Pricing and Optimization in Shared Vehicle Systems: An Approximation Framework

Abstract

Optimizing shared vehicle systems (bike-sharing/car-sharing/ride-sharing) is more challenging compared to traditional resource allocation settings due to the presence of complex network externalities - changes in the demand/supply at any location affect future supply throughout the system within short timescales. These externalities are well captured by steady-state Markovian models, which are therefore widely used to analyze such systems. However, using such models to design pricing/control policies is computationally difficult since the resulting optimization problems are high-dimensional and non-convex. To this end, we develop a general approximation framework for designing pricing policies in shared vehicle systems, based on a novel convex relaxation which we term elevated flow relaxation. Our approach provides the first efficient algorithms with rigorous approximation guarantees for a wide range of objective functions (throughput, revenue, welfare). For any shared vehicle system with n stations and m vehicles, our framework provides a pricing policy with an approximation ratio of 1+(n−1)/m. This guarantee is particularly meaningful when m/n, the average number of vehicles per station is large, as is often the case in practice. Further, the simplicity of our approach allows us to extend it to more complex settings: rebalancing empty vehicles, redirecting riders to nearby vehicles, multi-objective settings (such as Ramsey pricing), incorporating travel-times, etc. Our approach yields efficient algorithms with the same approximation guarantees for all these problems, and in the process, obtains as special cases several existing heuristics and asymptotic guarantees.