A symmetric equilibrium of a symmetric game has every player use the same strategy or the same distribution of strategies. Symmetry between players does not require identical behavior across contests with different prizes.
In a nondegenerate symmetric bimatrix game, player interchange pairs every nonsymmetric Nash equilibrium with a distinct swapped equilibrium. The equilibrium oddness theorem gives an odd total; the remaining fixed points, which are the symmetric equilibria, therefore also number an odd amount.
The continuous gain map takes a probability simplex to itself. The Brouwer fixed-point theorem supplies a fixed point, whose positive gains would contradict the zero strategy-weighted average payoff excess. Thus all gains vanish, giving a strategy that is a best response to itself. In a symmetric bimatrix game, this produces a symmetric equilibrium.
For a symmetric finite game, pure-action expected payoffs against independent identical opposing mixed strategies are continuous. The displayed map takes the probability simplex continuously into itself. At a Brouwer fixed-point theorem fixed point, put and . The fixed-point equations give . If , every supported action has strictly positive excess payoff, contradicting . Hence and every supported action is a best response, proving a symmetric Nash equilibrium.
Articles by others on the same topic
Symmetric equilibrium is a concept often used in game theory and economics. It refers to a situation in which all players in a game or economic model choose the same strategy, leading to an equilibrium where no player has an incentive to deviate unilaterally from that strategy. In symmetric equilibrium: 1. **Identical Strategies**: All players have access to the same strategies, and they choose the same one.