Defecting one round earlier dominates pure tit-for-tat, but defecting five rounds earlier doesn’t dominate pure tit-for-tat. Pure tit-for-tat is better against pure tit-for-tat. So there might be a nash equilibrium containing only strategies that play tit-for-tat until the last few rounds.
Defecting in the last x rounds is dominated by defecting in the last x+1, so there is no pure-strategy equilibrium which involves cooperating in any rounds. But perhaps you mean there could be a mixed strategy equilibrium which involves switching to defection some time near the end, with some randomization.
Clearly such a strategy must involve defecting in the final round, since there is no incentive to cooperate.
But then, similarly, it must involve defecting on the second-to-last round, etc.
So it should not have any probability of cooperating—at least, not in the game-states which have positive probability.
Right? I think my argument is pretty clear if we assume subgame-perfect equilibria (and so can apply backwards induction). Otherwise, it’s a bit fuzzy, but it still seems to me like the equilibrium can’t have a positive probability of cooperating on any turn, even if players would hypothetically play tit-for-tat according to their strategies.
(For example, one equilibrium is for players to play tit-for-tat, but with both players’ first moves being to defect.)
Defecting in the last x rounds is dominated by defecting in the last x+1, so there is no pure-strategy equilibrium which involves cooperating in any rounds. But perhaps you mean there could be a mixed strategy equilibrium which involves switching to defection some time near the end, with some randomization.
Clearly such a strategy must involve defecting in the final round, since there is no incentive to cooperate.
But then, similarly, it must involve defecting on the second-to-last round, etc.
So it should not have any probability of cooperating—at least, not in the game-states which have positive probability.
Right? I think my argument is pretty clear if we assume subgame-perfect equilibria (and so can apply backwards induction). Otherwise, it’s a bit fuzzy, but it still seems to me like the equilibrium can’t have a positive probability of cooperating on any turn, even if players would hypothetically play tit-for-tat according to their strategies.
(For example, one equilibrium is for players to play tit-for-tat, but with both players’ first moves being to defect.)
Yeah you’re right. I just realized that what I had in mind originally already implicitly had superationality.