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Concept

Shapley value

The rule that gives each player the average of what they add to the group already present, taken over every order in which the players could arrive. Four conditions — efficiency, symmetry, a player who adds nothing getting nothing, and additivity — leave it as the only possibility.

Named by 2 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 quietest loudest complaint in any two of three decide. The triangle of all splits of what a three-player group is worth, with the split minimising the largest excess marked, the average split beside it, and the loudest complaint named.

The objection nobody can make louder

When no split of the winnings survives every group's objection, the core is empty and the question changes — which split makes the loudest objection as quiet as it can be? Sorting the complaints and minimising them in dictionary order picks exactly one split, always, whether or not the core exists.

applied · the core

Named alongside it

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

CoalitionAllocationAxiomBalancednessCooperative gameCoreExcessExhaustive searchFairnessImputationLexicographic orderMarginal contribution

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