Ostracism. S. Nageeb Ali and David A. Miller UC-San Diego. June 9, 2012

Transcription

1 Ostracism S. Nageeb Ali and David A. Miller UC-San Diego June 9, 2012 Abstract We study the social norm of ostracism in which when a player cheats, others communicate about his defection, and she is then sanctioned by others while those innocent of cheating continue to cooperate. Ostracism features in most applications of community enforcement, and is implicit in reputational label mechanisms often described in the literature. To identify cheaters, a community relies on its ability to communicate and store information about past interactions; the informal story often invoked is that of a network of social and economic ties through which information spreads. Communication incentives have often not been studied in this literature, and we find that these incentives shapes how ostracism should be implemented. Our first result is negative: there exists no equilibrium in which ostracism is permanent and greater cooperation is sustained through community enforcement. The issue is that when cheated, the victim of cheating prefers to cheat herself than to complain. Our second result is positive: temporary ostracism equilibria which feature temporary punishments and permit guilty players to work towards redemption -can do better than permanent ostracism equilibria and even attain a benchmark of second best efficiency. Our results suggest a new perspective of the practice of forgiveness and redemption in social norms that arises from maintaining the incentive to communicate truthfully. We apply these results to buyer-seller networks and risk-sharing arrangements. Preliminary and incomplete. Not for circulation. snali@ucsd.edu and d9miller@ucsd.edu respectively. Address: Department of Economics, 9500 Gilman Drive #0508, La Jolla, CA We thank Dilip Abreu, Joyee Deb, Matt Elliott, Ben Golub, Alex Imas, Navin Kartik, Asim Khwaja, Bart Lipman, Meg Meyer, Markus Mobius, Adam Szeidl, Alex Wolitzky, and Peyton Young for helpful comments. Aislinn Bohren provided excellent research assistance. This work is financially supported by NSF grant SES Parts of this research were completed while Ali visited Microsoft Research and Harvard University, and he thanks them for their hospitality and support.

2 Contents 1 Introduction 1 2 Model Discussion of Model A Simple Example 11 4 Permanent Ostracism Mechanical Communication: A Benchmark Limits of Strategic Communication Extensions of Result Temporary Ostracism 24 6 Applications Buyer-Seller Networks Decentralized Risk-Sharing Conclusion 34 A Appendix 34 A.1 Proofs for Section A.2 Proofs for Section B Supplementary Appendix 44 B.1 Additional Proofs for Section B.2 Proofs for Section

3 1 Introduction At most places and at most times, individuals achieve cooperative relations through selfenforcing arrangements and without relying on legal enforcement. A vast literature on informal governance in the social sciences explores the significant role that community enforcement, third party punishment, and the networked pattern of social relations play in supporting cooperation. At its core is the idea that when an individual cheats any one of her partners, she recognizes that she may be punished by all of them, and is therefore less inclined to act opportunistically. Linking individuals bilateral relationships in this way allows a community to leverage social collateral towards greater cooperation. Our understanding of community enforcement at times invokes the notion of ostracism: a scheme in which an individual s reputation is linked to her past actions with others, and once an individual shirks in one relationship, punishments are targeted towards her. A tension fundamental to ostracism and community enforcement is that an individual s past behavior is observed only by her past partners and not the community at large, and so a mechanism is needed to link a player s rewards and punishments to her past actions. In other words, monitoring is private and knowledge of each individual s past is decentralized and distributed in society rather than held collectively. If an individual shirks, how can her future partners learn this so as to ostracize her? The literature on informal governance suggests a few answers to this question but perhaps the most prominent one is that information diffuses through a community s gossip and communication network: if an individual shirks, her victims tell others about it, and through these communications, her reputation diffuses to her future partners who are then able to punish her. This logic is at the core of important case studies of community enforcement (Greif 1989; Ostrom 1990; Ellickson 1991) and its role is highlighted in studies of how households and firms enforce risk sharing and lending arrangements in developing and transitional economies. 1 Even in the study of developed economies, the power to communicate and aggregate information so as to punish defectors has been used to motivate credit bureaus and rating systems in markets, and hierarchies in firms. 2 That communication plays an important role in cooperation has also been emphasized in studies outside of economics: 1 See Besley, Coate, and Loury (1993), McMillan and Woodruff (1999), Bloch, Genicot, and Ray (2008), Fafchamps (2011), and Kinnan and Townsend (2011). 2 See Williamson (1975), Klein and Leffler (1981), Padilla and Pagano (1997), Houser and Wooders (2006), and Resnick, Zeckhauser, Swanson, and Lockwood (2006). 1

4 Granovetter (1985) and Coleman (1988) discuss this in the context of social and economic networks, Posner (1995, 1996) discusses it in the context of the law, and even studies in evolutionary biology use a similar notion of image scoring to help understand indirect reciprocity (Brandt and Sigmund 2005; Nowak and Sigmund 2005). Yet, a basic question remains unanswered: do individuals have an incentive to communicate? It may be surprising that apart from a few exceptions, communication incentives have not been studied. Our theoretical understanding has been derived from frameworks with public monitoring or in which communication is assumed to be truthful. For example, much of the literature uses centralized reputational labels mechanisms (e.g. Kandori 1992; Okuno- Fujiwara and Postlewaite 1995; Tirole 1996) in which an individual s label is mechanically updated on the basis of her past interaction, and this is observed by all those who interact with her. Some other models explicitly model communication (e.g Raub and Weesie 1990; Dixit 2003a,b; Bloch et al. 2008; Fainmesser and Goldberg 2011) and offer illuminating analyses of how its speed, the distance that it travels, and its lag affect cooperation incentives, but do not focus on communication incentives. Our work aims to address this lacuna. We begin by establishing benchmarks in a setting with mechanical communication in which individuals are constrained to truthtelling and contrast two equilibria: private bilateral enforcement and permanent ostracism. The former corresponds to cooperation that is enforced without using the community at all; in our setting, the best bilateral enforcement is the standard grim trigger punishment. Permanent ostracism, by contrast, uses the web of social and economic connections by punishing a player in all of her relationships when she shirks on any partner. Naturally, it supports levels of cooperation that are manyfold those from private bilateral enforcement and indeed, no mutual effort equilibrium can support greater cooperation in any relationship. Permanent ostracism, thus, corresponds to first-best efficiency when communication is mechanical. Introducing communication incentives into this setting may appear straightforward: an individual would not wish to report on his own misdeeds but since others appear to have aligned interests in punishing her, they should have nothing to hide when she deviates. If victims and third parties were truthful, permanent ostracism could indeed still do better than bilateral private enforcement; the tension that we find is that victims and third parties may prefer to conceal evidence of guilty players and subsequently shirk themselves. With strategic communication, permanent ostracism equilibria not only fail to attain the first-best efficiency 2

5 benchmark but cannot even improve upon private bilateral enforcement. Communication incentives are satisfied in permanent ostracism only when permanent ostracism is altogether redundant! This negative result holds not only for every discount factor and interaction rate but also for every communication network, and even in a setting which features costless and partially verifiable communication. We defer the formal explanation to Section 4 but it has a surprisingly simple logic: suppose towards a contradiction that a permanent ostracism equilibrium supports greater cooperation in a pair of players, Bob and Carol, than they could if they had to rely on private bilateral enforcement. In such an equilibrium, Bob s incentives to work with Carol come from the bilateral enforcement within their relationship, and that Bob may be punished by others if he shirks. Now consider an off-path private history in which Bob observes everyone but Carol choosing to shirk. In this case, Bob privately knows that there is no one else who will punish him for shirking on Carol. Thus, even if Bob believes that Carol is innocent, he has a strict incentive to conceal the truth from her and shirk. The communication incentives are satisfied if and only if the stakes are no better than those that Bob and Carol obtain without relying on the community. Our finding suggests a tension between our theoretical understanding of ostracism and its prevalence. Yet, ostracism as seen in practice (Ellickson 1991; Ostrom 1990) is more nuanced than that captured in our models, and some of its subtle features might appear redundant in models with mechanical communication but play an important role when communication is strategic. We isolate one such feature: the provision of incentives through punishments that are temporary and involve guilty players being redeemed after repentance. We show that these simple equilibria can outperform both private bilateral enforcement and permanent ostracism. Moreover, for sufficiently patient players, temporary ostracism implements the level of cooperation that permanent ostracism would if victims and third-parties were mechanically constrained to communicate truthfully, and so adding the possibility for forgiveness and redemption can overturn the challenge of satisfying communication incentives. Our results offer a new understanding of how temporary punishments foster communication in community enforcement. Section 5 constructs a simple temporary ostracism equilibria by adapting important ideas from perfect monitoring environments (Fudenberg and Maskin 1986; Abreu 1988) to tackle issues that emerge in strategic communication of privately observed histories. We first con- 3

6 struct temporary ostracism equilibria using hard information, but later show how it can be supported using only cheap talk communication. The redemption that a guilty player makes once she knows that all know that she is guilty serves an important role. It extracts her future surplus from future cooperation and transfers that value to innocent players so as to offer an incentive incentive to communicate and work off the equilibrium path. By establishing that innocent individuals have a stronger incentive to work off the equilibrium path, we ensure that communication incentives off the equilibrium path are satisfied. The contrast between permanent and temporary ostracism is reminiscent of insights gleaned from imperfect monitoring environments: 3 the former burns continuation value of partnerships with guilty players whereas the latter transfers value to innocent players. Although monitoring is perfect within each relationship and punishments are not triggered on the path of play, transferring value is efficient because it subsidizes communication. We use these results to study two applications in which ideas from community enforcement have been useful. Section 6.1 studies networked markets of buyers and sellers in which participants on the same side of the market can share information with each other but have no payoff interactions. We show that if the incentive issue is two-sided, analogous results to those discussed above follow; in contrast, if the incentive issue is one-sided, then permanent ostracism can sustain more than private bilateral enforcement. Our results can thus be used to understand how an enforcement intermediary or institutional changes that eliminate incentive issues on one side of the market can complement rather than substitute for community enforcement. Section 6.2 studies ostracism equilibria in decentralized risk-sharing in which individuals may attempt to use communication to foster greater transfers and risk-pooling. These results on ostracism are derived in a novel framework of community enforcement that synthesizes repeated interaction and communication, which we describe in Section 2 (Section 3 describes a three player example). This framework is unconventional in several ways but affords a tractable analysis of community enforcement. The most notable difference to standard settings is our use of variable stakes partnerships in which players can adjust their cooperative arrangements based on their information. Apart from its inherent realism, this both affords a convenient way to compare the levels of cooperation across different equilibria for a fixed discount rate and allows us to study incentive issues in ostracism that would be obscured in standard prisoners dilemmas with fixed stakes. We discuss our reasons 3 See Abreu, Milgrom, and Pearce (1991), Fudenberg, Levine, and Maskin (1994), and Sannikov and Skrzypacz (2007, 2010). 4

7 for adopting this framework in greater detail in Remark 1. 4 Our focus is on community enforcement mechanisms that use information transmission, and feature punishments that are targeted towards defectors. A different approach is to study community enforcement in the absence of communication. The sharpest contrast in this regard is contagion (Kandori 1992; Ellison 1994) in which once any player shirks, her victims proceed to shirk in all of their relationships, and so on and so forth, spreading the punishment contagiously in the population. Thus, the community rather than a guilty individual is punished for an individual s defection. While contagion exposes the theoretical limits and scope of cooperation, 5 we believe that there are strong practical reasons to understand the extent to which communications permit punishments targeted towards the guilty. We interpret our negative result on permanent ostracism as uncovering a force that pushes towards either punishing innocent players (as in contagion) or maintaining cooperative ties with guilty players (as in temporary ostracism). For sufficiently patient players, temporary ostracism attains the same level of cooperative stakes that would be supported by the best contagion equilibrium in this setting, but has the appeal that if there were slight imperfections in the monitoring of each relationship, temporary ostracism would not require public correlations to maintain long-run cooperation. Only two papers to our knowledge have studied communication incentives in community enforcement. Lippert and Spagnolo (2011) study a setting in which each individual plays separate prisoners dilemmas with all of her partners simultaneously, and the payoffs from these prisoners dilemmas are heterogeneous. Community enforcement in their setting pools incentive constraints so as to use bilateral incentive constraints that hold with slack to subsidize other relationships, analogous to multimarket collusion (Bernheim and Whinston 1990). Our analysis is complementary in several respects, in addition to the many differences in our frameworks. First, they focus on network architectures and the pooling of incentive constraints whereas we focus on the distinction between permanent and temporary ostracism that arises with strategic communication. Second, community enforcement in their setting can support beyond private bilateral enforcement in only some but not all relationships; in contrast, because we do not use cross-subsidization, we study equilibria that supports cooperation beyond private bilateral enforcement in every relationship. 4 Variable stakes environments also feature in Sobel (1985), Ghosh and Ray (1996), Kranton (1996), and Watson (1999), but are used as a device to screen myopic types towards building a relationship over time. We find that this setting is useful even without heterogeneity in player types. 5 Indeed, we study it ourselves in Ali and Miller (2010). 5

8 Relative to the research on repeated games with private monitoring, it is already known that communication ameliorates private monitoring issues, especially as players become arbitrarily patient (Compte 1998; Kandori and Matsushima 1998). Our community enforcement setting differs from the games studied in prior work but the more important distinction is in our focus. The prior literature characterizes equilibrium payoffs as δ 1 whereas we study the communication incentives generated by a simple and intuitive class of community enforcement equilibria for a fixed level of patience. Our work is similar in spirit to Harrington and Skrzypacz (2011) who study communication and cooperation incentives in a particular class of equilibria that resonate with observed collusive arrangements between firms. 2 Model Society is composed of a finite set of players N = {1, 2,..., n + 1}, where n > 2. Individuals engage in bilateral repeated partnerships, and so there are G = n(n+1) 2 possible partnerships in society. We refer to a partnership between players i and j as a link ij in this network. We study both discrete-time and continuous-time settings. In the discrete-time setting, each period is of length : players have the opportunity to interact at times in T ( ) = {0,, 2,...}. In each period, society is either inactive in which case no link is selected, or it is active in which case a single link is selected; conditional on society being active, each link is selected with equal probability. The probability with which society is active is 1 e Gλ, where λ > 0 represents a Poisson arrival rate; we let p = 1 e Gλ denote the probability G with which a link is selected in each period. Our interest is in short time-periods ( 0). We also study the continuous-time game directly in which the time grid is T (0) = [0, ), and each link meets according to an independent Poisson recognition process of rate λ > 0. A feature common to both discrete and continuous time formulations is that the probability of two links being simultaneously selected is 0. When link ij meets, each of the two players chooses what to say to the other, what stakes to propose for their relationship, and whether to work or shirk. Payoffs from an interaction at real time t are discounted by e rt ; we write δ = e r for the per period discount factor. The extensive-form of each interaction is: 1. Communication Stage: Players send messages to each other sequentially. Independently of the past, with probability 1, player i sends a message first, and with proba- 2 bility 1, player j sends a message first. 2 6

9 2. Stake Selection Stage: Players propose stakes for their project. Player i proposes ˆφ t ij in [0, Φ], and the minimum of the two proposals, min{ ˆφ t ij, ˆφ t ji}, are the stakes φ t ij that are selected for that interaction. 3. Effort Stage: Each player simultaneously chooses to work (W) or shirk (S). If the stakes are φ, then their payoffs correspond to the prisoner s dilemma depicted in Figure 1. Player j W S Player W φ, φ V (φ), φ + T (φ) i S φ + T (φ), V (φ) 0, 0 Figure 1. The prisoners dilemma with stakes φ Mnemonically, T is the temptation to shirk and V is the loss from being the victim of someone else shirking. Both T and V are smooth, non-negative, and strictly increasing functions that satisfy T (0) = V (0) = 0. Central to our approach is the following assumption: Assumption 1 (Increasing Temptation). The temptation to shirk, T, is strictly convex and its slope satisfies T (0) = 0 and lim φ T (φ) =. We comment on the role of this assumption in Remark 1 at the end of this section. We also assume that the upper-bound on stakes Φ is sufficiently high so as to never bind in any of incentive conditions that we study; an upper-bound to feasible stakes precludes equilibria in which cooperation is sustained through an exploding sequence of stakes; restricting attention to equilibria in which stake proposals are uniformly bounded across all histories yields equivalent results. Information and Communication: We study an environment with private monitoring: when a pair ij interact at time t, players i and j are the only ones who directly observe that they have met, the messages that each shares with the other, the stakes that each proposes, and their effort choices. Our interest is understanding the extent to which players can use communication towards community enforcement. We study players who can communicate in a rich language that allows them to transmit their private histories to each other. An interaction between players 7

10 i and j at time t denoted by z t comprises the time t at which the pair meets, their names, the timing and contents of their communications to each other, the stakes that each announces, and their effort choices. Player i s private history at time t, h t i, is the set of all interactions that she has had strictly before time t. At history h t i, we let M(h t i) denote the set of available messages to player i. If communication were cheap talk, this would include the set of all possible histories up to time t. In evidentiary communication structures, a player can report or conceal only true interactions but cannot distort them or report interactions that did not take place. Using P(h) to denote the power set of history h, the following defines evidentiary communication: Definition 1. Communication is evidentiary if M(h t i) = P (h t i), and communication is cheap talk if M(h t i) includes every feasible private history at time t for player i. With evidentiary communication, a player s only strategic choice is to disclose or conceal interactions. Evidentiary communication corresponds to a setting with hard information whereas the cheap-talk setting is one in which all information is soft; in practice, one suspects that information lies between these two extremes. We focus on the setting with evidentiary communication for several reasons. First, since any setting in which information is soft has more deviations, our negative results on the strategic incentive to conceal the truth trivially extend to settings in which information is soft. Second, the evidentiary communication escapes any challenges of reports being conflicting, and allows us to exposit our result with the greatest transparency. Using evidentiary communication towards our negative results on permanent ostracism establishes it most strongly; in contrast, in our positive results on temporary ostracism, we explore the extent to which hard information is needed. The information contained in history h t i extends beyond the interactions that i has participated in first-hand up to that time; it also includes those that she learns about from others (including those interactions that they too have learned about through communication). 6 We expand the history h t i to the collection of all interactions recorded in it and denote this set of interactions by E(h t i). Strategies and Equilibria: A behavioral strategy is a function σ i = (σ M i, σ S i, σ A i ) such that in player i s time t interaction with player j, σ M i (j, h t i, m) is his message to player j 6 For example, these may include descriptions of all the interactions that k has had before some time τ at which point they are conveyed to j, which are then conveyed to i at time τ, for τ, τ < t. 8

11 σ A i where m = if player i communicates first, and m = m t j if player j communicates first, ( ) ( ) σi S j, h t i, m t i, m t j is his stakes proposal given the history and communications m t i, m t j, and ( j, h t i, m t i, m t j, ˆφ t ij, ˆφ ) t ji is his action choice. We study perfect extended-bayesian equilibria, 7 which we refer to throughout as equilibria. Our interest is in studying classes of simple equilibria that correspond to those studied in prior applications, and to understand questions of strategic communication in ostracism. In most of our analysis, we restrict attention to mutual effort equilibria, i.e., those in which all players work on the equilibrium path. In discussing the efficiency of equilibria, we use the notion of strongly efficiency (Jackson and Wolinsky 1996): the value of an equilibrium is the average across players of its equilibrium path expected payoffs, and an equilibrium is strongly efficient if its value is at least as high as that of any other equilibrium. 2.1 Discussion of Model Before analyzing this framework, we comment on two of its features. Remark 1 (Variable Vs. Fixed Stakes). Conventional analyses of repeated interaction in prisoners dilemmas measures the strength of enforcement by the minimal patience needed to sustain cooperation since the stakes are typically fixed. Our approach instead fixes patience and measures cooperation through the maximal level of stakes in each relationship under which working is incentive-compatible; Assumption 1 ensures that this is well-defined. This is a novel feature of our model that merits discussion and justification; we describe three reasons for us to analyze ostracism using this approach. First, we consider the flexibility inherent in variable-stakes to be a realistic feature of cooperative arrangements, whereby each pair of players in the community can adjust their relationship based on what they have observed in their relationship and in others. Second, the technological restrictions in a fixed-stakes environment are ill-suited towards understanding ostracism: as we discuss in the end of Section 4.1, because individuals cannot adjust the terms of their cooperation in a fixed-stakes environment, permanent ostracism equilibria necessarily fail to improve upon private bilateral enforcement even when communication is mechanical. A fixed-stakes model imposes a stark technological restriction and relaxing this restriction offers a glimpse of more subtle incentive issues that may impede ostracism. Third, the metric of 7 Perfect extended-bayesian equilibrium was introduced by Fudenberg and Tirole (1991) and further developed by Battigalli (1996): each player s behavior is sequentially rational given a belief system, where beliefs are derived via Bayes rule from a conditional probability system that satisfies strategic independence. 9

12 maximal stakes is technically convenient: we can compare and contrast different equilibria for the same vector of parameters (discount factor, interaction rates, period length), whereas a fixed stakes environment would require a comparison of sets of parameters (e.g. minimum discount factor) for which a strategy profile is an equilibrium. So as to permit such flexibility in the terms of cooperation, we develop and study a prisoners dilemma model with variable stakes. A similar analysis emerges if each player in each partnership chose actions from a continuous action space in which higher actions benefit her partner but are privately costly (as in e.g. Ghosh and Ray 1996). Indeed, we use analyses of this form in our application to buyer-seller markets in Section 6.1 and decentralized risksharing in Section 6.2. Remark 2 (Communication Protocol). We focus on a sequential communication protocol for tractability and to resonate with applications of interest. Our results on the limits of permanent ostracism emerge with the greatest clarity in this setting: when monitoring is private, individuals may be uncertain about what other individuals believe off-the-equilibrium path, and their communication incentives hinge on these beliefs. As the literature on repeated games with private monitoring has emphasized, analyzing belief-based equilibria is difficult even in standard repeated games, let alone in a community enforcement setting with a fixed level of patience. 8 Sequential communication sidesteps these challenges for at least one of the two parties in every interaction, which allows us to examine ex post communication incentives for at least one party. Our applied motivation for considering a sequential communication protocol is that we believe this protocol is closer to the gossip and information sharing that we envision playing a role in community enforcement. Nevertheless, in Section 4.3, we derive results on the possibilities and limits of permanent ostracism in various alternative communication protocols, including simultaneous communication. A separate issue is that our setting makes the timing of communication and interaction identical, and this appears at odds with the view that information diffuses at a rate faster than actions. We discuss in Section how the negative result on permanent ostracism continues to apply even when individuals have more opportunities to communicate than to interact in their dynamic partnership. 8 See Mailath and Morris (2002) and Ely, Hörner, and Olszewski (2005) for discussions of belief-based and belief-free equilibria in repeated games. Takahashi (2010) and Deb (2011) use belief-free constructions in community enforcement. 10

13 3 A Simple Example The important forces in our model can be illustrated through an example that involves three players, and in which T (φ) = φ 2 and V (φ) = φ + φ 2 for every stakes φ. Bob Ann Carol Figure 2 Private Bilateral Enforcement. Consider as a benchmark a setting in which all enforcement is private, and the community is not used to discipline individual relationships. Corresponding to bilateral grim trigger, a pair works if and only if they both have worked every prior time that they have met. A player prefers working to shirking if φ + T (φ) φ + δp φ. (Bilateral IC) 1 δ For each > 0, the existence of the highest stakes that satisfies this inequality is guaranteed by Assumption 1; we denote these stakes by φ( ). Taking limits for shorter time-periods, let φ(0) denote lim 0 φ( ). Using l Hopital s Rule, it follows that φ(0) solves T (φ) φ = λ r. For T (φ) = φ 2, it follows that φ( ) = δp 1 δ 0 λ r. and so each player s expected payoff is 2 ( λ r ) 2 in the continuous time limit. Permanent Ostracism. Community enforcement is useful when it supports cooperation at stakes greater than those that can be supported by private enforcement alone. A natural candidate is permanent ostracism: when a player shirks on another, the victim communicates this to the third-party at the first opportunity (when they meet in the future), and the guilty 11

14 party is punished permanently by all those who come to learn of the deviation, whereas innocent players continue to work together. To understand the incentives that this induces, consider a simple strategy profile in which on the path of play, Ann, Bob, and Carol are meant to work at stakes φ whenever any pair meets. Suppose that Ann and Bob interact, and consider Ann s incentives to shirk. Assuming that Bob will tell Carol, Ann can expect a further gain only if she can meet and shirk on Carol before Carol comes to learn of her deviation from Bob. With this in mind, Ann prefers to work with Bob if δp φ + T (φ) + 1 δ (1 2p ) (φ + T (φ)) φ + 2δp φ. (Effort IC) 1 δ The LHS of the above inequality reflects Ann s payoff from shirking immediately on Bob, and her discounted expected payoff from possibly being able to shirk on Carol before Carol meets Bob. 9 The RHS denotes today s and future cooperation payoffs from working with both partners. The highest symmetric stakes that satisfy this incentive condition assuming that T (φ) = φ 2 and for 0 is λ r ( r+4λ r+3λ), which indeed exceeds φ(0). If communication incentives are satisfied, then ostracism outperforms private enforcement for all 0 because it links the two relationships, and with positive probability, punishes a player s opportunistic behavior in one relationship through the other. Does Bob find it in his interest to disclose the truth to Carol? If Bob and Carol ostracize Ann, the pair expect to work only if their stakes in the continuation history are such that bilateral grim trigger deters each of them to not shirk on the other. In other words, the stakes in the continuation history must satisfy Bilateral IC and therefore cannot exceed φ( ). Since Bob s payoff from revealing Ann s deviation is increasing in these stakes, let us suppose that the pair selects stakes φ( ) in every subsequent interaction once they ostracize Ann. What is Bob s gain from concealing Ann s deviation from Carol? If Ann had already shirked on Carol, then Bob is indifferent between revealing and concealing the truth. On the other hand, if Carol reveals that Ann has not shirked on her, Bob knows that Carol thinks Ann is innocent. So Bob can proceed to shirk on Carol at the on path stakes of φ and gain an immediate payoff of φ + T (φ). Therefore, Bob finds truth telling to has an incentive 9 Ann s discounts payoffs that she gains from shirking on Carol by the time that she has to wait before meeting Carol and the probability that Carol has not yet met either Ann or Bob. As 0, this discounted probability converges to λ r+2λ. 12

15 to reveal the truth to Carol if and only if for each, φ + T (φ) φ( ) + δp φ( ). (Reporting IC) 1 δ The above inequality binds at φ( ) (by construction in Bilateral IC), and since T is increasing in φ, it follows that Bob has an incentive to reveal the truth to Carol if and only if φ is at most φ( ) and otherwise, he prefers to conceal the truth. In other words, community enforcement via permanent ostracism is an equilibrium only if it is redundant. Temporary Ostracism. Permanent ostracism burns the value that players derive from their relationship with Ann when she shirks; instead of burning that value, an ostracism equilibrium may do better by transferring value to Bob and Carol so as to subsidize communication incentives. In a temporary ostracism equilibrium, Ann redeems herself by working while permitting one of the two to shirk. Expecting Ann s future redemption, Bob and Carol continue to work with each other at the original equilibrium stakes. We construct the temporary ostracism equilibrium for the continuous time limit. Ann redeems herself by working one time at stakes φ R (while her partner shirks) and then she uses that evidence to return to equilibrium path cooperation at stakes φ with both Bob and Carol. The redemption stakes are set to extract Ann s future payoffs from cooperation: V (φ R ) = 2λ r φ. Once Ann knows that both Bob and Carol consider her guilty, she is indifferent between working towards redemption at stakes φ R and shirking. Whoever is Ann s partner when she works towards redeeming herself obtains a payoff of φ R + T ( φ R). Given this punishment, Ann s incentive to work on the equilibrium path is captured by Effort IC that we described in permanent ostracism: her only opportunity for gain after shirking on Bob is if she meets Carol before Bob meets Carol. The critical question is whether temporary ostracism is vulnerable to the same issue as permanent ostracism: will Bob tell Carol if Ann shirks on him? If Bob tells Carol, the pair will continue to work at the original stakes and both expect to return to future cooperation with Ann once she has redeemed herself. Moreover, Bob values the possibility of being the beneficiary of Ann s redemption in terms of being able to shirk while Ann works to redeem herself. Concealing the truth and shirking with Carol offers Bob 13

16 a payoff of φ + T (φ) whereas that from communicating the truth and working is φ + λ r φ + 2λ ( ) λ r + 2λ r φ + λ ( φ R + T (φ R ) ). r + 2λ This inequality is strictly satisfied at φ = φ(0), and therefore, temporary ostracism can support cooperation at stakes beyond permanent ostracism. Importantly, if r λ, then the inequality is also satisfied at every profile of stakes that satisfy Effort IC: temporary ostracism with strategic communication supports the same stakes that permanent ostracism would had victims been artificially constrained to reporting truthfully. 4 Permanent Ostracism Our interest is in understanding how communication incentives limit the possibility for cooperation in permanent ostracism equilibria. We begin with a benchmark in which communication is entirely mechanical and then introduce communication incentive constraints. 4.1 Mechanical Communication: A Benchmark Suppose that at the beginning of each interaction, every individual is constrained to truthfully revealing their past history: for every player i, and at every history h t i, the set of feasible messages, M(h t i), is the singleton {h t i}. Effectively, monitoring is perfect: if a player shirks on another, both of them reveal this to all third parties. Community enforcement is at its best: an individual is immediately punished in all her relationships when she shirks. At every history, players are partitioned into those who are guilty or innocent: all players are initially innocent but are deemed guilty once they shirk. Guilty players are punished by the stakes in their relationship being set to 0, while innocent players continue to work with each other at strictly positive stakes. To assess who is guilty, a player with history h considers its full record of interactions, E(h). For an interaction, z τ E(h), ρ(z τ ) is the pair of players interacting, and for player j ρ(z τ ), her effort choice is a τ j. Guilty players are in Ĝ(h) = { } j N : There exists z τ E(h) such that j ρ(z τ ) and a τ j = S, and the remaining players are considered innocent, denoted by Î(h). Definition 2. With mechanical communication, in a permanent ostracism strategy profile, at every time t and histories ( h t i, h t j), players i and j work with each other at strictly positive 14

17 stakes if {i, j} Î ( h t i h t j). Otherwise, each player announce stakes of 0 and works at those stakes. Innocent players who learn that others have shirked adjust their stakes so as to ensure that their effort incentives remain satisfied. Consider a history such that Î(h) includes players i and j. If the stakes between the pair ij depend only on the set of innocent players, then player i s incentive constraint at history h is: T (φ ij ) k Î(h)\{i} δp 1 δ φ ik. (Mech IC) By Assumption 1, there exists a symmetric stakes profile that satisfies Mech IC with equality for each. 10 We use this to construct a permanent ostracism equilibrium with mechanical communication in which each relationship sustains greater cooperation than it could through private bilateral enforcement. Proposition 1. With mechanical communication, there exists a permanent ostracism equilibrium in which on the equilibrium path, all players work at stakes φ( ) that solves T (φ) φ = nδp 1 δ 0 nλ r. (1) No mutual effort equilibrium supports mutual effort at stakes greater than φ( ) in any history. The above result characterizes the first best outcome that is possible without informational frictions. As the maximal stakes from private bilateral enforcement, φ( ), corresponds to n = 1 above, ostracism with mechanical communication clearly outperforms bilateral enforcement in every relationship. It is intuitive that φ( ) strictly increase in population size because being ostracized from more partnerships presents a harsher punishment for shirking. We prove that the above equilibrium is efficient among all mutual effort equilibria through a few elementary steps. First, every mutual effort equilibrium in which the off-path behavior does not correspond to permanent ostracism can be transformed to a permanent ostracism equilibrium since that is the worst possible punishment. Second, because the reward for working is increasing in all future stakes, for any equilibrium with asymmetric or nonstationary stakes on the equilibrium path, there exists a symmetric stationary stakes profiles 10 Recall that we assume throughout our analysis that Φ is sufficiently high that the feasibility constraint on stakes does not bind. 15

18 that attains strictly greater value. Thus, no mutual effort equilibrium can support mutual effort at higher stakes in any history than the highest symmetric stationary permanent ostracism equilibrium, which is what we construct in Proposition 1. This benchmark emphasizes the distinction between variable and fixed stakes. Were stakes fixed, permanent ostracism could do no better than bilateral enforcement even if communication is mechanical: players are either sufficiently patient to enforce mutual effort through bilateral grim trigger or unwilling to cooperate when only two innocent players remain. We view this as a technological restriction that may not be relevant to real-world ventures in which the level of cooperation can be adjusted. Importantly, relaxing this technological restriction permits us to analyze a more subtle issue of how communication in permanent ostracism is susceptible to manipulation. 4.2 Limits of Strategic Communication Mechanical communication requires Ann to report on others and also confess herself if she shirked in the past. Naturally, the first strategic challenge that comes to mind is that Ann has no incentive to confess since that results in her being permanently ostracized. Permanent ostracism with strategic thus relies on victims and third-parties to report when she shirks, but we show that this too fails. To see the incentive issues, we begin by formally defining permanent ostracism with strategic communication. For a private history for player i, h t i, and for time τ t, let m j (h t i, τ) denote the subset of interactions in E(h t i) that happened before τ in which player j was active. Player i then knows that player j has concealed interactions if for some interaction z τ, m τ j excludes an interaction from m j (h t i, τ). A player j is guilty in history h if there exists z τ E(h) such that j ρ(z τ ) and at least one of the following is true: 1. The action a τ j = S, 2. The message m τ j is a strict subset of m j (h, τ) and player j s proposed stakes are ˆφ τ j > 0. In history h, the set of guilty players is G(h) and those innocent are in its complement, I(h). Definition 3. With strategic communication, a strategy profile involves permanent ostracism if for every time t and history h t i, 1. If {i, j} I(h t i), then player i reveals history h t i. If m j (h t i, t) m t j, and j I ( h t i mj) t, then player i announces strictly positive stakes and works. 16

19 2. If j G(h t i), player i sends message m t i =, announces stakes of 0, and works. Permanent ostracism equilibria are within the class of mutual effort equilibria since an innocent player works and communicates with those she believes to be innocent. Someone who behaves opportunistically is punished by those who learn about her past behavior. Having shirked once, an individual s only opportunity for further gain is when she can shirk on other partners before they learn about her past defection. The threat of ostracism, thus, hinges on the rate at which news travels through the network (which happens when innocent players talk to each other). We first consider a simple class of strategy profiles: a permanent ostracism strategy profile is straightforward if the stakes for an innocent pair ij at time t are determined by I ( ) h t i h t j and are otherwise invariant to history and time. 11 To understand the rate at which news travels in a straightforward profile, consider a history h at which player i believes that all other players are innocent, and is considering a deviation. If she shirks on a partner j at time s, assuming that others will report truthfully about her deviation, let x (t) be the probability that another innocent player k thinks that player i is innocent at time s + t. Let X = δ t p x (t). t= The term X represents the viscosity of strategic communication: it measures the extent to which player i can deviate on others before information about his deviation spreads through the communications of victims and third-parties. Lemma 2 in the Appendix characterizes X and its limiting value as 0. Player i s incentives to work are thus captured by T (φ ij ) + k N \{i,j} (φ ik + T (φ ik )) X k N \{i} δp 1 δ φ ik. (IC Coop ij ) The distinction betweeen IC Coop ij and its analogue in the setting with mechanical communication (Mech IC on p. 15) is the second term above: mechanical communication has no viscosity, but with strategic communication, a player who shirks once can shirk on a third-party who does not learn that one is a guilty. Strategic communication necessarily cannot enforce as high stakes as mechanical communication, but nevertheless, can do better than private bilateral enforcement if victims and third-parties were truthful. The following 11 We borrow the label straightforward from Kandori (1992) who uses it to describe ostracism equilibria that condition on the minimal possible information; Okuno-Fujiwara and Postlewaite (1995) use norm equilibrium for a similar concept in their setting. 17

20 proposition characterizes the best possible stakes that satisfy equilibrium path incentives without considering the sequential rationality of communication off the equilibrium path. Proposition 2. With strategic communication, there exists a straightforward permanent ostracism strategy profile that satisfies IC Coop ij at stakes φ ( ) that solves T (φ) φ = ( in which on the path of play all players work ) ( ) 1 nδp 1 + X (n 1) 1 δ (n 1)X. (2) This is strongly efficient among all straightforward permanent ostracism strategy profiles that satisfy IC Coop ij. The stakes φ ( ) are less than the greatest with mechanical communication, φ( ), and exceed those from private enforcement, φ( ) for every ; this relationship holds in the continuous time limit, φ (0) ( φ(0), φ(0) ). This proposition shows that the absence of self-reporting also does not stop permanent ostracism with strategic communication from outperforming private bilateral enforcement. The above characterizes the analogue of the best straightforward Nash equilibrium: conditioning on victims and third-parties revealing information, the player is willing to work at stakes φ ( ) because she is punished in other relations when news spreads of her defection. Yet, this equilibrium is not sequentially rational insofar as there are histories off the path of play in which victims and third-parties prefer to conceal the defections of others. Theorem 1. No straightforward permanent ostracism equilibrium supports stakes in any partnership beyond φ( ), regardless of players patience and period length. Proof. We construct a history in which the communication incentive binds the equilibrium path stakes to be no greater than φ( ). Consider times t and τ = t + n, a pair of players {i, j} who meet at time t, and histories such that I ( h t i h t j) = N. Suppose that in the subsequent (n 1) periods, player i meets every player other than j who proceeds to shirk on him, and then players i and j meet at time τ. If player i reveals h τ i to player j, the best possible continuation play is cooperating at φ( ) forever with player j. If player i instead reports h t i and conceals h τ i \h t i, then he has the opportunity to shirk at the equilibrium path stakes, φ ij. Player i communicates truthfully if and only if φ ij satisfies Bilateral IC (on p. 11); since φ( ) are the highest stakes that satisfy Bilateral IC, it follows that φ ij φ( ). 18

21 The limits of straightforward permanent ostracism equilibrium has a clear intuition: if two players can support higher stakes in their partnership on the equilibrium path than off-theequilibrium path, other relationships serves as social collateral for each of them to cooperate. Yet, when one of them observes others choosing to defect, that social collateral is lost giving her the incentive to conceal information and shirk. This force causes permanent ostracism in even large societies to unravel to those stakes supportable by bilateral enforcement, for which community enforcement is entirely unnecessary. Theorem 1 pertains to straightforward equilibria, which correspond closest to that analyzed in prior work. Straightforward equilibria are particularly intuitive when one considers players distinguishing between innocent and guilty players, and communicating these labels to others. The following example shows how non-straightforward permanent ostracism equilibria do better in discrete time but these gains disappear as 0. Example 1. Consider the triad depicted in Figure 2 and a history-dependent stakes profile in which on the path of play, at time t, partnership ij works at stakes φ > φ( ) if one of them reveals an interaction at t in which both players worked, and φ( ) otherwise. A player has incentive to work on the equilibrium path if T (φ) + δp ( ) 2δp φ( ) + T (φ( )) 1 δ (1 2p ) 1 δ ( (1 δ(1 3p )) φ + δ(1 3δp )φ( ) For every > 0, the above inequality has slack at φ = φ( ) and so this history-dependent stakes profile can support working at stakes φ strictly greater than φ( ) when a player reports a cooperative interaction in the preceding period. Unlike the straightforward profile, off path communication incentives are also satisfied: when Ann shirks on Bob, and Bob subsequently meets Carol, Bob is indifferent between revealing and concealing the truth since in either case, he and Carol shall set stakes φ( ). support stakes that exceed those of bilateral enforcement. Thus, this form of permanent ostracism can Communication incentives nevertheless impose a limit that becomes severe as 0: to satisfy off-path communication incentives, on-path stakes that exceed φ( ) can only be supported when there was cooperation in the preceding round. Otherwise, Bob would have a strict incentive to conceal that Ann shirked on him if he knew that Carol did not know this. As 0, the likelihood of interactions in two successive periods vanishes, and so the payoffs from such an equilibrium collapse to bilateral enforcement. Formally, the payoff ). 19