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Concept

Equilibrium selection

The question of which equilibrium happens when a game has several and the definition prefers none. The standard tie-breakers - preferring the one that pays more, or the one that is safer against an unknown opponent - can disagree, and the safer one usually has the larger basin under any best-reply rule.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

A landscape nobody is looking at, and every move goes downhill on it. The 8 states of a congestion game with 3 participants and two resources, ordered by Rosenthal's potential, with every improving unilateral move drawn as an arrow. Every arrow points downward.

The landscape nobody is looking at

Letting participants move one at a time to whatever is currently better can cycle forever, and on a network of congestible roads it cannot. The reason is a single number attached to each state that falls by exactly what the mover saves.

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 A population that settles at 2/3. A contest over a prize worth 4 that costs 6 to fight for. Left: the growth rate of the share playing Hawk against that share, which is nought at 0, at 2/3 and at 1. Right: the share over time from 5 starting points, all converging on 2/3.

A mixture that is a population

A mixed equilibrium between two choosers is a knife-edge nobody has a reason to stand on. Read the same mixture as a population whose shares grow with how well they do, and it becomes a point every population is carried to — or one every population circles for ever without arriving.

applied · Equilibrium 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

Named alongside it

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

Best replyNash equilibriumExistence proofMixed strategyCongestionPotential functionCommitmentConditional probabilityCorrelated equilibriumDominant strategyEvolutionary stabilityExhaustive search

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