On equilibrium points in bimatrix games

We discuss sensitivity of equilibrium points in bimatrix games depending on small variances (perturbations) of data. Applying implicit function theorem to a linear complementarity problem which is equivalent to the bimatrix game, we investigate sensitivity of equilibrium points with respect to the perturbation of parameters in the game. Namely, we provide the calculation of equilibrium points derivatives with respect to the parameters.     

On equilibrium points of diagonaln-person games

In this paper, we compute explicitly the equilibrium points of diagonaln-person games when all of them have the same number of strategies. This number is arbitrary. A wide generalization of two-person games is immediately obtained.The author is grateful to Professor Joel Cohen who visited IMASL during the winter of 1987 and commented on the paper.     

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.     

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.     

Uniqueness of equilibrium points in bimatrix games

Bimatrix games are constructed having a given pair (x, y) as the unique equilibrium point within the class of all mixed strategy pairs whose nonzero components are the same as (resp., among) those of (x, y). In each case, necessary and sufficient conditions on (x, y) for the existence of such a game are obtained. All games having the first property are constructed. The work extends and complements recent (separate) works ofMillham [1972],Raghavan [1970] and the author. The methods and results are valid in the context of any ordered field.     

Bimatrix games with unique equilibrium points

A necessary and sufficient condition is found for a given pair of strategies to be the unique equilibrium point of some bimatrix game.     

Constructing bimatrix games with unique equilibrium points

We construct bimatrix games with prefixed equilibrium points in the mixed extension. The uniqueness conditions are studied and we obtain a wide class of games with unique arbitrary prefixed equilibrium points.     

FiniteN-person non-cooperative games with unique equilibrium points

A necessary and sufficient condition is found for a given completely mixed strategyN-tuple to be the unique equilibrium point of some finiteN-person non-cooperative game.     

Strictly perfect equilibrium points of bimatrix games

We consider the existence of strictly perfect equilibrium points for bimatrix games. We prove that an isolated and quasi-strong equilibrium point is strictly perfect. Our result shows that in a nondegenerate bimatrix game all equilibrium points are strictly perfect. Our proof is based on the labeling theory ofShapley [1974] for bimatrix games.     

On characterizing properties of the value sets and the equilibrium point sets of non-cooperative two-person games

In this paper, for certain subfamilies of the family of bounded measurable two-person games in normal form, the value sets are characterized by three properties, called the maximum property, the minimum property and the adjunction property. Furthermore, one of the papers of Vilkas is critically discussed. Finally, for bimatrix games, two systems of characterizing properties for the equilibrium point sets are given.     

On the constructability of solution to a pair of two person search games

Let B denote the collection of all subsets of [0, 1] having exactlyk points, and let R denote the collection of all (Lebesgue measureable) subsets of [0, 1] with total length exactly a < 1. Consider the following two games in which the opponents BLUE and RED chooseB from B andR from R respectively: In the Set Hider Game (SHG) the payoff to RED is 1 ifB ∩R is empty and 0 otherwise. In the Point Hider Game (PHG) the payoff to BLUE is 1 ifB ∩R is empty and 0 otherwise. In this paper we solve the SHG and show that the PHG has no reasonable solution fork>1.     

On generalized equilibrium points

Generalizations of the usual definition of saddle point and equilibrium point are introduced in this paper. The existence of these points is shown to be related to a class of functions that we call perturbed convex functions. First and second order conditions regarding the existence of these points are also proved.     

On a new class of nonzero-sum discounted stochastic games having stationary Nash equilibrium points

A new class of nonzero-sum Borel state space discounted stochastic games having stationary Nash equilibria is presented. Some applications to economic theory are also included. Received: January 2002/Revised: July 2002     

Regularized extraproximal method for finding equilibrium points in two-person saddle-point games

Three variants of the regularized extraproximal method are proposed for finding equilibrium points in two-person saddle-point games. Their convergence is examined, and a regularizing operator is constructed.     

On risk averse competitive equilibrium

We discuss risked competitive partial equilibrium in a setting in which agents are endowed with coherent risk measures. In contrast to social planning models, we show by example that risked equilibria are not unique, even when agents’ objective functions are strictly concave. We also show that standard computational methods find only a subset of the equilibria, even with multiple starting points.