In this scenario, dispatch would basically be a public utility, and drivers could trust the algorithm not to cheat. I.e. since Uber B and its competitors would be running any driver incentives, Uber A wouldn't even know about it. (And legally shall not know such things.)
I realize this is a half-baked pipe dream, but an interesting thought experiment.
But since Uber B has the brand recognition (through its UI), the question is if there will be a competitor. Nowadays, it seems that consumers converge more and more on a single brand for any service/product, which is probably due to the internet/social media.
This only seems true (at least over medium to longer times) for certain classes of product/service, namely those where network effects actually provide real value to consumers. Many products have tried to artificially integrate features that would initiate network affects (for example niche social networks for fitness trackers) but they end up making the product less usable because they're not part of the core value proposition consumers are looking for with that product.