Review of Economic Philosophy

Tag: Shapley value

  • Three Wives Problem and Shapley Value. Answer to Professor De Mesnard’s Criticism

    Abstract

    The solutions of the old problems of bankruptcy, rights arbitration, and contested garment were given in the Talmud without any justification. A paper published in the journal Synthese (Guiasu 2011) showed that these solutions may be obtained using the Shapley value from the n-person game theory if the characteristic functions of the games corresponding to these three problems essentially depend on the respective context. Three years later, a criticism of the paper from Synthese was posted on the internet (De Mesnard 2014). Here is the answer to this criticism.

    Classification JEL : B11, B16, C02, C71

    Outline

    1. Introduction
    2. Solving the Talmudic problems of bankruptcy and contested garment using the Shapley value
      1. Solution of the bankruptcy problem
      2. Solution of the contested garment problem
    3. Answer to Professor De Mesnard’s criticism
    4. Conclusion

    Keywords

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  • The 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”. This is a misstatement 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 creates a discontinuity in the distribution of the estate, which is not observed in the contested garment problem. 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.

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

    Keywords

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