On Maximal Vector Spaces of Finite Non-Cooperative Games

We consider finite non-cooperative 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 non-cooperative m1 ? m2 ? . . . ? mN - games in mixed strategies such that all games of a broader vector space of non- cooperative N person games on the product of unit (mi ? 1)-dimensional simpleces 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