# Nash theorem (in game theory)

A theorem on the existence of equilibrium points in a mixed extension of a finite 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).
It was established by J. Nash in . 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 Fixed-Point Theorem or the Brouwer Fixed-Point Theorem.