View source for Nash theorem (in game theory)

A theorem on the existence of equilibrium points in a mixed extension of a finite [[Non-cooperative game|non-cooperative game]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066040/n0660401.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066040/n0660402.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066040/n0660403.png" /> are the finite sets of players and their strategies, respectively, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066040/n0660404.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066040/n0660405.png" /> is the pay-off function of player <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066040/n0660406.png" /> (see also [[Games, theory of|Games, theory of]]). It was established by J. Nash in [[#References|[1]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066040/n0660407.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066040/n0660408.png" />, be the set of all probability measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066040/n0660409.png" />. Nash' theorem asserts that there is a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066040/n06604010.png" /> for which

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for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066040/n06604012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066040/n06604013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066040/n06604014.png" /> denotes the measure from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066040/n06604015.png" /> that results from replacing the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066040/n06604016.png" />-th component of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066040/n06604017.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066040/n06604018.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066040/n06604019.png" />. The known proofs of Nash' theorem rely on a fixed-point theorem.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Nash,   "Non-cooperative games"  ''Ann. of Math.'' , '''54'''  (1951)  pp. 286–295</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.N. Vorob'ev,   "Foundations of game theory. Non-cooperative games" , Moscow  (1984)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.N. Vorob'ev,   "Game theory. Lectures for economists and system scientists" , Springer  (1977)  (Translated from Russian)</TD></TR></table>