Correlated equilibrium

Formal definition

An N -player strategic game ( N , A i , u i ) is characterized by an action set A i and utility function u i for each player i . When player i chooses strategy a i ∈ A i and the remaining players choose a strategy profile described by the N − 1 -tuple a − i , then player i 's utility is u i ( a i , a − i ) .

A strategy modification for player i is a function ϕ i : A i → A i . That is, ϕ i tells player i to modify his behavior by playing action ϕ i ( a i ) when instructed to play a i .

Let ( Ω , π ) be a countable probability space. For each player i , let P i be his information partition, q i be i 's posterior and let s i : Ω → A i , assigning the same value to states in the same cell of i 's information partition. Then ( ( Ω , π ) , P i , s i ) is a correlated equilibrium of the strategic game ( N , A i , u i ) if for every player i and for every strategy modification ϕ i :

In other words, ( ( Ω , π ) , P i ) is a correlated equilibrium if no player can improve his or her expected utility via a strategy modification.

An example

Consider the game of chicken pictured. In this game two individuals are challenging each other to a contest where each can either dare or chicken out. If one is going to Dare, it is better for the other to chicken out. But if one is going to chicken out it is better for the other to Dare. This leads to an interesting situation where each wants to dare, but only if the other might chicken out.

In this game, there are three Nash equilibria. The two pure strategy Nash equilibria are (D, C) and (C, D). There is also a mixed strategy equilibrium where each player Dares with probability 1/3.

Now consider a third party (or some natural event) that draws one of three cards labeled: (C, C), (D, C), and (C, D), with the same probability, i.e. probability 1/3 for each card. After drawing the card the third party informs the players of the strategy assigned to them on the card (but not the strategy assigned to their opponent). Suppose a player is assigned D, he would not want to deviate supposing the other player played their assigned strategy since he will get 7 (the highest payoff possible). Suppose a player is assigned C. Then the other player will play C with probability 1/2 and D with probability 1/2. The expected utility of Daring is 0(1/2) + 7(1/2) = 3.5 and the expected utility of chickening out is 2(1/2) + 6(1/2) = 4. So, the player would prefer to Chicken out.

Since neither player has an incentive to deviate, this is a correlated equilibrium. The expected payoff for this equilibrium is 7(1/3) + 2(1/3) + 6(1/3) = 5 which is higher than the expected payoff of the mixed strategy Nash equilibrium.

The following correlated equilibrium has an even higher payoff to both players: Recommend (C, C) with probability 1/2, and (D, C) and (C, D) with probability 1/4 each. Then when a player is recommended to play C, she knows that the other player will play D with (conditional) probability 1/3 and C with probability 2/3, and gets expected payoff 14/3, which is equal (not less than) to the expected payoff when she plays D. In this correlated equilibrium, both players get 5.25 in expectation. It can be shown that this is the correlated equilibrium with maximal sum of expected payoffs to the two players.

Learning correlated equilibria

One of the advantages of correlated equilibria is that they are computationally less expensive than Nash equilibria. This can be captured by the fact that computing a correlated equilibrium only requires solving a linear program whereas solving a Nash equilibrium requires finding its fixed point completely. Another way of seeing this is that it is possible for two players to respond to each other's historical plays of a game and end up converging to a correlated equilibrium.