We consider ride-sharing networks served by human-driven vehicles (HVs) and autonomous vehicles (AVs). We propose a model for ride-sharing in this mixed autonomy setting for a multi-location network in which a ride-sharing platform sets prices for riders, compensations for drivers of HVs, and operates AVs for a fixed price with the goal of maximizing profits. When there are more vehicles than riders at a location, we consider three vehicle-to-rider assignment possibilities: rides are assigned to HVs first; rides are assigned to AVs first; rides are assigned in proportion to the number of available HVs and AVs. Next, for each of these priority possibilities, we establish a nonconvex optimization problem characterizing the optimal profits for a network operating at a steady-state equilibrium. We then provide a convex problem which we show to have the same optimal profits, allowing for efficient computation of equilibria, and we show that all three priority possibilities result in the same maximum profits for the platform. Next, we show that, in some cases, there is a regime for which the platform will choose to mix HVs and AVs in order to maximize its profit, while in other cases, the platform will use only HVs or only AVs, depending on the relative cost of AVs. For a specific class of networks, we fully characterize these thresholds analytically and demonstrate our results on an example.
