Concept

Core — where it appears

The set of ways of dividing what a group is worth that no smaller coalition can beat by walking out. It can be a polygon, a single point or empty, and its emptiness on games as simple as three-player majority is what forces rules such as the nucleolus to be defined instead.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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

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.

Coalition Allocation Balancedness Constraint Cooperative game Excess Exhaustive search Fairness Imputation Lexicographic order Nucleolus Polytope

All concepts