Matpic
mathematics, in pictures
Concept

Congestion

The situation in which a resource gets worse the more it is used, so what an option is worth depends on how many others take it. Games of this shape always have a pure equilibrium, and a search that lets participants move one at a time to whatever is currently better always finds one.

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

Also named here as equilibrium selection, potential function — the same set of essays touches all of them, so they are one junction rather than several.

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 replyEquilibrium selectionExistence proofNash equilibriumPotential functionConditional probabilityCorrelated equilibriumExhaustive searchGraphMixed strategyTermination

All concepts