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

Rui Yao, Xinyu Ma, Kenan Zhang

TL;DR

This paper formulates a coopetitive MaaS system as a multi-leader–multi-follower game in which a MaaS platform and service operators compete for traveler demand while travelers (followers) choose modes and routes under congestion. By introducing a virtual traffic operator, the authors transform the bi-level game into a single-level variational inequality ($VI$) with parallelizable solution opportunities, and they prove equilibrium existence for any viable wholesale capacity price. The follower response is governed by a perturbed utility Markovian choice model (PUMCM), yielding differentiable demand sensitivities that feed into the leaders’ optimization; a dual traffic-equilibrium structure (PUME) further anchors the link-time decisions. Numerical experiments on a small network and the Sioux Falls network show MaaS can create a win–win–win outcome: travelers and operators benefit while the MaaS platform remains profitable, with the Pareto-improving region depending on the wholesale price. The work demonstrates scalable solution procedures and provides insights into pricing, transfer penalties, and network effects, offering a tractable framework for planning and policy in real-world multi-modal systems.

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.

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

TL;DR

This paper formulates a coopetitive MaaS system as a multi-leader–multi-follower game in which a MaaS platform and service operators compete for traveler demand while travelers (followers) choose modes and routes under congestion. By introducing a virtual traffic operator, the authors transform the bi-level game into a single-level variational inequality () with parallelizable solution opportunities, and they prove equilibrium existence for any viable wholesale capacity price. The follower response is governed by a perturbed utility Markovian choice model (PUMCM), yielding differentiable demand sensitivities that feed into the leaders’ optimization; a dual traffic-equilibrium structure (PUME) further anchors the link-time decisions. Numerical experiments on a small network and the Sioux Falls network show MaaS can create a win–win–win outcome: travelers and operators benefit while the MaaS platform remains profitable, with the Pareto-improving region depending on the wholesale price. The work demonstrates scalable solution procedures and provides insights into pricing, transfer penalties, and network effects, offering a tractable framework for planning and policy in real-world multi-modal systems.

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.
Paper Structure (21 sections, 4 theorems, 26 equations, 12 figures, 4 tables, 1 algorithm)

This paper contains 21 sections, 4 theorems, 26 equations, 12 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Multi-modal network
  3. MaaS system as a multi-leader-multi-follower game
  4. Followers: Travelers
  5. Leader: Traffic operator
  6. Leader: MaaS platform
  7. Leaders: Mobility service operators
  8. Equilibrium and solution approach
  9. Equilibrium as a single-level VI problem
  10. Solution algorithm
  11. Numerical experiments
  12. Small network
  13. Setup
  14. Convergence of solution algorithm and robustness of equilibrium
  15. System outcomes with and without MaaS
  16. ...and 6 more sections

Key Result

Lemma 1

Suppose that i) the perturbation functions $F_s(\cdot), \forall s \in \mathcal{S}$ are essentially smooth and essentially convex, and ii) $T_{\pi} \mathbf{0}$ are bounded and non-positive. Then, there exists a unique fixed point $V^* \in \mathbb{R}^{|\mathcal{S}|}$ such that: In addition, the fixed point $V^*$ is optimal in the sense that $V^*\geq V_\pi, \forall \pi \in \Pi$.

Figures (12)

  • Figure 1: Multi-modal network (the operators serve both MaaS and non-MaaS travelers with the same fleet, and travelers face different fares in MaaS and non-MaaS)
  • Figure 2: Multi-leader-multi-follower game for a MaaS system.
  • Figure 3: Small network.
  • Figure 4: Solution algorithm convergence.
  • Figure 5: Gap in equilibria due to shift of initial solutions.
  • ...and 7 more figures

Theorems & Definitions (6)

  • Lemma 1
  • Lemma 2: yaoperturbed
  • Proposition 1
  • proof
  • Proposition 2: VNE existence
  • proof