On the Compatibility of Composition Axioms in Financial Networks

This article introduces composition down and composition up in financial networks, and analyzes their compatibility. The starting point for the analysis is the outcome that is prescribed by a rule. For example, a transfer rule prescribes, for each financial network, a clearing payment matrix that contains mutual payments between the agents to settle their mutual liabilities. However, as it turns out, the assets of some of the agents in the financial network are either undervalued or overvalued, making the proposed outcome infeasible to carry out. In such cases, one has the option to either apply the rule to the new situation, or reapply the rule to an appropriately adjusted situation that honors the initial outcome. A composition axiom requires that both options are equivalent to all the agents. In the context of financial networks, there can exist various adequate versions of the two composition axioms. Interestingly, the adequate versions of the two composition axioms that are compatible for allocation rules are not compatible for transfer rules in the sense that no transfer rule can satisfy them simultaneously. Nevertheless, we show that there exist alternative adequate versions of the two composition axioms that are compatible with respect to transfer rules. In fact, we show that the transfer rules that either always prescribe the bottom payment matrix, or the top payment matrix, satisfy the two composition axioms simultaneously.