Prisoner's Dilemma as perfect-information extensive-form game In MultiAgent Systems by Shoham and Leyton-Brown they mention (p. 121) that games in normal form don't always have an equivalent perfect-information game in extensive form, adding that experimentation with the Prisoner's Dilemma (PD) will show why this is the case.
My question is: what is the problem with the naive perfect-information extensive-form version of the PD where we start with, say, Prisoner 1 making a move and then Prisoner 2 responding to that? We still get that defection is best for both players so (naively) one could say: well, the outcomes are the same as in the normal form, so the extensive form here captures the same sort of situation as the normal form. Is it just that the corresponding normal form of the extensive form does not match with the original version?
For the record, the difference between normal form and extensive-form with perfect information is clear to me. I think it shows up nicely in Matching Pennies, for instance. I'm just not sure how to illustrate it on the PD.
 A: I think these authors give the answer themselves in the next sentences:

A little experimentation will convince the reader that there does not
exist a perfect-information game that is equivalent in the sense of
having the same strategy profiles  and the same payoffs. Intuitively,
the problem is that  perfect-information extensive-form games cannot
model simultaneity.

They are saying that the game from its normal form cannot be replicated in the perfect-information extensive form. You are saying that the set of Nash equilibria are the same in the normal and extensive form. Both statements are consistent, because you are talking about slightly different things.
Nash equilibrium is just one equilibrium concept that you can use to solve a game. Yet just because two different games have the same Nash equilibria (same strategies and outcomes) does not mean they are the same game. They are saying you cannot have the same game as the normal form PD if you are forced to represent it in the extensive form with perfect information.
An example: In the extensive form PD, as a second mover I could play the strategy "defect if the other guy defected, but cooperate if the other guy cooperated". This is simply not possible in the normal form PD, because you cannot condition on the other guy's move as the moves happen simulteneously. Hence, these are different games, because they allow for different strategies.
