The TAL-family of rules for bankruptcy problems

Résumé

This paper analyzes a family of solutions to bankruptcy problems that generalizes the Talmud rule (T) and encompasses both the con- strained equal-awards rule (A) and the constrained equal-losses rule (L). We study the structural properties of this family of rules and provide a characterization result that allows us to identify the distinctive features of these three reference rules, T,A,L. A bankruptcy problem describes a situation in which an arbitrator has to allocate a given amount E >0 of a perfectly divisible commodity, referred to as the estate, among a group N of agents, when the available amount is not enough to satisfy all their claims (ci)i�¸N. That is, E . X i�¸N ci. A solution to a bankruptcy problem is a procedure or .rule. that ex- hibits some desirable properties and determines, for each speciTc prob- lem, an allocation that satisTes two elementary restrictions: (i) No agent gets more than she claims nor less than zero; and (ii) The whole estate is distributed. Note that most rationing problems can be given this form. Since this is a well known problem and examples of these situations abound, we shall not insist on its relevance. The reader is referred to the works of Young (1994, ch. 4), Thomson (1995) and Moulin (2001) for a review of this literature. There are four classical solutions to the bankruptcy problem: the proportional solution, usually associated with Aristotle, the constrained equal-awards rule and the constrained equal-losses rule, that can be traced back to Maimonides, and the Talmud rule, that extends the an- cient .contested garment principle. [see Herrero & Villar (2001) for a comparative analysis of these solutions]. The Trst three of those rules implement the idea of equal division, with different reference variables (ratios, awards and losses, respectively). The Talmud rule is an alter- native procedure that combines the principles that identify the former three rules. It can be interpreted as implementing a protective criterion that ensures that all agents suffer a rationing that is .of the same sort. of that experienced by the whole society. The distribution procedure depends on whether the estate E is above or below one half of the ag- gregate claim P i.N ci. It can be justiÞed on the psychological principle of .more than half is like the whole, whereas less than a half is like nothing.. Thus, this rule looks at the size of the awards when they are below half of the aggregate claim (E = 12 P i.N ci) and at the size of the losses above that magnitude (E = 1 2 P i.N ci). The TAL-family generalizes this idea by applying exactly the same principle to all possible shares of the estate in the aggregate claim1. That is, for any given value of the parameter . . [0, 1], a rule in this family takes into account whether E is above or below . P i.N ci, and distributes the estate correspondingly. The rule associated with . = 1 2 is precisely the Talmud rule, as expected, and the extreme values . = 1 and . = 0 correspond to the constrained equal awards rule and the constrained equal losses rule, respectively. The proportional solution, however, is not part of this family. We show that the TAL-family is made exactly of those rules which are consistent, continuous, claims monotonic (both in gains and losses), and satisfy either independence of claims truncation or composition from minimal rights. In the Þrtst case we have . . £1 2 , 1 ¤ , whereas in the second one . . £ 0, 12 ¤ . Moreover, the characterization result is tight. We also show that these rules exhibit a precise duality relationship: the dual of the rule associated with the parameter . is that rule associated with the parameter (1 - .). 1