Two drivers race at one intersection. Alone, the best they reach is a coin-flip that sometimes crashes, or one bullying the other. But hang a traffic light between them — a shared random signal, privately whispering "go" to one and "stop" to the other — and no one wants to disobey the whisper. Aumann proved this in 1974: a correlation independent players cannot manufacture, that beats every Nash outcome. Terms go in, the mediator's distribution is tested, the verdict comes out. The blue team builds and defends; the red team tries to slip a fake past the check.
source R. J. Aumann, Subjectivity and Correlation in Randomized Strategies, Journal of Mathematical Economics 1 (1974) 67–96 — the first paper that journal ever ran. Stable copy: dklevine.com/archive/refs4389.pdf. Rendered, not quoted.
A mediator holds a distribution p over joint action pairs. It draws one pair and privately tells each player only their own coordinate as a recommendation. p is a correlated equilibrium iff obeying is a best response to the posterior your recommendation induces — for every player i, every recommended action a, every alternative b:
∑other p(a, other) · [ ui(a, other) − ui(b, other) ] ≥ 0
Linear in p, checked exactly. Below: the current distribution's tightest deviation gain per recommendation (negative = obedience strictly preferred).
| player · rec. | obey | best deviate | slack |
|---|
Take the-nash-equilibrium and hand the players a traffic light. Every Nash point survives — a product distribution over independent randomizers is always a correlated equilibrium (window 4 lets you load one and watch it pass).
But correlation reaches strictly further: outcomes independent players cannot build alone, because private randomizers can never be correlated. And this is the fixed target that no-regret learning walks toward — the-regret-matching converges into the CE set, never to a single Nash. Each sphere is the next one's premise.
The blue team's live check: re-verify the CE's constraints exactly, confirm every Nash is a CE, confirm the CE strictly beats every Nash total, and re-check convexity. If red tampers, this badge is where it shows.
Two drivers speed at each other; each picks C (chicken — swerve) or D (dare — hold straight). Aumann's payoff matrix (row, col):
| col C | col D | |
|---|---|---|
| row C | 6, 6 | 2, 7 |
| row D | 7, 2 | 0, 0 |
Two pure Nash equilibria — (D,C)→7,2 and (C,D)→2,7, each side wanting to be the bully — plus a mixed one (each dares w.p. 1/3) worth 14/3≈4.67 apiece. The crash (D,D) is what the light must route around. Feed a mediator distribution into the panel below.
Set the mediator's weight on each joint pair; the panel normalizes to a distribution and tests it live.
| col C | col D | |
|---|---|---|
| row C | ||
| row D |
Every number is computed from the constraints on the spot — never looked up. Push weight onto the crash and a deviation turns profitable; the verdict flips.
What the machine proves: the distribution putting 1/3 each on (C,C), (C,D), (D,C) — and zero on the crash — is a correlated equilibrium worth (5, 5), total 10. Every Nash tops out at total 28/3≈9.33. So the light delivers a payoff pair that sits strictly outside the convex hull of the Nash payoffs — a win independent play cannot reach.
The blue team's witness (left) confirms this live; the red team (right) tries to make a fake pass.
It also says nothing about which CE gets played (the set is infinite) or how a real light gets built without a referee. And it is a one-shot solution concept: it predicts a stable recommendation, not that anyone will learn or agree to it.
"Correlated equilibrium just means the players randomize together." Cut. Shared randomness is necessary but not enough — it is a CE only if no recommendation ever makes deviating profitable. The engine checks that constraint, not mere correlation.
"A CE can't beat Nash — equilibrium is equilibrium." Cut. On Chicken the CE scores 10 vs. the best Nash total 28/3≈9.33 — strictly outside the Nash hull, computed live.
"You need a fair coin." Kept, corrected. The signal need not be uniform — the 1/3-1/3-1/3-0 weights are exactly what makes both incentive constraints bind at once.
The red team's move: shove weight onto the crash to lift the total, then relabel that crash-weighted mess "the correlated equilibrium." The blue team's witness (window 7) is watching.
The forgery moves mass onto (D,D). Now a driver told "dare" would rather swerve — a strictly profitable deviation. The constraint check recomputes, exposes the deviation, and the witness turns red. Nothing is faked; the attack is real and it is caught.