Nash theorem (in game theory)

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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 [1]. 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.


[1] J. Nash, "Non-cooperative games" Ann. of Math. , 54 (1951) pp. 286–295
[2] N.N. Vorob'ev, "Foundations of game theory. Non-cooperative games" , Moscow (1984) (In Russian)
[3] N.N. Vorob'ev, "Game theory. Lectures for economists and system scientists" , Springer (1977) (Translated from Russian)
How to Cite This Entry:
Nash theorem (in game theory). Encyclopedia of Mathematics. URL:
This article was adapted from an original article by E.B. Yanovskaya (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article