Equilibria and Convergence in Fire Sale Games

TL;DR

This paper analyzes the existence, structure, and convergence of equilibria in a game-theoretic model of fire sales, revealing conditions under which stable outcomes occur and how dynamics approach these equilibria.

Contribution

It establishes existence and lattice structure of equilibria for convex price impact functions and analyzes convergence properties of best-response dynamics in fire sale games.

Findings

Equilibria exist for convex price impact functions and form a complete lattice.

Best-response dynamics often converge to socially optimal equilibria from natural initial states.

A simplified, less informational variant of dynamics also converges to the same equilibria.

Abstract

The complex interactions between algorithmic trading agents can have a severe influence on the functioning of our economy, as witnessed by recent banking crises and trading anomalies. A common phenomenon in these situations are fire sales, a contagious process of asset sales that trigger further sales. We study the existence and structure of equilibria in a game-theoretic model of fire sales. We prove that for a wide parameter range (e.g., convex price impact functions), equilibria exist and form a complete lattice. This is contrasted with a non-existence result for concave price impact functions. Moreover, we study the convergence of best-response dynamics towards equilibria when they exist. In general, best-response dynamics may cycle. However, in many settings they are guaranteed to converge to the socially optimal equilibrium when starting from a natural initial state. Moreover, we discuss a simplified variant of the dynamics that is less informationally demanding and converges to the same equilibria. We compare the dynamics in terms of convergence speed.