Computing Tarski Fixed Points in Financial Networks

2026-02-18

AI Summary

This study addresses the efficient computation of clearing states—specifically, the least and greatest Tarski fixed points—in generalized Eisenberg–Noe financial network models that incorporate default costs and general monotone piecewise-linear payment functions. The work presents the first strongly polynomial-time algorithm for exactly computing the least fixed point and provides an efficient method for computing the greatest fixed point in the presence of default costs. Furthermore, it introduces a combinatorial optimization framework to efficiently determine and construct Pareto-improving debt swaps. The proposed approach also enables rapid verification of the existence of fixed points within any specified interval in networks without default costs.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Tarski fixed points
• financial networks
• clearing
• claims trading
• default costs

Innovation

• Tarski fixed point
• financial networks
• clearing algorithms
• minimal clearing
• claims trading