On the Value of Some Infinite Matrix Games

It is shown that a zero-sum two-person noncooperative game A defined by a bounded infinite matrix in which each row converges to the same real number β and each column to the same real number α has a value V(A) if and only if α ≤ β, in which case α ≤ V(A) ≤ β. For any game defined by a bounded infinite matrix A = (a_ij), a necessary condition for V(A) to exist is that inf_j lim inf_ia_ij ≤ sup_i lim sup_ja_ij.