On the Linear Programming Model for Dynamic Stochastic Matching and Its Application to Pricing

TL;DR

This paper analyzes a linear programming model for dynamic stochastic matching in pricing problems, establishing concavity properties of the cost function, and develops an efficient MM algorithm with strong practical performance.

Contribution

It provides theoretical conditions for the concavity of the cost function and introduces a novel MM algorithm tailored for large-scale pricing applications.

Findings

Weak concavity holds under positive unmatched demand rates.

The MM algorithm outperforms gradient methods on real ridesharing data.

Conditions for concavity are characterized in the fluid LP model.

Abstract

Important pricing problems in centralized matching markets -- such as carpooling, food delivery and freight shipping platforms -- often exhibit a bi-level structure. At the upper level, the platform sets prices for heterogeneous demand types (e.g., rides across origin-destination pairs, food delivery orders across restaurant-customer pairs, or less-than-truckload shipments). The lower level subsequently matches converted demands to minimize operational costs; for example, by pooling riders into shared vehicles or consolidating multiple orders into single courier or trailer routes. Motivated by these applications, we study the optimal value (cost) function of a linear programming model with respect to demand arrival rates, originally proposed by Aouad and Saritac (2022) for cost-minimizing dynamic stochastic matching under limited time. In particular, we study the concavity properties of this cost function. We show that it suffices for every optimal basic feasible solution of the linear program to be nondegenerate in order to guarantee weak concavity. Leveraging this insight, we further establish that weak concavity holds when all demand types have strictly positive unmatched rates -- a natural condition in stochastic environments when demands have limited patience -- and characterize conditions under which this property is satisfied in the fluid linear program. Building on these theoretical insights, we develop a Minorization-Maximization (MM) algorithm that exploits the resulting difference-of-concave structure of the pricing problem. The algorithm requires little stepsize tuning and delivers substantial performance improvements over projected gradient methods on a large-scale, real-world ridesharing dataset with thousands of rider types (origin-destination pairs). This makes it a compelling algorithmic choice for solving such pricing problems in practice.