A uniformity principle for spatial matching

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.