  Modern financial networks are characterized by complex structures of mutual
obligations. Such interconnections may propagate and amplificate individual
defaults, leading in some cases to financial disaster. For this reason,
mathematical models for the study and control of systemic risk (the risk of
severe instabilities on the system as a whole, due to default of single
entities) have attracted considerable research attention in recent years. One
important line of research is concerned with mechanisms of clearing, that is,
the mechanism by which mutual debts are repaid, in the regular regime, or in a
default regime. One of the first models of a clearing mechanism was proposed by
Eisenberg and Noe and is based on the three rules: limited liability, the
priority of debt claims over the shareholders' interests, and the equal
priority of debts (pro-rata rule). These three principles naturally lead to the
concept of clearing vector (the vector of the entities' total payments). In
this paper, we propose a necessary and sufficient condition for the uniqueness
of clearing vector applicable to an arbitrary topology of the financial
network. Further, we show that the overall system loss can be reduced if one
relaxes the pro-rata rule and replaces the clearing vector by a matrix of
clearing payments. This approach shifts the focus from the individual interest
to the system, or social, interest, in order to control and contain the adverse
effects of cascaded failures.
