A uniformity principle for spatial matching

TL;DR

The paper investigates how to allocate a fixed total service-range budget across spatially distributed supply nodes to maximize demand fulfillment in a bipartite random geometric setting. It proves a uniformity principle: for fixed budget, making the service-range vector more uniform (in the majorization sense) increases the expected size of a maximum matching, with the objective characterized as Schur-concavity in the service ranges. The results hold for uniform demand/supply distributions in any dimension $k\ge1$, extend to Lipschitz-continuous distributions under certain conditions, and admit counterexamples illustrating limitations. For $k=1$, the authors develop a Markov-chain embedding to obtain exact or closed-form expressions for the heterogeneous-range case and propose a dual service-range model (inflexible vs flexible supply) with tractable bounds and an LLN-based characterization. The work offers theoretical guidance for ex-ante service-range design in ride-hailing, on-demand labor, and drone-delivery networks, with practical implications for incentive design and flexibility deployment, while highlighting open questions about radius-based modeling and two-stage/adaptive allocations.

Abstract

Platforms matching spatially distributed supply to demand face a fundamental design choice: given a fixed total budget of service range, how should it be allocated across supply nodes ex ante, i.e. before supply and demand locations are realized, to maximize fulfilled demand? We model this problem using bipartite random geometric graphs where $n$ supply and $m$ demand nodes are uniformly distributed on $[0,1]^k$ ($k \ge 1$), and edges form when demand falls within a supply node's service region, the volume of which is determined by its service range. Since each supply node serves at most one demand, platform performance is determined by the expected size of a maximum matching. We establish a uniformity principle: whenever one service range allocation is more uniform than the other, the more uniform allocation yields a larger expected matching. This principle emerges from diminishing marginal returns to range expanding service range, and limited interference between supply nodes due to bounded ranges naturally fragmenting the graph. For $k=1$, we further characterize the expected matching size through a Markov chain embedding and derive closed-form expressions for special cases. Our results provide theoretical guidance for optimizing service range allocation and designing incentive structures in ride-hailing, on-demand labor markets, and drone delivery networks.

Figures (13)

• Figure 1: Example of a non-uniform (a) and a uniform (b) allocation of service range. Orange circles denote demand nodes and blue squares denote supply nodes; shaded regions indicate supply service regions. Green segments show edges in the compatibility graph, and red segments highlight a maximum matching. The maximum matching size is $3$ in (a) and $4$ in (b).
• Figure 2: (Left) agents can choose their own level of flexibility. (Right) platform can provide incentives to influence service range. Images borrowed from TaskRabbit's blog post blog_taskrabbit.
• Figure 3: Demand and supply are uniformly distributed in their respective shaded regions.
• Figure 4: Markov dynamics of $\boldsymbol{\psi}(t)$ with dots and arrows in vector field notation: a dot represents the current location of $\boldsymbol{\psi}(t)$, and the arrow denotes its movement direction from that dot. For regions $\mathcal{B}$ and $\mathcal{E}$, the shaded sector represents the set of potential movement directions for $\boldsymbol{\psi}$.
• Figure 5: Illustrations used in proofs

Theorems & Definitions (49)

• Remark 1
• Definition 1: Majorization
• Theorem 1: Uniformity principle
• Corollary 1: Uniform allocation dominates
• proof
• Remark 2: Beyond uniform distribution
• Theorem 2
• Theorem 3
• Remark 3
• Remark 4: Bounds for the dual service range model