Glossary

Nash Equilibrium

A Nash equilibrium is a strategy profile $s^* = (s_1^*, s_2^*, \ldots, s_n^*)$ in which each player $i$ 's strategy $s_i^*$ is a best response to the strategies of all other players $s_{-i}^*$ . Formally, no player can increase their payoff $u_i$ by switching to any other strategy $s_i \in S_i$ while everyone else holds fixed. This mutual best-response requirement is equivalent to finding a fixed point of the best-response correspondence: the function that maps each player's opponents' strategies to that player's optimal reply. Nash (1950) used Kakutani's fixed-point theorem to prove that every finite game has at least one equilibrium — in pure or mixed strategies.

The plain-English reading: a Nash equilibrium is a stable resting point of strategic interaction. Once every player is playing their equilibrium strategy, nobody regrets their own choice given what the others did — there is no profitable unilateral deviation. Notice the qualifier: unilateral. A Nash equilibrium says nothing about coordinated deviations, and it need not be Pareto efficient. In the prisoner's dilemma, both players confessing is the unique Nash equilibrium, yet both would be better off if they could commit to silence — the equilibrium is Pareto dominated. When no pure-strategy equilibrium exists (as in matching pennies), Nash's theorem guarantees one in mixed strategies: each player randomizes over actions, and in a mixed equilibrium every action played with positive probability yields the same expected payoff — the indifference condition that pins down the mixing probabilities.

u_i(s_i^*, s_{-i}^*) \ge u_i(s_i, s_{-i}^*) \quad \text{for all } s_i \in S_i,\ \text{for every player } i

nash-equilibrium-condition

Nash equilibrium condition.

s_i^*

: player

i

's equilibrium strategy.

s_{-i}^*

: the equilibrium strategies of all other players, held fixed.

u_i

: player

i

's payoff function.

S_i

: player

i

's full strategy set. The inequality must hold for every player

i

simultaneously.

Where you'll see it

Nash equilibrium is the workhorse solution concept of non-cooperative game theory. In Cournot oligopoly, each firm's output choice is a best response to rivals' quantities, and the equilibrium pins down industry output and price. In Bertrand competition, firms undercut until price equals marginal cost — the unique equilibrium. The prisoner's dilemma is the canonical illustration of a Pareto-inefficient Nash equilibrium. Coordination games such as the Battle of the Sexes can have multiple pure-strategy equilibria and one mixed-strategy equilibrium. In auction theory, bidding strategies in a first-price sealed-bid auction form a Bayesian Nash equilibrium — the appropriate extension when players have private information. Virtually every result in industrial organization, political economy, and mechanism design builds on Nash equilibrium or a refinement of it (subgame perfection, sequential equilibrium, and so on).

Best response: The strategy $s_i^*$ that maximizes player $i$ 's payoff $u_i$ given the strategies $s_{-i}$ of all other players. A Nash equilibrium is precisely the profile where every player is simultaneously on their best-response correspondence — a mutual fixed point.
Mixed strategy: A probability distribution over a player's pure strategies. Pure strategies are the degenerate case where one action receives probability 1. In a mixed-strategy Nash equilibrium, each player must be indifferent across every action they play with positive probability; if one action were strictly better, the player would not be willing to randomize.
Dominant strategy: A strategy $s_i$ that is a best response to every strategy profile of the opponents — not just the equilibrium one. If every player has a dominant strategy, the dominant-strategy profile is a Nash equilibrium (and survives iterated deletion of dominated strategies). Most games have no dominant strategy, which is why Nash equilibrium is the more general concept.
Pareto efficiency: An outcome where no player can be made better off without making at least one other player worse off. A Nash equilibrium can be — and often is — Pareto inefficient: the prisoner's dilemma is the textbook example, where the equilibrium (confess, confess) is dominated by the cooperative outcome (silent, silent) that the players cannot sustain without a binding commitment.