Where will it drive its ass to? Since work hours and rush hour will still exist, there won't be demand for all the cars used between 7:30 - 9 and 4:30 - 6 to go take someone else on a trip. It'll just have to drive itself to a parking lot or garage, same as today.
And unless you want to wait every morning for your public taxi ride to show up [[pain in the ass), people will still want to buy their own cars. It's not like it's going to be profitable for some company to keep a huge fleet of cars roaming around the exurbs to respond on a moment's notice.
Also, keep in mind that a huge fleet of empty cars roving around everywhere so that you don't have to wait for pickup means, by definition, constant terrible traffic since you have both the cars with people in them you had before and now also all the empty cars required to quickly get to anybody who needs a ride rather than parked cars in garages. So either your travel time or your wait time just exploded compared to owning a personal car today, but at least you can look at your phone I guess.
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