# 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

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.

#### 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

This article was adapted from an original article by E.B. Yanovskaya (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article