Equilibria and Systemic Risk in Saturated Networks

TL;DR

This paper analyzes network equilibria with saturation nonlinearities, providing explicit formulas, conditions for uniqueness, and studying discontinuities, with applications to systemic risk in financial networks and strategic game behaviors.

Contribution

It extends existing models by deriving explicit equilibrium expressions, establishing conditions for uniqueness, and analyzing bifurcations in saturated network systems.

Findings

Explicit formulas for network equilibria

Necessary and sufficient conditions for equilibrium uniqueness

Identification of bifurcation phenomena affecting systemic risk and strategic sensitivities

Abstract

We undertake a fundamental study of network equilibria modeled as solutions of fixed point equations for monotone linear functions with saturation nonlinearities. The considered model extends one originally proposed to study systemic risk in networks of financial institutions interconnected by mutual obligations and is one of the simplest continuous models accounting for shock propagation phenomena and cascading failure effects. It also characterizes Nash equilibria of constrained quadratic network games with strategic complementarities. We first derive explicit expressions for network equilibria and prove necessary and sufficient conditions for their uniqueness encompassing and generalizing results available in the literature. Then, we study jump discontinuities of the network equilibria when the exogenous flows cross certain regions of measure 0 representable as graphs of continuous functions. Finally, we discuss some implications of our results in the two main motivating applications. In financial networks, this bifurcation phenomenon is responsible for how small shocks in the assets of a few nodes can trigger major aggregate losses to the system and cause the default of several agents. In constrained quadratic network games, it induces a blow-up behavior of the sensitivity of Nash equilibria with respect to the individual benefits.