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

Junlin Chen, Chiwei Yan, Hai Jiang

TL;DR

The paper addresses pricing in dynamic stochastic matching markets by studying a linear programming relaxation of the lower-level matching problem, deriving when the associated cost function $c(oldsymbol{blambda})$ is concave or weakly concave in demand arrival rates. It shows that weak concavity is guaranteed under nondegenerate optimal BFSs and under strictly positive unmatched rates, with further practical conditions for patience-homogeneous and heterogeneous settings. A Minorization-Maximization algorithm is developed to exploit the difference-of-concave structure of the pricing objective, delivering robust and scalable performance on large-scale datasets such as Chicago ridesharing, outperforming projected gradient methods, and requiring little stepsize tuning. The approach extends to practical extensions including limited supply, multi-product pricing, and matching disutility, and is connected to the MDP formulation through LP-based bounds, offering a scalable, effective tool for real-world pricing in dynamic matching platforms.

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.

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

TL;DR

The paper addresses pricing in dynamic stochastic matching markets by studying a linear programming relaxation of the lower-level matching problem, deriving when the associated cost function is concave or weakly concave in demand arrival rates. It shows that weak concavity is guaranteed under nondegenerate optimal BFSs and under strictly positive unmatched rates, with further practical conditions for patience-homogeneous and heterogeneous settings. A Minorization-Maximization algorithm is developed to exploit the difference-of-concave structure of the pricing objective, delivering robust and scalable performance on large-scale datasets such as Chicago ridesharing, outperforming projected gradient methods, and requiring little stepsize tuning. The approach extends to practical extensions including limited supply, multi-product pricing, and matching disutility, and is connected to the MDP formulation through LP-based bounds, offering a scalable, effective tool for real-world pricing in dynamic matching platforms.

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.

Paper Structure

This paper contains 42 sections, 34 theorems, 117 equations, 3 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Overview of Our Results
  3. Related Literature
  4. Dynamic matching with abandonment.
  5. Applications of dynamic matching and pricing.
  6. Organization
  7. Preliminaries
  8. Price Optimization Problem
  9. (Weak) Concavity of the Cost Function
  10. Homogeneous Patience Levels
  11. Analysis of One or Two Types of Demands
  12. Analysis of Multiple Types of Demands
  13. Heterogeneous Patience Levels
  14. Analysis of Two Types of Demands
  15. Analysis of Multiple Types of Demands
  16. ...and 27 more sections

Key Result

Proposition 1

When $\theta_i=0$, $\forall i \in [N]$, $c(\boldsymbol{\lambda})$ is convex and linear on $\Lambda$.

Figures (3)

  • Figure 1: Examples of upfront pricing in dynamic matching platforms.
  • Figure 2: Example showing the multimodality and non-differentiability of $g(\boldsymbol{\lambda})$ when $N=2$.
  • Figure EC.1: Markov chains when $N\leq 2$.

Theorems & Definitions (63)

  • Definition 1
  • Proposition 1
  • Proposition 2: Economies of scale
  • Definition 2: Weak concavity
  • Lemma 1
  • Proposition 3
  • Lemma 2
  • Proposition 4
  • Lemma 3
  • Proposition 5
  • ...and 53 more