This paper studies repeated games with imperfect public monitoring where the players are uncertain both about the payoff functions and about the relationship between the distribution of signals and the actions played. We introduce the concept of perfect public ex post equilibrium (PPXE), and show that it can be characterized with an extension of the techniques used to study perfect public equilibria. We develop identifiability conditions that are sufficient for a folk theorem; these conditions imply that there are PPXE in which the payoffs are approximately the same as if the monitoring structure and payoff functions were known. Finally, we define perfect type-contingently public ex post equilibria (PTXE), which allows players to condition their actions on their initial private information, and we provide its linear programming characterization.
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Ve consider the problem of designing a contract between a risk-averse agent and a risk-neutral principal when the agent's action is subject to moral hazard and the principal is free to propose a new contract after the agent has chosen his effort level but before the corresponding outcome is revealed.In this setting any optimal contract is equivalent to one that is "renegotiation-proof." A renegotiation-proof contract that induces the agent to choose high effort levels by promising a higher payment following good outcomes must also induce the agent to choose lower effort levels with sufficiently high probability that the contract would not be renegotiated.We show that for a range of utility functions for the agent, including exponential and logarithmic forms, the cost-minimizing renegotiation-proof contract for a given distribution of efforts is the same as the cost-minimizing contract for that distribution under commitment.Thus, the force of the renegotiation-proof constraint is not to change the way that given distributions are implemented, but rather to change which distributions are feasible.However, if the agent has constant relative risk aversion lower than one, the principal may prefer to give the agent an ex-ante rent in order to relax the renegotiation-proofness constraint, so that the optimal contract may differ from, that under commitment not only in the choice of distribution but also in the way that distribution is implemented.Our theory may shed some light on why compensation of managers and contractors is frequently insensitive to the information obtained after the relationship is terminated, and why executives have considerable discretion to adjust the riskiness of their compensation.1.
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 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. THAT STRATEGIC RIVALRY in a long-term relationship may differ from that of a one-shot game is by now quite a familiar idea. Repeated play allows players to respond to each other's actions, and so each player must consider the reactions of his opponents in making his decision. The fear of retaliation may thus lead to outcomes that otherwise would not occur. The most dramatic expression of this phenomenon is the celebrated for repeated games. An outcome that Pareto dominates the minimax point is called individually rational. The Folk Theorem asserts that any individually rational outcome can arise as a equilibrium in infinitely repeated games with sufficiently little discounting. As Aumann and Shapley [3] and Rubinstein [20] have shown, the same result is true when we replace the word Nash by (subgame) perfect and assume no discounting at all. Because the Aumann-Shapley/Rubinstein result supposes literally no discounting, one may wonder whether the exact counterpart of the Folk Theorem holds for equilibrium, i.e., whether as the discount factor tends to one, the set of equilibrium outcomes converges to the individually rational set. After all, agents in most games of economic interest are not completely patient; the no discounting case is of interest as an approximation. It turns out that this counterpart is false. There can be a discontinuity (formally, a failure of lower hemicontinuity) where the discount factor, 8, equals one, as we show in Example 3. Nonetheless the games in which discontinuities occur are quite degenerate, and, in the end, we can give a qualified yes (Theorem 2) to the question of whether the Folk Theorem holds with discounting. In particular, it always holds in two-player games (Theorem 1). This last result contrasts with the recent work of Radner-Myerson-Maskin [18] showing that, even in two-player games, the equilibrium set may not be continuous at 8 = 1 in
We develop a duopoly model in which exit occurs because of the existence of fixed costs or opportunity costs. Each firm enters the market knowing its own cost, but not that of its opponent. As times goes on, each firm becomes increasingly pessimistic about the cost of its remaining rival. The time of exit is the only strategic variable, so that our model is a of attrition. In contrast to the classic war of attrition, however, we assume that with positive probability each firm's costs may be low enough that staying in forever is a dominant strategy. Thus our model, unlike the classic one, has a unique equilibrium.
We characterize a generalization of discounted logistic choice that incorporates a parameter to capture different views the agent might have about the costs and benefits of larger choice sets. The discounted logit model used in the empirical literature is the special case that displays a “preference for flexibility” in the sense that the agent always prefers to add additional items to a menu. Other cases display varying levels of “choice aversion,” where the agent prefers to remove items from a menu if their ex ante value is below a threshold. We show that higher choice aversion, as measured by dislike of bigger menus, also corresponds to an increased preference for putting off decisions as late as possible.
The standard dual-self model of self-control, with a shorter-run self who cares only about the current period, is excessively sensitive to the timing of decisions and to the interpolation of additional "no-action" time periods in between the dates when decisions are made.We show that when the shorter-run self is not completely myopic, this excess sensitivity goes away.To accommodate the combination of short time periods and convex costs of self-control, we introduce a cognitive resource variable that tracks how the control cost depends on the self-control that has been used in the recent past.We consider models with both linear and convex control costs, illustrating the theory through a series of examples.We examine when opportunities to consume will be avoided or delayed, and we consider the way in which the marginal interest declines with delay.
We study the steady states of a system in which players learn about the strategies their opponents are playing by updating their Bayesian priors in light of their observations.Players are matched at random to play a fixed extensive -form game, and each player observes the realized actions in his own matches, but not the intended off -path play of his opponents or the realized actions in other matches.If lifetimes are long and players are very patient, the steady state distribution of actions approximates those of a Nash equilibrium.
Self-confirming equilibrium differs from Nash equilibrium in allowing players to have incorrect beliefs about how their opponents would play off of the equilibrium path.We provide several examples of ways that self -confirming and Nash equilibria differ.In games with "identified deviators , " all self -confirming equilibrium outcomes can be generated by extensive -form correlated equilibria.In two-player games, self -confirming equilibria with "unitary beliefs" are Nash.1.