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Correlated equilibrium

A distribution over the cells of a game such that neither party, told privately which of their own options was drawn, prefers to do something else. It generalises Nash equilibrium by dropping the requirement that the two randomisations be independent, and the set it defines is a polytope rather than a fixed point.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

A signal both can see, and neither wants to disobey. A two-by-two game with a distribution over its four cells, drawn as the weight on each. Obeying the recommendation is a best reply for both choosers, and the pair collects 21/2 between them.

A signal both can see

Two choosers who randomise privately can reach a set of outcomes that is smaller, and worse, than the set they reach when a device draws one cell and whispers each of them their half of it. Nothing is enforced and nobody is bound, and the arrangement is stable anyway.

applied · Equilibrium A signal both can see, and neither wants to disobey. A two-by-two game with a distribution over its four cells, drawn as the weight on each. Obeying the recommendation is a best reply for both choosers, and the pair collects 21/2 between them.

A signal both can see

Two choosers who randomise privately can reach a set of outcomes that is smaller, and worse, than the set they reach when a device draws one cell and whispers each of them their half of it. Nothing is enforced and nobody is bound, and the arrangement is stable anyway.

applied · Equilibrium Two equilibria, and two tests that disagree. The row chooser's expected payoff from each option against the column chooser's behaviour, for a joint effort worth more than a safe one. The lines cross at 0.750, which is the mixed equilibrium and the boundary between the two basins.

Two equilibria and no way to choose

A game can have two states nobody wants to leave, one paying more than the other, and the definition of an equilibrium has nothing to say about which happens. The two standard tie-breakers disagree, and the one that wins is usually the worse.

applied · Equilibrium

Named alongside it

The objects these essays reach for when they reach for this one.

Best replyConditional probabilityMixed strategyNash equilibriumExhaustive searchLinear programZero-sum gameCongestionEquilibrium selectionExistence proofPotential function

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