Impermanent types and permanent reputations

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Abstract

We study the impact of unobservable stochastic replacements for the long-run player in the classical reputation model with a long-run player and a series of short-run players. We provide explicit lower bounds on the Nash equilibrium payoffs of a long-run player, both ex-ante and following any positive probability history. Under general conditions on the convergence rates of the discount factor to one and of the rate of replacement to zero, both bounds converge to the Stackelberg payoff if the type space is sufficiently rich. These limiting conditions hold in particular if the game is played very frequently.

Introduction

The notion that commitment is valuable has long been a critical insight of non-cooperative game theory, and has deeply affected a number of social science fields, including macroeconomics, international finance and industrial organization.1 Existing reputation literature argues that in dynamic relationships, reputation concerns can substitute for a commitment technology. A patient long-run player who faces a sequence of short-run players who believe their opponent might be committed to a particular stage-game action benefits from such a perception, as shown by [11], [12].

However, [6], [7] show that if the long-run playerʼs actions are imperfectly observed by short-run players, reputation effects eventually disappear almost surely, at every equilibrium. This is particularly troubling since it shows that the original model cannot explain survival of reputation effects in environments where the agents have a long history.2 On the other hand, the commitment possibilities of a central bank or the managers of a firm may change over time, and market beliefs about the long run playerʼs commitment possibilities are progressively renewed. So the question is whether reputation effects are maintained perpetually if reputations are sown in the market only occasionally.

We model a long-run relationship as a repeated game between a long-run player and an infinite sequence of myopic opponents. The long-run player is either a normal type who takes actions optimally by considering the current and future consequences of his actions, or a commitment type who is committed to using a particular stage-game strategy in every interaction. The actions of the long-run player are imperfectly observed. At the beginning of every period, there is a positive probability that the long-run player is replaced by a new long-run player. The new player may also be a normal type, or he may be a commitment type. Neither the replacements nor the types of the long-run players are observed by the myopic players; hence, there is perpetual uncertainty about the long-run playerʼs type. However, in the course of the game the myopic players receive information regarding the type of their opponent through informative signals they observe about the long-run playerʼs actions.

Our main result is a pair of lower bounds on the Nash equilibrium payoffs of a normal type of long-run player as a function of the discount factor, the replacement rate, and the commitment type probability. The first bound is an ex-ante bound that is calculated at the beginning of the game. The second bound is on the long-run playerʼs equilibrium continuation payoffs at any positive probability history on the equilibrium path.

If replacements are arbitrarily infrequent and the long-run player is arbitrarily patient, our bound on the ex-ante payoff converges to the same bound as that established in [11], [12]. This shows that the introduction of infrequent replacements constitutes a small departure from the benchmark model.

When continuation payoffs are considered, replacements play both a positive and a negative role in the permanence of reputations. The negative effect is twofold. First, reputations naturally degrade and the short-run player doubts at every stage that he faces the same long-run player who played in previous stages. This makes reputation building less valuable in the long-run. Second, the long-run player anticipates that he might be replaced, and hence discounts the future more. In the extreme case where replacements occur at every stage, the long-run player doesnʼt care about future interactions and hence no reputation can be built. The positive effect is that, even if the long-run playerʼs reputation is severely damaged at some point, renewed doubt about his type in the mind of the short-run player offers the opportunity to rebuild a reputation.

We use our second bound to show that along a sequence of games with varying discount factors and replacement rates, if the discount factor goes to 1 at a faster rate than the rate at which the logarithm of the replacement rate goes to infinity,3 then the long-run player receives his highest possible commitment payoff after every equilibrium history of the repeated game. This shows that for a range of replacement rates (as a function of the discount factor), player 1 benefits from the positive effect of replacements without suffering significantly from the negative effects.

This result has a particularly natural interpretation in the study of frequently repeated games. Increasing the discount factor is sometimes interpreted as increasing the frequency with which a stage game is repeated.4 The conditions that our result requires are satisfied if the replacement events follow a Poisson distribution in real time with a constant hazard rate, and if the game is played in stages that become arbitrarily more frequent.5 No matter how rarely or frequently replacements occur in real time, they restore the persistency of reputation effects in frequently repeated games.

To derive our bounds, we calculate the expected discounted average of the one-period prediction errors of the short-run players, where the expectation is taken using the probability distribution function that is generated by conditioning on (i) player 1ʼs type being the commitment type at the beginning, and (ii) his type not changing. This idea is similar to the one introduced by [11], [12]. However, the active supermartingale approach used in their work is not naturally adapted in our model since the process that governs the beliefs of the short-run players has an extra drift due to replacements.

In our model, the probability that player 1 is a particular commitment type at every period is zero. In particular, there is no “grain of truth” in the fact that player 1 is of that particular commitment type at every stage. The “grain of truth” allows one to apply merging techniques such as [3] to models in which players have initial uncertainty about the behavior of other players, and to obtain the conclusion that players eventually predict the behavior of other players accurately (see, for example, [17], [13], [26]). It plays a central role in reputation models like those in [1], [11], [12].

We rely on an information theoretic tool called relative entropy (see [5], for an excellent introduction to the topic) to measure signal-prediction errors of player 2 more precisely, thus generalizing the approach in [14]. This allows us to measure the positive effects versus negative effects of replacements on reputation building. In [15], relative entropy is used to conveniently derive merging results.

Section snippets

Review of the literature

Reputation models were first introduced in the context of finitely repeated games in [18] and [23]. In infinitely repeated games, [11], [12] show that, under very weak assumptions on the monitoring technology, in any Nash equilibrium an arbitrarily patient long-run player obtains a payoff that is at least as much as the payoff he could get by publicly committing to playing any of the commitment type strategies. On the other hand, [6], [7], [29] show that for a class of stage games, all

Model

There is an infinite sequence of long-run and short-run players. At every period tN, a long-run player (player 1) plays a fixed, finite stage game G with a short-run player (player 2). The set of actions available to player i in G is Ai. Given any finite set X, Δ(X) represents the set of probability distributions over X, so the set of mixed stage-game strategies for player i is Si:=Δ(Ai).

The set of types available to player 1 is Ω={ω˜}Ωˆ. The type ω˜ is called player 1ʼs normal type. The

Main result

We define the infimum over all Nash equilibria of all continuation payoffs of player 1 at any history of the repeated game that is on the equilibrium path, as follows:v(μ,δ,ρ)=inf{π1,σ[h1] s.t. h1tH1,t,σ is a Nash equilibrium and Pσ(h1)>0}. We also consider the infimum of all Nash equilibrium payoffs of player 1 at the start of the game, following the initial history h1,1=.v1(μ,δ,ρ)=inf{π1,σ[] s.t. σ is a Nash equilibrium}.

Clearly, v1(μ,δ,ρ)v(μ,δ,ρ). Our main Theorem below offers bounds on

Proofs

The main idea of the proofs of both parts of Theorem 1 follow the classical argument of [11], [12]. Assume that at every stage player 1 follows the strategy corresponding to some commitment type ωˆ. The sequence of players 2 should eventually predict more and more accurately the distribution of signals induced by player 1ʼs actions; hence, each player 2 plays a best-response to a strategy of player 1 which is “not too far” from ωˆ. This provides a lower bound on player 1ʼs payoff while he plays

Concluding comments

Although the idea that impermanent types may restore reputation effects permanently is not entirely new, our paper is the first to show that this is true without imposing any assumptions on the stage game or without restricting the class of equilibrium strategies. Our main Theorem provides bounds on the equilibrium payoffs of the long-run player that hold uniformly after any history on the equilibrium path. We now briefly discuss upper bounds on equilibrium payoffs, continuation payoffs after

References (29)

  • H. Bar-Isaac et al.

    Seller reputation

    Found. Trends Microeconomics

    (2008)
  • D. Blackwell et al.

    Merging of opinions with increasing information

    Ann. Math. Statist.

    (1962)
  • H. Cole et al.

    Default, settlement, and signalling: Lending resumption in a reputational model of sovereign debt

    Int. Econ. Rev.

    (1995)
  • T.M. Cover et al.

    Elements of Information Theory

    (1991)
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    We are grateful to Umberto Garfagnini and Nuh Aygun Dalkiran for excellent research assistance. We also thank Alp Atakan and Satoru Takahashi for very helpful comments. Part of this research was conducted while Mehmet Ekmekci was visiting Cowles Foundation and Economics Department at Yale University.

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