Proportional dynamics, originated from peer-to-peer file sharing systems, models a decentralized price-learning process in Fisher markets. Previously, items in the dynamics operate independently of one another, and each is assumed to belong to a different seller. In this paper, we show how it can be generalized to the setting where each seller brings multiple items and buyers allocate budgets at the granularity of sellers rather than individual items. The generalized dynamics consistently converges to the competitive equilibrium, and interestingly relates to the auto-bidding paradigm currently popular in online advertising auction markets. In contrast to peer-to-peer networks, the proportional rule is not imposed as a protocol in auto-bidding markets. Regarding this incentive concern, we show that buyers have a strong tendency to follow the rule, but it is easy for sellers to profitably deviate (given buyers' commitment to the rule). Based on this observation, we further study the seller-side deviation game and show that it admits a unique pure Nash equilibrium. Though it is generally different from the competitive equilibrium, we show that it attains a good fairness guarantee as long as the market is competitive enough and not severely monopolized.

翻译：比例动态起源于点对点文件共享系统，用于模拟Fisher市场中的去中心化价格学习过程。以往研究中，该动态中的商品相互独立运作，且假定每件商品分属不同卖家。本文展示了如何将该动态推广至每个卖家提供多件商品、且买家以卖家而非单个商品为粒度分配预算的场景。推广后的动态始终收敛至竞争均衡，且有趣地与当前在线广告拍卖市场中流行的自动出价范式相关联。与点对点网络不同，比例规则在自动出价市场中并非强制协议。针对这一激励问题，我们证明买家有强烈动机遵循该规则，但卖家极易通过策略性偏离获利（在买家承诺遵守规则的前提下）。基于此发现，我们进一步研究卖家侧偏离博弈，证明其存在唯一纯策略纳什均衡。尽管该均衡通常不同于竞争均衡，但我们证明只要市场具有足够竞争性且未形成严重垄断，该均衡仍能提供良好的公平性保证。