We show that the proportional response dynamics, a utility based distributed dynamics, converges to the market equilibrium in the Fisher market with constant elasticity of substitution (CES) utility functions. By the proportional response dynamics, each buyer allocates his budget proportional to the utility he receives from each good in the previous time period. Unlike the tǎtonnement process and its variants, the proportional response dynamics is a large step discrete dynamics, and the buyers do not solve any optimization problem at each step. In addition, the goods are always cleared and assigned to the buyers proportional to their bids at each step. Despite its simplicity, the dynamics converges fast for strictly concave CES utility functions, matching the best upper-bound of computing the market equilibrium via the solution of a global convex optimization problem. © 2010 Elsevier B.V. All rights reserved.
Zhang, L. (2011). Proportional response dynamics in the Fisher market. In Theoretical Computer Science (Vol. 412, pp. 2691–2698). https://doi.org/10.1016/j.tcs.2010.06.021