Car rental companies are the most likely first bulk buyers of self-driving cars. They already buy, store, maintain, and clean large fleets of cars. They're all set up to handle the problems of a big fleet of cars used by many people. Uber doesn't have that infrastructure.
Car rental companies would like to have a car pick you up outside baggage claim and drive you away, rather than using a shuttle bus to get you to their car lot. Customers will like that, too. Even if the cars aren't self-driving on all roads, if they can get from rental car storage to the airport and onto the freeway, that's enough to justify the technology.
Avis already owns Zipcar, so they have app-based car services already deployed. So as soon as the technology is ready, Avis is there with the customer base and the infrastructure.
[1] https://www.bloomberg.com/news/articles/2017-06-26/alphabet-...
Burning money on autonomous now buys them little. They can't control it, they don't have key patents, they will have to buy it just like everyone else.
You don't have to panic, throw huge amounts of stock at a shifty development team, while ignoring due diligence (or do an incompetent job at it) all to do your own.