The Reverse Talmud Rule for Bankruptcy Problems

We introduce a new solution for bankruptcy problems that satisfies weaker versions of the Exemption en Exclusion properties from the literature. Although the principles of Exclusion and Exemption are appealing, the specific conditions under which an agent receives its claim, respectively nothing, seem arbitrary and are inconsistent in the sense that there is no bankruptcy rule that satisfies both. However, weakening these conditions (by putting lower boundaries on what is considered to be a ‘small claim’), there do exist rules satisfying both principles. In this paper we consider a Weak Exemption and a Weak Exclusion property such that there is a unique bankruptcy rule that satisfies these two properties together with Consistency and Weak Proportionality (i.e. a change in the estate effects the payoffs of agents with bigger claims more than the payoffs of agents with smaller claims). This rule turns out to be some kind of reverse of the famous Talmud rule. Moreover, we show that Weak Exemption and Weak Exclusion are each others dual, and that the Reverse Talmud rule also can be characterized as the unique Self-Dual solution that satisfies Weak Proportionality and either Weak Exemption or Weak Exclusion. Finally, we generalize the Reverse Talmud rule to a class of bankruptcy rules that all satisfy some Weak Exemption and some Weak Exclusion property (that are not necessarily each others dual), which contains the famous Constrained Equal Awards and Constrained Equal Losses rules as extreme cases.