  Shared Mobility Services (SMSs) are reshaping urban transportation systems by
providing flexible mobility options. With their capacity to decrease the number
of cars on the roads, these services can potentially improve the transportation
system's performance in terms of travel times and emissions. This emphasizes
the importance of analyzing and understanding their impacts on the system and
users' choices, especially when integrated into a complex multi-modal system,
including public transport (PT). Many studies overlook the synergies between
SMSs and PT, leading to inaccurate traffic estimations and planning. This
research offers an exhaustive review of multi-modal transportation system
models involving SMSs. We then introduce a traffic assignment analytical model
framed as a Mixed-Integer Quadratic Problem (MIQP). This model comprises
diverse travel possibilities, including SMSs, and handles intermodality by
allowing commuters to combine modes to optimize time and monetary expense. An
in-depth examination of commuters' behavior on two test cases and an analysis
of the price of anarchy highlights the disparities between user equilibrium and
system optimum in such intricate systems.
