Matching and pricing in ride-sharing: Optimality, stability, and financial sustainability

•Matching and pricing in dynamic ride-sharing.•Worst-case performance analysis of best stable matchings for existing intra-match pricing methods.•Proof of stability of an existing mechanism-design-based pricing method.•A new complete solution to a static problem proved to be system-wide optimal, stable, and deficit-free. The method runs in polynomial time.•Experiments in static and dynamic environments. In ride-sharing, a sustainable transportation mode, drivers share idle seats with riders. This paper addresses two key subproblems to find a ride-sharing solution: matching drivers and riders to achieve the system-wide optimality and allocating the resulting travel-cost saving among them by considering their self interests or the stability of the solution. Solutions to either subproblem significantly affect the efficiency of overall dynamic ride-sharing systems measured in total travel-cost saving and individual benefits of people involved. We evaluate the worst-case performance of the best stable solutions associated with four ridematching-defined pricing methods existing in the literature and widely used in practice. We then analyze a Vickrey-Clarke-Groves (VCG)-based pricing mechanism that lets the system achieve optimal system efficiency, prove its stability, and identify its drawbacks such as budget unbalance. We formulate the matching subproblem to maximize the total travel-cost saving and propose a simple and consistent pricing method to guarantee that the resulting ride-sharing solution not only is stable and system-wide optimal but also makes the system deficit-free and thus financially sustainable. Furthermore, such a ride-sharing solution can be obtained in polynomial time. In the numerical experiments, we evaluate our method against existing ones through a real-life instance and randomly generated instances in a dynamic environment.