In this article, the authors consider a two-person game in which the first player picks a row representative matrix M from a nonempty set 𝒜 of m×n matrices and a probability distribution x on {1,2,...,m} while the second player picks a column representative matrix N from a nonempty set ℬ of m×n matrices and a probability distribution y on {1,2,...,n}. This leads to the respective costs of xTMy and xTNy for these players. The authors establish the existence of an ∈-equilibrium for this game under the assumption that 𝒜 and ℬ are bounded. When the sets 𝒜 and ℬ are compact in ℝm’×n, 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.
