On solving mutual liability problems

Abstract

This paper introduces mutual liability problems, as a generalization of bankruptcy problems, where every agent not only owns a certain amount of cash money, but also has outstanding claims and debts towards the other agents. Assuming that the agents want to cash their claims, we will analyze mutual liability rules which prescribe how the total available amount of cash should be allocated among the agents. We in particular focus on bilateral -transfer schemes, which are based on a bankruptcy rule . Although in general a -transfer scheme need not be unique, we show that the resulting -transfer allocation is. This leads to the definition of -based mutual liability rules. For so called hierarchical mutual liability problems an alternative characterization of -based mutual liability rules is provided. Moreover it is shown that the axiomatic characterization of the Talmud rule on the basis of consistency can be extended to the corresponding mutual liability rule.

Decentralization and Mutual Liability Rules

“…Groote Schaarsberg et al (2018) present a method of solving mutual liability problems that makes use of so-called payment matrices. A payment matrix is a non-negative matrix P = (p ij ) i,j∈N where cell p ij indicates the payment of agent i to agent j.…”

“…As Example 2.1 shows, computing ϕ-transfer schemes can be complex in general as we typically deal with them on an ad hoc basis. Nonetheless, Groote Schaarsberg et al (2018) show that for a special subclass of mutual liability problems, so-called hierarchical mutual liability problems, ϕ-transfer schemes can be determined recursively. We will build on this recursive approach, albeit we extend it to general mutual liability problems and base it on a decentralized mechanism.…”

“…Had it not been for governments bailing out other financial institutions in financial distress, the financial system would have been on the verge of collapse. 1 The allocation issue described above can be modeled as a mutual liability problem as introduced by Groote Schaarsberg, Reijnierse, and Borm (2018). Mutual liability problems are a generalization of the well-studied notion of a bankruptcy problem as introduced by O' Neill (1982).…”

“…In fact, ϕ-based mutual liability rules will be the focus of this paper. A ϕ-based mutual liability rule put forward by Groote Schaarsberg et al (2018) prescribes that, for each mutual liability problem, the final allocation of the total estate should be the direct consequence of a ϕ-transfer scheme of bilateral payments that satisfies an internal consistency property.…”

“…This paper builds on the work of Groote Schaarsberg et al (2018) and provides new insights into ϕ-based mutual liability rules. The approach of Groote Schaarsberg et al (2018) can be regarded as a centralized mechanism, i.e. an independent authority is assigned the task to allocate the total estate among the agents.…”