A generalization of the Nash equilibrium theorem on bimatrix games

Abstract

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 ℬ 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 ɛ-equilibrium for this game under the assumption that$A$ and ℬ are bounded. When the sets$A$ and ℬ are compact in ℝ^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.