We consider finite noncooperative [Formula: see text] person games with fixed numbers [Formula: see text], [Formula: see text], of pure strategies of Player [Formula: see text]. We propose the following question: is it possible to extend the vector space of finite noncooperative [Formula: see text]-games in mixed strategies such that all games of a broader vector space of noncooperative [Formula: see text] person games on the product of unit [Formula: see text]-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.