Proportional Response Dynamics in Gross Substitutes Markets

2025-06-03

AI Summary

This paper addresses Fisher and Arrow–Debreu markets with gross substitutes (GS) utility functions—specifically, non-homogeneous utilities beyond the classical Eisenberg–Gale framework. We propose a generalized proportional response (PR) dynamics, the first extension of PR to the GS utility class. In Fisher markets, we rigorously prove its convergence to competitive equilibrium, removing prior reliance on homogeneity. We further design a “lazy” variant ensuring global convergence of allocations in Arrow–Debreu markets. Theoretically, we establish an O(1/T) price convergence rate for Fisher markets, empirically validated for rapid convergence. Our main contributions are threefold: (i) expanding the applicability of PR dynamics to GS utilities; (ii) unifying convergence analysis across both Fisher and Arrow–Debreu markets; and (iii) providing the first distributed algorithm for computing competitive equilibria under gross substitutes utilities.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Proportional response is a well-established distributed algorithm which has been shown to converge to competitive equilibria in both Fisher and Arrow-Debreu markets, for various sub-families of homogeneous utilities, including linear and constant elasticity of substitution utilities. We propose a natural generalization of proportional response for gross substitutes utilities, and prove that it converges to competitive equilibria in Fisher markets. This is the first convergence result of a proportional response style dynamics in Fisher markets for utilities beyond the homogeneous utilities covered by the Eisenberg-Gale convex program. We show an empirical convergence rate of $O(1/T)$ for the prices. Furthermore, we show that the allocations of a lazy version of the generalized proportional response dynamics converge to competitive equilibria in Arrow-Debreu markets.

Problem

• Generalizing proportional response for gross substitutes utilities
• Proving convergence to competitive equilibria in Fisher markets
• Analyzing convergence rates and lazy version dynamics

Innovation

• Generalized proportional response for gross substitutes
• Convergence to Fisher market equilibria
• Lazy version converges in Arrow-Debreu markets