Searching for Optimal Prices in Two-Sided Markets

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.

💡 Research Summary

The paper studies an online pricing problem in two‑sided markets where a platform repeatedly posts take‑it‑or‑leave‑it prices to a set of buyers and sellers, observes only binary accept/reject feedback, and aims to maximize either profit or gains‑from‑trade (GFT). The authors consider three increasingly expressive families of pricing mechanisms: (i) Single‑Price mechanisms that post a single price to all participants, (ii) Two‑Price mechanisms that post one price to all sellers and another to all buyers, and (iii) Segmented‑Price mechanisms that partition each side into up to two groups and assign a distinct price per group.

Profit maximization.
For profit, the natural benchmark is the optimal profit achievable by any Two‑Price mechanism. The authors design an algorithm that simultaneously learns the seller‑price and buyer‑price using a binary search on the union of the sellers’ cost set and the buyers’ value set, while respecting the budget‑balance constraint (seller price ≤ buyer price). By combining this search with volume‑reduction techniques that exploit observed trade volumes, they obtain a regret of (O(n^{2}\log\log T)), where (n) is the total number of traders and (T) is the time horizon. A matching lower bound shows this rate is optimal in (T).

GFT maximization – bilateral trade.
When there is a single seller and a single buyer, any price in the interval (