Advances in information and communication technology are enabling the growth of real-time ridesharing to improve system efficiency. Traffic congestion reflects the "price of anarchy" arising from individually optimal but decentralized decisions of trip departure time and ridesharing choices. Nonetheless, mobility sharing service providers have the ability to improve traffic conditions and minimize total vehicle travel time. This objective can be achieved when a mobility sharing service provider has the ability to collect origins, destinations, and flexible time windows of vehicle trips. The service provider can then optimize and assign ridesharing pairings, driver/passenger role designations, and trip departure, pickup, and dropoff times. Sharing vehicle trips and spreading departure times can reduce vehicle travel demand, thereby reducing congestion and enabling more ridesharing opportunities. Together, ridesharing, departure time spreading and improving traffic conditions form a positive feedback loop, leading to an optimal stage---minimal vehicle travel time. In this paper, the authors for the first time in literature formulate ridesharing pairing, driver designation, departure time assignment, and traffic assignment as a mixed integer nonlinear optimization problem. Considering the difficulty of solution due to the nonlinearity, the authors then introduce an approximation of the problem---an iterative combination of traffic assignment steps (T-step), and ridesharing-pairing, driver-designation, and departure-time-assignment steps (R-step). The authors also prove that the R-step can be simplified as a mixed integer linear programming (MILP) problem.