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I don't think there is an app out there that does the same. There are local ones, Kapten for instance, but nothing that works across so many territories and is as likely to be installed on so many visitors phones.
You're also assuming that the demand for taxis/minicabs is price inelastic which clearly isn't the case. (Although potentially the drivers may be working fewer but better paid hours but I suspect that would be unlikely.)
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Apps aren't difficult to programme. Investors will get behind a new, promising one if Uber goes tits up. Visitors will install them on their phones if they go to London, as which app to install will become widely known.
Where do I assume people don't look at price? Of course they do. However, I believe demand for minicab use will continue to go up for a number other reasons, not the least being that 'getting an Uber' is now a cultural institution and will continue even if Uber doesn't operate in London any more. Other factors are a steadily growing need to travel, a steadily growing perceived need to travel, various kinds of greenwash like electric cars or alternative fuels, etc.
(I think they will get a new licence and make the adjustments required, as London is one of their most important markets, but it would be good if they didn't continue here.)
I am convinced by it. The advantage that Uber had a few years ago because of its novel 'app' can now easily be caught up on by competitors. If all of its drivers are 'released', there will be someone else waiting for them.
I mean, I would personally prefer it if this kind of taxi service ended in London altogether, but it won't, and I'd prefer pretty much any smaller competitor over Uber, one of the worst companies around.