Not yet. A few years/decades after self-driving cars become common, though, I wouldn't be shocked to see manually driven cars banned or strictly limited for safety reasons. What then?
Then there will likely be several companies competing in the self-driving car market? It's not like when horse and buggy went out of fashion, we all had to rush out and buy a Ford.
Apple, Google, Microsoft, etc. compete in the smartphone world but all are privacy disasters in one way or another. I don't expect things like self-driving cars to buck that trend.
