I'm confused about defection becoming a dominant strategy.. Because the existence of a dominant strategy suggests to me that there should exist a unique Nash equilibrium here, which is not the case. Everyone defecting is a Nash equilibrium, but 50 people cooperating and 49 defecting is a Nash equilibrium as well, and a better one at that. Something (quite likely my intuition regarding Nash equilibria in games with more than 2 players) is off here. Also, it is of course possible to calculate the optimal probability that we should defect and I agree with FeepingCreature that this should be 0.5-e, where e depends on the size of the player base and goes to 0 when the player base becomes infinite. But I highly doubt that there's an elegant formula for it. It seems (in my head at least) that already for, say, n=5 you have to do quite a bit of calculation, let alone n=99.
I'm confused about defection becoming a dominant strategy.. Because the existence of a dominant strategy suggests to me that there should exist a unique Nash equilibrium here, which is not the case. Everyone defecting is a Nash equilibrium, but 50 people cooperating and 49 defecting is a Nash equilibrium as well, and a better one at that. Something (quite likely my intuition regarding Nash equilibria in games with more than 2 players) is off here. Also, it is of course possible to calculate the optimal probability that we should defect and I agree with FeepingCreature that this should be 0.5-e, where e depends on the size of the player base and goes to 0 when the player base becomes infinite. But I highly doubt that there's an elegant formula for it. It seems (in my head at least) that already for, say, n=5 you have to do quite a bit of calculation, let alone n=99.