Belief-free Equilibria Research Articles

Achieving Sustainable Cooperation in Generalized Prisoner's Dilemma with Observation Errors

A repeated game is a formal model for analyzing cooperation in long-term relationships, e.g., in the prisoner's dilemma. Although the case where each player observes her opponent's action with some observation errors (imperfect private monitoring) is difficult to analyze, a special type of an equilibrium called belief-free equilibrium is identified to make the analysis in private monitoring tractable. However, existing works using a belief-free equilibrium show that cooperative relations can be sustainable only in ideal situations. We deal with a generic problem that can model both the prisoner's dilemma and the team production problem. We examine a situation with an additional action that is dominated by another action. To our surprise, by adding this seemingly irrelevant action, players can achieve sustainable cooperative relations far beyond the ideal situations. More specifically, we identify a class of strategies called one-shot punishment strategy that can constitute a belief-free equilibrium in a wide range of parameters. Moreover, for a two-player case, the obtained welfare matches a theoretical upper bound.

Instability of belief-free equilibria

Various papers have presented folk theorem results for repeated games with private monitoring that rely on belief-free equilibria. I show that these equilibria are not robust against small perturbations in the behavior of potential opponents. Specifically, I show that essentially none of the belief-free equilibria is evolutionarily stable, and that in generic games none of these equilibria is neutrally stable. Moreover, in a large family of games (which includes many public good games), the belief-free equilibria fail to satisfy even a very mild stability refinement.

Equilibrium Characterization of Repeated Games with Private Monitoring

This paper studies sequential equilibria of repeated games with private monitoring. We recover a recursive structure by means of stochastic filtering. The set of complete, infinite-length outcome paths forms a filtered space, over which the filtrations represent the players' information sets. Finite-period histories are setwise identical to the filtered space, so that players face continuation games strategically identical to the original game. A player's first-order beliefs of her opponents' private histories along with her finite-period histories are sufficient statistics for continuation of play, and the value of continuation is a function of the two. In a pure-strategy sequential equilibrium profile, each player has a partition of her belief manifold. Over each submanifold, value functions, which indicate continuation values, are maximized when the histories and beliefs are consistent, that is, when the beliefs are reachable from initial beliefs through the histories. The mixed-strategy equilibria are the belief-free equilibria in the literature. Private beliefs are irrelevant in action choices and state transitions as all value functions are identical. However, beliefs affect the continuation values. We are able to characterize the pure-strategy equilibrium sets when the actions and signals are finite.

Coordination Failure in Repeated Games with Private Monitoring

Players coordinate continuation play in repeated games with public monitoring. This paper asks the robustness of such equilibrium play with respect to private monitoring perturbations that are ex-ante close to the public-monitoring structure. We show that, in two-player games with full support of public signals, no perfect public equilibrium is robust to private-monitoring perturbations under a regularity condition. This non-robustness result does not apply to belief-free equilibria, which violate the regularity condition. Indeed, we show that, in two-player games with an individual rank condition on public signals, every interior belief-free equilibrium is robust to private-monitoring perturbations. We also argue by means of an example that the non-robustness result is sensitive to the assumption that every private signal must be interpreted as some public signal with probability 1, and not with probability close to 1.

Coordination failure in repeated games with private monitoring

Abstract Players coordinate continuation play in repeated games with public monitoring. We investigate the robustness of such equilibrium behavior with respect to ex-ante small private-monitoring perturbations. We show that with full support of public signals, no perfect public equilibrium is robust if it induces a “regular” 2 × 2 coordination game in the continuation play. This regularity condition is violated in all belief-free equilibria. Indeed, with an individual full rank condition, every interior belief-free equilibrium is robust. We also analyze block belief-free equilibria and point out that the notion of robustness is sensitive to whether we allow for uninterpretable signals.

Belief Free Equilibria

The repeated game literature studies long run/repeated interactions, aiming to understand how repetition may foster cooperation. Conditioning future behavior on past play is crucial in this endeavor. For most situations of interest a given player does not directly observe the actions chosen by other players and must rely on noisy signals he receives about those actions. This is typically incorporated into models by defining a monitoring structure, that is, a collection of probability distributions over the signals each player receives (one distribution for each action profile players may play). Although this is simply meant to capture the fact that players don.t directly observe the actions chosen by others, constructed equilibria often depend on players precisely knowing the distributions, somewhat unrealistic in most problems of interest. This paper aims to show the fragility of belief free equilibrium constructions when one adds shocks to the monitoring structure in repeated games.

Defensive sniping and efficiency in simultaneous hard-close proxy auctions

A well-known myopic bidding strategy fails to support an equilibrium of simultaneous ascending proxy auctions for heterogeneous items when a hard-close rule is in place. This is because, in common with the single-auction case, last minute bidding (sniping) is a best response to naive behaviour. However, a modification to the myopic strategy in which all bidders submit an additional bid in the closing stages of the auction–a practice I call ‘defensive sniping’–is shown to yield an efficient, belief-free equilibrium of such environments. This equilibrium is essentially unique within the class of belief-free, efficient equilibria.

Repeated congestion games with bounded rationality

We consider a repeated congestion game with imperfect monitoring. At each stage, each player chooses to use some facilities and pays a cost that increases with the congestion. Two versions of the model are examined: a public monitoring setting where agents observe the cost of each available facility, and a private monitoring one where players observe only the cost of the facilities they use. A partial folk theorem holds: a Pareto-optimal outcome may result from selfish behavior and be sustained by a belief-free equilibrium of the repeated game. We prove this result assuming that players use strategies of bounded complexity and we estimate the strategic complexity needed to achieve efficiency. It is shown that, under some conditions on the number of players and the structure of the game, this complexity is very small even under private monitoring. The case of network routing games is examined in detail.

Individual Learning and Cooperation in Noisy Repeated Games

We investigate whether two players in a long-run relationship can maintain cooperation when the details of the underlying game are unknown. Specifically, we consider a new class of repeated games with private monitoring, where an unobservable state of the world influences the payoff functions and/or the monitoring structure. Each player privately learns the state over time but cannot observe what the opponent learned. We show that there are robust equilibria in which players eventually obtain payoffs as if the true state were common knowledge and players played a belief-free equilibrium. We also provide explicit equilibrium constructions in various economic examples. Copyright 2014, Oxford University Press.

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.

Weakly Belief-Free Equilibria in Repeated Games With Private Monitoring

Repeated games with imperfect private monitoring have a wide range of applications, but a complete characterization of all equilibria in this class of games has yet to be obtained. The existing literature has identified a relatively tractable subset of equilibria. The present paper introduces the notion of weakly belief-free equilibria for repeated games with imperfect private monitoring. This is a tractable class which subsumes, as a special case, a major part of the existing literature (the belief-free equilibria). It is shown that this class can outperform the equilibria identified by the previous work.

Belief-Free Equilibria in Games with Incomplete Information: Characterization and Existence

We characterize belief-free equilibria in infinitely repeated games with incomplete information with N \ge 2 players and arbitrary information structures. This characterization involves a new type of individual rational constraint linking the lowest equilibrium payoffs across players. The characterization is tight: we define a set of payoffs that contains all the belief-free equilibrium payoffs; conversely, any point in the interior of this set is a belief-free equilibrium payoff vector when players are sufficiently patient. Further, we provide necessary conditions and sufficient conditions on the information structure for this set to be non-empty, both for the case of known-own payoffs, and for arbitrary payoffs.

Belief-Free Equilibria in Games With Incomplete Information

We define belief-free equilibria in two-player games with incomplete information as sequential equilibria for which players' continuation strategies are best replies after every history, independently of their beliefs about the state of nature. We characterize a set of payoffs that includes all belief-free equilibrium payoffs. Conversely, any payoff in the interior of this set is a belief-free equilibrium payoff. The characterization is applied to the analysis of reputations.

A limit characterization of belief-free equilibrium payoffs in repeated games

The present paper studies repeated games with private monitoring, and characterizes the set of belief-free equilibrium payoffs in the limit as the discount factor approaches one and the noise on private information vanishes. Contrary to the conjecture by Ely et al. [J.C. Ely, J. Hörner, W. Olszewski, Belief-free equilibria in repeated games, Econometrica 73 (2005) 377–415], the equilibrium payoff set is computed by the same formula, no matter how many players there are. As an application of this result, a version of the folk theorem is established for N-player prisoner's dilemma games.

Purification in the Infinitely-Repeated Prisoners' Dilemma, Second Version

This paper investigates the Harsanyi (1973)-purifiability of mixed strategies in the repeated prisoners’ dilemma with perfect monitoring. We perturb the game so that in each period, a player receives a private payoff shock which is independently and identically distributed across players and periods. We focus on the purifiability of one-period memory mixed strategy equilibria used by Ely and V¨alim¨aki (2002) in their study of the repeated prisoners’ dilemma with private monitoring. We find that any such strategy profile is not the limit of one-period memory equilibrium strategy profiles of the perturbed game, for almost all noise distributions. However, if we allow infinite memory strategies in the perturbed game, then any completely-mixed equilibrium is purifiable. Keywords: Purification, belief-free equilibria, repeated games.