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.

翻译：平台在将空间分布的供给与需求进行匹配时面临一个基本的设计选择：给定固定的服务范围总预算，应如何在供给节点间进行事前分配（即在供给和需求位置实现之前），以最大化满足的需求？我们使用二部随机几何图对此问题进行建模，其中 $n$ 个供给节点和 $m$ 个需求节点均匀分布在 $[0,1]^k$（$k \ge 1$）上，当需求落在供给节点的服务区域内时形成边，该区域的体积由其服务范围决定。由于每个供给节点最多服务一个需求，平台性能由最大匹配的期望大小决定。我们建立了一个均匀性原则：当一种服务范围分配比另一种更均匀时，更均匀的分配会产生更大的期望匹配。这一原则源于范围扩展服务范围的边际收益递减，以及由于有限范围自然分割图而导致的供给节点间有限的干扰。对于 $k=1$，我们进一步通过马尔可夫链嵌入刻画了期望匹配大小，并推导了特殊情况的闭式表达式。我们的结果为优化网约车、按需劳动力市场和无人机配送网络中的服务范围分配和设计激励结构提供了理论指导。