Nash theorem (in game theory)
From Encyclopedia of Mathematics
A theorem on the existence of equilibrium points in a mixed extension of a finite non-cooperative game
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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
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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.
References
[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: http://encyclopediaofmath.org/index.php?title=Nash_theorem_(in_game_theory)&oldid=18406
Nash theorem (in game theory). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nash_theorem_(in_game_theory)&oldid=18406
This article was adapted from an original article by E.B. Yanovskaya (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article