The Commons · a rule older than the theory that explains it

Half of What Is Contested

A man dies owing 100, 200 and 300 to three claimants. The Mishnah says how to divide what he left, three times, and gives no reason. One of the three answers is neither equal nor proportional, and for most of two thousand years nobody could say what it was. In 1985 Robert Aumann and Michael Maschler showed it is the nucleolus. This page computes both sides, live, and lets you look for daylight between them.

Mishnah Ketubot 93a is not about commercial bankruptcy. It concerns a man with three wives whose marriage contracts (ketubot) promise 100, 200 and 300 zuz, who dies leaving an estate too small to honour all three. The text records what each wife receives, for three sizes of estate, and stops there. No principle is stated. Here is the table:

Read down the first column and the puzzle jumps out. At an estate of 100 the claims are ignored entirely and everyone takes a third. At 300 the claims are honoured exactly, half of each. In between, at 200, the smallest claimant gets 50 and the other two get 75 apiece, which is neither of those things and looks like nothing at all. Aumann and Maschler summarise two millennia of reading it: many authorities disagreed with the passage outright, others put the figures down to special circumstances the text does not state, a few tried to rationalise them directly and mostly got nowhere, and one modern scholar, unable to make sense of it, proposed an error in transcription (they cite Israel Lewy, 1908). It is none of those.

The base case: a garment two people are holding

A different tractate opens with a smaller problem. Bava Metzia 2a considers two people gripping one garment. If one says it is all mine and the other says half of it is mine, the ruling is that the first takes three quarters and the second one quarter.

The logic is a concession. The second claimant, by claiming only half, has conceded the other half outright; that half goes to the first without argument. What is left, the remaining half, is contested, and the contested part is split down the middle. So the first gets 1/2 + 1/4 = 3/4 and the second gets 1/4. Not proportional (that would be 2/3 and 1/3), not equal. Concede what you do not claim; halve what is contested.

That principle is stated for two claimants only. The estate table has three. Aumann and Maschler's move was to insist the two rules be the same rule: whatever a three-way division is, any two of the claimants, looking only at the money the two of them were handed between them, should be unable to object that the pair split it wrongly by the garment rule. That single demand, pairwise consistency, pins the answer down completely. The instrument below runs it.

The dismissal, and the answer to it

Three rows is three data points, and any number of rules can be bent through three points. That objection is correct and it is the reason the rest of this page exists. The claim is not that a rule fits three numbers. It is that a rule defined without any reference to the Mishnah, from a purely game-theoretic principle, reproduces the Mishnah's numbers and then keeps agreeing at every estate value in between and at every claims vector you care to invent.

The game-theoretic object is the nucleolus, introduced by David Schmeidler in 1969. Turn the claims problem into a cooperative game: for any coalition S of claimants, ask what S can guarantee itself if everyone outside S is paid in full first. That is

An allocation gives each coalition an excess, e(S,x) = v(S) − x(S): how much worse off S is than what it could guarantee. A high excess is a live grievance. The nucleolus is the allocation that makes the largest grievance as small as possible; then, subject to that, the second largest as small as possible; and so on down. It is a pure minimax, with no notion of concession, garments or halves anywhere in it. For any game whose set of imputations is non-empty and compact, which every bankruptcy game here is, it exists and it is unique (Schmeidler, 1969).

The panel below computes it. Not by table lookup: at each estate value it runs the lexicographic minimax as a sequence of linear programs, each solved exactly by enumerating the vertices of its feasible region. Then it plots the answer on top of the concede-and-halve curve.

The dots sit on the lines. With the Mishnah's claims loaded the sampler steps by 12.5, so there are dots exactly on the three Mishnah estates and on 412.5, and on every other point it lands on; the offline verifier checks the same identity at 137, where the sampler does not land, and at 212 further random games of its own. Turn on a rival rule and watch what disagreement actually looks like: proportional division is a straight fan through the origin and misses the middle row badly; the Shapley value, the other famous solution concept for cooperative games, is a perfectly respectable answer to the same game and is not this one. The nucleolus is not "the game-theory answer". It is one particular answer, and it is the one that matches.

The hinge at half the debt

Look at the curves near the middle and there is a visible kink, at exactly half the total owed. Below it, the rule is dividing gains: everyone is topped up equally until they hit half their claim, and anyone who reaches half stops. Above it, the rule flips and divides losses: everyone is docked equally, capped at half their claim. The two halves are mirror images, which is a real property with a name, self-duality: what you award at estate E equals the claim minus what you award at estate D − E. At estate 200 the awards are (50, 75, 75). At estate 400 the losses are (50, 75, 75).

This is easiest to believe as plumbing. Give each claimant a pair of tubes, each of height half their claim. The lower tubes stand on a common floor and are joined at the bottom, so poured money finds a common level and short tubes fill up and stop. The upper tubes hang from a common ceiling and are joined at the top, so the shortfall finds a common level from above. Pour the estate in. Below half the debt you are filling the floor basin; above it, the floor basin is full and you are drowning the ceiling basin from the bottom up. The kink is the moment the lower basin brims.

Check the consistency by hand

The rule's defining property is checkable without trusting anything on this page. Take any two claimants. Add up what the rule gave the two of them. Now forget the third claimant entirely and divide that pot between just those two by the garment rule of Bava Metzia. You get their two numbers back, unchanged. Every pair, every estate.

Try to beat the nucleolus

The minimax claim is falsifiable in the most direct way available: move money around and watch the sorted list of grievances get worse. Below, the first line is the excess vector at the rule's own allocation, sorted from worst grievance down. The second line is the excess vector after you shift money between two claimants. Lexicographic order is the test: read both lines left to right, and the first place they differ decides. If any transfer ever produced a lexicographically smaller line, Aumann and Maschler would be wrong.

Shaking one instance is weak evidence. The stronger version is a hunt: draw random claims vectors and random estates, compute the concede-and-halve division and the nucleolus by two completely different pieces of code, and look for a case where they part company. The button runs that hunt in your browser and reports the worst disagreement it found.