A Convex Programming Approach for Ridesharing User Equilibrium Under Fixed Driver/Rider Demand

With the proliferation of smartphone-based ridesharing apps around the world, traffic assignment with ridesharing is drawing increasing attentions in recent years. A number of ridesharing user equilibrium (RUE) models have been proposed, but most of them are formulated as mixed complementary problems based on presumed ridesharing price and inconvenience functions, thus are inconvenient to implement in reality. In this study, we propose an alternative approach to modeling the RUE when the driver- and rider-demand for each OD pair are fixed and given. By redefining the set of feasible driver trajectories, and introducing market clearing conditions to characterize drivers' net income at market equilibrium, we show that the resulting RUE conditions can be equivalently transformed into a convex programming problem, and establish the existence and uniqueness of RUE link flows under mild conditions. A subgradient algorithm with averaging is proposed to solve the problem. The dual subproblem has a similar structure as Beckmann's formulation, therefore enables the application of classical traffic assignment algorithms such as Frank-Wolfe. Numerical examples are provided to demonstrate the effectiveness of the model and algorithm