On Ibn Ezra's Procedure and Shapley Value

Abstract

We examine ibn Ezra's procedure (Rabinovitch 1973; O'Neill 1982) historically used to solve the Rights Arbitration problem in the general framework of bankruptcy problems. When the greatest claim is larger than or equal to the estate, the procedure is a maximal game (Aumann 2010). However, when the greatest claim is smaller than the estate, the axioms of efficiency (the whole estate is distributed) and satiation are difficult to satisfy simultaneously. We discuss both axioms to show that their importance and necessity are radically different. From then, for the part of the estate not covered by the greatest claim, we examine four possible procedures: the minimal overlap rule, Alcalde et al.'s (2005) iterative generalization, dictator game, and unanimity game. They pose many problems mainly because: (i) only the greatest claimant remains in the game; (ii) playing two different subgames is unrealistic; (iii) some claimants may receive more than they claim, which violates the satiation axiom; (iv) under the minimal overlap rule players simultaneously play a unanimity game and a dictator game; (v) Alcalde et al.'s generalization is is efficient but converges very slowly. This is why we propose an original and different rule, the series of nested unanimity games (SNUG). We show that it is guarantees efficiency and satiation for all claimants and compared to the other rules, SNUG is attractively homogenous --- only unanimity games are played.