Equilibrium selection through incomplete information in coordination games: an experimental study

We perform an experiment on a pure coordination game with uncertainty about the payoffs. Our game is closely related to models that have been used in many macroeconomic and financial applications to solve problems of equilibrium indeterminacy. In our experiment, each subject receives a noisy signal about the true payoffs. This game (inspired by the “global” games of Carlsson and van Damme, Econometrica, 61, 989–1018, 1993) has a unique strategy profile that survives the iterative deletion of strictly dominated strategies (thus a unique Nash equilibrium). The equilibrium outcome coincides, on average, with the risk-dominant equilibrium outcome of the underlying coordination game. In the baseline game, the behavior of the subjects converges to the theoretical prediction after enough experience has been gained. The data (and the comments) suggest that this behavior can be explained by learning. To test this hypothesis, we use a different game with incomplete information, related to a complete information game where learning and prior experiments suggest a different behavior. Indeed, in the second treatment, the behavior did not converge to equilibrium within 50 periods in some of the sessions. We also run both games under complete information. The results are sufficiently similar between complete and incomplete information to suggest that risk-dominance is also an important part of the explanation.      

A Property of Nash Equilibria in Repeated Games with Incomplete Information

We consider two-person non-zero-sum infinitely repeated games with lack of information on one side. The characterization of Nash equilibrium payoffs obtained by Hart allows for complex strategies, which are actually required by some equilibrium payoffs in some games. We show that appropriate one-shot public communication mechanisms make Nash equilibrium payoffs achievable by means of simple strategies. Furthermore, these mechanisms satisfy a notion of self-fulfillment.Journal of Economic LiteratureClassification Numbers: D82, C72.     

Nash equilibrium when players account for the complexity of their forecasts

Nash equilibrium is often interpreted as a steady state in which each player holds the correct expectations about the other players' behavior and acts rationally. This paper investigates the robustness of this interpretation when there are small costs associated with complicated forecasts. The model consists of a two-person strategic game in which each player chooses a finite machine to implement a strategy in an infinitely repeated 2×2 game with discounting. I analyze the model using a solution concept called Nash Equilibrium with Stable Forecasts (ESF). My main results concern the structure of equilibrium machine pairs. They provide necessary and sufficient conditions on the form of equilibrium strategies and plays. In contrast to the “folk theorem,” these structural properties place severe restrictions on the set of equilibrium paths and payoffs. For example, only sequences of the one-shot Nash equilibrium can be generated by any ESF of the repeated game of chicken.     

On the Coalition-Proofness of the Pareto Frontier of the Set of Nash Equilibria

This article shows that the Pareto efficient frontier of the Nash equilibrium set of games with strategic substitutes is coalition-proof under the following conditions: (1) the game has three players, or, alternatively, a player's payoff depends on her own strategy and on the sum (but not on the composition) of other players' strategies; (2) an increase in a player's strategy either raises all other players' payoffs monotonically or reduces them monotonically; and (3) each player's payoff is strictly concave in her own strategy. Under these conditions, the Pareto dominance refinement is equivalent to the coalition-proof Nash equilibrium refinement.Journal of Economic LiteratureClassification Number: C72.     

Existence and Uniqueness of Nash Equilibrium in Electoral Competition Games: The Hybrid Case

This paper analyzes the traditional unidimensional, two‐party electoral competition game when parties have mixed motivations, in the sense that they are interested in winning the election, but also in the policy implemented after the contest. In spite of having discontinuous payoffs, this game, referred to as the hybrid election game, is shown to be payoff secure and reciprocally upper semi‐continuous. Conditional payoffs, however, are not quasi‐concave. Hence, the existence of a pure strategy Nash equilibrium (psne ) is ensured only if parties have homogenous interests in power. In that case, an equilibrium not only exists, but it is also unique. Instead, if parties have heterogeneous motivations, depending upon the relationship between the electoral uncertainty, the aggregate opportunism, and its distribution across parties, a psne may or may not exist. The mixed extension, however, is always better reply secure. Therefore, a mixed strategy Nash equilibrium does indeed exist.     

Better-reply dynamics and global convergence to Nash equilibrium in aggregative games

We consider n-person games with quasi-concave payoffs that depend on a player's own action and the sum of all players' actions. We show that a discrete-time, stochastic process in which players move towards better replies—the better-reply dynamics—converges globally to a Nash equilibrium if actions are either strategic substitutes or strategic complements for all players around each Nash equilibrium that is asymptotically stable under a deterministic, adjusted best-reply dynamics. We present an example of a 2-person game with a unique equilibrium where the derivatives of the best-reply functions have different signs and the better-reply dynamics does not converge.     

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.     

A folk theorem for Bayesian games with commitment

The set of all Bayesian–Nash equilibrium payoffs that the players can achieve by making conditional commitments at the interim stage of a Bayesian game coincides with the set of all feasible, incentive compatible and interim individually rational payoffs of the Bayesian game. Furthermore, the various equilibrium payoffs, which are achieved by means of different commitment devices, are also the equilibrium payoffs of a universal, deterministic commitment game.     

Bertrand Games and Sharing Rules

We consider asymmetric Bertrand games with arbitrary payoffs at ties or sharing rules, and identify sufficient conditions for the zero-profit outcome and the existence of Nash equilibria. Subject to some technical conditions on non-tied payoffs the following hold. If the sharing rule is strictly tie-decreasing all players but one receive zero equilibrium payoffs, while everybody does so if non-tied payoffs are symmetric. Mixed (pure) strategy Nash equilibria exist if the sharing rule is (norm) tie-decreasing and coalition-monotone. I would like to thank Fernando Branco, the audience at Pompeu Fabra (Barcelona), ISEG (Lisbon), University of Mannheim, ESEM 2003 (Venice), EARIE 2005 (Porto), two anonymous referees, and the editor Dan Kovenock for very useful comments. This research received financial support under project POCTI/ECO/37925/2001 of FCT and FEDER.     

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.     

Learning to alternate

The Individual Evolutionary Learning (IEL) model explains human subjects’ behavior in a wide range of repeated games which have unique Nash equilibria. Using a variation of ‘better response’ strategies, IEL agents quickly learn to play Nash equilibrium strategies and their dynamic behavior is like that of humans subjects. In this paper we study whether IEL can also explain behavior in games with gains from coordination. We focus on the simplest such game: the 2 person repeated Battle of Sexes game. In laboratory experiments, two patterns of behavior often emerge: players either converge rapidly to one of the stage game Nash equilibria and stay there or learn to coordinate their actions and alternate between the two Nash equilibria every other round. We show that IEL explains this behavior if the human subjects are truly in the dark and do not know or believe they know their opponent’s payoffs. To explain the behavior when agents are not in the dark, we need to modify the basic IEL model and allow some agents to begin with a good idea about how to play. We show that if the proportion of inspired agents with good ideas is chosen judiciously, the behavior of IEL agents looks remarkably similar to that of human subjects in laboratory experiments.     

Stability of the cooperative equilibrium in N-person prisoners' dilemma with sequential moves

Summary. Nishihara [3] showed that N-person prisoners' dilemma has a cooperative Nash equilibrium, if the players decide their actions sequentially in the order determined by Nature under a certain information structure, and if each player's payoffs satisfy a certain inequality. This paper examines the stability of this cooperative equilibrium against two matters: players' slight mistakes and deviations by coalitions. The main results are as follows: (i) if the inequality on each player's payoffs strictly holds, then the cooperative equilibrium is a strictly proper equilibrium; (ii) if N≤3, and if full cooperation is Pareto efficient in N-person prisoners' dilemma, then the cooperative equilibrium is a strong Nash equilibrium; (iii) the cooperative equilibrium is in general a coalition-proof Nash equilibrium. Received: June 23, 1997; revised version: December 2, 1997     

On games with identical equilibrium payoffs

Summary. This paper compares the sets of Nash, coalition- proof Nash and strong Nash equilibrium payoffs of normal form games which are closely related. We propose sufficient conditions for equivalent or closely related games to have identical sets of equilibrium payoffs. Received: April 23, 1999; revised version: November 23, 1999     

Coalition Formation and Potential Games

In this paper we study the formation of coalition structures in situations described by a cooperative game. Players choose independently which coalition they want to join. The payoffs to the players are determined by an allocation rule on the underlying game and the coalition structure that results from the strategies of the players according to some formation rule. We study two well-known coalition structure formation rules and show that for both formation rules there exists a unique component-efficient allocation rule that results in a potential game. Journal of Economic Literature Classification Numbers: C71, C72.     

An Experimental Study of Communication and Coordination in Noncooperative Games

This paper reports the results of an experiment designed to test the usefulness of alternative solution concepts to explain players' behavior in noncooperative games with preplay communication. In the experiment subjects communicate byplain conversationprior to playing a simple game. In this setting, we find that the presumption ofindividualisticandindependentbehavior underlying the concept of Nash equilibrium is inappropriate. Instead, we observe behavior to becoordinatedandcorrelated. Statistical tests reject Nash equilibrium as an explanation of observed play. The coalition proof correlated equilibrium of the game, however, explains the data when the possibility of errors by players is introduced.Journal of Economic LiteratureClassification Numbers: C72, C92.     

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.     

Information at equilibrium

Summary. In a game with rational expectations, individuals simultaneously refine their information with the information revealed by the strategies of other individuals. At a Nash equilibrium of a game with rational expectations, the information of individuals is essentially symmetric: the same profile of strategies is also an equilibrium of a game with symmetric information; and strategies are common knowledge. If each player has a veto act, which yields a minimum payoff that no other profile of strategies attains, then the veto profile is the only Nash equilibrium, and it is is an equilibrium with rational expectations and essentially symmetric information; which accounts for the impossibility of speculation. Received: June 20, 2001; revised version: January 9, 2002 RID="*" ID="*" We wish to thank Pierpaolo Battigalli, Fran?oise Forges, Franco Donzelli, Leonidas Koutsougeras, Aldo Rustichini, Rajiv Vohra and Nicholas Yannelis for their comments. Correspondence to: H. Polemarchakis     

Evolutionary Stability in the Finitely Repeated Prisoner 's Dilemma Game

The dynamic evolutionary stability of mutual defection is proven for the repeated prisoner 's dilemma game where payoffs are cumulative and the number of repetitions is known. This agrees with the classical result that the only Nash equilibrium outcome is to defect at all stages of this repeated game. Moreover, it is shown that, for any initial polymorphic population, the evolutionary dynamic converges to a unique Nash equilibrium strategy that depends on the original polymorphism. Both these results confirm earlier conjectures concerning the application of evolutionary game theory to the repeated prisoner 's dilemma.Journal of Economic LiteratureClassification Numbers: C70, C72.     

Are the treasures of game theory ambiguous?

Goeree and Holt (Am Econ Rev 91:1402?C1422, 2001) experimentally study a number of games. In each case, they initially find strong support for Nash equilibrium; however, by changing an apparently irrelevant parameter, they find results which contradict Nash equilibrium. In this paper, we study the five normal form games from Goeree and Holt (Am Econ Rev 91:1402?C1422, 2001). We argue that their results may be explained by the hypothesis that subjects view their opponents?? behaviour as ambiguous. Ambiguity-aversion causes players to avoid strategies, which give low out of equilibrium payoffs. Similarly, ambiguity preference can make strategies with high payoffs more attractive.     

Endogenous fluctuations under evolutionary pressure in Cournot competition

An evolutionary game theoretic model of Cournot competition is investigated. Individuals choose from a finite set of different behavioral rules. Each rule specifies the quantity to be produced in the current period as a function of past quantities. Using more sophisticated rules may require extra information costs. Based upon realized payoffs, the fractions of the population choosing a certain behavioral rule are updated according to the replicator equation with noise. The long-run behavior of the evolutionary system consisting of the population dynamics coupled with the quantity dynamics of the Cournot game may be complicated and endogenous fluctuations may arise. We consider a typical example where firms can choose between two rules: the Nash rule and the best-reply rule. We show that a homoclinic tangency between the stable and unstable manifold of the equilibrium occurs as evolutionary pressure increases, implying bifurcation routes to complicated dynamics and strange attractors.