On the Convergence of T^atonnement for Linear Fisher Markets

2024-06-18

AI Summary

This paper investigates the convergence of the tâtonnement price-adjustment process in linear Fisher markets. Methodologically, it establishes, for the first time, an equivalence between tâtonnement and the Eisenberg–Gale dual program—showing that tâtonnement corresponds precisely to subgradient descent in the price space—and identifies and verifies a quadratic growth error bound in this space, a key structural condition enabling accelerated convergence analysis. Building on this, the paper rigorously proves that, under sufficiently small step sizes, the tâtonnement-generated price sequence converges linearly to an ε-approximate market equilibrium—a long-standing theoretical gap in the literature. The theoretical guarantees are comprehensively validated through systematic numerical experiments, demonstrating close alignment between predicted convergence rates and empirical behavior.

Technology Category

• Game Theory and Economic Paradigms: Auctions and Market-Based Systems
• Multiagent Systems: Mechanism Design
• Search and Optimization: Learning to Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

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.

Problem

• Convergence of tâtonnement in linear Fisher markets
• Analysis of price adjustment in market equilibrium
• Application of subgradient descent in convex programming

Innovation

• Tâtonnement as subgradient descent on convex program
• Convergence guaranteed with small step size
• Quadratic growth condition ensures price sequence stability