Matpic
mathematics, in pictures
Concept

Commitment

A move made and made binding before the other side chooses, which changes what the other side's best reply is. Its value comes from the other side's behaviour rather than from the move itself, so reducing one's own options can raise one's payoff — and everything it buys depends on the binding being real.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Announcing a mixture is worth 5/3 more than any equilibrium. The leader's payoff against the probability it announces for its first action, for a leader with a dominant action that is better off not being seen to play it, with the follower's reply switching where the follower is indifferent. The best announcement is worth 11/3; the best equilibrium of the simultaneous game is worth 2.

Worth more for being seen first

Moving first sounds like a disadvantage, since the other side gets to see the move and answer it. When the move is a mixture that is announced and believed, it is never a disadvantage, it is worth exactly nothing in a game of pure conflict, and in other games it is worth more than any equilibrium — sometimes by announcing an action that would never be played in secret.

applied · Equilibrium The payoffs repetition makes available in the prisoner's dilemma. A plot of the two choosers' average payoffs, with the stage game's four cells marked, their convex hull drawn, the two minmax values shown as lines, and the region above both shaded.

Patience instead of a contract

Commitment had to assume an announcement binds. Play the same game again tomorrow and the assumption is unnecessary — the future does the binding. What it costs is that nearly every outcome becomes an equilibrium, so a theory that could not choose between two now cannot choose between infinitely many.

applied · Equilibrium

Named alongside it

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

Best replyEquilibrium selectionMinimaxMixed strategyNash equilibriumZero-sum gameConvexityDominant strategy

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