We consider financial networks, where banks are connected by contracts such as debts or credit default swaps. We study the clearing problem in these systems: we want to know which banks end up in a default, and what portion of their liabilities can these defaulting banks fulfill. We analyze these networks in a sequential model where banks announce their default one at a time, and the system evolves in a step-by-step manner. We first consider the reversible model of these systems, where banks may return from a default. We show that the stabilization time in this model can heavily depend on the ordering of announcements. However, we also show that there are systems where for any choice of ordering, the process lasts for an exponential number of steps before an eventual stabilization. We also show that finding the ordering with the smallest (or largest) number of banks ending up in default is an NP-hard problem. Furthermore, we prove that defaulting early can be an advantageous strategy for banks in some cases, and in general, finding the best time for a default announcement is NP-hard. Finally, we discuss how changing some properties of this setting affects the stabilization time of the process, and then use these techniques to devise a monotone model of the systems, which ensures that every network stabilizes eventually.

中文翻译：

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金融网络中的顺序违约

我们考虑金融网络，通过债务或信用违约掉期等合同将银行连接起来。我们研究了这些系统中的清算问题：我们想知道哪些银行最终违约，这些违约银行可以承担哪些债务。我们以顺序模型分析这些网络，在这种模型中，银行一次公布其默认值，然后系统逐步发展。我们首先考虑这些系统的可逆模型，在这种模型中，银行可能会从违约中退回。我们显示，此模型中的稳定时间在很大程度上取决于公告的顺序。但是，我们还表明，在某些系统中，对于任何顺序的选择，该过程在最终稳定之前持续进行指数级的步骤。我们还表明，找到数量最少（或最大）的银行违约情况下的排序是一个NP难题。此外，我们证明了在某些情况下及早违约对银行而言可能是一种有利的策略，并且总的来说，为违约公告找到最佳时机非常困难。最后，我们讨论了更改此设置的某些属性如何影响过程的稳定时间，然后使用这些技术设计系统的单调模型，以确保每个网络最终稳定下来。