Dynamic Pricing and Matching for Two-Sided Queues

TL;DR

This paper studies dynamic pricing and dynamic matching in a two-sided queueing system represented by a bipartite graph of customer and server types. It develops a Markov decision process formulation and analyzes both fluid (η-large) and large-system (n,η) regimes, showing that fluid pricing with max-weight matching achieves an $O(\sqrt{\eta})$ profit loss and that a two-price policy with max-weight reduces this to $O(\eta^{1/3})$, with a matching lower bound of $\Omega(\eta^{1/3})$ proving near-optimality. Under complete resource pooling, the loss scales as $O(\sqrt{n})$ for static pricing and $O(n^{1/3})$ for two-price policies, with a corresponding lower bound $\Omega(n^{1/3})$; max-weight matching generally outperforms randomized matching, and state-space collapse under CRP enables delay-optimality results. The paper provides extensive theoretical results, supplemented by numerical experiments validating the performance of fluid and two-price policies across single-link and multi-type networks. Collectively, the findings offer a principled, scalable approach to joint pricing and matching in gig-economy and online marketplaces, achieving near-optimal profits in large-scale, multi-type two-sided queues.

Abstract

Motivated by applications from gig economy and online marketplaces, we study a two-sided queueing system under joint pricing and matching controls. The queueing system is modeled by a bipartite graph, where the vertices represent customer or server types and the edges represent compatible customer-server pairs. Both customers and servers sequentially arrive to the system and join separate queues according to their types. The arrival rates of different types depend on the prices set by the system operator and the expected waiting time. At any point in time, the system operator can choose certain customers to match with compatible servers. The objective is to maximize the long-run average profit for the system. We first propose a fluid approximation based pricing and max-weight matching policy, which achieves an $O(\sqrtη)$ optimality rate when all the arrival rates are scaled by $η$. We further show that a two-price and max-weight matching policy achieves an improved $O(η^{1/3})$ optimality rate. Under a broad class of pricing policies, we prove that any matching policy has an optimality rate that is lower bounded by $Ω(η^{1/3})$. Thus, the latter policy achieves the optimal rate with respect to $η$. We also demonstrate the advantage of max-weight matching with respect to the number of server and customer types $n$. Under a complete resource pooling condition, we show that max-weight matching achieves $O(\sqrt{n})$ and $O(n^{1/3})$ optimality rates for static and two-price policies, respectively, and the latter matches the lower bound $Ω(n^{1/3})$. In comparison, the randomized matching policy may have an $Ω(n)$ optimality rate.

Figures (19)

• Figure 1: Bipartite graph representation for two-sided queues.
• Figure 2: A ride hailing system with three regions.
• Figure 3: Optimal pricing policies under different values of penalty coefficients.
• Figure 4: Stationary distribution of queue length under different penalty coefficients.
• Figure 5: Performance of two-price and fluid pricing policy compared to the exact solution.

Theorems & Definitions (44)

• Definition 1
• Remark 1: Average waiting time.
• Proposition 1
• Definition 2
• Proposition 2
• Definition 3: Large-Scale Regime
• Definition 4: Profit Loss
• Theorem 1
• Proposition 3
• Definition 5