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Concept

Imputation

A division of what a group earns that hands out the whole of it and gives no member less than it could earn alone. The divisions satisfying every coalition's objection form a region inside that set, and the region is sometimes empty.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

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 Five certificates against any two of three decide. A table of every minimal balanced family on three players, what each demands of the game, and whether the grand coalition's value covers it — the complete test for whether a stable split exists.

Five weighings and the question is closed

Searching the triangle of splits can only ever fail to find a stable one, which is not the same as there being none. Weighing five families of coalitions against the whole settles the question outright — and the family that fails is the proof that nothing survives.

applied · The core The splits of any two of three decide that nobody can out-argue. The triangle of all splits of a joint gain, with the splits marked at which every player's loudest complaint against every other is matched by an equally loud complaint back.

An objection one player makes to another

The core lets a coalition object to everybody at once. Narrow it to one player objecting to one other, require every such objection to be met by an equally loud one coming back, and exactly one split survives — with no dictionary order anywhere in the argument.

applied · The core Three splits that between them beat every other. The triangle of splits of one pound among three players, with the three half-and-half splits marked and every other lattice point coloured by which of the three dominates it: 134, 134, 134 beaten by exactly one, 91 by more.

Three splits that answer every other

When any two of three people can take a pound, no division of it is safe from a pair walking off. Von Neumann and Morgenstern's answer was not a division but a set of them: three half-and-half splits that never beat one another and between them beat everything else. It is a solution — and so is the line on which one player is held at any fixed share below a half, so the same game has infinitely many, each describing a different settled way of treating the third player.

applied · The core What a left glove is worth as the market tips. For 20 players and L from 1 to 19: the core value of a left glove (1 below 10, any value at 10, 0 above) and the Shapley value, from 0.950 to 0.003.

One glove too many

Give some people left gloves and others right ones, and let any group sell the pairs it can make. If the two sides are equal, the core — the splits no group can beat by walking out — is every price at once. If one side has a single glove more, the core is one split: the scarce side takes the whole of every pair and the other side gets nothing, however large the market. The average over orders of arrival barely notices the difference, and the two rules disagree about almost everything a market is.

applied · The core

Named alongside it

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

CoalitionCoreBalancednessExcessExhaustive searchNucleolusShapley valueBallot problemConvex gameDominationDualityExistence proof

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