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Concept

Allocation

A way of dividing a fixed total between a fixed set of recipients, given as one number for each. It is the object every rule for sharing produces, and what distinguishes one rule from another is which allocations it is prepared to return.

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

Every order of arrival for three partners, and what each player adds. A table with one row per order in which the players could arrive, giving what each adds to the group already present, and the average of each column as that player's share.

The order everybody arrives in

Three people jointly earn nine, and the question is what each is owed. Ask instead what each adds on walking into a room the others are already in, average that over every order they could have arrived in, and four modest conditions leave no other answer.

applied · Shapley value The splits no group can beat, for three partners. The triangle of ways to split a fixed total between three players, with each coalition's demand drawn as a straight cut across it, and the region surviving every cut shaded.

A split nobody can walk away from

Every way of dividing what a group earns is a point of a triangle, and every coalition's threat to leave cuts a straight line across it. What survives all the cuts is the set of stable divisions — and for one three-player game there is nothing left.

applied · The core Maximising the product of two people's values. The frontier of value pairs from dividing 4 goods between two people, with the points maximising the product, the sum and the smaller value. The product's maximum is (65.0, 54.2) and is envy-free.

The product that makes a division fair

Divide goods to make the total happiness as large as possible and the result can be monstrously unfair; make the least happy person as happy as possible and it can waste. Multiply the people's values together and maximise the product instead, and something unexpected happens — nobody envies anybody when goods can be split, and nobody envies by more than one item when they cannot.

applied · Fair division An exchange box, and a core that shrinks as the market grows. Edgeworth box with endowment (3,1)/(1,3), utilities xy, contract curve the diagonal; core for 1, 2, 5 replicas: 1.7321–2.2679, 1.9161–2.0839, 1.9722–2.0278.

A market too large to bargain in

Two traders swapping goods can settle anywhere along a stretch of possible deals; nothing forces a price. Bring in a second pair of the same traders and some of those deals can be refused by a group of three. With thirty of each, the only deals no group can refuse lie within a few thousandths of the one a price would produce. Edgeworth guessed in 1881 that competition shrinks bargaining to prices; Debreu and Scarf proved it, and the shrinking can be computed.

applied · The core

Named alongside it

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

CoalitionConvexityCooperative gameCoreEfficiencyExhaustive searchFairnessAxiomConstraintDualityEnvy-freenessEquilibrium selection

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