Proportional Response Dynamics in Gross Substitutes Markets

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.