Profit-Aware Graph Framework for Cross-Platform Ride-Sharing: Analyzing Allocation Mechanisms and Efficiency Gains

Ride-hailing platforms (e.g., Uber, Lyft) have transformed urban mobility by enabling ride-sharing, which holds considerable promise for reducing both travel costs and total vehicle miles traveled (VMT). However, the fragmentation of these platforms impedes system-wide efficiency by restricting ride-matching to intra-platform requests. Cross-platform collaboration could unlock substantial efficiency gains, but its realization hinges on fair and sustainable profit allocation mechanisms that can align the incentives of competing platforms. This study introduces a graph-theoretic framework that embeds profit-aware constraints into network optimization, facilitating equitable and efficient cross-platform ride-sharing. Within this framework, we evaluate three allocation schemes -- equal-profit-based, market-share-based, and Shapley-value-based -- through large-scale simulations. Results show that the Shapley-value-based mechanism consistently outperforms the alternatives across six key metrics. Notably, system efficiency and rider service quality improve with increasing demand, reflecting clear economies of scale. The observed economies of scale, along with their diminishing returns, can be understood with the structural evolution of rider-request graphs, where super-linear edge growth expands feasible matches and sub-linear degree scaling limits per-rider connectivity.