If history is to repeat itself, Waymo will dominate on the whole planet excluding China, where Chinese self driving companies prosper but cannot reach outside.
I am not convinced. Many sovereign nations will think twice before handing a foreign corporation the keys to their mass transportation industry. They will have the motivation, and the means, to keep manufacturers in check, and play them off of each other. Hopefully this will prevent runaway monopolies from taking root.
I think it will be very hard to convince French regulators, for example, to allow any foreign corporation to have more visibility and control over the national transportation system than the French government itself.
And unlike ridesharing, which appeared practically overnight, self-driving car technology is under mainstream scrutiny several years before achieving meaningful scale. This means regulators have more time to prepare.
The car industry, and the national transportation infrastructure, is considered a matter of sovereignty, and a politically sensitive topic. This won't be anything like ridesharing, consumer electronics, or advertisement.
I think (and hope) that the result will be that Waymo will fail to get a winner-takes-all stranglehold on the global self-driving car industry. Instead it will be forced to compete on a relatively level playing field, against Didi, Apple, Amazon, or whoever decides to compete in the space. The more the better.
I am rooting for the regulators on this one. If they fail to protect our national transportation systems from Google-style walled gardens... Then that sends us down a very scary road, where almost every aspect of society and government ends up privatized, and managed from either the US west coast or China, with no accountability or representation.