Economically that seems like a wonderful idea, but politically it sounds doomed to failure. Taxi drivers tend to be pretty well organized politically, and in most cities even have laws in place (usually medallion systems) to limit competition. If they can keep these laws in place with their current very weak justifications, I wouldn't be surprised if they can shut down automated cars very easily, since the public would probably be a bit nervous about the idea anyways.
EDIT: Possibly I'm being too pessimistic. I expect automated cars could be really, really awesome and if so, and if they're allowed anywhere, eventually there'll be pressure in other cities to allow them too.
In many cities there's a distinction between medallion taxis—which you can hail off the street—and livery service—which you need to call for. Since you generally have to take whatever you get get when you hail a cab, the prices are regulated and, in return, the competition is limited.
Generally, the number of livery service drivers and cars is not limited.
That said, the Washington, DC taxi commission has been hassling the Internet-based Uber. Last month they ran a sting against an Uber-arranged driver and impounded his car.[1]
One of my concerns about an automated taxi system is that they'll become dirty and decrepit even faster than public transport because there's an increased level of privacy and nobody to maintain the vehicle between riders. How would you enjoy running late for something and the taxi pulls up, its seats covered in vomit? Eventually, all cars may be scratched up and 'tagged'.
Maybe this won't be as widespread a problem as I think it could be. Zipcar seems to be rather successful, although I've never used one.
EDIT: Possibly I'm being too pessimistic. I expect automated cars could be really, really awesome and if so, and if they're allowed anywhere, eventually there'll be pressure in other cities to allow them too.