With plentiful clean, modern and safe self-driving transportation options available, the need for outright car ownership, at least for urbanites, will become increasingly unattractive, not even mentioning insurance, fuel, maintenance & upkeep as well as parking space headaches and cost.
While big city dwellers who have never owned a car anyway, and who always use taxis, will be fine with public self-driving vehicles, I can't see a huge ownership shift happening with most of the previously private vehicle owners.
Especially to people with families that are used to storing lots of kid stuff in their own vehicles. Or heck, for almost anyone who's used to having their own car. Self-driving mode as an option, sure. Not owning the car? Nope.
There's too much freedom for non-city-dwellers in owning your own vehicle. From being able to leave (or delay) as much as you wish, to customization, to the above mentioned leaving junk in your car... or keeping it as clean as you wish. Not to mention that private ownership will likely still be cheaper for young adults who can afford (or are given) a used car, but cannot constantly afford paying a fare.
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Therefore, I think the primary urbanite users would be those who can no longer drive themselves. The elderly, those who've lost their license or are handicapped, etc. Yet any of those who travel a lot and are not poor, will likely want their own private vehicle for the same reasons noted above.
Now, once in a while I can see an urbanite saying, ah we need to carry a junior baseball team, let's call in some autonomous vans, but that would be the exception.
All IMO.