I guess that China is trying to be first. Since the government is very strong they can easily change the laws and regulate liability issues.
USA will try to follow to gain technical konowledge and for developing business cases. There the companies are very strong and can chnage regulation. Japan and EU will be last.
What Markus says makes sense but, I think it also depends on the suitability of the cities for autonomous driving. Autonomous driving can be really effective in places where the cities are small and are separated by some distance. This would result in less crowd and traffic while commuting thereby reducing the complexity for the systems and enabling the autonomous cars to work very efficiently. You can find such an environment only in Europe. And considering the technical expertise of Germany, Italy, Sweden and UK in making cars, I personally think Europe is best prepared for autonomous cars.
Europe, APAC (India and China) and USA are in the race. But who will roll-out first the full fledged one by resolving the burning concerns of technology robustness and required ecosystems to run, evolve and sustain for long is the market to see.