Maybe this is so obvious that I don't even need to write it. Maybe it's also still a bit too far into the future and too much science-fiction.
But autonomous (or self-driving) cars are slowly turning from an object of research into a reality of engineering. Currently, where it is allowed to drive them, it's only for purposes of research and development. But once a state or country will allow them for just transport, many others might follow in quick succession. (Some technologies are adopted very quickly, while others need a very long time until they are in widespread use or until they have displaced their predecessor technology. But that's a topic for another post.)
Now, let's just assume that self-driving cars will be available at some time near or far. Let's assume that they will be more expensive than people-driven cars, but that the price difference is not more than the relative difference between a current cheap and moderately expensive car. People will have the choice to buy and drive either one. And for many who can afford it the self-driving car will be the choice, because it gives them a lot of extra time to spend during their commute. Let's also assume that there will be a similar choice for taxis: you can get a self-driving one or get one with a driver. Since for the self-driving taxi, no driver needs to be paid, it will be cheaper than a person-driven taxi. Also, since there is no person to watch the passenger and make sure they are paying and not messing the car up, it's possible that self-driving taxis will require some sort of registration of the passengers just as is the case for car-sharing services now. Indeed, the difference between car sharing and taxis will blur when both are self-driving.
If one has to register to use a self-driving taxi service (for example, by requesting a RFID customer card which is linked to your payment details and name/photo), this is just like for car-sharing, only that a lot more people will do it. That's because unlike traditional car sharing where you need to go to a station to pick up your shared car and also leave it a station, the self-driving taxi (or next-generation shared car) can drive autonomously to pick you up and can also disappear on its own when you don't need it any more.
It's those last two features which previously have hindered the adoption of shared cars (because you need additional transport to get to stations) and which will make the self-driving shared car even more useful than your own car. First, because you don't need to have your own parking spot, which makes live a lot easier if you live in a urban setting without underground parking or other reserved parking space. Second, you don't need to find parking at your destination, which is two advantages at once: A - you can get off wherever a taxi could let you get off, which is often much closer to your destination than a parking spot would be, and B - you save all the time to find a parking spot, park the car, and pay for the parking (if needed).
So there: we'll have the luxury of taking a taxi at the price of taking a shared car, which is much cheaper than having your own car. Just think of it: the shared car will transport many other people over the course of the day, all getting the luxury of a taxi drive and contributing to the maintenance and purchase cost of the car. Since the car is used much more, it will be amortized much quicker, so shared cars will be upgraded to newer models and the latest technologies more often than people would buy a new private car.
So there you have it ‒ self-driving shared cars might replace a lot of privately-owned cars ‒ especially for people who don't have private parking space and even more so for people who already use a mix of public and private transportation. (Missed your last commuter train or need to do some shopping on the way home? Just take a self-driving taxi!)
I am already fantasizing what could be done to all this reconverted car parking space in our cities ‒ more green, more life, more socializing. Maybe even happier people?