On the Convergence of Tâtonnement for Linear Fisher Markets

TL;DR

This work tackles the long-standing question of whether discrete-time tâtonnement converges in Linear Fisher Markets (LFM). By recasting tâtonnement as last-iterate subgradient descent on the dual Eisenberg-Gale program, it establishes a quadratic-growth condition and bounded subgradients, leading to linear convergence to an $\epsilon$-neighborhood of the equilibrium for sufficiently small step $\eta$, with the neighborhood shrinking as $\eta\to 0$; the analysis extends to quasi-linear Fisher markets (QLFM) as well. The contributions include the first provable, near-linear convergence results for LFM, explicit bounds on price lower/upper bounds and excess-demand, and a clear demonstration of when convergence breaks (non-convergence cycles) for certain step sizes. Numerical experiments on synthetic and real data corroborate the theory, showing rapid initial convergence followed by small oscillations around the equilibrium that shrink with smaller $\eta$. Overall, the paper provides principled guarantees for decentralized price adjustment in LFMs and QLFMs, clarifying step-size roles and offering practical implications for online marketplace design.

Abstract

Tâtonnement 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 tâtonnement converges in this case. We show that, for a sufficiently small step size, the prices given by the tâtonnement 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 tâtonnement 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 tâtonnement is bounded above and away from zero. We also show that a similar convergence result holds for tâtonnement in quasi-linear Fisher markets. Numerical experiments are conducted to demonstrate that the theoretical linear convergence aligns with empirical observations.

Figures (19)

• Figure 1: Convergence of squared error norms on random generated instances ($v$ is generated from the exponential distribution with scale $1$) of different sizes under linear utilities. The bottom row zooms in on partial iterations to better show the initial linear convergence.
• Figure 2: Convergence of squared error norms on random generated instances of different sizes under QL utilities.
• Figure 3: Convergence of squared error norms on our instances generated from a movie rating dataset.
• Figure 4: Convergence of squared error norms on random generated instances ($v$ is generated from the uniform distribution $[0,1)$) of different sizes under linear utilities.
• Figure 5: Convergence of squared error norms on random generated instances ($v$ is generated from the uniform distribution $[0,1)$) of different sizes under quasi-linear utilities.

Theorems & Definitions (22)

• Lemma 1
• Lemma 2
• Lemma 3
• proof
• Theorem 1
• Corollary 1
• Theorem 2
• Corollary 2
• Lemma 4
• proof