Duality in financial networks

This article introduces the concept of duality in financial networks, which entails the division of losses versus gains. In bankruptcy problems, in which a bankrupt entity divides its non-negative assets among a group of claimants, duality is well understood. Financial networks generalize bankruptcy problems by allowing for multiple agents with individual assets interconnected by mutual claims. We argue that allowing for negative assets is imperative to adequately formulate duality for financial networks. We show that there is a linear relationship between transfer schemes based on claims rules in a financial network and transfer schemes based on the dual of those claims rules in the dual financial network. Moreover, dual financial networks enable us to define dual transfer rules and dual allocation rules. We show that both self-dual transfer rules and self-dual allocation rules have to be based on self-dual claims rules. On the other hand, we demonstrate that transfer rules based on self-dual claims rules need not be self-dual, whereas allocation rules based on self-dual claims rules are always self-dual.