(1) It’s a about the same kind of argument as saying “less interesting” :-P
(2) No, each round is not the same because the horizon is not unknown, it’s just that instead of a fixed number you have a probability distribution to deal with.
(3) We are not talking about the infinite-horizon case.
My main point is that the NE for the fixed-length IPD is “always defect”, but the fact that we really ought to be able to do much better than “always defect” is what makes that case particularly interesting.
(2 & 3) Sorry; I misunderstood. was thinking about the infinite-horizon case. If you have a probability distribution over possible lengths of the game then the problem is indeed more complex, but I don’t see that much benefit to it—it really doesn’t change things all that much.
In particular, if you still have a limit on the number of rounds, then the same reasoning by backwards induction still applies (i.e .the optimal strategy is to always defect), and so the same interesting aspect of the problem is still there.
Similarly, the optimal counter-strategy to TFT stays mostly the same. It will simply be “TFT, but always defect starting from round N” where N is some number that will depend on the specific probabilities in question.
The interesting aspect of the problem is still the part that comes from the finite limit, regardless of whether it’s fixed or has some kind of distribution over a finite number of possibilities.
then the same reasoning by backwards induction still applies (i.e .the optimal strategy is to always defect)
Not so fast.
Once a prisoner condemned to death was brought before the king to set the date of the execution. But the king was in a good mood, having just had a tasty breakfast, and so he said: “You will be executed next week, but to spare you the agony of counting down the hours of your life, I promise you that you will not know the day of your execution until the jailers come to take you to the gallows”.
The prisoner was brought back to his cell where he thought for a while and then exclaimed: “But I cannot be executed if the king is to keep his word! I cannot be executed on Sunday because if I’m alive on Saturday I’ll know the day of my execution and that breaks the king’s promise. And I cannot be executed on Saturday because I know I cannot be executed on Sunday so if Friday comes around and I’m still alive, I’ll know I have to be executed on Saturday. Etc., etc. This perfect reasoning by backwards induction says I cannot be executed during any day. And since the king always keeps his word, I’m safe!”. Much relieved, he lay down on his cot whistling merrily.
And so, when on Tuesday the guards came for him, he was very surprised. The king kept his word.
Sorry; I meant to say the “optimal” strategy is to defect. I don’t agree with the backwards induction; my point was that that argument is precisely what makes the problem interesting.
EDIT: By the way, I’m pretty sure I’ve come up with a strategy that is a Nash equilibrium (in IPD + simulations) and always cooperates with itself, so I very strongly agree that always defecting is not optimal.
(1) It’s a about the same kind of argument as saying “less interesting” :-P
(2) No, each round is not the same because the horizon is not unknown, it’s just that instead of a fixed number you have a probability distribution to deal with.
(3) We are not talking about the infinite-horizon case.
(4) Yes, and?
My main point is that the NE for the fixed-length IPD is “always defect”, but the fact that we really ought to be able to do much better than “always defect” is what makes that case particularly interesting.
(2 & 3) Sorry; I misunderstood. was thinking about the infinite-horizon case. If you have a probability distribution over possible lengths of the game then the problem is indeed more complex, but I don’t see that much benefit to it—it really doesn’t change things all that much.
In particular, if you still have a limit on the number of rounds, then the same reasoning by backwards induction still applies (i.e .the optimal strategy is to always defect), and so the same interesting aspect of the problem is still there.
Similarly, the optimal counter-strategy to TFT stays mostly the same. It will simply be “TFT, but always defect starting from round N” where N is some number that will depend on the specific probabilities in question.
The interesting aspect of the problem is still the part that comes from the finite limit, regardless of whether it’s fixed or has some kind of distribution over a finite number of possibilities.
Not so fast.
Once a prisoner condemned to death was brought before the king to set the date of the execution. But the king was in a good mood, having just had a tasty breakfast, and so he said: “You will be executed next week, but to spare you the agony of counting down the hours of your life, I promise you that you will not know the day of your execution until the jailers come to take you to the gallows”.
The prisoner was brought back to his cell where he thought for a while and then exclaimed: “But I cannot be executed if the king is to keep his word! I cannot be executed on Sunday because if I’m alive on Saturday I’ll know the day of my execution and that breaks the king’s promise. And I cannot be executed on Saturday because I know I cannot be executed on Sunday so if Friday comes around and I’m still alive, I’ll know I have to be executed on Saturday. Etc., etc. This perfect reasoning by backwards induction says I cannot be executed during any day. And since the king always keeps his word, I’m safe!”. Much relieved, he lay down on his cot whistling merrily.
And so, when on Tuesday the guards came for him, he was very surprised. The king kept his word.
Sorry; I meant to say the “optimal” strategy is to defect. I don’t agree with the backwards induction; my point was that that argument is precisely what makes the problem interesting.
EDIT: By the way, I’m pretty sure I’ve come up with a strategy that is a Nash equilibrium (in IPD + simulations) and always cooperates with itself, so I very strongly agree that always defecting is not optimal.