Proportional Dynamics in Exchange Economies

Simina Brânzei, Nikhil Devanur, Yuval Rabani

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

8 Scopus citations


We study the proportional dynamics in exchange economies, where each player starts with some amount of money and a good. Every day, players bring one unit of their good and submit bids on goods they like, each good gets allocated in proportion to the bid amounts, and each seller collects the bids received. Then every player updates their bids proportionally to the contribution of each good in their utility. This dynamic models a process of learning how to bid and has been studied in a series of papers on Fisher and production markets, but not in exchange economies. Our main results are as follows: (1) For all linear utilities, the dynamic converges to market equilibrium utilities and allocations, while the bids and prices may cycle. We give a combinatorial characterization of limit cycles for prices and bids. (2) We introduce a lazy version of the dynamic, where players may save money for later, and show this converges in everything: utilities, allocations, and prices. This answers an open question about exchange markets with linear utilities, where tatonnement does not converge to market equilibria, and no natural process leading to equilibria was known for all additive utilities. We also note this dynamics represents a process where the players exchange goods throughout time (in out-of-equilibrium states), while tatonnement only explains how exchange happens in the limit.

Original languageAmerican English
Title of host publicationEC 2021 - Proceedings of the 22nd ACM Conference on Economics and Computation
PublisherAssociation for Computing Machinery, Inc
Number of pages22
ISBN (Electronic)9781450385541
StatePublished - 18 Jul 2021
Event22nd ACM Conference on Economics and Computation, EC 2021 - Virtual, Online, Hungary
Duration: 18 Jul 202123 Jul 2021

Publication series

NameEC 2021 - Proceedings of the 22nd ACM Conference on Economics and Computation


Conference22nd ACM Conference on Economics and Computation, EC 2021
CityVirtual, Online

Bibliographical note

Publisher Copyright:
© 2021 ACM.


  • dynamical systems
  • exchange markets
  • learning to bid
  • proportional response


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