The core — the series

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.

part 1 · applied

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.

part 2 · applied

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.

part 3 · applied

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.

part 4 · applied

The core of a group worth the square of its size: the outline of its six arrival orders. Splits of 9 among three players; core corners (1, 5, 3), (5, 1, 3), (1, 3, 5), (5, 3, 1), (3, 1, 5), (3, 5, 1); arrival-order splits (1, 3, 5), (1, 5, 3), (3, 1, 5), (5, 1, 3), (3, 5, 1), (5, 3, 1); average 3, 3, 3.

The corners are the orders of arrival

Line the players up, let each join in turn, and pay each what it adds on arrival: every order gives a split. When a newcomer always adds at least as much to a bigger group, those splits are exactly the corners of the core — so the core is never empty, it is the outline of the orders, and the average over all of them lies inside it. For a group worth the square of its size the outline is a hexagon whose corners are the six orderings of 1, 3 and 5.

part 5 · applied

The cheapest tree for a remote user between two near ones, and each user paying for its own link. A source and 3 users with link costs source–A 2, source–B 9, source–C 2, A–B 1, B–C 1, A–C 2; the cheapest tree costs 4 and Bird's rule charges 2, 1, 1.

Each user pays for its own last link

Several users must be connected to a source, and the cheapest network that does it is a tree. Dividing its cost so that no group of users would rather build its own looks like a hard search, and it has a one-line answer: each user pays for the link that joins it to the tree on its way to the source. No group is ever overcharged — while the average over orders of arrival, the rule that settles so much else, can charge a pair more than its own connection costs.

part 6 · applied

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.

part 7 · applied

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.

part 8 · applied

An exchange box, and a core that shrinks as the market grows. Edgeworth box with endowment (3,1)/(1,3), utilities xy, contract curve the diagonal; core for 1, 2, 5 replicas: 1.7321–2.2679, 1.9161–2.0839, 1.9722–2.0278.

A market too large to bargain in

Two traders swapping goods can settle anywhere along a stretch of possible deals; nothing forces a price. Bring in a second pair of the same traders and some of those deals can be refused by a group of three. With thirty of each, the only deals no group can refuse lie within a few thousandths of the one a price would produce. Edgeworth guessed in 1881 that competition shrinks bargaining to prices; Debreu and Scarf proved it, and the shrinking can be computed.

part 9 · applied