Feasible Payoffs Research Articles

Repeated Games with Intervention: Theory and Applications in Communications

In communication systems where users share common resources, selfish behavior usually results in suboptimal resource utilization. There have been extensive works that model communication systems with selfish users as one-shot games and propose incentive schemes to achieve Pareto-optimal outcomes. However, in many communication systems, due to strong negative externalities among users, the sets of feasible payoffs in one-shot games are nonconvex. Thus, it is possible to expand the set of feasible payoffs by having users choose different action profiles in an alternating manner. In this paper, we formulate a model of repeated games with intervention. First, by using repeated games we can convexify the set of feasible payoffs in one-shot games. Second, by using intervention in repeated games we can achieve a larger set of equilibrium payoffs and loosen requirements for users' patience to achieve a target payoff. We study the problem of maximizing a welfare function defined on users' payoffs. We characterize the limit set of equilibrium payoffs. Given the optimal equilibrium payoff, we derive the sufficient condition on the discount factor and the intervention capability to achieve it, and design corresponding equilibrium strategies. We illustrate our analytical results with power control and flow control.

Symmetric Play in Repeated Allocation Games

We study symmetric play in a class of repeated games when players are patient. We show that, while the use of symmetric strategy profiles essentially does not restrict the set of feasible payoffs, the set of equilibrium payoffs is an interesting proper subset of the feasible and individually rational set. We also provide a theory of how rational individuals play these games, identifying particular strategies as focal through the considerations of Pareto optimality and simplicity. We report experiments that support many aspects of this theory.

Dynamic Sender-Receiver Games

We consider a dynamic version of sender-receiver games, where the sequence of states follows an irreducible Markov chain observed by the sender. Under mild assumptions, we provide a simple characterization of the limit set of equilibrium payoffs, as players become very patient. Under these assumptions, the limit set depends on the Markov chain only through its invariant measure. The (limit) equilibrium payoffs are the feasible payoffs that satisfy an individual rationality condition for the receiver, and an incentive compatibility condition for the sender.

Symmetric Play in Repeated Allocation Games

We study symmetric play in a class of repeated games when players are patient. We show that, while the use of symmetric strategy profiles essentially does not restrict the set of feasible payoffs, the set of equilibrium payoffs is an interesting proper subset of the feasible and individually rational set. We also provide a theory of how rational individuals play these games, identifying particular strategies as focal through the considerations of Pareto optimality and simplicity. We report experiments that support many aspects of this theory. JEL Code: C73, C92, D63

Belief-free equilibria in games with incomplete information: Characterization and existence

Abstract We generalize the results of Horner and Lovo (2009) [15] to N-player games with arbitrary information structure. First, we characterize the set of belief-free equilibrium payoffs under low discounting as the set of feasible payoffs that are individually rational, jointly rational, and incentive compatible. Second, we provide necessary and sufficient conditions on the information structure for this set to be non-empty.

Stochastic Stability in Finitely Repeated Two Player Games

We apply stochastic stability to undiscounted finitely repeated two player games without common interests. We prove an Evolutionary Feasibility Theorem as an analog to the Folk Theorem (Benoit and Krishna, 1985 and 1987). Specifically, we demonstrate that as repetitions go to infinity, the set of stochastically stable equilibrium payoffs converges to the set of individually rational and feasible payoffs. This derivation requires stronger assumptions than the Nash Folk Theorem (Benoit and Krishna, 1987). It is demonstrated that the stochastically stable equilibria are stable as a set, but unstable as individual equilibria. Consequently, the Evolutionary Feasibility Theorem makes no prediction more specific than the entire individually rational and feasible set.

On the Asymptotic Uniqueness of Bargaining Equilibria

This note reexamines the connection between the asymmetric Nash bargaining solution and the equilibria of strategic bargaining games. Several papers in the literature obtain the asymmetric Nash bargaining solution as the unique limit of subgame perfect equilibria in stationary strategies when the breakdown probability approaches zero. This note illustrates by means of two examples that this result depends crucially on the differentiability of the boundary of the set of feasible payoffs. In the first example the game has a unique stationary subgame perfect equilibrium that fails to converge to the asymmetric Nash bargaining solution. In the second example the game has two stationary subgame perfect equilibria that converge to two distinct limits as the breakdown probability vanishes. This example demonstrates that without differentiability of the set of feasible payoffs there is not even asymptotic uniqueness of stationary equilibria in the bargaining model.

Recursive Methods in Discounted Stochastic Games: An Algorithm forδ→ 1 and a Folk Theorem

We present an algorithm to compute the set of perfect public equilibrium payoffs as the discount factor tends to 1 for stochastic games with observable states and public (but not necessarily perfect) monitoring when the limiting set of (long-run players') equilibrium payoffs is independent of the initial state. This is the case, for instance, if the Markov chain induced by any Markov strategy profile is irreducible. We then provide conditions under which a folk theorem obtains: if in each state the joint distribution over the public signal and next period's state satisfies some rank condition, every feasible payoff vector above the minmax payoff is sustained by a perfect public equilibrium with low discounting.

Recursive Methods in Discounted Stochastic Games: An Algorithm for δ → 1 and a Folk Theorem

We present an algorithm to compute the set of perfect public equilibrium payoffs as the discount factor tends to one for stochastic games with observable states and public (but not necessarily perfect) monitoring when the limiting set of (long-run players’) equilibrium payoffs is independent of the initial state. This is the case, for instance, if the Markov chain induced by any Markov strategy profile is irreducible. We then provide conditions under which a folk theorem obtains: if in each state the joint distribution over the public signal and next period’s state satisfies some rank condition, every feasible payoff vector above the minmax payoff is sustained by a perfect public equilibrium with low discounting.

COLLUSION AND PREDATION UNDER THE CONDITION OF STOCHASTIC BANKRUPTCY

In this paper, on the basis of the framework of repeated games, we consider the strategic interaction between firms that may face financial constraints and go into bankruptcy when they earn low profit. We demonstrate that under asymmetric information concerning the financial constraints of the rival firm, firms adopt a pricing pattern wherein they switch to high (or collusive) prices after maintaining low prices for some periods. There are always such predatory pricing strategies sustained as equilibria when firms are sufficiently patient. We also show that the set of feasible and individual rational payoffs becomes narrower than that of the ordinal repeated games.

On the Asymptotic Uniqueness of Bargaining Equilibria

This note reexamines the connection between the asymmetric Nash bargaining solution and the equilibria of strategic bargaining games. Several papers in the literature obtain the asymmetric Nash bargaining solution as the unique limit of subgame perfect equilibria in stationary strategies when the breakdown probability approaches zero. This note illustrates by means of two examples that this result depends crucially on the differentiability of the boundary of the set of feasible payoffs. In the first example the game has a unique stationary subgame perfect equilibrium that fails to converge to the asymmetric Nash bargaining solution. In the second example the game has two stationary subgame perfect equilibria that converge to two distinct limits as the breakdown probability vanishes. This example demonstrates that without differentiability of the set of feasible payoffs there is not even asymptotic uniqueness of stationary equilibria in the bargaining model.

Bargaining with Non-Convexities

We show that in the canonical non-cooperative multilateral bargaining game, a subgame perfect equilibrium exists in pure stationary strategies, even when the space of feasible payoffs is not convex. At such an equilibrium there is no delay. We also have the converse result that randomization will not be used in this environment in the sense that all stationary subgame perfect equilibria do not involve randomization on the equilibrium path. Nevertheless, mixed strategy profiles can lead to Pareto superior payoffs in non-convex cases.

Non-Cooperative Support for the Asymmetric Nash Bargaining Solution

Our work contributes to the game-theoretic analysis of bargaining by providing additional non-cooperative support to the well-known Nash bargaining solution. In particular, in the present paper we study a model of non-cooperative multilateral bargaining with a very general proposer selection protocol and set of feasible payoffs. In each period of the bargaining game, one out of n players is recognized as the proposer according to an irreducible Markov process. The proposer offers a particular element of the convex set of feasible payoffs. If all players accept the offer, it is implemented. If a player rejects the offer, with some probability the negotiations break down and with the remaining probability the next period starts. We show that subgame perfect equilibria in stationary strategies exist and we fuly characterize the set of such equilibria. Our main result is that in the limit, as the exogenous risk of breakdown goes to zero, stationary subgame perfect equilibrium payoffs converge to the weighted Nash bargaining solution with the stationary distribution of the Markov proposer selection process as the weight vector.

Truthfulness in Coalitions: A Sequential Equilibrium Analysis

The paper integrates the formation of coalitions by a standard sequential offer procedure and the truthfulness - cheating strategies of their members, which support sequential equilibria. We derive the probabilities of truthfulness for players in a two - and three - member coalition in terms of the payoffs. It turns out that in a sequential equilibrium, these probabilities are determined by the tension between two basic incentives: the incentive to cheat and the incentive to defend against cheating. Both incentives are expressed in terms of payoffs. Among the results, we find that if the latter dominates the incentive to cheat, honesty is a pure strategy that supports a sequential equilibrium. Also, feasible payoff patterns are identified in which honesty and cheating do not matter; that is, whether the players are honest or not, the same coalitions emerge in a subgame perfect equilibrium. Other well defined payoff patterns are uncovered in which players' truthfulness is associated with different coalition sizes and structures so that honesty and cheating do matter in this sense.

A “Super” Folk Theorem for dynastic repeated games

We analyze dynastic repeated games. These are repeated games in which the stage game is played by successive generations of finitely-lived players with dynastic preferences. Each individual has preferences that replicate those of the infinitely-lived players of a standard discounted infinitely-repeated game. Individuals live one period and do not observe the history of play that takes place before their birth, but instead create social memory through private messages received from their immediate predecessors. Under mild conditions, when players are sufficiently patient, all feasible payoff vectors (including those below the minmax of the stage game) can be sustained by sequential equilibria of the dynastic repeated game with private communication. In particular, the result applies to any stage game with n ≥ 4 players for which the standard Folk Theorem yields a payoff set with a non-empty interior. We are also able to characterize fully the conditions under which a sequential equilibrium of the dynastic repeated game can yield a payoff vector not sustainable as a subgame perfect equilibrium of the standard repeated game. For this to be the case it must be that the players’ equilibrium beliefs violate a condition that we term “inter-generational agreement.”

On-the-job search and strategic bargaining

This paper studies wage bargaining in a simple economy in which both employed and unemployed workers search for better jobs. The axiomatic Nash bargaining solution and standard strategic bargaining solutions are inapplicable because the set of feasible payoffs is nonconvex. I instead develop a strategic model of wage bargaining between a single worker and firm that is applicable to such an environment. I show that if workers and firms are homogeneous, there are market equilibria with a continuous wage distribution in which identical firms bargain to different wages, each of which is a subgame perfect equilibrium of the bargaining game. If firms are heterogeneous, I characterize market equilibria in which more productive firms pay higher wages. I compare the quantitative predictions of this model with Burdett and Mortensen's [1998. Wage differentials, employer size and unemployment. International Economic Review 39, 257–273.] wage posting model and argue that the bargaining model is theoretically more appealing along important dimensions.

For Better or Forever: Formal versus Informal Enforcement

This article contrasts supporting partnerships through relational contracting and supporting partnerships through formal legal institutions. A large population of players interact in bilateral relationships. Efficiency requires cooperation, but cheating yields a higher short‐term payoff. There is a positive probability that the maximum feasible payoff available to a partnership decreases. Opportunistic behavior makes it impossible to realize the efficient outcome. A legal system can lead to efficient contracting. Without such a system, productive relationships arise in equilibrium if it is costly to initiate new relationships. This type of relational contracting tends to make partnerships last longer than is efficient.

Pareto Damaging Behaviors

This paper reports a rigorous experimental test of Pareto-damaging behaviors. We introduce a new graphical representation of dictator games with step-shaped sets of feasible payoffs to persons self and other on which strongly Pareto efficient allocations involve substantial inequality. The non-convexity and sharp nonlinearity of the Pareto frontier allow us systematically to classify Pareto-damaging allocations: as self-damaging or other-damaging and as inequality-increasing or inequality-decreasing. We find that self and other Pareto-damaging behaviors occur frequently even in circumstances - dictator games - that do not implicate reciprocity or strategic interaction. We also find patterns in this behavior, most notably that behavior that Pareto damages self always reduces inequality whereas behavior that Pareto damages other usually increases inequality.

Finitely repeated games: A generalized Nash folk theorem

This paper characterizes the set of feasible payoffs of finitely repeated games with complete information that can be approximated arbitrarily closely by Nash equilibria.

A folk theorem for minority games

We study a particular case of repeated games with public signals. In the stage game an odd number of players have to choose simultaneously one of two rooms. The players who choose the less crowded room receive a reward of one euro (whence the name “minority game”). The players in the same room do not recognize each other, and between the stages only the current majority room is publicly announced. We show that in the infinitely repeated game any feasible payoff can be achieved as a uniform equilibrium payoff, and as an almost sure equilibrium payoff. In particular we construct an inefficient equilibrium where, with probability one, all players choose the same room at almost all stages. This equilibrium is sustained by punishment phases which use, in an unusual way, the pure actions that were played before the start of the punishment.