Robust equilibria under non-common priors

This paper considers the robustness of equilibria to a small amount of incomplete information, where players are allowed to have heterogeneous priors. An equilibrium of a complete information game is robust to incomplete information under non-common priors if for every incomplete information game where each player's prior assigns high probability on the event that the players know at arbitrarily high order that the payoffs are given by the complete information game, there exists a Bayesian Nash equilibrium that generates behavior close to the equilibrium in consideration. It is shown that for generic games, an equilibrium is robust under non-common priors if and only if it is the unique rationalizable action profile. Set-valued concepts are also introduced, and for generic games, a smallest robust set is shown to exist and coincide with the set of a posteriori equilibria.     

Generalized potentials and robust sets of equilibria

This paper introduces generalized potential functions of complete information games and studies the robustness of sets of equilibria to incomplete information. A set of equilibria of a complete information game is robust if every incomplete information game where payoffs are almost always given by the complete information game has an equilibrium which generates behavior close to some equilibrium in the set. This paper provides sufficient conditions for the robustness of sets of equilibria in terms of argmax sets of generalized potential functions. These sufficient conditions unify and generalize existing sufficient conditions. Our generalization of potential games is useful in other game theoretic problems where potential methods have been applied.     

Infinite horizon common interest games with perfect information

We consider infinite horizon common interest games with perfect information. A game is a K-coordination game if each player can decrease other players' payoffs by at most K times his own cost of punishment. The number K represents the degree of commonality of payoffs among the players. The smaller K is, the more interest the players share. A K-coordination game tapers off if the greatest payoff variation conditional on the first t periods of an efficient history converges to 0 at a rate faster than K^−t as t→∞. We show that every subgame perfect equilibrium outcome is efficient in any tapering-off game with perfect information. Applications include asynchronously repeated games, repeated games of extensive form games, asymptotically finite horizon games, and asymptotically pure coordination games.     

An equilibrium closure result for discontinuous games

For games with discontinuous payoffs Simon and Zame (Econometrica 58:861–872, 1990) introduced payoff indeterminacy, in the form of endogenous sharing rules, which are measurable selections of a certain payoff correspondence. Their main result concerns the existence of a mixed Nash equilibrium and an associated sharing rule. Its proof is based on a discrete approximation scheme “from within” the payoff correspondence. Here, we present a new, related closure result for games with possibly noncompact action spaces, involving a sequence of Nash equilibria. In contrast to Simon and Zame (Econometrica 58:861–872, 1990), this result can be used for more involved forms of approximation, because it contains more information about the endogenous sharing rule. With such added precision, the closure result can be used for the actual computation of endogenous sharing rules in games with discontinuous payoffs by means of successive continuous interpolations in an approximation scheme. This is demonstrated for a Bertrand type duopoly game and for a location game already considered by Simon and Zame. Moreover, the main existence result of Simon and Zame (Econometrica 58:861–872, 1990) follows in two different ways from the closure result.     

Communication equilibrium payoffs in repeated games with imperfect monitoring

We characterize the set of communication equilibrium payoffs of any undiscounted repeated matrix-game with imperfect monitoring and complete information. For two-player games, a characterization is provided by Mertens, Sorin, and Zamir (Repeated games, Part A (1994) CORE DP 9420), mainly using Lehrer's (Math. Operations Res. (1992) 175) result for correlated equilibria. The main result of this paper is to extend this characterization to the n-player case. The proof of the characterization relies on an analogy with an auxiliary 2-player repeated game with incomplete information and imperfect monitoring. We use Kohlberg's (Int. J. Game Theory (1975) 7) result to construct explicitly a canonical communication device for each communication equilibrium payoff.     

Learning Dynamics in Games with Stochastic Perturbations

Consider a generalization of fictitious play in which agents′ choices are perturbed by incomplete information about what the other side has done, variability in their payoffs, and unexplained trembles. These perturbed best reply dynamics define a nonstationary Markov process on an infinite state space. It is shown, using results from stochastic approximation theory, that for 2 × 2 games it converges almost surely to a point that lies close to a stable Nash equilibrium, whether pure or mixed. This generalizes a result of Fudenherg and Kreps, who demonstrate convergence when the game has a unique mixed equilibrium. Journal of Economic Literature Classification Numbers: 000, 000, 000.     

An Experiment to Evaluate Bayesian Learning of Nash Equilibrium Play

We report on an experiment designed to evaluate the empirical implications of Jordan's model of Bayesian learning in games of incomplete information. A finite example is constructed in which the model generates unique predictions of subjects' choices in nearly all periods. When the “true” game defined by players' private information was one with a unique equilibrium in pure strategies, the experimental subjects' play converged to the equilibrium, as Jordan's theory predicts, even when the subjects had not attained complete information about one another. But when there were two pure strategy equilibria, the theory's predictions were not consistent with observed behavior. Journal of Economic Literature Classification numbers: D83, C72, C92.     

Coordination and Information in Critical Mass Games: An Experimental Study

We present experimental results on a repeated coordination game with Pareto-ranked equilibria in which a payoff from choosing an action is positive only if a critical mass of players choose that action. We design a baseline version of the game in which payoffs remain constant for values above the critical mass, and an increasing returns version in which payoffs keep increasing for values above the critical mass. We test the predictive power of security and payoff-dominance under different information treatments. Our results show that convergence to the payoff-dominant equilibrium is the modal limit outcome when players have full information about others' previous round choices, while this outcome never occurs in the remaining treatments. The paths of play in some groups reveal a tacit dynamic coordination by which groups converge to the efficient equilibrium in a step-like manner. Moreover, the frequency and speed of convergence to the payoff-dominant equilibrium are higher, ceteris paribus, when increasing returns are present. Finally, successful coordination seems to crucially depend on players' willingness to signal to others the choice of the action supporting the efficient equilibrium.     

Coordination of strategic responses to security threats: Laboratory evidence

Laboratory experiments are used to evaluate the extent to which players in games can coordinate investments that diminish the probability of losses due to security breaches or terrorist attacks. In this environment, economically sensible investments may be foregone if their potential benefits are negated by failures to invest in security at other sites. The result is a coordination game with a desirable high-payoff, high-security equilibrium and an undesirable low-security equilibrium that may result if players do not expect others to invest in security. One unique feature of this coordination situation is that investment in security by one player generates a positive externality such that all other players’ expected payoffs are increased, regardless of those other players’ investment decisions. Coordination failures are pervasive in a baseline experiment with simultaneous decisions, but coordination is improved if players are allowed to move in an endogenously determined sequence. In addition, coordinated security investments are observed more often when the largest single security threat to individuals is preventable by their own decisions to invest in security. The security coordination game is a “potential game,” and the success of coordination on the more secure equilibrium is related to the notion of potential function maximization and basin of attraction.      

Perfect communication equilibria in repeated games with imperfect monitoring

This paper introduces an equilibrium concept called perfect communication equilibrium for repeated games with imperfect private monitoring. This concept is a refinement of Myerson's [Myerson, R.B., 1982. Optimal coordination mechanisms in generalized principal agent problems, J. Math. Econ. 10, 67–81] communication equilibrium. A communication equilibrium is perfect if it induces a communication equilibrium of the continuation game, after every history of messages of the mediator. We provide a characterization of the set of corresponding equilibrium payoffs and derive a Folk Theorem for discounted repeated games with imperfect private monitoring.     

When less information is good enough: experiments with global stag hunt games

There is mixed evidence on whether subjects coordinate on the efficient equilibrium in experimental stag hunt games under complete information. A design that generates an anomalously high level of coordination, Rankin et al. (Games Econo Behav 32(2):315–337, 2000), varies payoffs each period in repeated play rather than holding them constant. These payoff “perturbations” are eerily similar to those used to motivate the theory of global games, except the theory operates under incomplete information. Interestingly, that equilibrium selection concept is known to coincide with risk dominance, rather than payoff dominance. Thus, in theory, a small change in experimental design should produce a different equilibrium outcome. We examine this prediction in two treatments. In one, we use public signals to match Rankin et al. (2000)’s design; in the other, we use private signals to match the canonical example of global games theory. We find little difference between treatments, in both cases, subject play approaches payoff dominance. Our literature review reveals this result may have more to do with the idiosyncrasies of our complete information framework than the superiority of payoff dominance as an equilibrium selection principle.     

Payoff dominance and risk dominance in the observable delay game: a note

We examine whether the payoff dominant sequential-move (Stackelberg) outcome is realized when timing is endogenized. We adopt the observable delay game formulated by Hamilton and Slutsky [Games Econ Behav 2(1):29–46, 1990]. We find that if one sequential-move outcome is payoff dominant, either (i) the outcome both players prefer is the unique equilibrium; or (ii) two sequential-move outcomes are equilibria and the one both players prefer is risk dominant. In other words, no conflict between payoff dominance and risk dominance in the observable delay game exists, in contrast to other games such as (non pure) coordination games. We also find that even if one of two sequential-move outcomes is the unique equilibrium outcome in the observable delay game, it does not imply that the equilibrium outcome is payoff dominant to the other sequential-move outcome.      

Learning by trial and error

A person learns by trial and error if he occasionally tries out new strategies, rejecting choices that are erroneous in the sense that they do not lead to higher payoffs. In a game, however, strategies can become erroneous due to a change of behavior by someone else. We introduce a learning rule in which behavior is conditional on whether a player experiences an error of the first or second type. This rule, called interactive trial and error learning, implements Nash equilibrium behavior in any game with generic payoffs and at least one pure Nash equilibrium.     

Computationally restricted unmediated talk under incomplete information

Summary. We show the role of unmediated talk with computational complexity bounds as both an information transmission and a coordination device for the class of two-player games with incomplete information and rational parameters. We prove that any communication equilibrium payoff of such games can be reached as a Bayesian-Nash equilibrium payoff of the game extended by a two phase universal mechanism of interim computationally restricted pre-play communication. The communication protocols are designed with the help of modern cryptographic tools. A familiar context in which our results could be applied is bilateral trading with incomplete information.Received: 9 September 2002, Revised: 14 March 2003, JEL Classification Numbers: C72. Correspondence to: Amparo UrbanoWe wish to thank financial aid from the Valencian Institute of Economic Research (IVIE) and partial support by DIGCYT under project PB95 - 1074. A previous version of this work appears as IVIE Working Paper WP-AD 99-07, under the title: "Unmediated talk under incomplete information".     

Endogenous transfers in the Prisoner's Dilemma game: An experimental test of cooperation and coordination

We test a two-stage compensation mechanism for promoting cooperation in Prisoner's Dilemma games. Players first simultaneously choose binding non-negative amounts to pay their counterparts for cooperating, and then play the induced game knowing these amounts. In our games, all payment pairs consistent with mutual cooperation in subgame-perfect equilibrium transform these games into coordination games, with both mutual cooperation and mutual defection as Nash equilibria in the second stage. When endogenous transfer payments are not permitted, cooperation is much less likely. Mutual cooperation is most likely when the (sufficient) payments are identical, and it is also substantially more likely with payment pairs that bring the mutual-cooperation payoffs closer together. Both the Fehr–Schmidt and Charness–Rabin models predict that transfers that make final payoffs closer are preferred; however, they do not explain why equal transfers are particularly effective. Transfers are also effective in sustaining cooperation even when they are imposed and not chosen.     

Learning with perfect information

For extensive form games with perfect information, consider a learning process in which, at any iteration, each player unilaterally deviates to a best response to his current conjectures of others' strategies; and then updates his conjectures in accordance with the induced play of the game. We show that, for generic payoffs, the outcome of the game becomes stationary, and is consistent with Nash equilibrium. In general, if payoffs have ties or if players observe more of each others' strategies than is revealed by plays of the game, the same result holds provided a rationality constraint is imposed on unilateral deviations: no player changes his moves in subgames that he deems unreachable, unless he stands to improve his payoff there. Moreover, with this constraint, the sequence of strategies and conjectures also becomes stationary, and yields a self-confirming equilibrium.     

Generalized assignment models: with an applicationto technology transfer

Summary.  We consider a generalized assignment model where the payoffs depend on the number of matchings that take place. We formulate a simple non-cooperative game and look for subgame perfect equilibrium of this model. Existence is established for a wide class of games. We also look at a refinement criterion which, for the standard assignment model, selects the -optimal outcome as the unique equilibrium. We then apply these concepts to a model of technology transfer between domestic and foreign firms. Received: June 24, 1994; revised version October 12, 1995     

Chores

We analyze situations where the provision of each of c public goods must be voluntarily assumed by exactly one of n private agents in the absence of transfer schemes or binding contracts. We model this problem as a complete information, potentially infinite horizon game where n agents simultaneously wage c wars of attrition. Providing a public good commits an agent not to take on the provision of another public good for a fixed period. We explore the strategic trade-offs that this commitment ability and the multiplicity of tasks provide. Subgame perfect equilibria (SPEs) are characterized completely for games with two agents and two public goods. For games with two identical agents and c > 1 identical public goods, we establish that an equilibrium that yields a surplus-maximizing outcome always exists and we provide sufficient conditions under which it is the unique equilibrium outcome. We show that under mild conditions, the surplus-maximizing SPE is the unique symmetric SPE. Journal of Economic Literature Classification Number: H41, C72, D13.     

UNIQUENESS OF EQUILIBRIUM PAYOFFS IN FINITELY REPEATED GAMES WITH IMPERFECT MONITORING*

This paper examines when a finitely repeated game with imperfect monitoring has a unique equilibrium payoff vector. This problem is nontrivial under imperfect monitoring, because uniqueness of equilibrium (outcome) in the stage game does not extend to finitely repeated games. A (correlated) equilibrium is equilibrium minimaxing if any player's equilibrium payoff is her minimax value when the other players choose a correlated action profile from the actions played in the equilibrium. The uniqueness result holds if all stage game correlated equilibria are equilibrium minimaxing and have the same payoffs. The uniqueness result does not hold under weaker conditions.     

Equilibrium price formation in markets with differentially informed agents

We consider a pure exchange economy with a finite set of types of agents which have incomplete and asymmetric information on the states of nature. Our aim was to describe the equilibrium price formation and analyze how the lack of information may affect the allocation of resources. To do so, we adapt to an asymmetric information scenario a variant of the Shapley–Shubik game introduced by Dubey and Geanakoplos (J Math Econ 39:391–400, 2003 ).