On Maximal Vector Spaces of Finite Noncooperative Games

We consider finite noncooperative N person games with fixed numbers mi, i = 1,…,N, of pure strategies of Player i. We propose the following question: is it possible to extend the vector space of finite noncooperative (m1 × m2 ×⋯ × mN)-games in mixed strategies such that all games of a broader vector space of noncooperative N person games on the product of unit (mi − 1)-dimensional simplices have Nash equilibrium points? We get a necessary and sufficient condition for the negative answer. This condition consists of a relation between the numbers of pure strategies of the players. For two-person games the condition is that the numbers of pure strategies of the both players are equal.