Mobility-as-a-service (MaaS) system as a multi-leader-multi-follower game: A single-level variational inequality (VI) formulation

2026-01-27

AI Summary

This study addresses the multi-agent interactions among Mobility-as-a-Service (MaaS) platforms, transport operators, and travelers by formulating a multi-leader–multi-follower game: the MaaS platform procures mobility capacity from operators and sells multimodal itineraries to users, while travelers dynamically choose routes based on prices and congestion. To efficiently solve this complex hierarchical game, the authors propose a single-level variational inequality (VI) model based on virtual transport operators, transforming the original multi-level structure into a parallelizable single-level formulation. By integrating a traffic assignment dual method with parallel algorithms, the approach enables large-scale computation. Numerical experiments on both a small network and an extended multimodal Sioux Falls network demonstrate the model’s effectiveness, showing that appropriately set wholesale capacity prices can simultaneously enhance platform profits, operator revenues, and user welfare—achieving a Pareto improvement for all three stakeholders.

Technology Category

• Multiagent Systems: Mechanism Design
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Game Theory and Economic Paradigms: Cooperative Game Theory

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Economics, Online Markets and Human Computation: The sharing economy
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

This study models a Mobility-as-a-Service (MaaS) system as a multi-leader-multi-follower game that captures the complex interactions among the MaaS platform, service operators, and travelers. We consider a coopetitive setting where the MaaS platform purchases service capacity from service operators and sells multi-modal trips to travelers following an origin-destination-based pricing scheme; meanwhile, service operators use their remaining capacities to serve single-modal trips. As followers, travelers make both mode choices, including whether to use MaaS, and route choices in the multi-modal transportation network, subject to prices and congestion. Inspired by the dual formulation for traffic assignment problems, we propose a novel single-level variational inequality (VI) formulation by introducing a virtual traffic operator, along with the MaaS platform and multiple service operators. A key advantage of the proposed VI formulation is that it supports parallel solution procedures and thus enables large-scale applications. We prove that an equilibrium solution always exists given the negotiated wholesale price of service capacity. Numerical experiments on a small network further demonstrate that the wholesale price can be tailored to align with varying system-wide objectives. The proposed MaaS system demonstrates potential for creating a"win-win-win"outcome -- service operators and travelers are better off compared to the"without MaaS"scenario, meanwhile the MaaS platform remains profitable. Such a Pareto-improving regime can be explicitly specified with the wholesale capacity price. Similar conclusions are drawn from the experiment of an extended multi-modal Sioux Falls network, which also validates the scalability of the proposed model and solution algorithm.

Problem

• Mobility-as-a-Service
• multi-leader-multi-follower game
• variational inequality
• equilibrium modeling
• multi-modal transportation

Innovation

• multi-leader-multi-follower game
• single-level variational inequality
• Mobility-as-a-Service (MaaS)
• virtual traffic operator
• coopetitive pricing