(1− a)− a(1− p)

]δ +

[(1− a+ α)− (a+ α)(1− p)

]P

≤ 0, for 1− p ≥ 1− a+ α

a+ α>

1− aa

, and

DS =[ω + (a− 1)(ω − δ) + aω + ω + a(1− p)(x− P − δ)

]−

−[ω + (a− 1)ω + aω + ω + a(1− p)x

]=

[(1− a)− a(1− p)

]δ − a(1− p)P

≤ 0.

We save to show the cases where either an IA or S player drops con-tribution to 0 in round T − 1, which will also lead to negative benefitsgiven gTIA = 0 follows in the next round.

- Other pooling equilibria with more than one round (including the finalround) where gt = (gtIA, g

tS) < ω are prone to deviations. To give

an example, suppose an equilibrium path with gtIA = gtS = ω,∀t <T and t 6= k; gkIA = gkS = z < ω; gTIA = x, gTS = 0. After a deviationto z + δ in round-k, since any round between k + 1 and T will notaccommodate gti = ω − P + δ with P < δ, we relay the matching andpunishing period to round-T , and require the post-deviation belief to:

- stay unchanged, if players adopt P ∈ [0, a(1−p)−(1−a)a(1−p) δ]; or

- believe the deviator is inequity-averse if P ∈ [a(1−p)−(1−a)a(1−p) δ, δ].

With this belief update, equilibrium plays after the deviation is almost

46

unchanged. The deviation benefits are:

DIA =[ω + (a− 1)(z + δ) + az + (T − k − 1)2aω +

+ω + (a− 1)(x− P ) + a(1− p)(x− P + δ)− αp(x− P + δ)]−

−[ω + (2a− 1)z + (T − k − 1)2aω +

+ω + (a− 1)x+ a(1− p)x− αpx]

=[(a− α− 1) + (a+ α)(1− p)

](δ − P )

≥ 0, for 1− p ≥ 1− a+ α

a+ α, P ≤ δ;

DS =[ω + (a− 1)(z + δ) + az + (T − k − 1)2aω +

+ω + a(1− p)(x− P + δ)]−

−[ω + (2a− 1)z + (T − k − 1)2aω + ω + a(1− p)x

]= (a− 1)δ + a(1− p)(δ − P )

> 0 for 1− p > 1− aa

, P ≤ a(1− p)− (1− a)

a(1− p)δ.

≤ 0 for P ≥ a(1− p)− (1− a)

a(1− p)δ.

So for both z and x smaller than ω, players do deviate and beliefs areupdated correctly, and an contradiction is reached. In case x = ω,P = δ is the only possibility and an IA player becomes indifferentbetween deviation and not. That is the second kind of pooling equilib-rium that we proposed.

Beliefs can of course be updated in a different way, for example onealways believes that a deviator is selfish. In that case the final roundswill be played differently. We find then that either the belief is correctand an S player does deviate, or neither type deviates. The Bayes’ ruledoes not apply for the latter and economists normally accept such a be-lief update, which leads to the acceptance of more equilibria. However,in this paper allowing such a belief update is comparable to imposinga harshest punishment in the final rounds for any deviation. Since wehave imposed LPD, I tend to equally rule out the belief being updatedtowards one being selfish, unless it has to.

47

(c) When 1 − p ≥ (1−a)2+aαa(1−a)+aα , there exists one type of separating equilibrium

with both types contributing fully until T − 2 round.

- The proposed path can be realized in an equilibrium:

• Easy to show that the final round level y is small enough to preventan S player from mimicking an IA one, meanwhile large enoughto prevent IA from mimicking S in round T − 1.

• No δ-deviations in any round-t. As in (b), suppose belief doesnot change if gti = ω − δ, and after punishment and matching inround t+ 1, the play follows the original equilibrium path. I showt = T − 1 here. Situations with t < T − 1 are more obvious.

For 1− p ≥ 1−a+αα+β

, deviation benefits are:

DIA = {ω + (a− 1)(ω − δ) + a(1− p)ω + ω + (1− p)[[(a− 1)(y − P ) + a(y − P − δ)]− αp(ω − δ)} −

−[ω + (a− 1)ω + a(1− p)ω + ω + (1− p)(2a− 1)y − αpω

]=

[(1− a+ α)− (a+ α)(1− p)

]δ +

[(1− 2a)(1− p)

]P

≤ 0

DS =[ω + (a− 1)(ω − δ) + a(1− p)ω + ω + (1− p)a(y − P − δ)

]−

−[ω + a(1− p)ω + ω

]= (a− 1)(ω − δ) + a(1− p)(y − δ)− a(1− p)P

< 0 for y ≤ 1− aa

ω

1− p

The negative deviation profit does not change for 1− p < 1−a+αα+β

.

- There do exist other separating equilibria with less-than-maximal contri-butions before round-T . However, there can be at most one more suchround before the final. To give an example, suppose an equilibriumpath with gtIA = gtS = ω,∀t < T − 1, t 6= k; gkIA = gkS = z < ω;gT−1S = gTS = 0, gT−1IA = ω, gTIA = y if gT−1 = (ω, ω) and 0 otherwise.Following (b), let δ-deviations be punished and matched in round-T ,with the belief being not updated, and separation still happening at

48

round T − 1. For an IA player, the profit for deviating in round-k is:

DIA =[ω + (2a− 1)z + (a− 1)δ + (T − k − 2)2aω + ω + (a− 1)ω +

+a(1− p)ω + ω + (1− p)[(a− 1)(y − P ) + a(y − P + δ)]− αp(ω + δ)]−

−[ω + (2a− 1)z + (T − k − 2)2aω + ω + (a− 1)ω +

+a(1− p)ω + ω + (1− p)(2a− 1)y − αpω]

=[− (1− a+ α) + (a+ α)(1− p)

]δ − (2a− 1)(1− p)P

This is positive only if P ≤ (a+α)(1−p)−(1−a+α)(2a−1)(1−p) δ. So LPD alone is not

sufficient to guarantee the happening of such a deviation, and as a resultthe proposed equilibrium can manage to exist. The case of k = T − 1has the same conclusion. However, k must be unique, in the sensethat any equilibrium with two or more less-than-maximal contributionrounds before the final one is prone to profitable deviations. The profitis ensured by one’s matching behavior in the latter non-maximal con-tribution round to any deviations in the earlier one.

- There does not exist equilibria which separate types in a round t < T − 1.Suppose in an equilibrium types are separated in round t < T − 2 withgtIA = z and gtS = 0. A selfish player can mimic an inequity-averseplayers’ action in this round for a benefit:

DS = {ω + (a− 1)z + a(1− p)z + (1− p)[(T − t− 1)2aω + ω + ax

]+

+p[ω + az + (T − t− 2)2aω + ω + 2(1− a)ω]} −− {ω + a(1− p)z + (1− p)

[ω + (a− 1)z + (T − t− 2)2aω + ω

]+

+p[(T − t)ω]}= (2a− 1)[pz + p(T − t− 2)ω + (1− p)ω] + 2p(1− a)ω + a(1− p)x> 0

The post-deviation game is played following early complete informationpropositions. A revealed selfish player always clears contribution differ-ences before playing cooperation again with a revealed inequity-averseplayer. The qualitative result will not change for t = T − 2.

49

B.3.2 Elaborating the Reputation Equilibrium in Proposition 6

In this section we complete details for the reputation equilibrium of Proposi-tion 6. There is no formal proof, as the elaboration already shows the validityof this equilibrium. We will start backwardly from the final round.

Denote {εt} as a series of small positive numbers with values uniquely de-termined according to rules that we will shortly define. Denote an inequity-averse player by IA, the selfish player who contributes ω in round-k by SR,and a selfish player who contributes 0 in round-k by SUR. R for Revealedand UR for UnRevealed. Further denote the probability that SR contributesω in round t by zt, with t > k. Denote the probability that an unknowntype player contributes ω in round-t by qt, the belief about an unknown typeplayer being IA by 1 − pt, and the probability that SUR contributes ω inround-t by rt.

Suppose that if 0 is contributed by both players in round-k, or by eitherplayer in a round following k, (0, 0) will be realized for the public goods tillthe end of game. So for each round t we study below, the history is restrictedto gk = (ω, 0), and gτ = (ω, ω),∀k < τ < t.

- IA is willing to contribute ω in each round t > k. This is the only way tokeep equal contribution with the other member meanwhile having a highestoverall payoff in the equilibrium.

- SR contributes 0 in round T , so SUR also contributes 0 in round T − 1, forit is irrelevant for the future. In round T − 1, SR contributes ω for sure if:

ω + (a− 1)ω + aqT−1ω + ω + aqT−1ω

> ω + aqT−1ω + ω

⇒ qT−1 >1− aa

.

Let qT−1 = 1−aa

+ εT−1.

50

- In round T − 2, SUR randomizes between contributing ω and 0 if:

ω + (a− 1)ω + azT−2ω + 2ω + azT−2ω

= ω + azT−2ω + 2ω

⇒ zT−2 =1− aa

This means SR is randomizing between contributing ω and 0 in this round,which in turn requires:

ω + (a− 1)ω + aqT−2ω + qT−2V T−1R + (1− qT−2)2ω

= ω + aqT−2ω + 2ω

⇒ qT−2qT−1 =1− aa

1 + qT−2

2,

where V T−1R represents SR’s continuity utility in round T − 1 after gT−2 =

(ω, ω), presented as the left hand side of his inequity condition for round T−1.

- In round T − 3, SUR randomizes if:

ω + (a− 1)ω + azT−3ω + zT−3V T−2UR + (1− zT−3)3ω

= ω + azT−3ω + 3ω

⇒ zT−3zT−2 =1− aa

.

Similarly, V T−2UR represennts SUR’s continuity utility in round T − 2 after

gT−3 = (ω, ω). Since zT−2 = 1−aa

, there must be zT−3 = 1, which correspondsto SR contributing ω for sure in this round:

ω + (a− 1)ω + aqT−3ω + qT−3V T−2R + (1− qT−3)3ω

> ω + aqT−3ω + 3ω

⇒ qT−3qT−2 >1− aa

.

Let qT−3qT−2 = 1−aa

+ εT−3.

- In round T − 4, SUR randomizes if:

ω + (a− 1)ω + azT−4ω + zT−4V T−3UR + (1− zT−4)4ω

= ω + azT−4ω + 4ω

⇒ zT−4 =1− aa

,

51

where zT−3 = 1 has been applied to the last line. It corresponds to SRrandomizes in this round:

ω + (a− 1)ω + aqT−4ω + qT−4V T−2R + (1− qT−4)4ω

= ω + aqT−4ω + 4ω

⇒ qT−4qT−3 =1− aa

1 + qT−4

1 + qT−2.

- Rolling backward like this, we have the following systems, in which n ∈ N:

{zt} : zT−(2n−1) = 1 {qt} : qT = 1

zT−2n = 1−aa

qT−(2n−1)qT−(2n−2) = 1−aa

+ εT−(2n−1)

qT−2nqT−(2n−1) = 1−aa

1+qT−2n

1+qT−(2n−2)

Here every zt is uniquely defined, so is every qt given the set {εt}.

- Suppose round-k is a round T − (2n− 1). In this round, an S randomizesbetween contributing ω to become SR, and contributing 0 to become SUR.Denote the probability that a player contributes ω in this round by δk. Theselfish player’s problem is:

ω + (a− 1)ω + aδkω + δk(2n− 1)ω + (1− δk)[V k+1R

](12)

= ω + aδkω + δk[V k+1UR

]+ (1− δk)(2n− 1)ω.

Meanwhile an inequity-averse player contributes 0 for certain:

ω + (a− 1)ω + aδkω + δk(2n− 1)ω + (13)

+(1− δk)[ω + aqk+1ω + (2n− 2)ω − α(1− qk+1)ω]

≤ ω + aδkω + δk[V k+1IA

]+ (1− δk)(2n− 1)ω.

In the LHS of (13), we have assumed that, whenever the situation is for SRto randomize, it is not attractive for IA to contribute, due to the latter’sfear for disadvantageous inequities that associate with making contributions.V k+1IA is IA’s continuity utility at round k+1, which corresponds to matching

the other’s action in the previous round. To give an example:

V T−2IA = ω + (a− 1)ω + azT−2ω + zT−2

[ω + (a− 1)ω + azT−1ω +

+zT−1(ω + (a− 1)ω) + (1− zT−1)(ω)]

+ (1− zT−2)2ω,

52

in which we have known that zT−2 = 1−aa

and zT−1 = 1.

The two conditions (12) and (13) can be reduced to: δk ≤ α(1−qk+1)

α(1−qk+1)+2(1−a)( 1−aa

)(n−1)

δk = aqk+1−(1−a)aqk+1+(1−a)

(14)

where we used k = T − (2n− 1). A necessary condition for (14) is:

qk+1 ≤α + (1− a)(1−a

a)(n−1)

α + a(1−aa

)(n−1)

One can show that this necessary condition can be satisfied by {qt} throughcarefully choosing {εt}.

(14) then uniquely determines δk with qk+1. The probability that a self-ish player contributes ω in this round is obtained as rk = δk/pk. To have afeasible rk ∈ [0, 1], pk needs to be large, which is in line with our assumptionthat 1− p is small in the current section.

- Throughout the game, the belief is updated according to the Bayes’ rule.That is:

1− pt+1 =1− pt

qt.

1 − pt is updated from 1 − pt−1 and qt−1. The T − 1 round belief is thenupdated all the way from the prior belief in the following way:

1− pT−1 =1− p

q1q2...qT−2(15)

We know 1 − pT−1 = qT−1 = 1−aa

+ εT−1. Also knowing qt = 1,∀t < k, (15)will uniquely define n (equivalently, k) based on the prior belief p and {εt},

- Finally, with {qt} defined and the beliefs 1− pt obtained according to (15),SUR’s strategy in terms of probabilities of contributing ω in each round, rt,are obtained out of the relationship: qt = (1− pt) + pt · rt.

We have hence completely elaborated the proposed reputation equilib-rium and at the same time validated its existence.

53