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.