Computing Tarski Fixed Points in Financial Networks

TL;DR

This paper introduces efficient algorithms for computing minimal and maximal fixed points in complex financial networks, enhancing understanding of systemic risk and clearing mechanisms in interconnected institutions.

Contribution

It presents the first strongly polynomial algorithm for minimal fixed points and a polynomial-time method for maximal fixed points in generalized Eisenberg-Noe models.

Findings

Efficient algorithms for minimal fixed points in broad financial network models.

Polynomial-time computation of maximal fixed points under certain conditions.

Decidability results for fixed point existence in networks without default costs.

Abstract

Modern financial networks are highly connected and result in complex interdependencies of the involved institutions. In the prominent Eisenberg-Noe model, a fundamental aspect is clearing -- to determine the amount of assets available to each financial institution in the presence of potential defaults and bankruptcy. A clearing state represents a fixed point that satisfies a set of natural axioms. Existence can be established (even in broad generalizations of the model) using Tarski's theorem. While a maximal fixed point can be computed in polynomial time, the complexity of computing other fixed points is open. In this paper, we provide an efficient algorithm to compute a minimal fixed point that runs in strongly polynomial time. It applies in a broad generalization of the Eisenberg-Noe model with any monotone, piecewise-linear payment functions and default costs. Moreover, in this scenario we provide a polynomial-time algorithm to compute a maximal fixed point. For networks without default costs, we can efficiently decide the existence of fixed points in a given range. We also study claims trading, a local network adjustment to improve clearing, when networks are evaluated with minimal clearing. We provide an efficient algorithm to decide existence of Pareto-improving trades and compute optimal ones if they exist.