[atrifix]

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I think #5 is incorrect, they don't know when the game is going to end.

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IMHO, if we are talking about real-life situations, then all of the assumptions are strictly false, except perhaps #1. I.e., players do not behave rationally, they do not have CKR, perfect information/recall, etc. Some, like perfect recall, may be approximately true, but "approximately true" means "literally false". I think the interesting thing is trying to create a model that can predict behavior, though, and for the model, we'll have to make certain assumptions that are literally false.

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However, even if they did, if they were both rational, then they would cooperate every round. This is especially true if they will be playing multiple games with multiple other people, all of whom know how each other have played in previous games. The best strategy is a tit-for-tat, and knowing this, it is best to cooperate. A constant defector may fair a smidgeon better than the tit-for-tat player in one game (since on the first round, the defector will gain more than the cooperator), but when playing multiple games, the defector will end up costing himself a lot -- as everyone else will soon be defecting against him, but cooperating with those that are cooperating -- thus, the cooperators will be gaining more than the defector.

Anyway, even in a single game, the defector will know that the tit-for-tat player will defect on every round after the first if the defector defects on the first round. However, if he cooperates on the first round, then he will gain more on subsequent rounds, and thus will maximize his gain over the course of the game. This is the most rational play that both can make.

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If our definition of rationality includes not playing dominated strategies, then this is not really correct. I suppose we could have an alternative definition of rationality that doesn't include not playing dominated strategies, but I'm not sure what that would be.

It's true that TFT is strictly dominated by always defecting (ALL D), but more importantly, it's strictly dominated by strategy TFT-1, e.g., TFT until last round, then defect. Now if all the assumptions hold, then no one will play TFT, because they'd do better to play TFT-1. Similarly, both players know that they're rational, they know that TFT won't be played. Since both players realize that TFT-1 is dominated by TFT-2, they both know that they won't play TFT-1...and so on. Hence if all the assumptions hold they'll defect on every round.