A Long-Run Collaboration on Long-Run Games

The following sections are included:

• ACKNOWLEDGMENTS

• CONTENTS

• INTRODUCTION

We show that subgame-perfect equilibria of infinite-horizon games arise as limits, as the horizon grows long and epsilon small, of subgame-perfect epsilon-equilibria of games which are truncated after a finite horizon. A number of applications show that this result provides a useful technique for analyzing the existence and uniqueness of infinite-horizon equilibria. We extend our result to the sequential equilibrium concept.

We provide a necessary and sufficient condition for equilibria of a game to arise as limits of ε-equilibria of games with smaller strategy spaces. As the smaller games are frequently more tractable, our result facilitates the characterization of the set of equilibria.

If players are small, one might expect that optimal reactions to one-player deviations are negligible, so that the open- and closed-loop equilibria are approximately the same. We investigate the circumstances in which this is true.

The following sections are included:

• Introduction

• The Model

• A Game of Averages

• The General Case

• Existence of Finite Games

• References

The philosophy of equilibrium refinements is that the analyst, if he knows things about the structure of the game, can reject some Nash equilibria as unreasonable. The word “know” in the preceding sentence deserves special emphasis. If in a fixed game the analyst can reject a particular equilibrium outcome, but he cannot do so for games arbitrarily “close by,” then he may have second thoughts about rejecting the outcome. We consider several notions of distance between games, and we characterize their implications for the robustness of equilibrium refinements.

We examine games played by a single large player and a large number of opponents who are small, but not anonymous. If the play of the small players is observed with noise, and if the number of actions the large player controls is bounded as the number of small players grows, the equilibrium set converges to that of the game where there is a continuum of small players. This paper extends previous work on the negligibility of small players by dropping the assumption that small players' actions are “anonymous.” That is, we allow each small player's actions to be observed separately, instead of supposing that the small players' actions are only observed through their effect on an aggregate statistic.

A single long-run player plays a simultaneous-move stage game against a sequence of opponents who play only once, but observe all previous play. Let the “Stackelberg strategy” be the pure strategy to which the long-run player would most like to commit himself. If there is positive prior probability that the long-run player will always play the Stackelberg strategy, then his payoff in any Nash equilibrium exceeds a bound that converges to the Stackelberg payoff as his discount factor approaches one. When the stage game is not simultaneous move, this result must be modified to account for the possibility that distinct strategies of the long-run player are observationally equivalent.

This paper studies reputation effects in games with a single long-run player whose choice of stage-game strategy is imperfectly observed by his opponents. We obtain lower and upper bounds on the long-run player's payoff in any Nash equilibrium of the game. If the long-run player's stage-game strategy is statistically identified by the observed outcomes, then for generic payoffs the upper and lower bounds both converge, as the discount factor tends to 1, to the long-run player's Stackelberg payoff, which is the most he could obtain by publicly committing himself to any strategy.

The following sections are included:

• INTRODUCTION

• THE MODEL

• AN IMPATIENT PLAYER 2

• A PATIENT PLAYER 2

• EXAMPLE

• APPENDIX

• REFERENCES

In traditional reputation models, the ability to build a reputation is good for the long-run player. In [Ely, J., Valimaki, J., 2003. Bad reputation. NAJ Econ. 4, 2; http://www.najecon.org/v4.htm. Quart. J. Econ. 118 (2003) 785–814], Ely and Valimaki give an example in which reputation is unambiguously bad. This paper characterizes a class of games in which that insight holds. The key to bad reputation is that participation is optional for the short-run players, and that every action of the long-run player that makes the short-run players want to participate has a chance of being interpreted as a signal that the long-run player is “bad.” We allow a broad set of commitment types, allowing many types, including the “Stackelberg type” used to prove positive results on reputation. Although reputation need not be bad if the probability of the Stackelberg type is too high, the relative probability of the Stackelberg type can be high when all commitment types are unlikely.

When either there are only two players or a “full dimensionality” condition holds, any individually rational payoff vector of a one-shot game of complete information can arise in a perfect equilibrium of the infinitely-repeated game if players are sufficiently patient. In contrast to earlier work, mixed strategies are allowed in determining the individually rational payoffs (even when only realized actions are observable). Any individually rational payoffs of a one-shot game can be approximated by sequential equilibrium payoffs of a long but finite game of incomplete information, where players' payoffs are almost certainly as in the one-shot game.

We study repeated games in which players observe a public outcome that imperfectly signals the actions played. We provide conditions guaranteeing that any feasible, individually rational payoff vector of the stage game can arise as a perfect equilibrium of the repeated game with sufficiently little discounting. The central condition requires that there exist action profiles with the property that, for any two players, no two deviations—one by each player—give rise to the same probability distribution over public outcomes. The results apply to principal-agent, partnership, oligopoly, and mechanismdesign models, and to one-shot games with transferable utilities.

We present a general algorithm for computing the limit, as δ → 1, of the set of payoffs of perfect public equilibria of repeated games with long-run and short-run players, allowing for the possibility that the players' actions are not observable by their opponents. We illustrate the algorithm with two economic examples. In a simple partnership we show how to compute the equilibrium payoffs when the folk theorem fails. In an investment game, we show that two competing capitalists subject to moral hazard may both become worse off if their firms are merged and they split the profits from the merger. Finally, we show that with short-run players each long-run player's highest equilibrium payoff is generally greater when their realized actions are observed.

This paper studies repeated games in which players are imperfectly informed about the uncertain consequences of their opponents stage game actions. We show that if the game is informationally connected, the set of sequential equilibrium payoffs includes the enforceable mutually punishable set, if the intertemporal criterion is (i) the lim inf time average, or (ii) the limit of ε-equilibria with discount factor δ, and ε → 0 as δ → 1. We also explore the link between these two criterion functions.

We show that the use of communications to coordinate equilibria generates a Nash-threats folk theorem in two-player games with “almost public” information. The results generalize to the n-person case. However. the two-person case is more difficult because it is not possible to sustain equilibria by comparing the reports of different players. and using these “third parties” to effectively enforce contracts.

We provide a characterization of the limit set of perfect public equilibrium payoffs of repeated games with imperfect public monitoring as the discount factor goes to one. Our result covers general stage games including those that fail a “full-dimensionality” condition that had been imposed in past work. It also provides a characterization of the limit set when the strategies are restricted in a way that endogenously makes the full-dimensionality condition fail, as in the strongly symmetric equilibrium studied by Abreu [Abreu, D., 1986. Extremal equilibria of oligopolistic supergames. J. Econ. Theory 39, 191–228] and Abreu et al. [Abreu, D., Pearce, D., Stacchetti, E., 1986. Optimal cartel equilibria with imperfect monitoring. J. Econ. Theory 39, 251–269]. Finally, we use our characterization to give a sufficient condition for the exact achievability of first-best outcomes. Equilibria of this type, for which all continuation payoffs lie on the Pareto frontier, have a strong renegotiation-proofness property: regardless of the history, players can never unanimously prefer another equilibrium.

In a repeated game with imperfect public information, the set of equilibria depends on the way that the distribution of public signals varies with the players' actions. Recent research has focused on the case of “frequent monitoring,” where the time interval between periods becomes small. Here we study a simple example of a commitment game with a long-run and short-run player in order to examine different specifications of how the signal distribution depends upon period length. We give a simple criterion for the existence of efficient equilibrium, and show that the efficiency of the equilibria that can be supported depends in an important way on the effect of the player's actions on the variance of the signals, and whether extreme values of the signals are “bad news” of “cheating” behavior, or “good news” of “cooperative” behavior.

The following sections are included:

• ERRATUM