Set Of Payoffs Research Articles

Repeated games with incomplete information and discounting

We analyze discounted repeated games with incomplete information, such that the players' payoffs depend only on their own type (known-own payoff case). We describe an algorithm for finding all equilibrium payoffs in games for which there exists an open set of belief-free equilibria of Hörner and Lovo (2009). This includes generic games with one-sided incomplete information and a large and important class of games with multisided incomplete information. When players become sufficiently patient, all Bayesian Nash equilibrium payoffs can be approximated by payoffs in sequential equilibria in which information is revealed finitely many times. The set of equilibrium payoffs is typically larger than the set of equilibrium payoffs in repeated games without discounting and is larger than the set of payoffs obtained in belief-free equilibria. The results are illustrated in bargaining and oligopoly examples.

Online concealed correlation and bounded rationality

Correlation of players' actions may evolve in the common course of the play of a repeated game with perfect monitoring (“online correlation”). In this paper we study the concealment of such correlation from a boundedly rational player. We show that “strong” players, i.e., players whose strategic complexity is less stringently bounded, can orchestrate the online correlation of the actions of “weak” players, where this correlation is concealed from an opponent of “intermediate” strength. The feasibility of such “online concealed correlation” is reflected in the individually rational payoff of the opponent and in the equilibrium payoffs of the repeated game.This result enables the derivation of a folk theorem that characterizes the set of equilibrium payoffs in a class of repeated games with boundedly rational players and a mechanism designer who sends public signals.The result is illustrated in two models, bounded recall strategies and finite automata.

Strongly Symmetric Equilibria in Bandit Games

This paper studies strongly symmetric equilibria (SSE) in continuous-time games of strategic experimentation with Poisson bandits. SSE payoffs can be studied via two functional equations similar to the HJB equation used for Markov equilibria. This is valuable for three reasons. First, these equations retain the tractability of Markov equilibrium, while allowing for punishments and rewards: the best and worst equilibrium payoff are explicitly solved for. Second, they capture behavior of the discrete-time game: as the period length goes to zero in the discretized game, the SSE payoff set converges to their solution. Third, they encompass a large payoff set: there is no perfect Bayesian equilibrium in the discrete-time game with frequent interactions with higher asymptotic efficiency.

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.

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.

Cooperative and non-cooperative Nash solution for linear supply function equilibrium game

In this paper, the competition among suppliers in an oligopolistic market is studied using supply function equilibrium model. Since the decision of each supplier affects other suppliers’ profit, we have a game problem rather than optimization. In the proposed supply function equilibrium game, both of the cooperative and non-cooperative behaviors of the players are studied. In non-cooperative case, the Nash equilibrium point of the game is obtained for n-player in constant and price sensitive demand cases. In cooperative game, the concept of Nash bargaining solution is utilized to study the collusive behavior of the suppliers. The convexity and closeness of the players’ payoff set in supply function equilibrium model is proved as the necessary and sufficient condition for existence and uniqueness of the solution and then, the mathematical formulation of Nash bargaining solution is derived. In addition, the cooperative and non-cooperative behavior of players is compared over a wide range of parameters in a case study.

Stability of marketable payoffs with long-term assets

We consider a stochastic financial exchange economy with a finite date-event tree representing time and uncertainty and a financial structure with possibly long-term assets. We exhibit a sufficient condition under which the set of marketable payoffs depends continuously on the arbitrage free asset prices. This generalizes previous results of Angeloni–Cornet and Magill–Quinzii involving only short-term assets. We also show that, under the same condition, the useless portfolios do not depend on the arbitrage free asset prices. We then provide an existence result of financial equilibrium for long term nominal assets for any given state prices with assumptions only on the fundamental datas of the economy.

An algorithm for two-player repeated games with perfect monitoring

Consider repeated two-player games with perfect monitoring and discounting. We provide an algorithm that computes the set V ∗ of payoff pairs of all pure-strategy subgame perfect equilibria with public randomization. The algorithm provides significant efficiency gains over the existing implementations of the algorithm from Abreu, Pearce and Stacchetti (1990). These efficiency gains arise from a better understanding of the manner in which extreme points of the equilibrium payoff set are generated. An important theoretical implication of our algorithm is that the set of extreme points E of V ∗ is finite. Indeed, |E |≤ 3|A|, where A is the set of action profiles of the stage game.

Does Money Illusion Matter?: Comment

This paper experimentally investigates whether money illusion generates substantial nominal inertia. Building on the design of Fehr and Tyran (2001), we find no evidence that agents choose high nominal payoffs over high real payoffs. However, participants do select prices associated with high nominal payoffs within a set of maximum real payoffs as a heuristic to simplify their decision task. The cognitive challenge of this task explains the majority of the magnitude of nominal inertia; money illusion exerts only a second-order effect. The duration of nominal inertia depends primarily on participants' best response functions, not the prevalence of money illusion. (JEL C91, D21, D83, E31, E41, E52, L11)

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.

Risk-neutral equilibria of noncooperative games

Game-theoretic solution concepts such as Nash and Bayesian equilibrium start from an assumption that the players’ sets of possible payoffs, measured in units of von Neumann–Morgenstern utility, are common knowledge, and they go on to define rational behavior in terms of equilibrium strategy profiles that are either pure or independently randomized and which, in applications, are often taken to be uniquely determined or at least tightly constrained. A mechanism through which to obtain a common knowledge of payoff functions measured in units of utility (or common priors over predetermined sets of such functions) is not part of the model. This paper describes an operational method of constructing a state of common knowledge of the key parameters of the players’ utility functions in terms of conditional small bets on the game’s outcome. When the rationality criterion of joint coherence (no arbitrage) is applied in this setting, the solution of a game is typically characterized by a convex set of correlated equilibria. In the most general case, where players are risk averse, the parameters of the equilibria are risk-neutral probabilities, interpretable as products of subjective probabilities and relative marginal utilities for money, as in financial markets. Risk aversion generally enlarges the set of equilibria and may present opportunities for Pareto-improving modifications of the rules of the game.

Bayesian Repeated Games and Reputation

Under appropriate assumptions (private values and uniform punishments), the Nash equilibria of a Bayesian repeated game without discounting are payoff-equivalent to tractable, completely revealing, equilibria and can be achieved as interim cooperative solutions of the initial Bayesian game. This characterization does not apply to discounted games with patient players. In a class of public good games, the set of Nash equilibrium payoffs of the undiscounted game can be empty, while limit (perfect Bayesian) Nash equilibrium payoffs of the discounted game, as players become infinitely patient, do exist. These equilibria share some features with the ones of multi-sided reputation models.

Stochastic Games in Continuous Time: Persistent Actions in Long-Run Relationships, Second Version

This paper studies a class of continuous-time stochastic games in which the actions of a long-run player have a persistent effect on payoffs. For example, the quality of a firm's product depends on past as well as current effort, or the level of a policy instrument depends on a government's past choices. The long-run player faces a population of small players, and its actions are imperfectly observed. I establish the existence of Markov equilibria, characterize the Perfect Public Equilibria (PPE) pay-offset as the convex hull of the Markov Equilibria payoff set, and identify conditions for the uniqueness of a Markov equilibrium in the class of all PPE. The existence proof is constructive: it characterizes the explicit form of Markov equilibria payoffs and actions, for any discount rate. Action persistence creates a crucial new channel to generate intertemporal incentives in a setting where traditional channels fail, and can provide permanent non-trivial incentives in many settings. These results offer a novel framework for thinking about reputational dynamics of firms, governments, and other long-run agents.

The Renegotiation-Proofness Principle and Costly Renegotiation

We study contracting and costly renegotiation in settings of complete, but unverifiable information, using the mechanism-design approach. We show how renegotiation activity is best modeled in the fundamentals of the mechanism-design framework, so that noncontractibility of renegotiation amounts to a constraint on the problem. We formalize and clarify the Renegotiation-Proofness Principle (RPP), which states that any state-contingent payoff vector that is implementable in an environment with renegotiation can also be implemented by a mechanism in which renegotiation does not occur in equilibrium. We observe that the RPP is not valid in some settings. However, we prove a general monotonicity result that confirms the RPP’s message about renegotiation opportunities having negative consequences. Our monotonicity theorem states that, as the costs of renegotiation increase, the set of implementable state-contingent payoffs becomes larger.

A primal condition for approachability with partial monitoring

In approachability with full monitoring there are two types of conditions that are known to be equivalent for convex sets: a primal and a dual condition. The primal one is of the form: a set $\mathcal{C}$ is approachable if and only all containing half-spaces are approachable in the one-shot game. The dual condition is of the form: a convex set $\mathcal{C}$ is approachable if and only if it intersects all payoff sets of a certain form. We consider approachability in games with partial monitoring. In previous works [5，7] we provided a dual characterization of approachable convex sets and we also exhibited efficient strategies in the case where $\mathcal{C}$ is a polytope. In this paper we provide primal conditions on a convex set to be approachable with partial monitoring. They depend on a modified reward function and lead to approachability strategies based on modified payoff functions and that proceed by projections similarly to Blackwell's (1956) strategy. This is in contrast with previously studied strategies in this context that relied mostly on the signaling structure and aimed at estimating well the distributions of the signals received. Our results generalize classical results by Kohlberg [3] (see also [6]) and apply to games with arbitrary signaling structure as well as to arbitrary convex sets.

First Price Package Auction with Many Traders

This paper studies a first price package auction with many sellers in addition to many buyers in a decentralized networked market. We first show that the set of equilibrium payoffs with profit-target strategies is equal to a bidder optimal core. We then show that for buyers, the equilibrium payoff vector is Pareto-dominated by the VCG payoff vector, but Pareto-dominates theWalrasian competitive equilibrium payoff vector. This contrasts with a package auction with a single seller, in which it is shown that the unique equilibrium payoff vector is equal to the VCG payoff vector supported by a Walrasian pricing if goods are substitute. Finally, we discuss coalition-proofness of equilibria and stability of networks.

Dynamic sender–receiver games

Abstract 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.

The Role of Commitment in Bilateral Trade

This paper solves for the set of equilibrium payoffs in bargaining with interdependent values when the informed party makes all offers, as discounting vanishes. The seller of a good is informed of its quality, which affects both his cost and the buyer's valuation, but the buyer is not. To characterize this payoff set, we derive an upper bound, using mechanism design with limited commitment. We then prove that this upper bound is tight, by showing that all its extreme points are equilibrium payoffs. Our results shed light on the role of different forms of commitment on the bargaining process. In particular, we show that it is the buyer's inability to commit to a contract before observing the terms of trade that precludes efficiency.

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.

The All-Pay Auction with Complete Information and Identity-Dependent Externalities

We derive a necessary and sufficient condition for the existence of equilibria with only two active players in the all-pay auction with complete information and identity-dependent externalities. This condition shows that the generic equilibrium of the standard all-pay auction is robust to the introduction of small identity-dependent externalities. In general, however, the presence of identity-dependent externalities invalidates well-established qualitative results concerning the set of equilibria of the first-price all-pay auction with complete information. With identity-dependent externalities equilibria are generally not payoff equivalent, and identical players may earn different payoffs in equilibrium. These observations show that Siegel’s (2009) results characterizing the set of equilibrium payoffs in all-pay contests, including the all-pay auction as a special case, do not extend to environments with identity-dependent externalities. We further compare the all-pay auction with identity-dependent externalities to the first-price winner-pay auction with identity-dependent externalities. We demonstrate that the equilibrium payoffs of the all-pay auction and winner-pay auction cannot be ranked unambiguously in the presence of identity-dependent externalities by providing examples of environments where equilibrium payoffs in the all-pay auction dominate those in the winner-pay auction and vice versa.