Robustness to strategic uncertainty

We introduce a criterion for robustness to strategic uncertainty in games with continuum strategy sets. We model a player's uncertainty about another player's strategy as an atomless probability distribution over that player's strategy set. We call a strategy profile robust to strategic uncertainty if it is the limit, as uncertainty vanishes, of some sequence of strategy profiles in which every player's strategy is optimal under his or her uncertainty about the others. When payoff functions are continuous we show that our criterion is a refinement of Nash equilibrium and we also give sufficient conditions for existence of a robust strategy profile. In addition, we apply the criterion to Bertrand games with convex costs, a class of games with discontinuous payoff functions and a continuum of Nash equilibria. We show that it then selects a unique Nash equilibrium, in agreement with some recent experimental findings.     

Brown–von Neumann–Nash dynamics: The continuous strategy case

Brown and von Neumann introduced a dynamical system that converges to saddle points of zero sum games with finitely many strategies. Nash used the mapping underlying these dynamics to prove existence of equilibria in general games. The resulting Brown–von Neumann–Nash dynamics are a benchmark example for myopic adjustment dynamics that, in contrast to replicator dynamics, allow for innovation, but require less rationality than the best response dynamics. This paper studies the BNN dynamics for games with infinitely many strategies. We establish Nash stationarity for continuous payoff functions. For negative semidefinite games (that include zero sum games), we generalize the results of Brown and von Neumann. In addition, we show that evolutionarily robust Nash equilibria are asymptotically stable. A complete stability analysis for doubly symmetric games is also obtained.     

Testing the TASP: An experimental investigation of learning in games with unstable equilibria

We report experiments studying mixed strategy Nash equilibria that are theoretically stable or unstable under learning. The Time Average Shapley Polygon (TASP) predicts behavior in the unstable case. We study two versions of Rock-Paper-Scissors that include a fourth strategy, Dumb. The unique Nash equilibrium is identical in the two games, but the predicted frequency of Dumb is much higher in the game where the NE is stable. Consistent with TASP, the observed frequency of Dumb is lower and play is further from Nash in the high payoff unstable treatment. However, Dumb is played too frequently in all treatments.     

A One-Stage Model of Link Formation and Payoff Division

In this paper we introduce a strategic form model in which cooperation structures and divisions of the payoffs are determined simultaneously. We analyze the cooperation structures and payoff divisions that result according to Nash equilibria, strong Nash equilibria, and coalition proof Nash equilibria. We find that no cycle will be formed if a player claims a positive amount for the formation of one of its links and that a player does not necessarily profit from a central position in a cooperation structure. Journal of Economic Literature Classification Numbers: C71, C72.     

Endogenous timing in the presence of non-monotonicities

The aim of this paper is to generalize the endogenous timing game proposed by Hamilton and Slutsky (Games and Economic Behavior, 1990, 2, pp. 29–46) by allowing the payoff or the marginal payoff of a player to become non-monotonic with respect to the strategy of the opponent. We propose a taxonomy of the subgame-perfect Nash equilibria based on the characteristics of the payoff functions proposed by Eaton (Canadian Journal of Economics, 2004, 37, pp. 805–29). We determine under which conditions of the initial payoff functions commitment has a social value and when the simultaneous-move Nash equilibrium is commitment robust and discuss its Pareto efficiency.     

CORRELATED EQUILIBRIA,GOOD AND BAD: AN EXPERIMENTAL STUDY*

We report results from an experiment that explores the empirical validity of correlated equilibrium, an important generalization of Nash equilibrium. Specifically, we examine the conditions under which subjects playing the game of Chicken will condition their behavior on private third‐party recommendations drawn from publicly announced distributions. We find that when recommendations are given, behavior differs from both a mixed‐strategy Nash equilibrium and behavior without recommendations. In particular, subjects typically follow recommendations if and only if (1) those recommendations derive from a correlated equilibrium and (2) that correlated equilibrium is payoff‐enhancing relative to the available Nash equilibria.     

Equivalence of strong and coalition-proof Nash equilibria in games without spillovers

Summary This paper examines the conditions which guarantee that the set of coalition-proof Nash equilibria coincides with the set of strong Nash equilibria in the normal form games withoutspillovers. We find thatpopulation monotonicity properties of the payoff functions, when the payoff of a player changes monotonically when the size of the group of players choosing the same strategy increases, are crucial to obtain the equivalence of these two solution concepts. We identify the classes of games, satisfying population monotonicity properties, which yield the equivalence of the set of coalition-proof Nash equilibria and the set of strong Nash equilibria. We also provide sufficient conditions for the equivalence result even when the population monotonicity assumptions are relaxed.We wish to thank Mamoru Kaneko, Akihiko Matsui, Tomoichi Shinotsuka, Benyamin Shitoviz, Tayfun Sonmez, William Thomson, the participants of the Southeastern Economic Theory Meeting in Charlottesville and the seminars at CORE and University of Tsukuba for useful discussions and comments. Our special thanks due anonymous referee for the suggestion to add a section addressing the issue of existence of a strong Nash equilibrium.     

Finding all Nash equilibria of a finite game using polynomial algebra

The set of Nash equilibria of a finite game is the set of nonnegative solutions to a system of polynomial equations. In this survey article, we describe how to construct certain special games and explain how to find all the complex roots of the corresponding polynomial systems, including all the Nash equilibria. We then explain how to find all the complex roots of the polynomial systems for arbitrary generic games, by polyhedral homotopy continuation starting from the solutions to the specially constructed games. We describe the use of Gröbner bases to solve these polynomial systems and to learn geometric information about how the solution set varies with the payoff functions. Finally, we review the use of the Gambit software package to find all Nash equilibria of a finite game.     

The power of ESS: An experimental study

Abstract. Our experimental design mimics a traditional evolutionary game framework where players are matched pairwise to play a symmetric 33 bimatrix game that has two Nash equilibria. One equilibrium is an evolutionary stable state, or ESS; the other is an equilibrium in dominated strategies. Our primary experimental result is the observation that the ESS becomes extremely attractive when subjects have minimal information about the payoff functions, although the dominated equilibrium assures the highest equilibrium payoff. The attractiveness of the ESS is only moderate when players are completely informed about the 33 payoff matrix. Correspondence to: S.K. Berninghaus     

Adaptation and complexity in repeated games

The paper presents a learning model for two-player infinitely repeated games. In an inference step players construct minimally complex inferences of strategies based on observed play, and in an adaptation step players choose minimally complex best responses to an inference. When players randomly select an inference from a probability distribution with full support the set of steady states is a subset of the set of Nash equilibria in which only stage game Nash equilibria are played. When players make ‘cautious’ inferences the set of steady states is the subset of self-confirming equilibria with Nash outcome paths. When players use different inference rules, the set of steady states can lie between the previous two cases.     

Evolutionary stability and lexicographic preferences

We explore the interaction between evolutionary stability and lexicographic preferences. To do so, we define a limit Nash equilibrium for a lexicographic game as the limit of Nash equilibria of nearby games with continuous preferences. Nash equilibria of lexicographic games are limit Nash equilibria, but not conversely. Modified evolutionarily stable strategies (Binmore and Samuelson, 1992. J. Econ. Theory 57, 278–305) are limit Nash equilibria. Modified evolutionary stability differs from “lexicographic evolutionarily stability” (defined by extending the common characterization of evolutionary stability to lexicographic preferences) in the order in which limits in the payoff space and the space of invasion barriers are taken.     

Continuity of the payoff functions

Summary. In his Nash equilibrium paper, Glicksberg states that the payoff functions are continuous. Such a function is defined on the product of mixed strategies, which are the Borel probability measures on a compactum, endowed with the product of the weak topologies. The continuity property is used in proving the existence of Nash equilibria. This note proves that the payoff functions are continuous, which is not immediate to establish. Received: June 10, 1999; revised version: June 24, 1999     

Equilibria in large games with continuous procedures

We study existence of equilibria in large games where players use boundedly rational procedures. The equilibria are different from Nash equilibria; the difference persists even when players use procedures for which the observations gathered in any period are used to evaluate the payoff from different actions.     

Nash equilibria in ∞-dimensional spaces: an approximation theorem

Summary. We show, by employing a density result for probability measures, that in games with a finite number of players and ∞-dimensional pure strategy spaces Nash equilibria can be approximated by finite mixed strategies. Given ε>0, each player receives an expected utility payoff ε/2 close to his Nash payoff and no player could change his strategy unilaterally and do better than ε. Received: July 15, 1997; revised version: February 6, 1998     

A Non-cooperative Approach to Bankruptcy Problems

We propose an elementary game form that allows to obtain the allocations proposed by any acceptable bankruptcy rule as the unique payoff vector of the corresponding Nash equilibria.     

Directed Networks with Spillovers

We study noncooperative network formation in two types of directed networks. In the first type, called the model with global spillovers, the payoff of a player depends on the number of links she forms as well as the total number of links formed by all other players. In the second type, called the model with local spillovers, the payoff of a player depends on the number of links she forms and the total number of links formed by her immediate neighbors, as well as the number of links formed by players outside her neighborhood. For both classes of games we investigate the existence of pure strategy Nash equilibria and characterize the Nash networks under a number of different second order conditions on the payoff function.     

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.     

Measuring Strategic Uncertainty in Coordination Games

This paper proposes a method to measure strategic uncertainty by eliciting certainty equivalents analogous to measuring risk attitudes in lotteries. We apply this method by conducting experiments on a class of one-shot coordination games with strategic complementarities and choices between simple lotteries and sure payoff alternatives, both framed in a similar way. Despite the multiplicity of equilibria in the coordination games, aggregate behaviour is fairly predictable. The pure or mixed Nash equilibria cannot describe subjects' behaviour. We present two global games with private information about monetary payoffs and about risk aversion. While previous literature treats the parameters of a global game as given, we estimate them and show that both models describe observed behaviour well. The global-game selection for vanishing noise of private signals offers a good recommendation for actual players, given the observed distribution of actions. We also deduce subjective beliefs and compare them with objective probabilities.     

Risk attitudes and the stag-hunt game

A stag-hunt game (with the risky and safe actions) has two pure Nash equilibria that are Pareto-rankable. The risky action leads either to the Pareto-superior equilibrium (high payoff) or to out of equilibrium (low payoff) depending on the opponent’s action. Both players may want to obtain high payoff but uncertainty about the opponent’s action may prevent them to take such strategic risk. This paper investigates how information about the risk attitude of an opponent affects a player’s action choice in the stag-hunt game. I find that although a subject’s propensity to choose the risky action depends on her opponent’s risk attitude, it does not depend on her own risk attitude.