Stochastic Online Fisher Markets: Static Pricing Limits and Adaptive Enhancements

TL;DR

This work introduces online Fisher markets with sequentially arriving users who have privately known budgets and linear utilities, and analyzes the performance of static versus adaptive pricing. It proves a fundamental $\Omega(\sqrt{n})$ lower bound for static pricing, and develops adaptive posted-price methods that rely on either knowledge of the distribution or revealed preferences, achieving $O(\log(n))$ regret with constant constraint violation in discrete-distribution settings and $O(\sqrt{n})$ (or $O(n^{2/5})$ via two-stage updates) in revealed-preference scenarios. A general feasibility framework ensures capacity constraints can be respected with minimal impact on regret, and extensive numerical experiments validate theoretical guarantees while illustrating practical efficacy against benchmarks. The results highlight the value of adaptive pricing in online Fisher markets, including privacy-preserving mechanisms and feasible implementations for real-time resource allocation.

Abstract

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.

Figures (6)

• Figure 1: Log-log plots comparing the regret of the feasible variants of the revealed preference algorithms, one with a fixed step size and another with a two-stage adjustment in the step size of the price updates, and the three benchmarks presented in Section \ref{['sec:benchmark-overview']} for instances 1 (left) and 2 (right).
• Figure 2: Validation of Lemma \ref{['lem:lipshitzness']} for an instance with $n = 10,000$ users, where all users have a fixed budget of one, and two goods, each with a capacity of $c_j = n = 10,000$.
• Figure 3: Comparison between the static expected equilibrium pricing algorithm and its dynamic counterpart (Algorithm \ref{['alg:AlgoProbKnownDiscrete']}) on regret and constraint violation metrics.
• Figure 4: Comparison between Algorithm \ref{['alg:PrivacyPreserving']} that has an additive price update step to a corresponding algorithm with a multiplicative price update step on regret and constraint violation metrics.
• Figure 5: Numerical validation of the positivity of prices during the operation of Algorithm \ref{['alg:PrivacyPreserving']} in two market settings: (i) the market instance in the proof of Theorem \ref{['thm:lbStatic']} (left), and (ii) instance two described in Section \ref{['sec:experimental-setup-details']} (right). The y-axis denotes the minimum price across all goods across 300 problem instances, i.e., 300 runs of Algorithm \ref{['alg:PrivacyPreserving']} on different instances drawn from the specified distribution corresponding to each market setting.

Theorems & Definitions (20)

• Definition 1
• Theorem 1
• Corollary 1
• Theorem 2: Regret and Constraint Violation Bounds for Algorithm \ref{['alg:AlgoProbKnownDiscrete']}
• Theorem 3: Regret and Constraint Violation Bounds for Algorithm \ref{['alg:PrivacyPreserving']}
• Theorem 4: Regret and Constraint Violation for Two-Stage Revealed Preference Algorithm
• Theorem 5: Generic Regret Bound for Feasible Algorithms
• Corollary 2: Regret of Feasible Variant of Algorithm \ref{['alg:PrivacyPreserving']} with a Fixed Step Size
• Corollary 3: Regret of Feasible Variant of Algorithm \ref{['alg:PrivacyPreserving']} with a Two-Stage Step Size Adjustment
• Corollary 4: Regret Guarantee of Feasible Variant of Algorithm \ref{['alg:AlgoProbKnownDiscrete']}