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Concept

Nash equilibrium

A choice by every participant such that none of them can do better by changing their own while the rest stay fixed. Its existence for finite games follows from a fixed-point theorem, and finding one is a different matter from knowing it is there.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

The value of a 2×3 zero-sum game, named from both sides. The row chooser's expected payoff against each column as a line over the mixing probability, with the lower envelope and its maximum, beside the same construction from the column chooser's side. Both give 19/15.

The value from both sides

Two choosers move at the same instant, and each asks the cautious question — how much can be guaranteed, whatever the other does. With pure choices the two answers are usually different numbers; allow a probability and they are forced to be the same one.

applied · Equilibrium The link that makes every traveller later. Four nodes and two routes, with the equilibrium flow and travel time before a zero-cost link is added between A and B and after. The travel time rises from 10 to 12.

The road that makes everyone later

An equilibrium is a state nobody can improve alone, which is a much weaker thing than a state anybody would choose. Adding a link that costs nothing to use makes every traveller in this network strictly slower, and the arithmetic says by exactly how much.

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 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

Named alongside it

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

Best replyMixed strategyCorrelated equilibriumExistence proofConditional probabilityCongestionConvexityDominant strategyEquilibrium selectionExhaustive searchGraphLinear program

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