correlated equilibrium

Definition

Let $\Gamma = (N, (S^i)_{i \in N}, (u^i)_{i \in N})$ be a finite $N$-person game in strategic normal form, with $N$ denoting the set of players, $S^i$ the set of strategies, $u^i: \prod_{i \in N} S^i \to \mathbb{R}$ is player $i$'s payoff function. Generic element of $S$ is $s = (s^i)_{i \in N}$, $s^{-i} = (s^{i'})_{i' \neq i}$ is strategy combination of all players except $i$.

A probability distribution $\psi$ on $S$ is a correlated equilibrium of $\Gamma$ if, $\forall i \in N$, $\forall j \in S^i$, $\forall k \in S^i$, we have $$\sum_{s \in S: s^i = j} \psi(s) [u^i(k,,s^{-i}) - u^i(s)] \leq 0$$

Define a correlated $\epsilon$-equilbrium if the right-hand side above is replaced by $\epsilon$.

Notes

Every Nash equilibrium is a correlated equilibrium, the special case where $\psi$ is a product measure i.e. the play of different players is independent.

References

1. Hart S, Mas-Colell A. A Simple Adaptive Procedure Leading to Correlated Equilibrium. Econometrica, 2000; 68(5): 1127-1150.