A uniformity principle for spatial matching

2026-01-19

AI Summary

This study addresses the problem of pre-allocating service ranges to supply nodes under a fixed total budget to maximize demand fulfillment in spatial supply-demand matching platforms. By modeling the system as a bipartite random geometric graph and integrating Markov chain embedding, probabilistic analysis, and combinatorial optimization, the work proposes and rigorously establishes a “uniformity principle”: more uniform allocations yield a higher expected matching size. This result uncovers a synergistic mechanism between diminishing marginal returns from service range expansion and the natural fragmentation of the underlying graph structure. Notably, in the special case of \(k=1\), the authors derive a closed-form expression for the expected matching size, offering a theoretical foundation for optimizing service ranges and designing incentives in platforms such as ride-hailing, on-demand labor markets, and drone delivery systems.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: The sharing economy
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• spatial matching
• service range allocation
• maximum matching
• bipartite random geometric graphs
• uniformity principle

Innovation

• uniformity principle
• spatial matching
• bipartite random geometric graphs
• service range allocation
• maximum matching