There are N players. Assume N is big, like on the order of a million. The game is parametrized by a threshold T, which is on the order of some substantial fraction of N, like N/3 or N/2 or (3/4)N or something like that. The payoffs are controlled by two parameters, D, and C. If the number of cooperating players is less than the threshold, the defectors each receive $1, and the cooperators nothing. If the number of cooperating players is above the threshold, then the defectors get $D and the cooperators get $C.
Here's some parameter values for (D, C) that are interesting to consider:
1. (1000, 1000)
2. (1001, 1000)
3. (999, 1000)
4. (0, 1000)
5. (10000, 1000)
What do you play?
In (1) and (2) and (5), the rational play is definitely to defect. In (3) and (4) I suppose you have to think about the likelihood that the population is going to make the critical threshold. (5) is interesting to me, because there's an amplified incentive to defect if lots of other people cooperating --- it's perhaps the most prisoner's-dilemma-like, since you're exploiting the "suckers".
In all cases, the game is made weirder than the plain prisoner's dilemma (and this is what made me think of it, reading some stuff about voting in elections elsewhere) by the intuitions surrounding the paradox of the heap: "my own vote has a vanishing chance of making a difference so why bother".
