### 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.

$(S, \sigma, R, M)$-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.

#### Keywords

Berge equilibrium; Nash equilibrium; game organization; reflexive Banach spaces; existence of solutions; invariance

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