Elsevier

Journal of Economic Theory

Volume 157, May 2015, Pages 553-605
Journal of Economic Theory

Reputation in the long-run with imperfect monitoring

https://doi.org/10.1016/j.jet.2015.01.012Get rights and content

Abstract

We study an infinitely repeated game where two players with equal discount factors play a simultaneous-move stage game. Player one monitors the stage-game actions of player two imperfectly, while player two monitors the pure stage-game actions of player one perfectly. Player one's type is private information and he may be a “commitment type,” drawn from a countable set of commitment types, who is locked into playing a particular strategy. Under a full-support assumption on the monitoring structure, we prove a reputation result for stage games with a strong Stackelberg action: if there is positive probability that player one is a particular type whose commitment payoff is equal to player one's highest payoff, consistent with the players' individual rationality, then a patient player one secures this type's commitment payoff in any Bayes–Nash equilibrium of the repeated game. In contrast, if the type's commitment payoff is strictly less than player one's highest payoff consistent with the players' individual rationality, then the worst perfect Bayesian equilibrium payoff for a patient player one is equal to his minimax payoff.

Introduction

A patient player's reputation concerns are the dominant incentives that determine equilibrium payoffs in repeated games where a patient player faces a myopic opponent. And this is true regardless of the monitoring structure.1 Building a reputation when facing an equally patient opponent, however, is more difficult. A patient opponent might be willing to sacrifice short-term payoffs to test whether the player, who is trying to build a reputation, will go through with his threats or promises. This makes it prohibitively expensive to build a reputation in certain repeated simultaneous-move games played against a patient opponent if stage-game actions are perfectly monitored (Cripps and Thomas [11]). In this paper, we instead focus on repeated simultaneous-move games played by equally patient players where the opponent's stage-game actions are imperfectly monitored. We show that reputation effects are prominent under imperfect monitoring even in certain repeated games where reputation effects are absent under perfect monitoring.

Specifically, suppose that player one's type is private information and that he may be one of many finite-automaton “commitment types” each of whom is locked into playing a particular repeated-game strategy. We explore whether an uncommitted or “normal” player can exploit his opponent's uncertainty to establish a reputation and thereby guarantee an advantageous equilibrium payoff. Our central finding is a lower bound on player one's (he) equilibrium payoff for repeated games where he observes only an imperfect public signal of his opponent's stage-game action while his opponent perfectly monitors player one's actions. This lower bound is tight if the set of commitment types includes a pure-strategy finite-automaton commitment type with no shortfall, i.e., a type with a commitment payoff that is equal to player one's highest payoff compatible with the players' individual rationality (player one's highest IR payoff).2 In particular, if a finite automaton with no shortfall is available, then a patient player one can guarantee his highest IR payoff in any Bayesian–Nash equilibrium of the repeated game. Player one can guarantee his highest IR payoff by simply mimicking the strategy of the finite automaton with no shortfall, even if player two believes that player one is another finite automaton with arbitrarily higher probability.

The lower bound that we establish implies a reputation result for certain repeated games where there exists a pure stage-game action (a strong Stackelberg action) which has a commitment payoff equal to player one's highest IR payoff.3 In a repeated game where there is a strong Stackelberg action, the commitment type which plays the strong Stackelberg action in each period of the repeated game is a pure strategy finite automaton with no shortfall. Therefore, in such a repeated game, player one can guarantee his highest IR payoff by simply mimicking this type, that is by playing the strong Stackelberg action in each period of the repeated game.

We turn next to the question of whether player one can still benefit from a reputation even if the shortfall for the only available commitment type is positive (i.e., the type's commitment payoff is less than player one's highest IR payoff). In this case, we show that a patient player one's worst equilibrium payoff is equal to his minimax. Therefore, player one guarantees only his lowest payoff if he compromises by mimicking a commitment type which has positive shortfall. Taken in conjunction with our reputation result, this implies that reputation building against an equally patient rival is an all-or-nothing phenomenon: in such a repeated game, player one guarantees either his best or only his worst repeated-game payoff.

Whether player one can successfully build a beneficial reputation by mimicking a pure strategy finite automaton hinges on whether there is a commitment type that has no shortfall which player one can mimic. Moreover, if player one has a strong Stackelberg action, then such a finite automaton always exists because the commitment type which plays the strong Stackelberg action in each period is a finite automaton with no shortfall. Now suppose instead that there is no strong Stackelberg action for player one. In this case, we show that there is no pure strategy finite automaton with no shortfall either, i.e., there exists a pure strategy finite automaton with no shortfall if and only if player has a strong Stackelberg action. Therefore, if there is no strong Stackelberg action, then player one's worst equilibrium payoff is equal to his minimax regardless of which pure strategy finite automaton he mimics. In other words, it is not possible for player one to successfully build a beneficial reputation by mimicking a finite automaton unless he has a strong Stackelberg action. Conversely, whenever player one can successfully build a reputation, he can do so by mimicking the least complex commitment type that plays the strong Stackelberg action in each period. In other words, added complexity does not improve a patient player one's worst payoff as long as the complexity is still finite.

One of our key assumptions is that player one does not observe player two's intended action, but only sees an imperfect signal of it, as in a model of moral hazard. We also assume that the support of the distribution of signals is independent of how player two plays; we call this the full-support imperfect-monitoring assumption. This assumption is indispensable and, intuitively, ensures that every reward and punishment in player one's strategy will occasionally be triggered, so that player two will learn how player one responds to all sequences of public outcomes. We discuss this assumption in more detail in Section 3.2.

We obtain our reputation result by calculating a lower bound, which holds across all equilibria, on player one's payoff when he mimics a commitment type that plays a pure strategy (as in Fudenberg and Levine [16]). In this context, our assumption that player one's stage-game actions are perfectly monitored greatly aids our analysis. This is because the perfect-monitoring assumption simplifies the dynamics of how player one's reputation evolves. In particular, because player two perfectly monitors player one's stage-game actions and because the commitment type plays a pure strategy, player one's reputation level weakly increases – but only as long as player two observes him play the same stage-game action as the action the commitment type would have played; otherwise, his reputation level collapses to zero.4 If we relax the assumption that player one's actions are perfectly monitored, then a technically challenging statistical learning problem arises. Whether an appropriate statistical learning technique can be developed or applied for this framework remains an open question beyond the scope of this paper.5,6

For our results, we assume that all the commitment types are finite automata. This assumption plays two crucial roles in our analysis: First, it allows us to provide a sharp characterization of the stages games in which player one can successfully build a reputation. Second, the assumption that all the other types are also finite automata in conjunction with the full support imperfect monitoring assumption allows us to establish a statistical learning result which we then use to establish our reputation bound. If we did not restrict the commitment types to finite automata, then both our characterization of stage games as well as our learning result can fail. We discuss these issues further in Section 5. There is also an economic rationale for restricting attention to finite automata as it excludes cases where player one builds a reputation by mimicking a more intricate commitment type which plays an infinitely complex strategy. Our results show that player one need not mimic a type more complicated than the simplest automaton for repeated games where there is a strong Stackelberg action.7

Lastly, the reputation results in games with asymmetric discounting (Fudenberg and Levine [16], [17] or Celentani et al. [6]) are robust to the introduction of two-sided uncertainty, while the reputation result that we present in this paper is not. In order to obtain our one-sided reputation result, we allow for only one-sided uncertainty. In other words, we replace asymmetric discount factors as in Fudenberg and Levine [16], [17] or Celentani et al. [6] with one-sided asymmetric information.

This paper is most closely related to work on reputation effects in repeated simultaneous-move games with equally patient agents (see Cripps and Thomas [11], Cripps et al. [8], and Chan [7]).8 We make three main contributions to this literature. First, we provide the first reputation result for all games with a strong Stackelberg action.9 Previous reputation results are for only a strict subset of stage games with a strong Stackelberg action: those with strictly conflicting interests (Cripps et al. [8]) or strictly-dominant-action stage games (Chan [7]).10 Second, we are the first to explore reputation effects under imperfect monitoring. Previous work assumed perfect monitoring. Finally, our work highlights the role that full-support imperfect monitoring plays for a reputation effect in repeated games. Without full-support imperfect monitoring, our reputation result may fail to obtain for repeated common-interest games (Cripps and Thomas [11] and Chan [7]).

This paper also relates to work on reputation effects in repeated games where a patient player one faces a nonmyopic, but arbitrarily less patient, opponent (Schmidt [21], Celentani et al. [6], Aoyagi [1], Cripps et al. [9], Evans and Thomas [15], and Cripps and Thomas [12]). In repeated games where a patient player faces a less patient opponent, Celentani et al. [6] and Aoyagi [1] establish reputation results under full-support imperfect monitoring. Although the results in repeated games with a less patient opponent are similar in spirit to the results we establish here, we should point out two important differences. First, against a less patient opponent, player one can build a reputation by mimicking a commitment type with positive shortfall, i.e., player one can guarantee a compromise payoff (Celentani et al. [6] and Cripps et al. [9]). In contrast, this is not possible when player one faces an equally patient opponent. Second, with equally patient agents, the limitation on the types that facilitate reputation building to those with no shortfall implies a restriction on the class of stage games (i.e., those with a strong Stackelberg action). Again, this contrasts with the case where player one faces a less patient opponent, as in Celentani et al. [6]. Because player one can guarantee a compromise payoff against a less patient opponent, Celentani et al. [6] are able to establish a reputation result which applies to all stage games when there is full-support imperfect monitoring.

This paper is also closely related to Atakan and Ekmekci [2], which proves a reputation result for repeated extensive-form games of perfect information with equally patient players. The three main differences between the two papers are as follows: First, in this paper we study the Bayesian equilibria of repeated simultaneous-move games whereas the focus of Atakan and Ekmekci [2] is on the perfect Bayesian equilibria of a repeated game where the two players never move simultaneously. In particular, the reputation result of Atakan and Ekmekci [2] leverages the particular form of sequential rationality, implied by perfect Bayesian equilibrium for games where the two players move sequentially, in a way that one cannot if the two players move simultaneously or if the focus is on Bayesian equilibria. Two, this paper assumes imperfect monitoring whereas Atakan and Ekmekci [2] assume that both players' moves are perfectly monitored. Three, here we assume that the commitment types are finite automata but we place no restriction on player two's prior. In contrast, the reputation result in Atakan and Ekmekci [2] depends on the set of other commitment types having sufficiently low prior probability.

Section snippets

The model

We consider an infinitely repeated game in which a finite, two-player, simultaneous-move stage game Γ is played in periods t{0,1,2,}. The players discount payoffs using a common discount factor δ[0,1). For any set X, Δ(X) denotes the set of all probability distribution functions over X. The set of pure actions for player i in the stage game is Ai, and the set of mixed stage-game actions is Δ(Ai). After each period, player two's stage-game action is imperfectly observed through a public

Reputation effects

In this section we present our main reputation result (Theorem 1) which applies to stage games that both have a strong Stackelberg action and have no gap.

Theorem 1

Suppose that Γ satisfies Assumption FS, has a strong Stackelberg action, and has no gap; and suppose that all the commitment types are finite automata. Let ω denote a commitment type which is an irreducible pure-strategy finite automaton with no shortfall. If μ(ω)>0, then U1NE(μ)g¯1.

Proof

Lemma 1 implies that an irreducible pure-strategy finite

Nonreputation results

Now we turn our focus to the following two questions: Which types, if available, facilitate reputation building for player one? In which strategic situations (i.e., for which class of stage games) can player one successfully build a reputation? In addressing these questions, we restrict attention to pure-strategy finite automata.26 Under this restriction, we show that (i) only types with no shortfall facilitate

The role of finite automata

In our analysis, we assume that all the commitment type's are finite automata. We need this assumption for two different reasons when proving our reputation result. First, this assumption allows us to prove a learning result that we present as Lemma A.2 in Appendix A. Second, it allows us to provide a sharp link between the class of games and shortfall via Lemma 1. Also, when proving our non-reputation results we assume that player one mimics a commitment type which is a finite automaton. Below

Conclusion

In this paper we studied an infinitely repeated game where two players with equal discount factors play a simultaneous-move stage game under Assumption FS imperfect monitoring. Our first main result established a reputation result for stage games with a strong Stackelberg action: if there is positive probability that player one is a finite automaton whose commitment payoff is equal to player one's highest IR payoff, then a patient player one secures his highest IR payoff in any Nash equilibrium

References (23)

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    On the non-existence of reputation effects in two-person infinitely repeated games

    (2000)
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    We would like to thank Martin Cripps, Eddie Dekel, Eduardo Faingold, Johannes Horner, Christoph Kuzmics, Heng Liu, and Larry Samuelson for useful discussions. This paper has also benefited from the comments of seminar participants at Northwestern University, Princeton University, Yale University, University of Pennsylvania, Koç University, Stanford University, Bilkent University. Part of this research was conducted while Mehmet Ekmekci was visiting the Cowles Foundation and Economics Department at Yale University.

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