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.
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