Generator

coalition

A generator in the applied library, called 13 times across 2 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

coalition is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

`show: "core"`

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.

`show: "axioms"`

Three ways to split a joint gain, and the conditions each one breaks. A grid of three sharing rules against three small games, with the conditions each rule violates named in the cell, and one rule that violates none.

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

a rational is a whole numerator over a non-zero whole denominator ×1
a rational is never divided by zero ×1
and at least one of the other rules fails one ×1
averaging what each player adds passes every condition on every game tested ×1
every ordering ends with everybody in ×1
some coalition objects to every split drawn here ×1
the coalitions together demand more than there is to give ×1
the core is drawn for three players ×1
the empty coalition is worth nothing ×1
the exact arithmetic stays inside the safe integer range ×1
the game is one of partnership, majority, gloves, dummy ×1
the game names a value for every coalition ×1
the marginal contributions along one ordering add up to the whole ×1
the resolution of the exhaustive sweep is a whole number between 12 and 120 ×1
the shares add up to what the whole group is worth ×1
the sweep and the balancedness condition agree about whether any split survives ×1
the view is one of axioms, core ×1
two to four games this family knows ×1

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Applied

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 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.

The whole library · What the figures prove