The Three Musketeers. Old Solutions to Bankruptcy Problems

This paper concentrates on a comparative analysis of three basic rules to solve bankruptcy problems from an axiomatic viewpoint. These rules are: (i) The proportional rule, that divides the estate proportionally to agents' claims. (ii) The constrained equal-awards rule, that divides equally the estate among the agents under the condition that nobody gets more than her claim. (iii) The constrained equal-losses rule, that divides equally the diference between the aggregate claims and the budget, provided no agent ends up with a negative award. The comparative analysis of the three aforementioned rules aims at clarifying the class of real life problems for which each of these solutions is better. With this purpose in mind and following the recommendations given above, we concentrate on those characterizations that permit an easy comparison of these three rules. In particular, we focus on a family of results that characterize each of these rules by three independent axioms, two of which are common to all of them. As the Three Musketeers were four so are our three rules. The Talmud rule here will play the role of D'Artagnan. This is an appealing allocation rule that amounts to solve bankruptcy problems by combining the constrained equal awards rule and the constrained equal losses rule. We start by formally introducing the family of bankruptcy problems and the three basic rules. Then we present several appealing properties for bankruptcy rules. We offer a joint characterization of the three bankruptcy rules in terms of some of those properties, as well as independent characterizations. Previous properties help us to analyze also the contested garment rule, and its consistent extension, the Talmud rule (the rule playing here the role of D'Artagnan). Variants and extensions of the aforementioned rules are also analyzed. The paper ends by providing noncooperative support of the constrained equal-awards rule and of the constrained equal-losses rule.