With Uber, as it is today, you have the strong forces of a network effect (where drivers are stuck because the passengers are there, and the passengers are stuck because the drivers are there). This effect is historically very difficult to break, a canonical example would be craigslist.
While rolling out self driving cars, simply requires money. If for example, San Francisco has 1000 Uber drivers active at any time today, all you need is to roll out 1000 cars yourself in the city, and the service you provide is equivalent to Uber.
In other words, if self driving technology becomes common, the market is going to be commoditized to the point where there is very little profit margin for any player (such as in the airline industry).
In a commodity market you win with combination of price and brand.
When it comes to price, a player with a fleet of a million cars can wage a pricing war for longer than a player with a fleet of 10 thousand cars because economies of scale mean that it costs them less, per car, to provide the service.
If this is a new entrant, the incumbents also have an accumulated war chest from past revenues they can use to outprice the competition.
And they also have more money to out-advertise the new guy. GM spends $3 billion a year on advertising which is why, not counting Tesla, the youngest surviving car company was founded in 1925.
It's also why Kellog dominates in cornflake space, coca-cola in soda business etc.
Which is why there will be a mad rush to grab the market share early on. Once someone reaches significant scale, it'll be virtually impossible to catch up with them.
It's not so bad (or maybe wouldn't be if cars themselves weren't so expensive.)
Assuming some even distribution of avaialable vehicles in an area, time-to-pickup scales like (area/vehicles), so the returns to scale diminish fairly quickly.
And "evenly spaced vehicles" might be the worst case depending on user preferences. It might be better to have fast pickups in some places and really slow pickups in others than average times everywhere, so long as users know the pattern -- slow from my house, fast from my work etc.
There are people who only buy Volvos because of the reputation for safety they've built up.
This is why I think driverless taxis will be much bigger than many people expect, but also why this industry is going to get monopolized like none other. Bigger players will be able to get taxis to you faster, offer them cheaper, and still show bigger profit margins than smaller players. And without scaling labor costs... whoever wins this industry is going to be making a completely ridiculous amount of money. Consider that in 2016 3.2 trillion miles were driven in the US alone. 10% of those miles is 320 billion. A profit margin of a nickle above operating costs is a net of $16 billion. That's at operating on a nickle per mile, with 10% of 2016's mileage, in a single country.
On the other hand I expect all the players will realize this, so we should see and benefit from some insane competition as long as the market remains relatively free.