A Nash equilibrium makes players randomize independently — separate private dice, never touching. A correlated equilibrium lets one shared device whisper a different private recommendation to each player. Nobody is forced to listen. That one change in the plumbing is often worth real payoff — and sometimes lands outside everything Nash can reach. Thirteen games below, from two players up to four.
A shared coin everybody watches is already a kind of mediator, and it is genuinely useful — it settles arguments about which equilibrium to play. What it can never do is reach a payoff that is not some average of equilibria. The thing that breaks that ceiling is not correlation but secrecy: each player learning their own instruction and being left uncertain about everyone else's. Two games, worked all the way through.
The mediator's plan, ⅓ on each shaded cell
| Left | Right | |
|---|---|---|
| Top | ⅓5, 1 | 0, 0 |
| Bottom | ⅓4, 4 | ⅓1, 5 |
Equilibria pay 5, 1 · 1, 5 · 2½, 2½ — totals of 6, 6 and 5. The best is 6, and an average can never beat its best ingredient, so no public coin clears 6. The plan pays 3⅓ each, a total of 6⅔, and it does it by using the 4, 4 cell — which is an equilibrium of nothing.
| Row hears | Then believes | Obey | Switch | Does |
|---|---|---|---|---|
| Top | Left, certainly | 5 | 4 | obeys |
| Bottom | Left or Right, 50/50 | 2.5 | 2.5 | obeys |
Told Bottom, Row cannot tell which of the two Bottom cells was drawn. Obeying is ½(4) + ½(1) = 2.5; switching to Top is ½(5) + ½(0) = 2.5. Dead level — and level is enough.
Half the time, both are told to swerve
| Swerve | Straight | |
|---|---|---|
| Swerve | 50%6, 6 | 25%2, 7 |
| Straight | 25%7, 2 | 0, 0 |
Equilibria pay 2, 7 · 7, 2 · 4⅔, 4⅔ — totals of 9, 9 and 9⅓. The plan pays 5¼ each, a total of 10½, beating every mixture of equilibria. The crash cell is never drawn at all, and mutual swerving — worth the most to the pair, and stable under nothing — carries half the weight.
| Row hears | Then believes | Obey | Switch | Does |
|---|---|---|---|---|
| Swerve | Swerve ⅔, Straight ⅓ | 4.67 | 4.67 | obeys |
| Straight | Swerve, certainly | 7 | 6 | obeys |
Being told to swerve is partial bad news: it makes the other swerving more likely, but not certain. Charging anyway pays ⅔(7) + ⅓(0) = 4.67, exactly what swerving pays. The ⅓ chance of a wreck is what holds the line.
Suppose the signal is public. Everyone sees it, so after it lands each player knows precisely what every other player is about to do — there is no residual uncertainty to hold a belief about.
A profile where everyone knows exactly what the others are doing and still nobody wants to move is, word for word, the definition of a Nash equilibrium. So whatever a public device produces, it produces an equilibrium in each of its branches, and its payoff is an average of equilibrium payoffs. That average lies in the convex hull, and an average never exceeds its largest ingredient.
The ceiling is therefore not about how clever the coin is. Weight it however you like, add as many faces as you like: the ceiling is the best equilibrium, and it does not move.
Now each player sees only their own line. Row hearing "Bottom" learns something about Column — but not everything, and different instructions leave Row with different beliefs. That is the degree of freedom a public coin does not have.
Obedience is then checked belief by belief rather than cell by cell. A cell that would be indefensible if it were common knowledge can survive inside a bundle, because the player cannot tell which cell of the bundle they are in. Aumann's 4, 4 survives by being confused with 1, 5; Chicken's 6, 6 survives by being confused with 2, 7.
Privacy is not a detail of how the mediator is built. It is the entire source of the extra payoff: the mediator is selling calibrated ignorance.
Defection is better than cooperation against Cooperate (5 beats 3) and better against Defect (1 beats 0). Better against everything, so better against every belief.
There is no ignorance worth manufacturing, because no belief the mediator could induce would change anyone's mind. The correlated set collapses onto the single Nash point, and the whole apparatus buys exactly nothing. A mediator needs your best reply to depend on what you think — dominance is precisely the case where it does not.
A traffic light is public: both drivers see the same signal, and it works. It works because its plan only ever puts weight on cells that are already equilibria — one goes, one waits — so the cap on public devices is not binding.
What the light buys is a fair, crash-free split rather than extra payoff. That is the ordinary case, and it is worth seeing clearly before the exotic one: most of the time correlation buys coordination, not surplus. Aumann's example and Chicken are the games where it buys both.
Randomize on your own. My dice and your dice never touch.
You may deviate to any action at all, knowing only the others' mixes. Nothing is ever whispered to you, so there is nothing to condition on.
Obey the whisper. You hear your own instruction and nobody else's.
You may deviate with a different reply to every instruction — swerve when told to swerve, but go straight when told to go straight. Each such swap has to fail.
Commit before you listen. Decide now, hear nothing, play it.
You may deviate only to one fixed action, chosen in advance — the same whatever you would have been told. That is the whole of the coarseness.
Coarse is a statement about the deviations, not about the plans. A correlated equilibrium must survive every reply-to-the-whisper a player could invent — a whole function from instructions to actions. A coarse correlated equilibrium only has to survive the constant functions. Fewer deviations to rule out is a weaker demand, so the coarse set is bigger, and it contains the correlated one. The reason to care: a player who ignores the advice and just plays the same thing every time is exactly a player with no regret, and that is what simple learning algorithms converge to. Coarse correlated equilibrium is where repeated play actually lands, without anyone hiring a mediator.
—
A recommendation is only advice. Each player sees their own instruction, infers what the others were probably told, and asks: can I do better by ignoring it? A correlated equilibrium is exactly a plan where the honest answer is always no. Slack is how much payoff obedience beats the best disobedience by.
Both concepts choose a joint distribution over action profiles — a plan saying how often each cell comes up. The difference is which plans are even available. Nash may only pick plans that factor into independent per-player dice, a curved surface through the space. A correlated equilibrium may pick anything satisfying the obedience inequalities, which is a flat-sided convex body. The surface is thin; the body is solid, and it contains every Nash equilibrium.
Each tick is one round under the mediator, shaded by the total payoff that round produced.