The concepts of $(S, \sigma)$-invariance and $(S, \sigma, R, M)$-invariance are introduced and are used to prove two existence theorems of equilibrium in the sense of Berge [2] and Nash [1, 2] using fixed point arguments. Radjef's results [8] have been extended. Conditions under which these equilibria are Nash are also shown.

Assuming that each player's strategy set is a subset of a reflexive Banach space and that the strategies can be partitioned in such a way that the argmax of each player's objective over an element of the considered partition is unique and satisfies one of the invariance properties, equilibria exist. Similar results are obtained for games with an infinite number of players.