Fisher markets are one of the most fundamental models for resource allocation. However, the problem of computing equilibrium prices in Fisher markets typically relies on complete knowledge of users' budgets and utility functions and requires transactions to happen in a static market where all users are present simultaneously. Motivated by these practical considerations, we study an online variant of Fisher markets, wherein users with privately known utility and budget parameters, drawn i.i.d. from a distribution, arrive sequentially. In this setting, we first study the limitations of static pricing algorithms, which set uniform prices for all users, along two performance metrics: (i) regret, i.e., the optimality gap in the objective of the Eisenberg-Gale program between an online algorithm and an oracle with complete information, and (ii) capacity violations, i.e., the over-consumption of goods relative to their capacities. Given the limitations of static pricing, we design adaptive posted-pricing algorithms, one with knowledge of the distribution of users' budget and utility parameters and another that adjusts prices solely based on past observations of user consumption, i.e., revealed preference feedback, with improved performance guarantees. Finally, we present numerical experiments to compare our revealed preference algorithm's performance to several benchmarks.

翻译：费舍尔市场是资源分配中最基础的模型之一。然而，费舍尔市场中均衡价格的计算问题通常依赖于对用户预算和效用函数的完全了解，并要求交易发生在所有用户同时存在的静态市场中。基于这些实际考量，我们研究了一种在线变体的费舍尔市场，其中用户具有私有的效用和预算参数，这些参数从某个分布中独立同分布地抽取，并按顺序到达。在此设定下，我们首先研究了静态定价算法的局限性，该算法为所有用户设定统一价格，并从两个性能指标进行评估：（i）遗憾，即在线算法与拥有完全信息的先知在艾森伯格-盖尔规划目标上的最优性差距；以及（ii）容量违规，即商品相对于其容量的过度消耗。鉴于静态定价的局限性，我们设计了自适应公布定价算法，一种算法了解用户预算和效用参数的分布，另一种算法则仅基于过去观察到的用户消费（即显示偏好反馈）来调整价格，并提供了改进的性能保证。最后，我们通过数值实验，将我们的显示偏好算法与若干基准算法的性能进行了比较。