On-demand mobility-as-a-Service platform assignment games with guaranteed stable outcomes

•Developed a model for Mobility-as-a-Service platform equilibrium.•Captures on-demand mobility, coopetition between operators, user equilibrium.•Algorithm converges to global optimum when outcomes are non-empty.•Stability guarantee is ensured in algorithmic design that may introduce subsidy.•Tests on 748-link network can output firm market entry, inequities, and congestion. Mobility-as-a-Service (MaaS) systems are two-sided markets, with two mutually exclusive sets of agents, i.e., travelers/users and operators, forming a mobility ecosystem in which multiple operators compete or cooperate to serve customers under a governing platform provider. This study proposes a MaaS platform equilibrium model based on many-to-many assignment games incorporating both fixed-route transit services and mobility-on-demand (MOD) services. The matching problem is formulated as a convex multicommodity flow network design problem under congestion that captures the cost of accessing MOD services. The local stability conditions reflect a generalization of Wardrop's principles that include operators’ decisions. Due to the presence of congestion, the problem may result in non-stable designs, and a subsidy mechanism from the platform is proposed to guarantee local stability. A new exact solution algorithm to the matching problem is proposed based on a branch and bound framework with a Frank-Wolfe algorithm integrated with Lagrangian relaxation and subgradient optimization, which guarantees the optimality of the matching problem but not stability. A heuristic which integrates stability conditions and subsidy design is proposed, which reaches either an optimal MaaS platform equilibrium solution with global stability, or a feasible locally stable solution that may require subsidy. For the heuristic, a worst-case bound and condition for obtaining an exact solution are both identified. Two sets of reproducible numerical experiments are conducted. The first, on a toy network, verifies the model and algorithm, and illustrates the differences local and global stability. The second, on an expanded Sioux Falls network with 82 nodes and 748 links, derives generalizable insights about the model for coopetitive interdependencies between operators sharing the platform, handling congestion effects in MOD services, effects of local stability on investment impacts, and illustrating inequities that may arise under heterogeneous populations.