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]] $ \Gamma \stackrel{\text{df}}{=} \langle J,(S_{i})_{i \in J},(H_{i})_{i \in J} \rangle $, where
* $ J $ is a finite set of players,
* $ (S_{i})_{i \in J} $ is their strategy profile, and
* $ H_{i}: S \stackrel{\text{df}}{=} \prod_{i \in J} S_{i} \to \mathbb{R} $ is a pay-off function for player $ i $, for each $ i \in J $ (see also [[Games, theory of|Games, theory of]]).
It was established by J. Nash in [[#References|[1]]]. For each $ i \in J $, let $ M_{i} $ denote the set of all probability measures on $ S_{i} $. Nash’s theorem then asserts that there exists a measure $ \mu^{*} \in M \stackrel{\text{df}}{=} \prod_{i \in J} M_{i} $ such that
$$
\forall i \in J, ~ \forall \mu_{i} \in M_{i}: \qquad
{H_{i}}(\mu^{*}) \geq {H_{i}}(\mu^{*} \| \mu_{i}),
$$
where $ \mu^{*} \| \mu_{i} $ denotes the measure on $ M $ that results from replacing the $ i $-th component of the vector $ \mu^{*} $ by $ \mu_{i} $, and $ {H_{i}}(\mu) \stackrel{\text{df}}{=} \mathsf{E}(H_{i},\mu) $. All known proofs of Nash’s theorem rely on a fixed-point theorem, such as the [[Kakutani theorem|Kakutani Fixed-Point Theorem]] or the [[Brouwer theorem|Brouwer 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. {{ZBL|0045.08202}}</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>