It would be interesting to attempt to express mathematically expected hail times based on driver & rider density. Obviously the gains to adding density start out very big but then, as you say, decrease. But when exactly do you get past the elbow on the curve?
There's no elbow, the curve looks the same all the way down. And it's a square-root relationship. That is, you need to increase number of drivers by a factor of four in order to reduce the expected distance between a driver and a passenger by a factor of two.
Even leaving aside the assumption of uniform distribution, hail time isn't only a function of distance. Once you get below a 5-minute expected hail (which happens regularly with Uber in the middle of DC), a substantial portion is eaten up by the driver simply orienting himself after accepting a hail, then trying to find the passenger when he arrives.
It doesn't actually assume uniform distribution, it assumes that the distribution pattern doesn't change as the number of drivers goes up. Which from my experience is a good assumption over reasonable amounts of changes in driver numbers. It might change if you start to recruit some new demographic of drivers.
Drivers who are close to you having a hard time getting to you is definitely one of the tough algorithmic problems in this space! At Flywheel, we kept mentioning it, like, "Okay, what if we tried to take into account one-way streets and driver heading," and ultimately we were (when I was there at least) just too resource constrained to try to take on such a fiddly complex data-intensive problem.
One thing I don't really like about Uber is that they have All The Money In The World and yet I feel like they don't have a lot of ambition to deliver a really good app experience. Like, I get a weird loading screen every time I bg/fg my app. What's up with that? This is Android, the app can just fuckin' run while it's bged. And they aren't tackling the "driver who can get to you soonest" problem (as far as I can tell from the perspective of a passenger) or the "how to help a driver figure out who you are in a crowd" problem or any of the other cool stuff in this space.
Like, what's the point of being a $all the billions company if you don't deliver a really good app?
