In a Fisher market, agents (users) spend a budget of (artificial) currency to buy goods that maximize their utilities while a central planner sets prices on capacity-constrained goods such that the market clears. However, the efficacy of pricing schemes in achieving an equilibrium outcome in Fisher markets typically relies on complete knowledge of users' budgets and utilities and requires that transactions happen in a static market wherein all users are present simultaneously. As a result, we study an online variant of Fisher markets, wherein budget-constrained users with privately known utility and budget parameters, drawn i.i.d. from a distribution $\mathcal{D}$, enter the market sequentially. In this setting, we develop an algorithm that adjusts prices solely based on observations of user consumption, i.e., revealed preference feedback, and achieves a regret and capacity violation of $O(\sqrt{n})$, where $n$ is the number of users and the good capacities scale as $O(n)$. Here, our regret measure is the optimality gap in the objective of the Eisenberg-Gale program between an online algorithm and an offline oracle with complete information on users' budgets and utilities. To establish the efficacy of our approach, we show that any uniform (static) pricing algorithm, including one that sets expected equilibrium prices with complete knowledge of the distribution $\mathcal{D}$, cannot achieve both a regret and constraint violation of less than $\Omega(\sqrt{n})$. While our revealed preference algorithm requires no knowledge of the distribution $\mathcal{D}$, we show that if $\mathcal{D}$ is known, then an adaptive variant of expected equilibrium pricing achieves $O(\log(n))$ regret and constant capacity violation for discrete distributions. Finally, we present numerical experiments to demonstrate the performance of our revealed preference algorithm relative to several benchmarks.

翻译：在Fisher市场中，代理人（用户）花费预算（虚拟货币）购买能最大化自身效用的商品，而中央规划者对容量受限的商品设置价格，以实现市场出清。然而，在Fisher市场中通过定价方案达到均衡结果的有效性通常依赖于对用户预算和效用的完全了解，并且要求交易发生在所有用户同时并存的静态市场中。因此，我们研究了一种在线变体的Fisher市场：具有已知预算和效用参数（从分布$\mathcal{D}$中独立同分布抽取）的用户按序进入市场。在此设定下，我们开发了一种仅基于用户消费观测（即显示偏好反馈）来调整价格的算法，并实现了$O(\sqrt{n})$的遗憾值与容量违反量，其中$n$为用户数量且商品容量按$O(n)$缩放。这里的遗憾度量是Eisenberg-Gale规划目标中在线算法与具备用户预算和效用完全信息的离线预言机之间的最优性差距。为验证方法的有效性，我们证明任何均匀（静态）定价算法（包括完全已知分布$\mathcal{D}$并设定期望均衡价格的方法）都无法同时实现低于$\Omega(\sqrt{n})$的遗憾值与约束违反量。尽管我们的显示偏好算法无需分布$\mathcal{D}$的先验知识，但结果表明：若$\mathcal{D}$已知，对于离散分布，期望均衡定价的自适应变体可实现$O(\log(n))$的遗憾值与常数级容量违反量。最后，我们通过数值实验展示了所提显示偏好算法相对于多个基准的性能。