PHOENIX — Three weeks into his new job as Arizona’s governor, Doug Ducey made a move that won over Silicon Valley and paved the way for his state to become a driverless car utopia.

It was January 2015 and the Phoenix area was about to host the Super Bowl. Mr. Ducey learned that a local regulator was planning a sting on Lyft and Uber drivers to shut down the ride-hailing services for operating illegally. Mr. Ducey, a Republican who was the former chief executive of the ice cream chain Cold Stone Creamery, was furious.

“It was the exact opposite message we should have been sending,” Mr. Ducey said in an interview. “We needed our message to Uber, Lyft and other entrepreneurs in Silicon Valley to be that Arizona was open to new ideas.” If the state had a slogan, he added, it would include the words “open for business.”