Three Wives Problem and Shapley Value

Abstract

We examine the Talmudic three wives problem, which is a generalization of the Talmudic contested garment problem solved by Aumann and Maschler (1985) using coalitional procedure. This problem has many practical applications. In an attempt to unify all Talmudic methods, Guiasu (2010, 2011) asserts that it can be explained in terms of “run-to-the-bank”, that is, of Shapley value in a “cumulative game”. It can be challenged because the coalitional procedure yields the same result as the nucleolus, which corresponds to a “dual game”. As Guiasu’s solution is paradoxical (it has all the appearances of truth), my contribution consists in explaining the concepts, particularly truncation, that play a central role in the demonstration, and then analyzing in what way Guiasu’s argument is misleading. After recalling what the Talmudic division problem is, how it is solved by Aumann and Maschler’s coalitional procedure (i.e., the nucleolus or the Shapley value of the dual game), and how Guiasu solves it by the Shapley value of a cumulative game, I show that (i) Guiasu omitted to truncate the data (claims exceeding the value of the estate must be reduced to the available level of estate), while truncation is required in the context. (ii) He attributes the surplus (obtained after sharing out the estate) to all applicants equally: this contradicts the contested garment solution. (iii) This implies that the estate cannot exceed the greatest claim, which is obviously false. (iv) Guiasu’s approach violates the axiom of continuity of payoffs. I conclude that Guiasu’s attempt to explain the three wives problem in terms of “run-to-the-bank” is unsuccessful and actually contradicts the contested garment problem.

Keywords: Shapley value, contested garment, three wives, Talmudic division

JEL Classification: D31, D63, D71, B1, B4