Fixed points, Nash games and their organizations

Abstract

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.