Saturday, December 17, 2005

A number of people have asked me to write something about Nobel laureate Yisrael Aumann's work on game theoretic ideas in the gemara. Here's a very brief synopsis of his most important work in that area. It concerns the Mishnah in Kesubos 93a.

The Mishnah considers a case in which a man has left behind three wives (or, more generally, creditors) to whom he owes 100, 200 and 300, respectively. The estate, however, is inadequate to cover all the debts, so that some mechanism is needed for distributing the limited funds equitably. The Mishnah offers three examples of what it regards as the proper distribution. If the estate consists of 100, then it is divided equally, that is, each woman receives 33 1/3. If it consists of 300, it is divided proportionally, that is, each woman receives exactly half of her claim. If it consists of 200, then the woman whose claim is 100 receives 50 and the other two women receive 75 each.

The gemara notes that the Mishnah reflects the view of Rabbi Nathan but that Rabbi Yehuda HaNasi disagrees. Some opinions in the gemara further argue that even Rabbi Nathan’s rules only apply in special cases. Nevertheless, we might reasonably ask if there is some discernible principle that underlies the three cases. On the face of it, either equal distribution (as in the case of an estate of 100) or proportional distribution (as in the case of an estate of 300) makes good sense, but it is not clear why the one method should be used in one case and the other method in the other. Moreover, the case of the estate of 200 follows neither method and appears altogether mysterious.

Aumann noted that game theory provides the appropriate tools for analyzing the three cases of the Mishnah in order to find some general principle that could be applied to any size estate and any number of claimants with any size claims. In his original paper, written with Michael Maschler and published in an economics journal, Aumann shows that the solution offered by the Mishnah uniquely satisfies a particular set of constraints and can best be described in terms of a game-theoretic concept known as the nucleolus. In a later paper, written for an audience knowledgeable in Talmud but not necessarily in mathematics, Aumann sidesteps the game-theoretic technicalities that motivated the earlier results and explains the same ideas in terms of well-known Talmudic concepts.

The central idea, as presented in the latter paper, can be easily grasped. There is a well-known principle for dividing a disputed sum between two claimants. According to this method, described at the very beginning of Bava Metzia, each claimant first gets that part conceded by the other claimant and then what remains is divided equally. Thus, in the famous case of one claiming ½ and the other claiming the whole, the second claimant receives the ½ conceded by the first and the remaining half is split. This leaves the first claimant with ¼ and the second with ¾, as indicated in the Mishnah there. This same principle appears in several other places in the gemara.

The question that arises is: can this principle be generalized to more than two claimants? Aumann notes a remarkable fact about the three cases in the Mishnah in Kesubos: In each case, if one isolates any two of the three women, the total that the Mishnah gives the two collectively is divided between them according to the two-claimant principle of Bava Metzia. Consider, for example, the difficult case of the estate of 200 and the women who claim 100 and 200, respectively. The Mishnah gives the two women 125 collectively (50 and 75, respectively). Of this 125, the first concedes 25 (she only claims 100) and the second concedes nothing (her claim is greater than the total), so the second gets the conceded 25 and the remaining 100 is divided equally. This leaves the first with 50 and the second with 75, exactly as stated in the Mishnah. The same holds for all other pairs of women in all the cases discussed in the Mishnah.

Aumann concludes that the Mishnah in Kesubos is a generalization of the two-claimant principle in Bava Metzia. Most remarkably, Aumann proves that this generalization is unique. No solution to the three-claimant problem of Kesubos, other than the one offered, would be consistent in this manner with the two-claimant solution offered in Bava Metzia. In fact, there is only one such solution – and a constructive method for computing it – for any size estate and any number of women with any size claims.

Aumann’s generalization of the Mishnah’s problem is by no means the only possible one, (see R. Brody, “A Talmudic Principle of Distribution" in Higayon 1 for a different generalization) but it certainly serves as an outstanding example of the use of modern mathematical tools in the service of Torah study.


Anonymous Anonymous said...

Is Brody's paper available on the web?

4:20 AM  
Anonymous daat y said...

yes.And your discussion is an excellent exaaple of clarity and conciseness.

10:38 AM  
Blogger Ben said...

I haven't found Brody's paper on the web.

11:40 PM  

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