On purification of equilibrium in Bayesian games and expost Nash equilibrium

Treating games of incomplete information, we demonstrate that the existence of an ex post stable strategy vector implies the existence of an approximate Bayesian equilibrium in pure strategies that is also expost stable. Through examples we demonstrate the ‘bounds obtained on the approximation’ are tight. The main results of this paper first appeared in University of Warwick Department of Economics Discussion Paper #710. 2004.     

Resource pricing games on graphs: existence of Nash equilibria

In this letter, we consider a non-cooperative resource pricing game on a graph where sellers (i.e., players) set the prices for their own resources to maximize the payoffs and buyers migrate to seek the least expensive resources. We present a model for the resource pricing game and prove the existence of Nash equilibria on regular and hierarchical graphs. The results obtained are applicable to the study of market economies, social networks and computer networks where individuals trade resources in a spatially extended environment.     

On Nash equilibrium points and games of imperfect information

This paper considers a class of two-player, nonzero-sum games in which the players have only local, as opposed to global, information about the payoff functions. We study various modes of behavior and their relationship to different stability properties of the Nash equilibrium points.     

Nash equilibrium in compact-continuous games with a potential

If the preferences of the players in a strategic game satisfy certain continuity conditions, then the acyclicity of individual improvements implies the existence of a (pure strategy) Nash equilibrium. Moreover, starting from any strategy profile, an arbitrary neighborhood of the set of Nash equilibria can be reached after a finite number of individual improvements.     

On the existence of Nash equilibria in large games

We study the existence of Nash equilibria in games with an infinite number of players. We show that there exists a Nash equilibrium in mixed strategies in all normal form games such that pure strategy sets are compact metric spaces and utility functions are continuous. The player set can be any nonempty set.     

Uniqueness of Nash equilibrium for linear-convex stochastic differential games

The uniqueness of Nash equilibria is shown for a class of stochastic differential games where the dynamic constraints are linear in the control variables. The result is applied to an oligopoly.This paper benefitted from comments by two anonymous referees and by L. Blume and C. Simon.     

Behavior and deliberation in perfect-information games: Nash equilibrium and backward induction

Doxastic characterizations of the set of Nash equilibrium outcomes and of the set of backward-induction outcomes are provided for general perfect-information games (where there may be multiple backward-induction solutions). We use models that are behavioral, rather than strategy-based, where a state only specifies the actual play of the game and not the hypothetical choices of the players at nodes that are not reached by the actual play. The analysis is completely free of counterfactuals and no belief revision theory is required, since only the beliefs at reached histories are specified.     

Characterization of the existence of a pure-strategy Nash equilibrium

In this work, we provide a necessary and sufficient condition for the existence of a pure-strategy Nash equilibrium for non-cooperative games in topological spaces.     

On computational search for Nash equilibrium in hexamatrix games

The problem of numerical finding of a Nash equilibrium in a 3-player polymatrix game is considered. Such a game can be completely described by six matrices, and it turns out to be equivalent to the solving a nonconvex optimization problem with a bilinear structure in the objective function. Special methods of local and global search for the optimization problem are proposed and investigated. The results of computational solution of the test game are presented and analyzed.     

Directed graphical structure,Nash equilibrium,and potential games

This paper considers the directed graphical structure of a game, called influence structure, where a directed edge from player $i$ to player $j$ indicates that player $i$ may be able to affect $j$’s payoff via his unilateral change of strategies. We give a necessary and sufficient condition for the existence of pure-strategy Nash equilibrium of games having a directed graph in terms of the structure of that graph. We also discuss the relationship between the structure of graphs and potential games.     

Topological degree and number of Nash equilibrium points of bimatrix games

In this paper, using the topological degree, we give a new proof of a well-known result: the number of Nash equilibrium points of a nondegenerate bimatrix game is odd. The calculation of the topological degree allows the localization of the whole set of non-degenerate equilibrium points.     

On some properties of Nash equilibrium points in two-person games

In modern game theory, a lot of attention is paid to the concept of Nash equilibrium. The paper is devoted to the study of some properties of the set A of Nash equilibrium points in two-person games. In particular, the character of possible complexity of the set A is investigated, and the stability of the set A under small perturbations of payoff functions is analyzed.     

A generalization of the Nash equilibrium theorem on bimatrix games

In this article, we consider a two-person game in which the first player picks a row representative matrixM from a nonempty set $A$ ofm ×n matrices and a probability distributionx on {1,2,...,m} while the second player picks a column representative matrixN from a nonempty set ? ofm ×n matrices and a probability distribution y on 1,2,...,n. This leads to the respective costs ofx ^t My andx ^t Ny for these players. We establish the existence of an ?-equilibrium for this game under the assumption that $A$ and ? are bounded. When the sets $A$ and ? are compact in ?^mxn, the result yields an equilibrium state at which stage no player can decrease his cost by unilaterally changing his row/column selection and probability distribution. The result, when further specialized to singleton sets, reduces to the famous theorem of Nash on bimatrix games.