Nash theorem (in game theory)
A theorem on the existence of equilibrium points in a mixed extension of a finite non-cooperative game
where and are the finite sets of players and their strategies, respectively, and : is the pay-off function of player (see also Games, theory of). It was established by J. Nash in . Let , , be the set of all probability measures on . Nash' theorem asserts that there is a measure for which
for all , , where denotes the measure from that results from replacing the -th component of the vector by , and . The known proofs of Nash' theorem rely on a fixed-point theorem.
|||J. Nash, "Non-cooperative games" Ann. of Math. , 54 (1951) pp. 286–295|
|||N.N. Vorob'ev, "Foundations of game theory. Non-cooperative games" , Moscow (1984) (In Russian)|
|||N.N. Vorob'ev, "Game theory. Lectures for economists and system scientists" , Springer (1977) (Translated from Russian)|
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