To be clear, uncertainty about the number of iterations isn’t enough. You need to have positive probability on arbitrarily high numbers of iterations, and never have it be the case that the probability of p(>n rounds) is so much less than p(n rounds) that it’s worth defecting on round n regardless of the effect of your reputation. These are pretty strong assumptions.
So cooperation is crucially dependent on your belief that all the way from 10 rounds to Graham’s number of rounds (and beyond), the probability of >n rounds conditional on n rounds is never lower than e.g. 20% (or whatever number is implied by the pay-off structure of your game).
To be clear, uncertainty about the number of iterations isn’t enough. You need to have positive probability on arbitrarily high numbers of iterations, and never have it be the case that the probability of p(>n rounds) is so much less than p(n rounds) that it’s worth defecting on round n regardless of the effect of your reputation. These are pretty strong assumptions.
So cooperation is crucially dependent on your belief that all the way from 10 rounds to Graham’s number of rounds (and beyond), the probability of >n rounds conditional on n rounds is never lower than e.g. 20% (or whatever number is implied by the pay-off structure of your game).