A generalization of the Nash equilibrium theorem on bimatrix games

摘要

In this article, we consider a two-person game in which the first player picks a row representative matrixMfrom a nonempty set$$A$$ofm×nmatrices and a probability distributionxon {1,2,...,m} while the second player picks a column representative matrixNfrom a nonempty set ℬ ofm×nmatrices and a probability distribution y on 1,2,...,n. This leads to the respective costs ofxtMyandxtNyfor 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 ℝmxn, 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 bima