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Computing Tarski Fixed Points in Financial Networks

Leander Besting, Martin Hoefer, Lars Huth

TL;DR

An efficient algorithm to compute a minimal fixed point that runs in strongly polynomial time applies in a broad generalization of the Eisenberg-Noe model with any monotone, piecewise-linear payment functions and 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.

Computing Tarski Fixed Points in Financial Networks

TL;DR

An efficient algorithm to compute a minimal fixed point that runs in strongly polynomial time applies in a broad generalization of the Eisenberg-Noe model with any monotone, piecewise-linear payment functions and 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.
Paper Structure (21 sections, 15 theorems, 31 equations, 1 figure, 1 algorithm)

This paper contains 21 sections, 15 theorems, 31 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Results and Techniques
  3. Further Related Work
  4. Model and Preliminaries
  5. Network Model
  6. Payment Functions
  7. Clearing States
  8. Claims Trades
  9. Computing Clearing States
  10. Minimal Clearing State
  11. Flooding SCCs
  12. Solving LP \ref{['eq:FloodLP']}
  13. Increasing External Assets
  14. Solving LP \ref{['eq:IncreaseLP']}
  15. Correctness and Polynomial Time
  16. ...and 6 more sections

Key Result

Lemma 1

At the beginning of each iteration of the main while-loop, $\mathbf{b}^{(x)} \le \mathbf{a}^{(x)}$ and $\mathbf{b}$ is a minimal clearing state of the financial network $\mathcal{F}$ with external assets $\mathbf{b}^{(x)}$.

Figures (1)

  • Figure 1: Consider the financial network displayed at the top. Payments of bank $v$ on each edge for edge-ranking payment functions with $(v,w)$ ranked first (left); proportional payment functions (middle); piecewise-linear functions with interval borders $0, 50,55,90,100,\infty$ (right).

Theorems & Definitions (36)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • proof
  • Definition 2
  • Definition 3
  • Lemma 3
  • proof
  • Lemma 4
  • proof : Proof of Lemma \ref{['lem:invariant']}
  • ...and 26 more