Nash equilibrium
= Nash equilibrium
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{title2=$\sigma^*$}
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= Nash equilibria
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A <Nash equilibrium> is a <strategy profile> in which every player's strategy is a <best response> to the others. For a <finite game> and <mixed strategies>, its precise condition is
$$
U_i(\sigma_i^*,\sigma_{-i}^*)\geq U_i(\tau_i,\sigma_{-i}^*)
\quad\text{for every player }i\text{ and every }\tau_i\in\Delta(S_i).
$$
The opponents' strategies $\sigma_{-i}^*$ are held fixed while testing each <unilateral deviation>. Equality is allowed: a player may have other equally good <best responses>. Because <expected payoff> is a <linear function> of the player's own mixture, it suffices to test every <pure strategy>. Equivalently, every choice in the <support of a mixed strategy> must attain the largest pure-choice <expected payoff>, and no unused choice may exceed it.
The two worked examples illustrate different uses of this condition. <Matching pennies> has no <pure strategy> <Nash equilibrium>, but its unique <mixed strategy> <Nash equilibrium> removes each player's vulnerability to predictable play. The <asymmetric coordination example> has three <Nash equilibria>, and its mixed one fails <Pareto efficiency>. A <Nash equilibrium> tests individual incentives, whereas <Pareto efficiency> compares feasible outcomes that may require several players to change together. In the <prisoner's dilemma>, for example, mutual defection is a <Nash equilibrium> even though mutual cooperation gives both players a larger <payoff> because $R>P$.
Existence in a <finite game> is guaranteed by <Nash's theorem>. The theorem does not select one <Nash equilibrium>, promise <Pareto efficiency>, or assert convergence of a learning process. Nash's https://pmc.ncbi.nlm.nih.gov/articles/PMC1063129/[1950 existence paper]{ref} used the <Kakutani fixed-point theorem>; his https://cs.uwaterloo.ca/~klarson/teaching/F04-886/papers/nash51.pdf[1951 paper, section Existence of Equilibrium Points]{ref} gave a proof using the <Brouwer fixed-point theorem>. The connection to a <fixed point> captures the idea of choices that remain mutually optimal when considered together.