Note on non-cooperative convex game

Introduction. Nash's equilibrium-point theorem for many-person games can be approached by two methods: first, the Kakutani-type fixed-point theorem 1 is very useful for this game problem; second, in case of finite-dimensional multilinear payoffs, J. Nash himself has given an elegant procedure In a previous paper [10] one of us proved a general minimax theorem in making use of a procedure analogous to that of Nash. The present note is a continuation of this paper, and its main purpose is to offer further improvements of Nash's method so as to treat noncooperative many-person games played over infinite-dimensional convex sets, based on a generalization of von Neumann's symmetrization method 2 of game matrices. The results thus obtained contain further weakening of (especially topological) assumptions of the equilibriumpoint theorem.