Online Learning in Fisher Markets with Unknown Agent Preferences

In a Fisher market, agents (users) spend a budget of (artificial) currency to buy goods that maximize their utilities, and producers set prices on capacity-constrained goods such that the market clears. The equilibrium prices in such a market are typically computed through the solution of a convex program, e.g., the Eisenberg-Gale program, that aggregates users' preferences into a centralized social welfare objective. However, the computation of equilibrium prices using convex programs assumes that all transactions happen in a static market wherein all users are present simultaneously and relies on complete information on each user's budget and utility function. Since, in practice, information on users' utilities and budgets is unknown and users tend to arrive over time in the market, we study an online variant of Fisher markets, wherein users enter the market sequentially. We focus on the setting where users have linear utilities with privately known utility and budget parameters drawn i.i.d. from a distribution $\mathcal{D}$. In this setting, we develop a simple yet effective algorithm to set prices that preserves user privacy while achieving a regret and capacity violation of $O(\sqrt{n})$, where $n$ is the number of arriving users and the capacities of the goods scale as $O(n)$. Here, our regret measure represents the optimality gap in the objective of the Eisenberg-Gale program between the online allocation policy and that of an offline oracle with complete information on users' budgets and utilities. To establish the efficacy of our approach, we show that even an algorithm that sets expected equilibrium prices with perfect information on the distribution $\mathcal{D}$ cannot achieve both a regret and constraint violation of better than $\Omega(\sqrt{n})$. Finally, we present numerical experiments to demonstrate the performance of our approach relative to several benchmarks.