Incentive-based strategies for road–waterway city logistics: A single-leader multi-follower game-theoretic framework

Urban freight transportation contributes substantially to congestion, pollution, and operational inefficiencies, particularly in dense urban areas where delivery activities remain strongly dependent on road transport. To address these impacts, policymakers are increasingly exploring incentive-based mechanisms that promote modal shifts toward alternative freight services within multimodal city logistics systems, while this study focuses specifically on the shift from road delivery to inland waterway transport. The effectiveness of such policies critically depends on the strategic interactions among multiple stakeholders. We therefore propose a single-leader multi-follower game-theoretic framework involving a transportation authority, logistics service providers (LSPs), and end customers. The transportation authority acts as the leader and designs the waterway subsidy policy to promote road-to-waterway modal shift under fiscal constraints. The LSPs act as followers and maximize their own profits by determining service prices and road–waterway delivery allocations. Customer demand is modeled endogenously through a discrete choice model, and the joint response of the LSPs is characterized by a generalized Nash equilibrium. Finally, three implementation strategies for deploying road-to-waterway modal-shift policies are examined and compared in a case study of Amsterdam, providing insights into their relative efficiency and practical implications.