  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. An expanded
Sioux Falls network test 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.
