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A Bayesian Nash equilibrium consists of type-dependent strategies such that every type maximizes its conditional expected payoff given the other players' strategies and the type distribution. It is a Nash equilibrium in the strategic description whose actions are whole contingent strategy functions. Auction incentive comparisons can test a deviation by imitating another type's equilibrium bid.

For a symmetric bimatrix game with payoff matrices $P,P^{\mathsf{T}}$, any nonzero $x$ satisfying these equations gives the symmetric Nash equilibrium $(x/{\sum }_i{x}_i,x/{\sum }_i{x}_i)$. Every positive coordinate attains the same maximal payoff. This one-vector construction differs from a two-vector Lemke-Howson algorithm path, which can return asymmetric equilibria.

Complementary pivoting follows adjacent bases while maintaining all but one label of a complementarity system. The Lemke-Howson algorithm for a bimatrix game follows the duplicated label between two tableaux until it restores the dropped label at a nonzero completely labelled pair. Normalization then gives a Nash equilibrium.

Starting from the full finite bimatrix game, sequentially remove actions weakly dominated by current mixed strategies, with a strict improvement somewhere. At least one Nash equilibrium survives on the final action sets. Take an Nash equilibrium of the reduced game and restore actions in reverse. A restored action is no better against the fixed surviving opponent strategy than its dominating mixture of already available actions, so it introduces no profitable deviation. Any self-weight in a dominating mixture can be removed and renormalized because strict improvement forces that weight below one. Using dominance tested only against an arbitrary initially restricted opponent set does not guarantee an Nash equilibrium of the original full game.

The Lemke-Howson algorithm follows an almost completely labeled path in the product of two best-response polytopes of a bimatrix game. Drop one label at the artificial zero pair, then alternately pivot away the duplicated label until the dropped label returns. A nonzero completely labeled endpoint, after normalization, gives a Nash equilibrium. The two players' vectors must be kept separate even in a symmetric game.

A probability distribution over joint action profiles implements a convex combination of their two-player payoff vectors. The attainable expected payoffs form the convex hull of the pure joint payoffs. The distribution need not factor into independent mixed strategies, and a freely enforceable lottery need not satisfy the incentive constraints of a Nash equilibrium.

If contest rewards and costs add, a fixed deviation's expected payoff depends on each rival's per-contest marginal distributions, rather than on dependence between that rival's efforts across contests. Replacing the conditional independent sampling of two active efforts by another copula with the same conditional marginals therefore preserves every deviation payoff. Consequently it preserves the Nash equilibrium, even when the new joint strategy distribution is different. This explains why unique participation probabilities and bid marginals need not imply a unique full mixed strategy.

Nash's theorem states that every finite game has at least one Nash equilibrium in mixed strategies.