Ride-sharing is not an area that can be monopolized. As soon as companies like Uber and Lyft stop subsidising rides, competitors will spawn up like mushrooms.
Unless Uber/Lyft manage to create fully automated self-driving cars AND find a way to OWN all those cars, they will never have a monopoly.
The people who own the cars will have the freedom to switch to any ride-sharing network they like (or even multiple networks).
Then there will be new kinds of ride-finder apps which will allow people to select between multiple providers/networks - A bit like how Expedia lets you select between multiple airlines.
The Airline business is very competitive and low-margin these days, I think it will be the same for companies like Lyft and Uber.