Why would you drop prices when there is no competition? Makes no sense. Basic supply and demand.
My experience with Uber was essentially: (1) "Hey, it sounds great, but that's a bit pricey. It can't really be worth it, can it?" (2) Decided to take someone to dinner via Uber because it was more impressive than a cab. (3) "Hey, this is actually pretty cool and totally worth it." (4) I'm now a repeat customer, even when traveling alone.
In my eyes, the key is that they need to find a step (2) for a more general case. Having a day where there are no cabs might be a small push, but having a day where there are no cabs and Uber is the price you would have paid anyway is absolutely killer.
Short answer: Long-term thinking is better for them than short-term thinking.