The debts the Mishna divided

The Mishna, the code of Jewish law compiled around the year 200, has a short passage in the tractate Ketubot about a man who dies leaving three wives, each owed a sum under her marriage contract: 100, 200 and 300. The estate does not cover the 600 he owes. The passage gives the division for three sizes of estate. If the estate is 100, each wife takes $33\tfrac13$. If it is 300, they take 50, 100 and 150. If it is 200, they take 50, 75 and 75.

The first row is equal division. The third is proportional division. The second is neither, and no simple principle seems to produce all three. For about eighteen centuries commentators tried and failed to find one; some proposed that the middle row was a copying error. In 1985 Robert Aumann and Michael Maschler showed that the whole table follows from one rule, that the rule is the extension to three people of a judgement the Mishna makes elsewhere about two people and a garment, and that it coincides with the nucleolus of a cooperative game — the division that makes the loudest objection as quiet as possible, a concept that David Schmeidler had defined in 1969 with no thought of ancient law.

Two people and a garment #

The key is in a different tractate, Bava Metzia, which opens with two people holding a garment. One says the whole of it is hers; the other says half of it is his. The Mishna awards three quarters to the first and one quarter to the second. The reasoning is explicit in the commentary: the second person does not claim half of the garment at all, so that half goes to the first without dispute; the other half is claimed by both, so it is split equally.

Stated generally, for two creditors with claims $a$ and $b$ and an estate $E$: each first receives what the other concedes, $\max(0, E - b)$ and $\max(0, E - a)$, and whatever remains, the part both claim, is divided equally. A claim larger than the estate is treated as a claim to the whole estate, because no one can be conceded less than nothing.

It is the bankruptcy counterpart of one cuts and the other chooses: a two-person procedure whose fairness is visible without any theory. Below 50, neither claim leaves anything uncontested, so the estate is halved. Between 50 and 100 the smaller claimant’s whole claim is contested and she receives half of it, 25, while the larger claimant takes everything else. Above 100 both concessions are positive and the two are, in effect, sharing the losses equally. The rule is neither equal division nor proportional division, and for two people it looks like the obvious fair thing: give away what is not in dispute, and split what is.

Every pair splits like the garment #

Aumann and Maschler’s observation was that the Mishna’s table for three people has a property that ties it to the garment. Take any row and any two of the widows. Add up what the two received, and imagine that sum as an estate to be divided between just those two, with their original claims. Divide it by the garment rule. The answer is exactly what the table gave them.

At an estate of 200, the widows owed 100 and 200 receive 125 between them. The second concedes nothing to the first, because her claim of 200 already exceeds 125; the first concedes 25 to the second; the remaining 100 is split, giving 50 and 75, the table’s values. The widows owed 200 and 300 share 150; each claim exceeds 150, so nothing is conceded and 150 is halved to 75 and 75, as in the table. Every pair in every row works in the same way.

This property, consistency, is strong. It says that no two creditors, looking only at what they have received together, could reallocate it between themselves by the two-person rule and get anything different. Aumann and Maschler proved that for any number of creditors and any estate there is exactly one division consistent with the garment rule. So the Mishna’s three rows are not three separate judgements that happen to look odd. They are the only possible extension of the garment rule to three people.

The rule written out #

The consistent division has a closed form, and it explains the shape of the table. Treat each creditor as holding two half-claims. If the estate is at most half the total debts, award it on the first half-claims by equal division, giving everyone the same amount except those whose half-claim is smaller, who get exactly their half-claim. If the estate exceeds half the total, everyone first gets her full half-claim, and the shortfall is assigned to the second half-claims in the same way: equal losses, except that no one loses more than half her claim.

At an estate of 100, half-claims are 50, 100 and 150, and equal division of 100 gives $33\tfrac13$ each, below every half-claim. At 200, equal division would give $66\tfrac23$, which exceeds the first widow’s half-claim of 50; she is held at 50 and the other two share the remaining 150 equally, 75 each. At 300, exactly half the debts, every creditor receives exactly half her claim — 50, 100, 150 — which happens to be proportional. Above 300 the rule runs in reverse, with losses in place of awards, and the paths in the figure are symmetric about the middle. The rule treats what is missing from the estate exactly as it treats what is there.

The same table as a game #

The game-theoretic reading needs one more idea. Think of the creditors as players who may form coalitions, and ask what each coalition can guarantee itself. A coalition $S$ can count on whatever is left after everyone outside it is paid in full, if anything is: Barry O’Neill defined this bankruptcy game in 1982 by

$$v(S) = \max\Big(0,\; E - \sum_{i \notin S} d_i\Big).$$

The corners are the orders of arrival showed that such games are convex, so their core — the divisions that every coalition accepts, the splits nobody can walk away from — is never empty, and usually it is large.

The nucleolus picks one point of the core by a sequence of fairness comparisons. For each division, list every coalition’s excess, how far short of its guarantee the division leaves it, and sort the list from largest to smallest. The nucleolus is the division whose sorted list is smallest in dictionary order: the worst-off coalition is as well off as possible, then the next worst, and so on. For the estate of 200 the figure computes it without any formula, by trying every division in steps of one twelfth of a unit and comparing the sorted lists exactly. It lands on 50, 75 and 75. The same search on the other two estates lands on $33\tfrac13$ each and on 50, 100, 150. Proportional division, at $33\tfrac13$, $66\tfrac23$ and 100, is a different point of the same core.

Tested on problems the Mishna never posed #

Three rows of an ancient table could agree with a rule by accident. The figure takes 80 problems with random claims and random estates and computes the nucleolus of each by exhaustive search, independently of the closed form, which is evaluated separately. The grid of twelfths is fine enough, because the equal level in the closed form is a remainder divided among at most three half-claims, each a multiple of a half. In all 80 problems the two agree exactly. Proportional division differs from both by up to 4.2 and by 1.05 on average.

The proof that this holds for any number of creditors goes through consistency. The nucleolus of any game is consistent in a suitable sense, with respect to the games that a group of players faces after the others are paid off — a property A. I. Sobolev established in 1975; for two creditors the nucleolus of the bankruptcy game is easily computed to be the garment rule; and the uniqueness of the consistent extension forces the nucleolus for $n$ creditors to equal the Talmud rule.

Other solutions that land in the same place #

The nucleolus is not the only solution concept of cooperative games, and for bankruptcy games several of them agree. The kernel, built from objections one player makes to another and balanced counter-objections, can be a large set in general; Michael Maschler, Bezalel Peleg and Lloyd Shapley showed in 1972 that for convex games it shrinks to a single point, the nucleolus. Since bankruptcy games are convex, the Mishna’s division is also the unique division at which no widow has a stronger claim against another than the other has against her, in the kernel’s precise sense. And the core is never empty here, so the five weighings that test whether a game has a core always pass; what needs choosing is a point inside it, not whether one exists.

The Shapley value, by contrast, is defined by four conditions about adding and symmetrising games, and nothing in those conditions mentions pairs. It is a reasonable rule and it is not the Mishna’s.

A dial between the rules #

The rules compared below can be arranged on a single dial, in the same way that the divisor methods of apportionment are one formula with a parameter. Young showed that the rules which are consistent, symmetric and continuous are exactly the ones that award each creditor $f(\lambda, d_i)$ for some function increasing in a common level $\lambda$, chosen so that the awards add up to the estate. Proportional division takes $f = \lambda d_i$. Constrained equal awards takes $\min(\lambda, d_i)$. The Talmud rule’s function rises like equal awards until it reaches half the claim, waits there while the level climbs, and then rises again like equal losses on the other half: one function with two kinks. In apportionment the parameter decides how fractions of a seat are rounded; here it decides how a shortfall is felt by creditors of different sizes.

Four rules, four philosophies #

The Talmud rule is one of a family, and the comparisons show what it chose. Proportional division is the rule most modern bankruptcy law uses and that Aristotle recommended: every creditor loses the same fraction of her claim. Constrained equal awards gives everyone the same amount, capped at her claim; Maimonides, in the twelfth century, argued for it in this very setting, and the Talmud rule uses it on half-claims. Constrained equal losses makes everyone lose the same amount, as far as her claim allows, and the Talmud rule uses it on the other half-claims. And the Shapley value of the bankruptcy game — what each creditor would receive on average if the creditors were paid in full in a random order of arrival — is a fourth rule, which O’Neill studied and which in general differs from the Talmud rule once there are three creditors.

The Talmud rule is the one that treats small and large claimants as the garment does. A creditor whose claim is small is protected by the cap at half her claim: at an estate of 200 the widow owed 100 gets exactly half of what she is owed, which is more than proportional division would give her. And the half-claims are a dividing line for everyone at once: when the estate is less than half the debts no creditor receives more than half her claim, and when it is more, every creditor receives at least half.

A modern concept in an ancient text #

The surprising connection here is historical as much as mathematical. The nucleolus was invented in 1969 to pick a single solution out of the core of a cooperative game, a concept of mid-twentieth-century economics. The bankruptcy game was invented in 1982. Neither has anything to do with marriage contracts, and nobody writing in the Mishna had the words for either. And yet the oldest written answer to a three-person bankruptcy problem is the nucleolus, and was not recognised as a coherent rule until both ideas existed. Aumann, who shared the Nobel prize in economics in 2005, often used the case as an example of how a precise modern concept can be the key to a text that the people who wrote it understood intuitively.

It also cuts the other way. The garment principle — concede what is not in dispute, split what is — reads as plain good sense for two people, and the consistency requirement is plain good sense for groups. Together they determine the nucleolus for every bankruptcy problem. The fairness that the concept was built to capture can be stated in terms a judge two thousand years ago would recognise.

Exact on twelfths, and only for three creditors #

The search for the nucleolus is exact only on a grid, and it is exact here because the grid of twelfths provably contains the answer for three creditors with whole-number data. For four or more creditors, or for fractional claims, a finer grid would be needed, and a search over a finer grid quickly becomes slow; the general case is proved, not searched. The 80 random problems are evidence that the closed form has been implemented correctly and that the theorem holds where it was tested, not a proof that it holds everywhere.

Nor can the pictures say what the authors of the Mishna intended. The Gemara, the later commentary, records disagreement with the passage, and the case for reading it through the garment rule rests on the consistency of the result, not on any statement in the text.

The rule computed, Aumann and Maschler quoted #

Computed here: the Talmud rule in closed form; the garment rule and its agreement with the Talmud rule for two creditors at 301 estates; the nine pairs of the table; the core of the bankruptcy game on a grid of divisions of 200; the nucleolus of the Mishna’s three problems and of 80 random problems by exhaustive lexicographic search in twelfths, in exact integer arithmetic; and the comparison with proportional division, which is checked to differ from the nucleolus.

Quoted: the texts of Ketubot 93a and Bava Metzia 2a; Aumann and Maschler’s theorems of 1985; Schmeidler’s nucleolus; O’Neill’s bankruptcy game; the consistency of the nucleolus; the positions of Aristotle and Maimonides; and the Gemara’s dissent.

Still open: what the Mishna meant #

Whether the editors of the Mishna had the consistent division in mind, or reached the three rows by some other principle that happens to agree with it, cannot be settled from the text, and it is debated. Aumann and Maschler’s explanation is the only one known that produces all three rows from one rule, and it uses only a principle stated elsewhere in the same body of law. But the first row has also been read as plain equal division and the third as plain proportional division, each justified separately, with the middle row remaining the puzzle it always was.

On the mathematical side, the question of which two-person rules have a consistent extension to any number of people has been answered — H. Peyton Young characterised them in 1987 — and the answer is a large family. What is less settled is which members of that family people actually regard as fair. Experiments that ask people to divide estates find no single rule chosen by everyone, and the choice shifting with how the claims arose and how the question is framed; why, is an open question in the study of fairness rather than in mathematics.

Concede, then split #

The Mishna’s division of an estate among three creditors is the unique division in which every pair of creditors shares its combined award as two people share a contested garment — conceding what is not claimed by the other and halving the rest. That division is equal awards on half-claims up to half the debts and equal losses on half-claims beyond, and it is the nucleolus of O’Neill’s bankruptcy game. Searching every division in twelfths finds the nucleolus on the Mishna’s three rows and on 80 random problems, and it never agrees with proportional division except where the estate is exactly half the debts.