It reminds me of a problem noticed in Iterated Prisoner’s Dilemma. Conventional wisdom says the best thing to do is to cooperate on a tit-for-tat basis – that is, we both keep cooperating, because if we don’t the other person will punish us next turn by defecting.

But it has been pointed out there’s a flaw here. Suppose we are iterating for one hundred games. On Turn 100, you might as well defect, because there’s no way your opponent can punish you later. But that means both sides should always play (D,D) on Turn 100. But since you know on Turn 99 that your opponent must defect next turn, they can’t punish you any worse if you defect now. So both sides should always play (D,D) on turn 99. And so on by induction to everyone defecting the entire game. I don’t know of any good way to solve this problem, although it often doesn’t turn up in the real world because no one knows exactly how many interactions they will have with another person. Which suggests one possible solution to the original problem is for nobody to know the exact number of people.

Edit: 1) This is from a ssc article, I can't remember which, I didn't write the above paragraph, I just copied it when I read it and thought it interesting. 2) I'll award the bounty, I just have to think about the answers a little bit.