This paper builds on the recent work of Groote Schaarsberg et al. (Math Methods Oper Res 87(3):383–409, 2018) on mutual liability problems. In essence, a mutual liability problem comprises a financial network in which agents may have both monetary individual assets and mutual liabilities. Here, mutual liabilities reflect rightful monetary obligations from past bilateral transactions. To settle these liabilities by reallocating the individual assets, mutual liability rules are analyzed that are based on centralized bilateral transfer schemes which use a certain bankruptcy rule as its leading allocation mechanism. In this paper we derive a new characterization of mutual liability rules by taking a decentralized approach instead, which is based on a recursive individual settlement procedure. We show that for bankruptcy rules that satisfy composition, this decentralized procedure always leads to the same allocation as the one prescribed by the corresponding mutual liability rule based on centralized bilateral transfer schemes. Finally, we introduce a new reduction method for mutual liability problems and prove that any bankruptcy-rule-based mutual liability rule is invariant with respect to such a reduction.

本文以Groote Schaarsberg等人的最新工作为基础。（Math Methods Oper Res 87（3）：383–409，2018）关于共同责任问题。本质上，共同责任问题包括一个金融网络，在这个网络中，代理人可能同时拥有货币性的个人资产和共同责任。在这里，共同负债反映了过去双边交易产生的合法货币义务。为了通过重新分配单个资产来清算这些负债，分析了基于集中式双边转移计划的共同赔偿责任规则，该计划使用某种破产规则作为其主要分配机制。在本文中，我们通过采用基于递归个人结算程序的分散方法，推导出了相互责任规则的新特征。我们表明，对于满足组成要求的破产规则，这种分散的程序总是导致与基于集中式双边转移计划的相应共同责任规则所规定的分配相同的分配。最后，我们为共同责任问题引入了一种新的减少方法，并证明了任何基于破产规则的共同责任规则对于这种减少都是不变的。