TALMUDIC BANKRUPTCY PROBLEM : SPECIAL AND GENERAL SOLUTIONS

Abstract

An elementary solution is presented to the special bankruptcy problem described in the Babylonian Talmud. The presented solution pertains to the numerical data presented in the Tractate of Kethuboth 93a. General solution for any combination of data is also given. Solution of the special case appears to be suitable for inclusion in courses on economy, as well as in group study environments. 1 Introduction Modern solutions of division problems are based, primarily on the idea of proportionality, advocated by Aristotle: just act necessarily involves at least four terms: two persons for whom it is in fact just, and two shares in which its justice is exhibited. And there will be the same equality between the shares as between the persons, because the shares will be in the same ratio to one another as the persons what is just in this sense, then, is what is proportional, and what is unjust is what violates the proportion. Oppenheim (17) interprets the Aristotelian approach as follows: Aristotle himself enlarged the criterion of egalitarianism to include rules which allot shares to equals; i.e. equal shares of some specified kind to all who are equal with respect to some specified characteristic. Conversely, a rule is inegalitarian when either equals are awarded unequal shares or unequals equal shares. However, is not necessarily synonymous with justice. Impressive examples of fair division which do not use proportionality in the usual setting are given in the Mishna (24, 25), a 1800-year old document that forms the basis for Jewish civil, criminal, and religious law. This paper deals with a division problem discussed in the Mishna, tractate Kethuboth (28). The particular allocations mentioned there, in the terminology of Aumann and Maschler (2) look mysterious; but whatever they may mean, they do not fit any obvious extension of either equal or proportional division. Over two millennia, this Mishna has spawned a large literature. Many authorities disagree with it outright. Others attribute the figures to special circumstances not made explicit in the A few have attempted direct rationalization of the figures as such, mostly with little success. One modern scholar, exasperated by his inability to make sense of the text, suggested errors in transcription. In brief, the passage is notoriously difficult. For this problem, which we have not yet introduced, Aumann and Maschler (2) provided an algorithm for solution in their breakthrough paper via the game-theoretic approach, utilizing the nucleolus concept. They presented, in their own words, three different justifications of the solution to the bankruptsy problem that the nucleolus prescribes in terms that are independent of each other and of game theory, and that were well within the reach of the sages of the Mishna. A remarkable fact is that the non-game-theoretic approaches were identified by Aumann and Maschler (2) after they had found the solution via game-theoretic analysis; again in their own words, only after realizing that the numbers in the Mishna correspond to the nucleolus did we find independent rationales. Without the 2000 Mathematics Subject Classification. 01A07 ethnomathematics, 91B mathematical economics, 97D education and instruction in mathematics.