A GENERALIZATION OF THE NASH EQUILIBRIUM THEOREM ON BIMATRIX GAMES

In this article, we consider a two-person game in which the first play er picks a row representative matrix M from a nonempty set A of m x n matrices and a probability distribution x on {1, 2,..., m} while the s econd player picks a column representative matrix N from a nonempty se t B of m x n matrices and a probability distribution y on {1,2,..., n} . This leads to the respective costs of x(T)My and x(T)Ny for these pl ayers. We establish the existence of an epsilon-equilibrium for this g ame 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 wh ich stage no player can decrease his cost by unilaterally changing his row/column selection and probability distribution. The result when fu rther specialized to singleton sets, reduces to the famous theorem of Nash on bimatrix games.