Two equivalence results for two-person strict games

A game is strict if for both players, different profiles have different payoffs. Two games are best response equivalent if their best response functions are the same. We prove that a two-person strict game has at most one pure Nash equilibrium if and only if it is best response equivalent to a strictly competitive game, and that it is best response equivalent to an ordinal potential game if and only if it is best response equivalent to a quasi-supermodular game.     

Reinterpreting mixed strategy equilibria: a unification of the classical and Bayesian views

We provide a new interpretation of mixed strategy equilibria that incorporates both von Neumann and Morgenstern's classical concealment role of mixing, as well as the more recent Bayesian view originating with Harsanyi. For any two-person game, G, we consider an incomplete information game, in which each player's type is the probability he assigns to the event that his mixed strategy in G is “found out” by his opponent. We show that, generically, any regular equilibrium of G can be approximated by an equilibrium of in which almost every type of each player is strictly optimizing. This leads us to interpret i's equilibrium mixed strategy in G as a combination of deliberate randomization by i together with uncertainty on j's part about which randomization i will employ. We also show that such randomization is not unusual: for example, i's randomization is nondegenerate whenever the support of an equilibrium contains cyclic best replies.     

Sowing Doubt Optimally in Two-Person Repeated Games

Consider a two-person repeated game, where one of the players, P1, can sow doubt, in the mind of his opponent, as to what P1's payoffs are. This results in a two-person repeated game with incomplete information. By sowing doubt, P1 can sometimes increase his minimal equilibrium payoff in the original game. We prove that this minimum is maximal when only one payoff matrix, the negative of the payoff matrix of the opponent, is added (the opponent thus believes that he might play a zero-sum game). We obtain two formulas for calculating this maximal minimum payoff. Journal of Economic Literature Classification Numbers: C7, D8.     

A Bayesian Model of Voting in Juries

We model voting in juries as a game of incomplete information, allowing jurors to receive a continuum of signals. We characterize the unique symmetric equilibrium of the game, and give a condition under which no asymmetric equilibria exist under unanimity rule. We offer a condition under which unanimity rule exhibits a bias toward convicting the innocent, regardless of the size of the jury, and give an example showing that this bias can be reversed. We prove a “jury theorem” for our general model: As the size of the jury increases, the probability of a mistaken judgment goes to zero for every voting rule except unanimity rule. For unanimity rule, the probability of making a mistake is bounded strictly above zero if and only if there do not exist arbitrarily strong signals of innocence. Our results explain the asymptotic inefficiency of unanimity rule in finite models and establishes the possibility of asymptotic efficiency, a property that could emerge only in a continuous model. Journal of Economic Literature Classification Numbers: C72, D72.     

Arms Races and Negotiations

Two players simultaneously decide whether or not to acquire new weapons in an arms race game. Each player's type determines his propensity to arm. Types are private information, and are independently drawn from a continuous distribution. With probability close to one, the best outcome for each player is for neither to acquire new weapons (although each prefers to acquire new weapons if he thinks the opponent will). There is a small probability that a player is a dominant strategy type who always prefers to acquire new weapons. We find conditions under which the unique Bayesian–Nash equilibrium involves an arms race with probability one. However, if the probability that a player is a dominant strategy type is sufficiently small, then there is an equilibrium of the cheap-talk extension of the game where the probability of an arms race is close to zero.     

Equilibrium play in matches: Binary Markov games

We study two-person extensive form games, or “matches,” in which the only possible outcomes (if the game terminates) are that one player or the other is declared the winner. The winner of the match is determined by the winning of points, in “point games.” We call these matches binary Markov games. We show that if a simple monotonicity condition is satisfied, then (a) it is a Nash equilibrium of the match for the players, at each point, to play a Nash equilibrium of the point game; (b) it is a minimax behavior strategy in the match for a player to play minimax in each point game; and (c) when the point games all have unique Nash equilibria, the only Nash equilibrium of the binary Markov game consists of minimax play at each point. An application to tennis is provided.     

Basins of attraction and equilibrium selection under different learning rules

A deterministic learning model applied to a game with multiple equilibria produces distinct basins of attraction for those equilibria. In symmetric two-by-two games, basins of attraction are invariant to a wide range of learning rules including best response dynamics, replicator dynamics, and fictitious play. In this paper, we construct a class of three-by-three symmetric games for which the overlap in the basins of attraction under best response learning and replicator dynamics is arbitrarily small. We then derive necessary and sufficient conditions on payoffs for these two learning rules to create basins of attraction with vanishing overlap. The necessary condition requires that with probability one the initial best response is not an equilibrium to the game. The existence of parasitic or misleading actions allows subtle differences in the learning rules to accumulate.     

Instability in the implementation of Walrasian allocations

The existence of game forms which implement Walrasian allocations as Cournot (Nash) equilibrium outcomes is well known. However, if the equilibria are also required to be locally dynamically stable, at least for environments with unique Walrasian allocations, this paper shows that the requisite game forms do not exist. Our definition of a game form entails certain regularity conditions, and requires the Cournot equilibrium to be unique when the Walrasian equilibrium is unique. The main result is that for such a game form, there does not exist a continuous-time strategy adjustment process which ensures the local stability of Cournot equilibria throughout a certain class of environments having unique Walrasian equilibria. Each trader adjusts his strategy in response to his own characteristics and the observed current strategies of others; but the direction and magnitude of adjustments are not constrained by any behavioral assumptions. The definitions permit the inclusion of an artificial player, such as an auctioneer, so the well-known tatonnement instability emerges as a special case.     

p-Dominance and Equilibrium Selection under Perfect Foresight Dynamics

This paper studies equilibrium selection based on a class of perfect foresight dynamics and relates it to the notion of p-dominance. A continuum of rational players is repeatedly and randomly matched to play a symmetric n×n game. There are frictions: opportunities to revise actions follow independent Poisson processes. The dynamics has stationary states, each of which corresponds to a Nash equilibrium of the static game. A strict Nash equilibrium is linearly stable under the perfect foresight dynamics if, independent of the current action distribution, there exists a consistent belief that any player necessarily plays the Nash equilibrium action at every revision opportunity. It is shown that a strict Nash equilibrium is linearly stable under the perfect foresight dynamics with a small degree of friction if and only if it is the p-dominant equilibrium with p<1/2. It is also shown that if a strict Nash equilibrium is the p-dominant equilibrium with p<1/2, then it is uniquely absorbing (and globally accessible) for a small friction (but not vice versa). Set-valued stability concepts are introduced and their existence is shown. Journal of Economic Literature Classification Numbers: C72, C73.     

The Envelope Theorem and Comparative Statics of Nash Equilibria

The Envelope Theorem for Nash equilibria shows that the strategic reaction of the other players in the game is important for determining how parameter perturbations affect a given player's indirect objective function. The fundamental comparative statics matrix of Nash equilibria for theithplayer in anN-player static game includes the equilibrium response of the otherN−1players in the game to the parameter perturbation and is symmetric positive semidefinite subject to constraint. This result is fundamental in that it holds for all sufficiently smooth Nash equilibria and is independent of any curvature or stability assumptions imposed on the game.Journal of Economic LiteratureClassification Numbers: C72, C61.     

On the existence of pure strategy Nash equilibria in two person discrete games

We construct a generalized two-person discrete strategy static game of complete information where continuity, convexity and compactness cannot be invoked to show the existence of pure strategy Nash equilibrium. We show that, when best responses are unique from both sides, a condition of Minimal Acyclicity is necessary and sufficient for the existence of pure strategy Nash equilibria.     

A game of timing and visibility

We consider the following abstraction of competing publications. There are n players in the game. Each player i chooses a point x_i in the interval [0,1], and a player's payoff is the distance from its point x_i to the next larger point, or to 1 if x_i is the largest. For this game, we give a complete characterization of the Nash equilibrium for the two-player game, and, more important, we give an efficient approximation algorithm to compute numerically the symmetric Nash equilibrium for the n-player game. The approximation is computed via a discrete version of the game. In both cases, we show that the (symmetric) equilibrium is unique. Our algorithmic approach to the n-player game is non-standard in that it does not involve solving a system of differential equations. We believe that our techniques can be useful in the analysis of other timing games.     

Egalitarianism and Efficiency in Repeated Symmetric Games

We analyze the symmetric equilibria of repeated symmetric games where there is a conflict of interests over equilibria—the battle-of-the-sexes or the hawk–dove game are key examples. If one restricts attention to symmetric equilibria, efficient equilibria must be egalitarian. For finitely repeated games, and generic discount factors, there is a unique outcome path which ensures efficiency within the class of symmetric equilibria. This is also true for the infinitely repeated games if the players are sufficiently impatient. Journal of Economic Literature Classification Numbers: C72, C73.     

Overcoming the coordination problem: Dynamic formation of networks

We analyze a multi-period entry game among privately informed agents who differ with respect to the number of agents who must enter in order for their own entry to be profitable. In each period agents who have not yet joined decide whether to subscribe to a network. There exists a unique equilibrium that approximates any symmetric equilibrium arbitrarily closely as the discount factor approaches one. This resolves the coordination problem. Ex-post efficiency is necessarily achieved asymptotically as the population size grows large. These results do not hold if subscribers can reverse their decisions without cost.     

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.     

Correlated Equilibrium in Stochastic Games

We study the existence of uniform correlated equilibrium payoffs in stochastic games. The correlation devices that we use are either autonomous (they base their choice of signal on previous signals, but not on previous states or actions) or stationary (their choice is independent of any data and is drawn according to the same probability distribution at every stage). We prove that any n-player stochastic game admits an autonomous correlated equilibrium payoff. When the game is positive and recursive, a stationary correlated equilibrium payoff exists. Journal of Economic Literature Classification Numbers: C72, C73.     

Beliefs and Public Good Provision with Anonymous Contributors

We analyze a static game of public good contributions where finitely many anonymous players have heterogeneous preferences about the public good and heterogeneous beliefs about the distribution of preferences. In the unique symmetric equilibrium, the only individuals who make positive contributions are those who most value the public good and who are also the most pessimistic; that is, according to their beliefs, the proportion of players who most like the public good is smaller than it would be according to any other possible belief. We predict whether the aggregate contribution is larger or smaller than it would be in an analogous game with complete information and heterogeneous preferences, by comparing the beliefs of contributors with the true distribution of preferences. A trade‐off between preferences and beliefs arises if there is no individual who simultaneously has the highest preference type and the most pessimistic belief. In this case, there is a symmetric equilibrium, and multiple symmetric equilibria occur only if there are more than two preference types.     

How Proper Is Sequential Equilibrium?

A strategy profile of a normal form game is proper if and only if it is quasi-perfect in every extensive form (with that normal form). Thus, properness requires optimality along a sequence of supporting trembles, while sequentiality only requires optimality in the limit. A decision-theoretic implementation of sequential rationality, strategic independence respecting equilibrium (SIRE), is defined and compared to proper equilibrium, using lexicographic probability systems. Finally, we give tremble-based characterizations, which do not involve structural features of the game, of the rankings of strategies that underlie proper equilibrium and SIRE.Journal of Economic LiteratureClassification Numbers: C70, C72.     

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.     

Evolution with Delay

This paper analyses evolutionary models with delays in players’ observations. I apply best response dynamics to symmetric 2 × 2 games, in particular the coordination game and the game with no symmetric pure strategy equilibrium. When the dynamics have no delay, their solutions converge to symmetric Nash equilibria monotonically. However when I introduce delays to the dynamics, although these dynamics represented by differential difference equations are seemingly simple, it shows that they have periodic solutions. JEL Classification Number: C79.