This is exactly Prisoner's Dilemma and all relevant literature applies. Iterate a few times with a large population and the optimal strategy goes to all sorts of weird places.

Well, in the Prisoner's Dilemma is that you can't communicate or trust each other: you're assuming everyone acts in self-interest.

...which is exactly the opposite of the assumption I make with my friends, especially when the stakes are this low.

I can imagine two friends who love each others company and desperately want to hang out keep canceling plans because they have the app set to let the other one out.

They each end up spending time with more demanding people who don’t care what they want.

(Because this is the internet: kidding. The app can work, just be honest with the app at least)