Doesn't the number of cars driving around depend on density of riders? Just getting rid of the driver would make every car have one extra seat available for passengers, assuming cars maintain the same number of seats.
So what I'm more curious about is if self-driving enables more options for ride sharing, just as Uber had Uber pool, at a cheaper price, perhaps it could open up more options (at different prices) between riding with 50 strangers on a bus or taking a 4-seat car by myself?
Overall, I don't know if cars on road at time X will go up or down. Heck, if there aren't cars parking on the streets, then some roads in cities may even add two extra lanes for traffic, which could reduce traffic further.
But as others have pointed out, ridesharing companies with drivers haven't really reduced traffic, so maybe self-driving cars won't either, however I seem to think it'll be more of a step change when cars drive themselves.