Nash equilibrium

A strategy profile in a game where no player can improve their outcome by changing strategy alone, given the other players’ strategies. It need not maximize total welfare or be unique.

Best response: A strategy that gives a player the highest payoff given the other players’ choices. An equilibrium occurs when every player’s chosen strategy is a best response to the others.
Prisoner's dilemma: A two-player game in which individually rational choices produce a worse outcome for both than mutual cooperation. Defection by both players is an equilibrium, despite being jointly worse than cooperation.
Pareto efficiency: A state is Pareto efficient if no feasible change can improve someone’s outcome without worsening another’s. A Nash equilibrium may be Pareto inefficient, as the Prisoner’s Dilemma demonstrates.
Price of anarchy: A measure comparing the social cost of a worst equilibrium with the optimal social cost. It quantifies how much decentralized equilibrium behavior can worsen system performance.
Payoff matrix: A table listing each player’s payoff for every combination of strategies in a finite game. In two-player games, equilibrium candidates can be found by comparing payoffs across each row and column.
Battle of the sexes (game theory): A coordination game in which two players prefer meeting but disagree over which shared activity to choose. Its multiple equilibria show how players can coordinate on outcomes that favor different people.
Correlated equilibrium: A probability distribution over strategy profiles where no player benefits from deviating after receiving a private recommendation. It permits shared signals to coordinate behavior beyond independent mixed strategies.
Mechanism design: The study of designing rules and institutions to achieve desired outcomes when participants have private information and incentives. Designers often seek rules whose equilibria implement socially desirable outcomes.
Dominant strategy: A strategy that yields a player at least as high a payoff as every alternative, whatever others choose. Equilibrium does not require dominant strategies; each choice need only respond optimally to the actual profile.
Chicken (game): A game in which each player prefers yielding if the other stands firm, while mutual stubbornness is disastrous. Its asymmetric equilibria illustrate how strategic conflict can settle without a mutually best outcome.

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