# Let the Rabbi Split the Pie

## Talmudic wisdom applied to bankruptcy.

A man dies, leaving more debts than assets. How should the estate be divided among his creditors? Two thousand years ago, the sages of the Babylonian Talmud addressed this question in a mysterious way--by offering a series of numerical examples with no hint of the general underlying principle. According to two Israeli scholars, the reasoning of the ancient rabbis is best understood in the light of modern economic theory.

Take a concrete example. Suppose three creditors are owed $100, $200, and $300, respectively--a total of $600 in debts--but there is less than $600 to distribute. Who gets how much? The Talmud (Kethubot 93a) makes the following prescriptions:

1) If there is $100 to distribute, then everyone gets an equal share; that is, everyone gets $33.33.

2) If there is $200 to distribute, then the first creditor gets $50, while the other two get $75 each.

3) If there is $300 to distribute, then the first creditor gets $50, the second gets $100, and the third gets $150. (In this case, the payouts are proportional to the original claims.)

Where do these numbers come from, and how should we behave if there is, say, $400 or $500 to distribute? The Talmud does not tell us. But certain patterns are evident. Apparently the rabbis reasoned that nobody can legitimately claim more than the value of the entire estate. Thus when the estate contains only $100, the claims to $100, $200, and $300 are treated as equal. When the estate contains only $200, the claims to $200 and $300 are treated as equal (but superior to the claim of $100).

Another clue can be found elsewhere in the Talmud (Baba Metzia 2a): "Two hold a garment; one claims all, the other claims half. Then the one is awarded 3/4, the other 1/4." The rabbinical reasoning seems to have gone something like this: "Both claim half the garment, while only one claims the other half. So we'll split the disputed half equally and give the undisputed half to its undisputed owner." Elsewhere in the Talmud, the rabbis apply similar reasoning to settle a case where one claims all and the other claims a third.

Now we've stated two principles: First, claims cannot exceed 100 percent of the estate, and second, we should follow the contested-garment rule. With these, we can prescribe the division of any bankrupt estate, provided there are just two creditors. Here's an example: Suppose the estate consists of $125, and two creditors claim $100 and $200, respectively. By the first principle, the $200 claim is immediately reduced to $125. Now there is $100 in dispute and $25 undisputed. According to the contested-garment principle, the $100 is divided equally. Therefore the $100 claimant gets $50, and the $125 claimant gets the remaining $75.

But what should we do when there are three or more creditors? According to Professors Robert Aumann and Michael Maschler of the Hebrew University in Jerusalem, we can solve this problem by introducing just one more principle, which they call *consistency*. According to the consistency principle, any pair of creditors must divide their collective share according to the principles we've already enunciated. To see what consistency means in practice, think again about a $200 estate, to be divided among creditors who claim $100, $200, and $300. The Talmud awards $50 to the first and $75 to the second; thus the first two creditors have a collective share of $125. And this $125 is divided between them exactly as we prescribed in the preceding paragraph. So the Talmudic prescription satisfies the consistency principle in this instance. It's not hard to confirm the same would be true if you started with the first and third creditors, or the second and third.

But wait! All we've done is checked that the first two creditors divided their collective share of $125 appropriately; we haven't explained why their collective share is $125 in the first place. Aumann and Maschler have an answer: Any division other than 50-75-125 would be inconsistent. (That is, with any other division, some pair of creditors would have its collective share divided incorrectly.) In fact, they have proved more generally that every bankruptcy problem has *exactly one* consistent solution. Once you've found a consistent division, you can be sure that no other is possible.

So perhaps the Talmudists proceeded by trial and error, considering various divisions and rejecting each one as inconsistent until they hit upon the unique consistent division of 50-75-125. Or maybe they had a more systematic approach. Systematic approaches are possible but a bit complicated. Click for an explanation of the simplest.

W hatever method the rabbis used, they appear to have used it--pardon the pun--consistently. It's not hard to check that *all* the Talmudic examples always satisfy the consistency principle. And the consistency principle gives a complete explanation for each example, in the sense that, in each case, only one consistent solution is possible, and we can imagine that the rabbis kept trying until they found it. The consistency principle is both universally applicable (because a consistent solution can always be found) and universally unambiguous (because there is never more than one consistent solution).

Suppose, for example, that an estate of $400 is to be divided among creditors who claim $100, $200, and $300. A consistent solution is to award them $50, $125, and $225. (Click for help on seeing why this is consistent.) But from Aumann and Maschler's work, we know that if you've found one consistent solution, you've found them all. So this is the *only* division that obeys all the principles we've stated. Although the ancient rabbis failed to consider this particular example, Aumann and Maschler express confidence that if they *had* considered it, they would have endorsed this unique consistent solution.

Why is the consistent solution the right solution? Aumann and Maschler argue that consistency appeals to our intrinsic sense of fairness. But, in the Talmudic tradition, if you don't like that argument, Aumann and Maschler have another.

Imagine that all the creditors are put in a room and told to agree among themselves on a division of the estate; if they can't agree, nobody gets anything. Suppose also that any creditor who is offered 100 percent of his claim (by a consensus among the others) is required to accept it and leave the room. What would the bargaining process look like, and what would the outcome be?

There is a branch of economics called "bargaining theory" that attempts to answer such questions; unfortunately, the answers turn out to depend rather heavily on auxiliary assumptions. But Aumann and Maschler have proved that in the case of the bankruptcy negotiation, it follows from reasonable assumptions that the creditors would eventually agree to divide the estate in accordance with the consistency principle. Thus, according to Aumann and Maschler, all the Talmudic prescriptions coincide with what the creditors themselves would have agreed to, given appropriate bargaining rules and sufficient time.

** If you missed the systematic way to solve the bankruptcy problem, click. If you'd like to review why the $50, $125, and $225 distribution of the $400 estate is consistent,.**