Right now, I can't create Newber because it would be hard to convince drivers and users to switch. You need good driver availability at all hours of the day and night and drivers won't want to be on your service earning nothing if you don't yet have users. Many companies like Sidecar went bust hoping to hit a critical mass and overcome that barrier to entry. With self-driving cars, it's easy for me to pay, say, $400/mo/vehicle and blanket a city even before there are users. It isn't cheap, but it also isn't so expensive. 100 vehicles would only be $40,000/mo. That's certainly not such high start-up costs for anything with a bit of funding. Plus, the economics work out nicely. $400/mo comes out to $13/day. Let's say you only charge $3.25 per ride. You'd only need to do 4 rides per day to justify that $400/mo - and 4 rides per 24-hour period is extremely low. Even when you start thinking about costs beyond the fixed price of the car (fuel, maintenance), it's still pretty good. You're going to get at least 4 rides per gallon in a reasonably fuel-efficient car. Make them Prius vehicles and get 10 5-mile rides on a $2-3 gallon. Tires cost money, but if the average ride is 5 miles you'll get around 10k rides for a set of tires adding $0.20 per ride?
Point being, the only reason Uber doesn't have a lot of competition today is that it's very hard to overcome the network effect. Once self-driving cars come into the picture, it will be easy to create many Uber competitors that can overcome the network effect by blanketing a city quickly and easily and scale quite reasonably by ordering new self-driving vehicles as demand increases. That will put a lot of downward pressure on pricing. I think Uber has found that if they're more expensive than competitors like Lyft and Fasten, riders will switch. Self-driving cars will make that pressure only more intense.
At some point, Uber is likely to face competition from car manufacturers themselves. If you're GM, you can see the writing on the wall that car buying might be going away. But you could always offer people a car subscription. GM would certainly have a cost advantage on the vehicles compared to Uber. Likewise, they might face competition from logistics companies like FedEx or Amazon. Why not create vans that can be fitted with seats for the morning commute and create ad-hoc bus routes as users request them (think of Uber Pool on steroids or a bus route that conveniently goes from your house to your work on your schedule). During the passenger lull during the working day, put packages in them and have a quad-copter or robot move them from van to stoop.
The thing is that self-driving vehicles just looks like a lot of competition for Uber. Yes, it will mean lower costs, but I don't see how they'll fend off a legion of new competitors where they have no barrier to entry with any good margins. The only thing Uber could do is keep margins so low that they won't make a profit. But that's no good for them. If they ever try to raise the margins, others will swoop in. Plus, if Amazon has shown us anything, it's that a lot of competition will stick around even if your margins are tiny.
I think there's a big market that will come from self-driving vehicles. Car ownership might not last. Deliveries might be automated. Traditional public transit might be replaced by ad-hoc routes with 15-passenger vans. But it's hard to see Uber lacking competition here. Self-driving vehicles will simply make it too easy to compete and leave Uber without its traditional barrier to entry.