We study the Proportional Response dynamic in exchange economies, where each player starts with some amount of money and a good. Every day, the 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 the 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: - For 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. - 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. - For CES utilities in the substitute range $[0,1)$, the dynamic converges for all parameters. This answers an open question about exchange economies with linear utilities, where tatonnement does not converge to market equilibria, and no natural process leading to equilibria was known. We also note that proportional response is a process where the players exchange goods throughout time (in out-of-equilibrium states), while tatonnement only explains how exchange happens in the limit.

翻译：我们研究交换经济中的比例响应动态，其中每个参与者初始持有一定数量的货币和一种商品。每日，参与者提供一单位其商品，并对喜爱的商品进行竞价，每种商品根据竞价金额按比例分配，卖方收集所获竞价款。随后，每个参与者根据每种商品对其效用的贡献比例更新竞价。该动态建模了学习如何竞价的过程，已在关于Fisher市场和生产市场的系列论文中得到研究，但尚未应用于交换经济。我们的主要结果如下：- 在线性效用下，动态收敛至市场均衡效用与分配，但竞价和价格可能循环。我们给出价格与竞价极限循环的组合特征描述。- 我们提出一种惰性版本动态，其中参与者可储蓄货币以备后续使用，并证明该版本在效用、分配和价格上均收敛。- 对于替代范围$[0,1)$内的CES效用，动态对所有参数均收敛。这回答了关于线性效用交换经济的一个未解决问题：在该场景下，试错法不收敛于市场均衡，且此前未知任何可导向均衡的自然过程。我们还指出，比例响应是一个参与者随时间（在非均衡状态下）进行商品交换的过程，而试错法仅解释极限状态下的交换如何发生。