Dynamics of Distributed Updating in Fisher Markets

TL;DR

This paper proves new convergence results for distributed algorithms in Fisher markets with CES utilities, introducing novel convex formulations and notions of strong Bregman convexity, with implications for algorithm efficiency.

Contribution

It introduces new convex and convex-concave formulations for Fisher markets and establishes convergence rates for generalized Proportional Response algorithms using novel Bregman convexity concepts.

Findings

Linear convergence for a broad class of CES utilities.

Empirical O(1/T) convergence rate when including linear and Leontief utilities.

New notions of strong Bregman convexity and convex-concave functions.

Abstract

A major goal in Algorithmic Game Theory is to justify equilibrium concepts from an algorithmic and complexity perspective. One appealing approach is to identify natural distributed algorithms that converge quickly to an equilibrium. This paper established new convergence results for two generalizations of Proportional Response in Fisher markets with buyers having CES utility functions. The starting points are respectively a new convex and a new convex-concave formulation of such markets. The two generalizations correspond to suitable mirror descent algorithms applied to these formulations. Several of our new results are a consequence of new notions of strong Bregman convexity and of strong Bregman convex-concave functions, and associated linear rates of convergence, which may be of independent interest. Among other results, we analyze a damped generalized Proportional Response and show a linear rate of convergence in a Fisher market with buyers whose utility functions cover the full spectrum of CES utilities aside the extremes of linear and Leontief utilities; when these utilities are included, we obtain an empirical O(1/T) rate of convergence.