Robert Aumann, commenting on a passage in the Talmud regarding a man who dies with three debts and insufficient funds to pay them all, came up with a game theoretic understanding of how the Talmud says his estate should be divided. This problem is also discussed here and referenced here (both over on Judaism.SE). (There is discussion about whether Bob actually was the first to come up with this theory, but that's for another time.)
This article describes the concept behind it. The cases in question range from an estate of \$50 to one of \$600 and debts of \$100, \$200, and \$300. TL;DR, the theory is as follows:
- Divide the money between all creditors until the lowest creditor receives exactly half of his claim. At this point, freeze the lowest creditor's account.
- Repeat step 1 with the remaining creditors, until no creditors are left.
- Continue awarding money to the highest creditor, until the difference between the creditor's claim and the amount he's receiving equals the difference between the creditor's claim and the next-highest creditor's claim. At this point, unfreeze the next-highest creditor's account and begin splitting the remaining money amongst them.
- Repeat step 4 with the remaining creditors, until all creditors' accounts are unfrozen.
- Continue splitting the money amongst all creditors until no money is left in the estate. (Obviously, you would stop long before this step if no money is left at that point.)
This can be summarized in the following chart:
My question is threefold.
One, is there a single math equation that summarizes all of this? (Or at least one math equation per claimant.)
Two, the article repeatedly says that Bob used game theory to come to this conclusion. Where is the game theory behind this? It sounds to me like logic and algebra stuffed into a game matrix. Is there anything more to this mathematical game he's set up?
Third, which most likely will answer the previous two questions. Where is his original article?