Let the Rabbi Split the Pie
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.
Whatever 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,
.
