Perfect Equilibrium Payoffs Research Articles

Persistence in a dynamic moral hazard game

This paper explores how the persistence of past choices creates incentives in a continuous time stochastic game involving a large player (e.g., a firm) and a sequence of small players (e.g., customers). The large player faces moral hazard and her actions are distorted by a Brownian motion. Persistence refers to how actions impact a payoff‐relevant state variable (e.g., product quality depends on past investment). I characterize actions and payoffs in Markov perfect equilibria (MPE) for a fixed discount rate, show that the perfect public equilibrium (PPE) payoff set is the convex hull of the MPE payoff set, and derive sufficient conditions for a MPE to be the unique PPE. Persistence can serve as an effective channel for intertemporal incentives in a setting where traditional channels fail. Applications to persistent product quality and policy targeting demonstrate the impact of persistence on equilibrium behavior.

Stochastic Control Approach to Reputation Games

Through a stochastic-control-theoretic approach, we analyze reputation games, where a strategic long-lived player acts in a sequential repeated game against a collection of short-lived players. The key assumption in our model is that the information of the short-lived players is nested in that of the long-lived player. This nested information structure is obtained through an appropriate monitoring structure. Under this monitoring structure, we show that, given mild assumptions, the set of perfect Bayesian equilibrium payoffs coincides with Markov perfect equilibrium payoffs, and hence, a dynamic programming formulation can be obtained for the computation of equilibrium strategies of the strategic long-lived player in the discounted setup. We also consider the undiscounted average-payoff setup, where we obtain an optimal equilibrium strategy of the strategic long-lived player under further technical conditions. We then use this optimal strategy in the undiscounted setup as a tool to obtain a tight upper payoff bound for the arbitrarily patient long-lived player in the discounted setup. Finally, by using measure concentration techniques, we obtain a refined lower payoff bound on the value of reputation in the discounted setup. We also study the continuity of equilibrium payoffs in the prior beliefs.

A complete folk theorem for finitely repeated games

This paper analyzes the set of pure strategy subgame perfect Nash equilibria of any finitely repeated game with complete information and perfect monitoring. The main result is a complete characterization of the limit set, as the time horizon increases, of the set of pure strategy subgame perfect Nash equilibrium payoff vectors of the finitely repeated game. This model includes the special case of observable mixed strategies.

On the Limit Equilibrium Payoff Set in Repeated and Stochastic Games

This paper provides a dual characterization of the limit set of perfect public equilibrium payoffs in stochastic games (in particular, repeated games) as the discount factor tends to one. As a first corollary, the folk theorems of Fudenberg, Levine and Maskin (1994), Kandori and Matsushima (1998) and Horner, Sugaya, Takahashi and Vieille (2011) obtain. As a second corollary, in the context of repeated games, it follows that this limit set of payoffs is a polytope (a bounded polyhedron) when attention is restricted to equilibria in pure strategies. We provide a two-player game in which this limit set is not a polytope when mixed strategies are considered.

Implementation of Nash bargaining solutions with non-convexity

Nash solutions for two-player bargaining problems with non-convexity are shown to be dictatorial selections of Nash product maximizers in recent literature. In this paper we show that these solutions are implementable as unique subgame perfect equilibrium payoff allocations of a sequential game.

Comparison of information structures in stochastic games with imperfect public monitoring

This paper studies the impact of an improvement of information structure upon the perfect public equilibrium payoff set in discounted stochastic games with imperfect public monitoring. We first suggest three partial orders on information structures in stochastic games. Although each of them reduces to the notion of garbling in repeated games (Kandori in Rev Econ Stud 59:581–593, 1992), we find that an improvement of information in terms of our two garbling notions does not imply an expansion of the equilibrium payoff set for some games. We also show that more informativeness in terms of our third notion of garbling is sufficient for the expansion, thereby extending the well-known monotonicity result in Kandori (1992) to stochastic games.

Subgame perfect equilibria in majoritarian bargaining

We study the division of a surplus under majoritarian bargaining in the three-person case. In a stationary equilibrium as derived by Baron and Ferejohn (1989), the proposer offers one third times the discount factor of the surplus to a second player and allocates no payoff to the third player, a proposal which is accepted without delay. Laboratory experiments show various deviations from this equilibrium, where different offers are typically made and delay may occur before acceptance. We address the issue to what extent these findings are compatible with subgame perfect equilibrium and characterize the set of subgame perfect equilibrium payoffs for any value of the discount factor. We show that for any proposal in the interior of the space of possible agreements there exists a discount factor such that the proposal is made and accepted. We characterize the values of the discount factor for which equilibria with one-period delay exist. We show that any amount of equilibrium delay is possible and we construct subgame perfect equilibria such that arbitrary long delay occurs with probability one.

Discounted stochastic games with voluntary transfers

This paper studies discounted stochastic games with perfect or imperfect public monitoring and the opportunity to conduct voluntary monetary transfers and possibly burn money. This generalization of repeated games with transfers is ideally suited to study relational contracting in applications with long-term investments and also allows to study collusive industry dynamics. We show that for all discount factors every perfect public equilibrium payoff can be implemented with a class of simple equilibria that have a stationary structure on the equilibrium path and optimal penal codes with a stick-and-carrot structure. We develop an algorithm for perfect monitoring to compute the set of equilibrium payoffs and find simple equilibria that implement these payoffs.

Subgame perfect equilibrium in a bargaining model with deterministic procedures

Two players, A and B, bargain to divide a perfectly divisible pie. In a bargaining model with constant discount factors, $$\delta _A$$ and $$\delta _B$$ , we extend Rubinstein (Econometrica 50:97–110, 1982)’s alternating offers procedure to more general deterministic procedures, so that any player in any period can be the proposer. We show that each bargaining game with a deterministic procedure has a unique subgame perfect equilibrium (SPE) payoff outcome, which is efficient. Conversely, each efficient division of the pie can be supported as an SPE outcome by some procedure if $$\delta _A+\delta _B\ge 1$$ , while almost no division can ever be supported in SPE if $$\delta _A+\delta _B < 1$$ .

A strategic implementation of the sequential equal surplus division rule for digraph cooperative games

We provide a strategic implementation of the sequential equal surplus division rule (Beal et al. in Theory Decis 79:251–283, 2015). Precisely, we design a non-cooperative mechanism of which the unique subgame perfect equilibrium payoffs correspond to the sequential equal surplus division outcome of a superadditive rooted tree TU-game. This mechanism borrowed from the bidding mechanism designed by Perez-Castrillo and Wettstein (J Econ Theory 100:274–294, 2001), but takes into account the direction of the edges connecting any two players in the rooted tree, which reflects some dominance relation between them. Our proofs rely on interesting properties that we provide for a general class of bidding mechanisms.

Uniqueness of stationary equilibrium payoffs in the Baron–Ferejohn model with risk‐averse players

AbstractI study a multilateral sequential bargaining model among risk‐averse players in which the players may differ in their probability of being selected as the proposer and the rate at which they discount future payoffs. For games in which agreement requires less than unanimous consent, I characterize the set of stationary subgame perfect equilibrium payoffs. With this characterization, I establish the uniqueness of the equilibrium payoffs. For the case where the players have the same discount factor, I show that the payoff to a player is non‐decreasing in his probability of being selected as the proposer. For the case where the players have the same probability of being selected as the proposer, I show that the payoff to a player is non‐decreasing in his discount factor. This generalizes earlier work by allowing the players to be risk averse.

Subgame perfect equilibria in discounted stochastic games

This paper considers policies and payoffs corresponding to subgame perfect equilibrium strategies in discounted stochastic games with finitely many states. It is shown that a policy is induced by an equilibrium strategy if and only if it can be supported with the threat of reverting to the induced policy that gives the least equilibrium payoff for the deviator. It follows that the correspondence of subgame perfect equilibrium payoffs is the largest fixed-point of a correspondence-valued operator defined by the players' incentive compatibility conditions. Moreover, the fixed-point iteration converges to the equilibrium payoff correspondence.

Characterizing the limit set of perfect and public equilibrium payoffs with unequal discounting

We study repeated games with imperfect public monitoring and unequal dis- counting. We characterize the limit set of perfect and public equilibrium payoffs as discount factors converge to 1 with the relative patience between players fixed. We show that the pairwise and individual full rank conditions are sufficient for the folk theorem. In this paper, we characterize the equilibrium payoffs in repeated games with imperfect public monitoring and unequal discounting as discount factors converge to 1 with rel- ative patience fixed. In particular, we show that the pairwise and individual full rank conditions are sufficient for the folk theorem. Lehrer and Pauzner (1999) (henceforth LP) analyze two-player repeated games with perfect monitoring and unequal discounting. They define the set of feasible and sequen- tially individually rational (henceforth SIR) payoffs and show that in two-player games with perfect monitoring, the limit set of subgame perfect equilibrium payoffs coincides with that of SIR payoffs as discount factors converges to 1 with the relative patience fixed (the folk theorem). Recently, Chen and Takahashi (2012) extend the result to n-player games with perfect monitoring. This paper extends their results to imperfect public monitoring. While the proofs of both Lehrer and Pauzner (1999 )a ndChen and Takahashi (2012) are constructive, we em- ploy a nonconstructive approach using the recursive structure of the perfect and public equilibrium (henceforth PPE). Specifically, we attain a characterization of the set of PPE payoffs as discount factors converge to1. In addition, we characterize SIR payoffs. Given these characterizations, we show that if the pairwise and individual full rank conditions are satisfied, these two sets coincide, that is, the folk theorem holds.

Bargaining in Standing Committees with an Endogenous Default: Figure 1

Committee voting has mostly been investigated from the perspective of the standard Baron–Ferejohn model of bargaining over the division of a pie, in which bargaining ends as soon as the committee reaches an agreement. In standing committees, however, existing agreements can be amended. This article studies an extension of the Baron–Ferejohn framework to a model with an evolving default that reflects this important feature of policymaking in standing committees: In each of an infinite number of periods, the ongoing default can be amended to a new policy (which is, in turn, the default for the next period). The model provides a number of quite different predictions. (i) From a positive perspective, the key distinction turns on whether the quota is less than unanimity. In that case, patient enough players waste substantial shares of the pie each period and the size principle fails in some pure strategy Markov perfect equilibria. In contrast, the unique Markov perfect equilibrium payoffs in a unanimity committee coincide with those in the corresponding Baron–Ferejohn framework. (ii) If players have heterogeneous discount factors then a large class of subgame perfect equilibria (including all Markov perfect equilibria) are inefficient.

Repeated Sequential Prisoner's Dilemma: The Stackleberg Variant

We study the Stackleberg variant of the repeated Sequential Prisoner's Dilemma (SPD). The game goes in two stages, and the two players, the leader and the follower, are asymmetric in both stages. In the first stage of the game, the leader chooses a strategy (for the repeated SPD of the second stage), which is immediately known to the follower. In the second stage, they play repeated SPD: In each round the follower moves after observing the leader's action. Assuming complete rationality, we find some extraordinary properties of this model. (i) The (subgame perfect) equilibrium payoff profile is unique, which lies on the corner of the region predicted by classical folk theorems: It is best for the leader and at the same time worst for the follower, (ii) the leader has simple optimal strategies that are one-step memory and stationary. These features are in great contrast with classical results, where either uniqueness cannot be guaranteed and equilibrium strategies are often quite complicated, or bounded rationality is required. Although full cooperation, i.e., the outcome is always (cooperate, cooperate), is not attainable in our model, at least a half of the optimal social welfare can be guaranteed. We also do a non-equilibrium analysis which makes the usual equilibrium analysis more convincing.

Reputation in the long-run with imperfect monitoring

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.

Public Information in Markov Games

In a Markov game, players engage in a sequence of games determined by a Markov process. In this setting, this paper investigates the impact of varying the informativeness of public information, as defined by Blackwell 1951 and 1953, pertaining to the games that will be played in future periods. In brief, when a curvature condition on payoffs is satisfied, the finding is that, for any fixed discount factor, the set of strongly symmetric subgame perfect equilibrium payoffs of a Markov game with more informative signals is contained in this set of equilibrium payoffs if the Markov game is played with any less informative signals. The second result shows that larger equilibrium payoffs are possible with less informative signals when the curvature condition fails, but only for some discount factors. The third result strengthens the curvature condition, but generalizes the first result to all subgame perfect equilibrium payoffs. Finally, a collusion application is presented to illustrate the curvature condition.

Public information in Markov games

In a Markov game, players engage in a sequence of games determined by a Markov process. In this setting, this paper investigates the impact of varying the informativeness of public information, as defined by Blackwell [8,9], pertaining to the games that will be played in future periods. In brief, when a curvature condition on payoffs is satisfied, the finding is that, for any fixed discount factor, the set of strongly symmetric subgame perfect equilibrium payoffs of a Markov game with more informative signals is contained in this set of equilibrium payoffs if the Markov game is played with any less informative signals. The second result shows that larger equilibrium payoffs are possible with less informative signals when the curvature condition fails, but only for some discount factors. The third result strengthens the curvature condition, but generalizes the first result to all subgame perfect equilibrium payoffs. Finally, a collusion application is presented to illustrate the curvature condition.

Justifiable punishments in repeated games

In repeated games, subgame perfection requires all continuation strategy profiles must be effective to enforce the equilibrium; they serve as punishments should deviations occur. It does not require whether a punishment can be justified for the deviation, which creates a great deal of freedom in constructing equilibrium strategies, resulting the well-known folk theorem. We introduce justifiable punishments in repeated games. After one player deviates, the corresponding continuation or punishment is justifiable if either the deviation is bad to the other player or the continuation itself is good to the other player. We characterize the set of payoff vectors that can be supported by subgame perfect equilibria with justifiable punishments, as the discount factor goes to one. This limiting set of equilibrium payoffs can be quite different from the set of subgame perfect equilibrium payoffs. Any efficient, feasible, and strictly individually rational payoff can be supported by equilibrium with justifiable punishments.

On the limit perfect public equilibrium payoff set in repeated and stochastic games

This paper provides a dual characterization of the existing ones for the limit set of perfect public equilibrium payoffs in a class of finite stochastic games (in particular, repeated games) as the discount factor tends to one. As a first corollary, the folk theorems of Fudenberg et al. (1994), Kandori and Matsushima (1998) and Hörner et al. (2011) obtain. As a second corollary, it is shown that this limit set of payoffs is a convex polytope when attention is restricted to perfect public equilibria in pure strategies. This result fails for mixed strategies, even when attention is restricted to two-player repeated games.