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

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.

💡 Research Summary

This paper tackles a fundamental bi‑level optimization problem that arises in modern centralized matching platforms such as ridesharing, food‑delivery, and less‑than‑truckload (LTL) freight services. At the upper level the platform chooses upfront prices for a large set of heterogeneous demand types (e.g., origin‑destination pairs). Prices affect the Poisson arrival rates λ = (λ_i) through a one‑to‑one, invertible demand‑price relationship; the revenue from type i is λ_i p_i(λ_i), which is assumed concave in λ_i. At the lower level the platform must match the realized demands before their patience (exponential waiting time with rate θ_i) expires, incurring a matching cost c(i,j) when a pair (i,j) is matched and a solo cost c(i) when a demand abandons. The lower‑level cost minimization is modeled by the fluid linear program (LP) introduced by Aouad and Sarıtaç (2022), which provides a tight lower bound on the optimal cost of the underlying continuous‑time Markov decision process (MDP).

The central theoretical contribution is a rigorous analysis of how the optimal LP cost function c(λ) varies with the arrival‑rate vector λ. Classical parametric LP theory does not apply because λ appears both in the constraint coefficients and in the right‑hand sides. The authors first prove that if every optimal basic feasible solution of the LP is non‑degenerate (i.e., all basic variables are strictly positive), then c(λ) is weakly concave: for any λ^1, λ^2 and any α∈