16.10.05

R. A. N. E. S. T. K. S. Game theory older than law and economics? Consider Robert Aumann and Michael Mashler, "Game Theoretic Analysis of a Bankruptcy Problem from the Talmud," 36 J. Econ. Theory 195 (1985). Here's what one observer has to say.
The Babylonian Talmud is the compilation of ancient law and tradition set down during the first five centuries A.D. which serves as the basis of Jewish religious, criminal and civil law. One problem discussed in the Talmud is the socalled marriage contract problem: a man has three wives whose marriage contracts specify that in the case of this death they receive 100, 200 and 300 respectively. The Talmud gives apparently contradictory recommendations. Where the man dies leaving an estate of only 100, the Talmud recommends equal division. However, if the estate is worth 300 it recommends proportional division (50,100,150), while for an estate of 200, its recommendation of (50,75,75) is a complete mystery. This particular Mishna has baffled Talmudic scholars for two millennia. In 1985, it was recognised that the Talmud anticipates the modern theory of cooperative games. Each solution corresponds to the nucleolus of an appropriately defined game.
The article puts into perspective the famous Solomonic divide-the-baby solution (1 Kings 3:16) that many economists refer to as an early manifestation of incentive compatibility (the true mother prefers to give the baby to the false claimant rather than have him killed.) Apparently the rules of division predate Solomon. Somebody who pays closer attention to this than I notes,

[Solomon] even pretended to apply the well-known law of dividing disputed property. If two people come to court holding on to the ends of a piece of clothing, and each claims it to be his, the court divides it and gives each one half. King Solomon seemed to pretend to be ignorant of the many complicated details of this law, and to think that it applied to babies as well, which would have been ridiculously simpleminded. No judge would ever make such a foolish mistake. Yet, he succeeded in convincing the two women that he was serious.

Nonetheless, he was careful not to let the trick go too far. He specifically commanded his servants to bring the sword to him, not to give it to one of the guards.

Presumably the guards hadn't figured out incentive compatibility. The complications the commentator refers to, however, are complications that can be cracked using game theory. There are three conflicting principles to resolve. First, if the smallest complaint exceeds the property to be divided, it is divided equally. Second, only the disputed part of the property is to be divided (thus, if one person claim the whole cloth and the other half, one shall be given 3/4 and the other 1/4; Solomon's baby follows immediately.) Third, some disputed properties can be divided in proportion to the claims. Equal division comes into conflict with proportional division. The contribution of the cited paper is to suggest a theoretically consistent basis for allocating a property among contested claims in such a way that (loosely) no coalition of claimants is capable of obtaining a more favorable division.

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