Actually, I think such a system would kill public transportation because it would become cheaper to use the cars than public transport.
I don't know the exact breakdown but my guess is that salaries of drivers are much more expensive than buses and gas, especially given those are often unionized jobs with very costly benefits (pension etc.).
That's a good point. I still think public transportation would be replaced.
I believe the main reason local government provides public transportation is because to achieve low prices it needs to be subsidized. No private enterprise is interested in competing in a market where you can't make profit.
At the same time, you don't see local government e.g. opening grocery shops: this need is well filled by private enterprise.
The car rental business would develop despite public transportation because it covers a much bigger needs. However, when it reaches scale and the low price that comes with it, it might make public transportation no longer needed as it'll also cover that use case.
Even if it's slightly more expensive, would you rather pay $2, wait 15mins, not get exactly where you want to go and hope no one will punch you in your face and steal your iphone (true story!) or pay $5, wait 5min and get exactly where you wanted to go?
If enough people start doing that, at some point the fixed cost of running public transportation will be too high to justify low usage by public.
Not to mention I do expect for profit enterprise to adopt driverless cars much faster than government agency, hence giving them a cost advantage for quite a few years.
I don't know the exact breakdown but my guess is that salaries of drivers are much more expensive than buses and gas, especially given those are often unionized jobs with very costly benefits (pension etc.).