I learned an interesting example from game theory. A man dies, leaving debts totalling more than his estate. Should the estate be divided among the creditors? Let’s consider three creditors, with debts of 100, 200, and 300. And let’s consider three scenarios corresponding to estates of 100, 200,
and 300. The Hebrew Mishna specifies the divisions shown in the attached picture, and this solution is quite fascinating. For 100, the division is equal: each creditor receives 33 1/3. For 300, the division is proportional: the first creditor gets 50, the second receives 100, and the third gets 150. But what about the second case? (50, 75, 75)?
Auman and Mashler devised a solution using pre-nucleolus. Of course, the Torah does not discuss this. However, it reveals an interesting principle of division illustrated by another Mishna, “Two hold a garment; one claims it all, the other half. Then one is awarded 3/4, the other 1/4.” The principle is straightforward. The lesser claimant relinquishes half the garment to the greater claimant. Only the remaining half is contested; this half is therefore split equally. Notice that this differs significantly from proportional division.
Your claim is 100, right? Take 50 and leave.
Incidentally, this is an excellent tactic for bargaining or business negotiations: always ask for more than you expect to receive without
a struggle.
This explains the 50/75/75 in the second row of the attached table. Each creditor demands at least 100, and the estate is worth 100 too, so for this scenario, equal division functions well. If the estate is worth 200, the first creditor receives 50% of her claim, and the other creditors will receive at least the same amount, totaling 50+50+50=150. We have 200, so the remaining 200-150 will be divided among the creditors. Not equally, the first 25 is half of (50% of what she claims, 0.5*200=100) and (what she gets, which is 50), making it 75. The third creditor also takes 75, as a remainder.
The third scenario, 300, follows the same logic. If you give the first creditor all 100, it is unfair, as the 2nd and 3rd creditors won’t receive their due 200 and 300. So the first creditor will take 50, and the others will also take 50 each, totaling 150. The remainder, 300-150=150. The second creditor desires 200, half of this is 100, and she has already received 50. Adding another 50 to the first 50 results in 100 for her, which is 50% of her expected amount, while the remainder goes to the third creditor.
The detailed paper is here: http://www.cs.cmu.edu/~arielpro/15896s15/docs/paper8.pdf
Hi, I'm Rauf Aliev. 