Dynamic game-theoretic pricing for urban rail transit and feeder ride-hailing services: A case study in Hangzhou

Amid the rise of widespread ride-hailing and the declining mode share of urban rail transit, this study proposes an integrated mobility solution that combines urban rail with ride-hailing feeder services. Using game theory, we developed a bi-level dynamic pricing model to coordinate rail and ride-hailing operations. This framework characterizes strategic interactions among stakeholders as follows: (1) upper-level operators (rail transit and ride-hailing platforms) achieve joint profit maximization through pricing optimization and (2) passengers choose a mode based on utility trade-offs considering cost, travel time, and personal attributes. Mathematical proofs establish conditions for the existence and uniqueness of Nash equilibrium solutions, and sensitivity analyses facilitate efficient computational resolution. Empirical case studies reveal two key findings: (1) in the absence of government intervention, coordinated pricing-reduction strategies can attract passenger flow to the feeder mode across all time periods; (2) policy interventions effectively enhance feeder mode share, increasing the passenger share for urban rail transit while enabling moderate strategies that generate net government revenue during specific time windows.