Bear with me here, trying to come at this from the other end. Why do Uber and Lyft have to develop their own self driving tech? If what they sell is their routing algorithm for drivers, they can simply lease self driving cars from the companies directly and run them on their network. The car companies would be happy to get a cut of the profits. What advantage would they get by developing their own tech? They have to…
They haven't succeeded in their initial goal of exterminating the existing cab companies, and that leaves them in a situation where developing self-driving technology could be a matter of life and death for them. The incumbent cab companies are much better positioned to profit from self-driving cars. They've already got all the infrastructure, expertise and staff it takes to manage fleets of vehicles. The bigger ones already have their own apps (e.g., Curb). If self-driving cars hit the general market first, basically all they have to do to run Uber and Lift out of town is buy some and factor the fact that they aren't paying cab drivers anymore into their prices.
By contrast, most of Uber and Lyft's basic business models right now are predicated on the idea of hiring humans, who in turn manage their own cars. That has deep implications: It means they have zero expertise in fleet management. They don't have any of their own garages for storing cars and handling maintenance. They do have apps, but the most valuable parts of their apps are the parts for managing humans. Incentive mechanisms designed to try and ensure that supply of rides and demand for rides remain relatively stable, for example. Their existing business operations are heavily built around recruiting humans - both riders and drivers. None of that is particularly useful in a world with self-driving cars, and the bits that are - basically just the mechanics of enabling people to hail rides and pay for them - are drastically easy to copy.