Nashequilibria

Nashequilibria, usually written as Nash equilibria, are a central concept in game theory named after John Nash. A Nash equilibrium is a strategy profile in which no player can improve their payoff by unilaterally changing their strategy, given the strategies chosen by the others. In finite games, equilibria may be pure or mixed; mixed equilibria involve players randomizing over strategies with probabilities that make opponents indifferent among their best responses.

Nash's existence theorem states that every finite normal-form game has at least one Nash equilibrium in mixed strategies. Pure strategy equilibria are special cases. A game may have multiple equilibria, including several pure equilibria, several mixed equilibria, or none in pure form. Equilibria can be sensitive to the precise payoff structure and do not specify a unique outcome when several exist.

Computing equilibria uses the best-response concept: a profile is an equilibrium if each player's strategy is a best response to the others. For two-player games, algorithms such as the Lemke-Howson method can find one or all equilibria; in general, computing equilibria is computationally challenging (PPAD-complete).

Classic examples illustrate the variety: Prisoner's Dilemma has a single pure Nash equilibrium at (Defect, Defect); Battle of the Sexes has two pure equilibria and one mixed equilibrium; Rock-Paper-Scissors has only a mixed equilibrium with each option played with probability 1/3.

Extensions and related concepts include correlated equilibrium, Bayesian Nash equilibrium for games with incomplete information, and evolutionary interpretations such as evolutionarily stable strategies. Nash equilibria provide a foundational predictive notion in economics, political science, biology, and computer science, though they do not always predict which equilibrium will be selected in practice.