Matpic
mathematics, in pictures
Concept

Nucleolus

The division of a group's winnings whose loudest complaint is quietest, with remaining ties broken by the second loudest and so on. It exists and is unique for every game, agrees with the stable divisions whenever any exist, and reproduces a division table from the Mishnah.

Named by 3 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 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 widows and the Mishna's table. The Mishna's division of estates of 100, 200 and 300 among claims of 100, 200 and 300: 33⅓ each; 50, 75, 75; 50, 100, 150.

The debts the Mishna divided

A man dies owing his three wives 100, 200 and 300. The Mishna divides an estate of 100 equally, of 300 in proportion, and of 200 as 50, 75 and 75, and for nearly two thousand years nobody could say why. The table is the nucleolus of a cooperative game, and the only division in which every pair splits its share as two people split a contested garment.

applied · The core

Named alongside it

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

CoreCoalitionExcessExhaustive searchImputationBalancednessBankruptcy problemConsistencyCooperative gameFairnessLexicographic orderShapley value

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