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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.

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.

A bimatrix game is nondegenerate if every mixed strategy with $k$ positive coordinates has at most $k$ pure best responses for the other player. This condition prevents ties in the leaving-variable choice along the usual Lemke-Howson algorithm path and allows unique continuation once the dropped label has been selected.