Nash Equilibrium

A profile of strategies where no player can profit by unilateral deviation.

Ask the Game theory assistant1 min read · Updated September 9, 2026

Definition

A strategy profile $s^*=(s_1^*,\ldots,s_n^*)$ is a Nash equilibrium if $u_i(s_i^*,s_{-i}^*)\ge u_i(s_i,s_{-i}^*)$ for every player $i$ and every $s_i\in S_i$ .

Equivalently, $s_i^*\in\arg\max_{s_i\in S_i} u_i(s_i,s_{-i}^*)$ for each $i$ — every player best-responds to everyone else.

Intuition

No player regrets their choice given what the others did — it's a self-enforcing agreement.

It is a fixed point of the joint best-response map: nobody has a profitable unilateral deviation.

Worked example

In the Prisoner's Dilemma the unique pure Nash equilibrium is mutual defection $(D,D)$ , even though $(C,C)$ Pareto-dominates it.

Matching Pennies has no pure equilibrium but a unique mixed one in which each player randomizes $(1/2,1/2)$ .

The math

Nash (1950) proved existence in every finite game by applying Kakutani's fixed-point theorem to the best-response correspondence on the compact convex set $\Delta(S)=\prod_i\Delta(S_i)$ .

Equivalently, equilibria are the zeros of the Nash map where each player minimizes their regret $\max_{s_i'}u_i(s_i',s_{-i})-u_i(s)$ .

Where it is used

Central to oligopoly theory, routing games, auction design, and evolutionary biology.

Provides the baseline benchmark for the Price of Anarchy and mechanism design guarantees.

Go deeper

GuideFind a Nash Equilibrium with Best Responses

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