On the Convergence of T\^atonnement for Linear Fisher Markets

TL;DR

This paper proves that the t extsuperscript{atonnement} process converges to equilibrium prices in linear Fisher markets with a small step size, using convex optimization techniques and error bound conditions, supported by numerical experiments.

Contribution

It establishes the convergence of t extsuperscript{atonnement} in linear Fisher markets by linking it to convex optimization and error bounds, a problem previously unresolved.

Findings

Convergence guaranteed with small step size

Convergence up to a small approximation radius

Numerical results match theoretical convergence rates

Abstract

T\^atonnement is a simple, intuitive market process where prices are iteratively adjusted based on the difference between demand and supply. Many variants under different market assumptions have been studied and shown to converge to a market equilibrium, in some cases at a fast rate. However, the classical case of linear Fisher markets have long eluded the analyses, and it remains unclear whether t\^atonnement converges in this case. We show that, for a sufficiently small step size, the prices given by the t\^atonnement process are guaranteed to converge to equilibrium prices, up to a small approximation radius that depends on the stepsize. To achieve this, we consider the dual Eisenberg-Gale convex program in the price space, view t\^atonnement as subgradient descent on this convex program, and utilize last-iterate convergence results for subgradient descent under error bound conditions. In doing so, we show that the convex program satisfies a particular error bound condition, the quadratic growth condition, and that the price sequence generated by t\^atonnement is bounded above and away from zero. We also show that a similar convergence result holds for t\^atonnement in quasi-linear Fisher markets. Numerical experiments are conducted to demonstrate that the theoretical linear convergence aligns with empirical observations.