Correlated Equilibrium

An equilibrium in which players obey recommendations from a common randomization device.

Ask the Game theory assistant 1 min read · Updated September 9, 2026

Definition

A distribution $\mu\in\Delta(S)$ is a correlated equilibrium if $\sum_{s_{-i}}\mu(s_i,s_{-i})\bigl(u_i(s_i,s_{-i})-u_i(s_i',s_{-i})\bigr)\ge 0$ for all $i,s_i,s_i'$ .

Given the recommendation $s_i$ , following it weakly dominates any deviation $s_i'$ in expectation over $s_{-i}\mid s_i$ .

Intuition

A trusted signal coordinates play — think of a traffic light rotating priorities.

Every Nash equilibrium is correlated, but correlated equilibria can achieve strictly higher welfare.

Worked example

In Chicken with payoffs $(0,0),(4,1),(1,4),(-5,-5)$ , correlated play with $(D,S),(S,D),(S,S)$ each with probability $1/3$ gives expected payoff $10/3$ — higher than any Nash.

Traffic lights implement a correlated equilibrium between arriving drivers.

The math

Aumann (1974): the set of correlated equilibria is convex, contains all Nash equilibria, and can be computed via linear programming.

No-regret dynamics converge to the set of correlated equilibria in every finite game (Hart-Mas-Colell).

Where it is used

Explains social norms and conventions as coordination devices.

Computationally attractive and learnable via simple algorithms.

More in Game theory

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