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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:

  1. 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.
  2. Repeat step 1 with the remaining creditors, until no creditors are left.
  3. 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.
  4. Repeat step 4 with the remaining creditors, until all creditors' accounts are unfrozen.
  5. 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:

Talmud 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?

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The above representations of some Talmud bankruptcy problems are formally bankruptcy situations. Therefore, there is no game theory behind such a representation, because a bankruptcy situation is not game. However, from such a situation a cooperative game (TU game) can be derived.

By doing so, note first that a bankruptcy situation can be formalized as an ordered pair $(E,\mathbf{d})$, where $E \in \mathbb{R}$ is the bankrupt estate and $\mathbf{d}=\{d_{1},\ldots,d_{n}\} \in \mathbb{R}^{n}$ is a claims or debts vector such that $d_{k} \ge 0$ for all $k \in N$ and $0 \le E \le \sum_{k=1}^{n}\,d_{k}$ is given. This problem is called a bankruptcy situation, since the bankrupt estate is insufficient to meet all claims simultaneously. From this situation a corresponding transferable utility game, a bankruptcy game $\langle N, v_{E,d} \rangle $, can be derived through \begin{equation} \label{eq:bankr} v_{E,d}(S):= \max\bigg(0, E - \sum_{k \in N\backslash S} \, d_{k}\bigg) \qquad\text{for all}\quad \emptyset \neq S \subseteq N, \end{equation} with the convention that $v_{E,d}(\emptyset) = 0$ and $N$ is the set of creditors. This game class has been introduced by B. O'Neill (1982). A coalition of $s$-creditors in $S$ gets either zero or what remains from the estate $E$ after the opponents in coalition $N\backslash S$ are payed in accordance with their claims in $\mathbf{d}$.

There is no single equation that summarizes all of these solutions. Nevertheless, it is well known that the unique numbers presented in the Talmud to solve particular bankruptcy problems coincide with the nucleolus of the corresponding bankruptcy game. The details can be found in the following articles/book:

O'Neill, B. (1982), A Problem of Rights Arbitration from the Talmud, pp. 345-371, Mathematical Social Sciences.

Aumann, R. J. and Maschler, M. (1985), Game Theoretic Analysis of a Bankruptcy Problem form the Talmud, pp. 195-213, Journal of Economic Theory.

Driessen, T.S.H. (1988), Cooperative Games, Solutions and Applications, Theory and Decision Library, Kluwer.

Thomson, W. (2002), Axiomatic and Game-Theoretic Analysis of Bankruptcy and Taxation Problems: A Survey, pp. 249.297, Mathematical Social Sciences.

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