Matpic
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Concept

Cooperative game

A list of players together with a number for every subset of them, saying what that group could secure on its own. It carries no moves and no order of play; what is asked of it is how the value of the whole group should be divided.

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 Drop one condition, and something else satisfies the rest. A column for each of the four conditions, holding a sharing rule that breaks that one and keeps the other three, with the split each rule gives on a stated four-player game.

None of the four conditions is spare

Four conditions pick out one sharing rule. The half that is usually shown is that they are enough; the other half is that each is needed — drop any one and a different rule satisfies the rest, so the list cannot be shortened.

applied · shapley value Every order of arrival for three users of one shared capacity, 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.

Sharing a cost that is not the sum of its parts

Three users need capacities three, six and twelve of one shared thing, and serving any group costs the largest of them. Averaging what each adds over every order of arrival divides the bill — and for this family the average collapses to a rule anybody could apply by hand.

applied · shapley value

Named alongside it

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

FairnessMarginal contributionShapley valueAllocationCoalitionExhaustive searchSymmetryAdditivityAirport problemAxiomAxiomatic characterisationClosed form

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