Matpic
mathematics, in pictures
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 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as coalition, cooperative game — the same set of essays touches all of them, so they are one junction rather than several.

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

Named alongside it

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

CoalitionCooperative gameExhaustive searchFairnessAxiomConstraintCoreMarginal contributionPermutationPolytopeShapley valueSimplex

All concepts