Concept

Price of anarchy

The ratio between the worst cost of a stable outcome and the cost of the best outcome a planner could impose. Stable means that no one can gain by changing course alone, and the ratio measures how much a system loses by leaving choices to the individuals in it.

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

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

Where sellers stand when nobody wants to move. 2 sellers: 1 equilibria (e.g. 12,12 of 24); 3 sellers: 0 equilibria; 4 sellers: 3 equilibria (e.g. 6,6,17,18 of 24); 5 sellers: 4 equilibria (e.g. 4,4,12,19,20 of 24); 6 sellers: 25 equilibria (e.g. 3,3,9,14,20,21 of 24).

Sellers on a street stand back to back

Put two ice-cream sellers on a beach and they end up side by side in the middle, though customers would walk half as far if they stood at the quarter marks. Three sellers never settle at all: whoever is squeezed in the middle leaps outside, for ever. Four stand in pairs, five in pairs with one alone in the centre, six in endlessly many ways. Let them set prices too, with travel costing the square of the distance, and the conclusion reverses: the two sellers move as far apart as the beach allows.

applied · Equilibrium

Named alongside it

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

Best reply Mixed strategy Nash equilibrium Braess paradox Convexity Dominant strategy Exhaustive search Graph Median voter Voronoi diagram

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