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Two cars reach a crossing at the same moment. If both go, they crash; if both wait, they sit there. A traffic light solves this without changing anyone’s incentives. It just tells each driver, privately, what to do, and both are happy to obey.

Correlated equilibrium

Nash equilibrium lets players randomize on their own, independently. But a shared signal, a light, a mediator, a public event, can correlate their choices, and that opens outcomes no independent play can reach. In Chicken, a device that whispers a different instruction to each driver gives both a higher expected payoff than the symmetric mixed equilibrium. Its total payoff also exceeds either pure equilibrium, though it does not make the favoured player in a pure equilibrium better off.

From earlier lessons

Two quick ones before we start.

You mix your own moves to...

A focal point selects among equilibria using...

The game: Chicken

Two drivers race head-on. Each can Swerve or go Straight. If both go Straight they crash (worst for both). If one swerves and the other does not, the one who held their nerve wins. Find the pure equilibria.

Two pure equilibria, and each hands the win to one driver: not much help when you are the one told to swerve. There is a symmetric mixed equilibrium too, but it risks the crash and pays only about $4.67$ each.

Predict, then play

Now add a mediator. Each round it sends you a private recommendation, Swerve or Straight, drawn so that the pair never both go Straight. The other driver follows their own signal. Should you follow yours?

Why did that happen?

Take a guess, then reveal the reason

Independent mixing sometimes lands you both on Straight, which is the crash. A shared signal removes exactly that case: it recommends the three safe cells only, never the disaster. And you obey not because anyone forces you, but because, given what your recommendation tells you about the other driver, following it is your best reply. The correlation, not any enforcement, is what buys the extra payoff.

The math

Mixed Nash first. Let each driver go Straight with probability $q$ . The opponent is indifferent when $6 - 4q = 7 - 7q$ , so $q = \tfrac13$ , and the value is $6 - 4\cdot\tfrac13 = \tfrac{14}{3} \approx 4.67$ each.

Now the mediator draws uniformly from the three cells that are not (Straight, Straight): $(\text{Sw},\text{Sw})$ , $(\text{Sw},\text{St})$ , $(\text{St},\text{Sw})$ , and privately tells each player their part. Expected payoffs are

\tfrac13(6) + \tfrac13(2) + \tfrac13(7) = 5 \quad\text{for each player},

above the mixed Nash. For this to be an equilibrium, obeying must be a best response given what the recommendation reveals about the opponent.

Definition, correlated equilibrium

A correlated equilibrium is a probability distribution over outcomes, run by a device that privately recommends a move to each player, such that no player can gain by deviating from their recommendation given the belief it induces about the others. Every Nash equilibrium is one (an uninformative device). Some correlated equilibria improve on a mixed Nash equilibrium or raise total welfare, but they need not Pareto-dominate every Nash equilibrium.

Work it out

Check that obeying is a best response

Told Swerve: this happens in $(\text{Sw},\text{Sw})$ and $(\text{Sw},\text{St})$ , so the opponent is equally likely to Swerve or go Straight. Obeying (Swerve) pays $\tfrac12(6) + \tfrac12(2) = 4$ ; deviating (Straight) pays $\tfrac12(7) + \tfrac12(0) = 3.5$ . Obey.

Told Straight: this happens only in $(\text{St},\text{Sw})$ , so the opponent surely swerves. Obeying (Straight) pays $7$ ; deviating (Swerve) pays $6$ . Obey.

Both recommendations are self-enforcing, so the device is a correlated equilibrium, and it averages $(5,5)$ . That beats the mixed Nash $4.67$ for both players, and beats the pure equilibria on total payoff ( $10$ against $9$ ), though the driver who would have held firm at $(7,2)$ does worse.

Key idea

Let a shared signal correlate players’ choices and you can reach outcomes independent randomization cannot. Obedience needs no enforcement, only that following the recommendation is each player’s best reply.

Check yourself

Check yourselfapply

The mediator averages (5,5), above the mixed Nash 4.67. Where does the extra payoff come from?

Check yourselfanalyse

You are told Swerve. Why is obeying rational rather than grabbing Straight for the higher 7?

Check yourselfbrings back: mixed strategies

How does a correlated equilibrium relate to the mixed-strategy Nash equilibria you already know?

Sources

Aumann, R. J. (1974). Subjectivity and correlation in randomized strategies. Journal of Mathematical Economics, 1(1), 67-96.