Searching for Optimal Prices in Two-Sided Markets

TL;DR

This work delivers a comprehensive theory for online pricing in two-sided markets, establishing tight regret bounds across three pricing mechanisms (Single-Price, Two-Price, Segmented-Price) for both gains-from-trade (GFT) and profit objectives. A key insight is that learnability dramatically improves with modest increases in pricing expressiveness: constant regret is possible in bilateral markets, but general two-sided markets encounter fundamental hardness due to the mismatch phenomenon unless segmentation is employed. The authors introduce Optimistic Binary/Conservative search strategies, optimistic fictitious markets, and Segment-based pricing to circumvent these barriers, achieving regret bounds such as $O(1)$ (GFT, bilateral), $O(\log\log T)$ (GFT, bilateral with profit focus, and one-to-many settings), and $O(n^2 \log\log T + n^3)$ (GFT, general two-sided with Segmented-Price). In contextual settings, the results extend via ellipsoid and Steiner-polynomial techniques to yield regret bounds that scale with the context dimension $d$, e.g., $O(n^2 d^2 \log T)$ for GFT and $O(n^2 d^2 \log T)$ for profit, highlighting the power of structured learning in high-dimensional pricing. Overall, the paper delineates sharp boundaries between learnable and unlearnable regimes in online two-sided pricing and demonstrates how richer pricing expressiveness can overcome fundamental hardness barriers with practical implications for real-world platforms.

Abstract

We investigate online pricing in two-sided markets where a platform repeatedly posts prices based on binary accept/reject feedback to maximize gains-from-trade (GFT) or profit. We characterize the regret achievable across three mechanism classes: Single-Price, Two-Price, and Segmented-Price. For profit maximization, we design an algorithm using Two-Price Mechanisms that achieves $O(n^2 \log\log T)$ regret, where $n$ is the number of traders. For GFT maximization, the optimal regret depends critically on both market size and mechanism expressiveness. Constant regret is achievable in bilateral trade, but this guarantee breaks down as the market grows: even in a one-seller, two-buyer market, any algorithm using Single-Price Mechanisms suffers regret at least $Ω\!\big(\frac{\log\log T}{\log\log\log\log T}\big)$, and we provide a nearly matching $O(\log\log T)$ upper bound for general one-to-many markets. In full many-to-many markets, we prove that Two-Price Mechanisms inevitably incur linear regret $Ω(T)$ due to a \emph{mismatch phenomenon}, wherein inefficient pairings prevent near-optimal trade. To overcome this barrier, we introduce \emph{Segmented-Price Mechanisms}, which partition traders into groups and assign distinct prices per group. Using this richer mechanism, we design an algorithm achieving $O(n^2 \log\log T + n^3)$ regret for GFT maximization. Finally, we extend our results to the contextual setting, where traders' costs and values depend linearly on observed $d$-dimensional features that vary across rounds, obtaining regret bounds of $O(n^2 d \log\log T + n^2 d \log d)$ for profit and $O(n^2 d^2 \log T)$ for GFT. Our work delineates sharp boundaries between learnable and unlearnable regimes in two-sided dynamic pricing and demonstrates how modest increases in pricing expressiveness can circumvent fundamental hardness barriers.

Theorems & Definitions (61)

• Example 1.1: Two-sided pricing in online retail marketplaces
• Example 1.2: Two-sided pricing in ride-hailing platforms
• Example 1.3: Uniform pricing in wholesale electricity markets
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 4.1
• Lemma 4.1
• proof