Concept

Balancedness

A condition on a game's values saying that no schedule of overlapping coalitions, each player fully occupied, earns more than the whole group working together. It holds exactly when some division of the winnings survives every coalition's objection, which is what makes it a finite test for a question about infinitely many divisions.

Named by 2 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

Named alongside it

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

Coalition Core Imputation Duality Excess Exhaustive search Existence proof Lexicographic order Linear program Nucleolus Shapley value

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