This is an analysis which to me seems trivial, but which I very rarely see brought up in any discussion of games like The Prisoner's Dilemma or The Centipede Game which are well known for having 'counterintuitive' 'rational' strategies (the only similar thing I have seen is here). I apologize if any of the below is unclear- I'm not especially familiar with the terminology of game theory.
In each of these games, as they are typically formulated, it is assumed that the players are perfectly rational and that they understand the game. It seems to me as though a rational player would assume that any other rational player would make exactly the same choice they would, since they are themselves rational, and therefore in symmetric games like the Prisoner's Dilemma the only rational choice is to behave cooperatively, as both players choosing to behave cooperatively gives better results for each of them than both of them choosing to behave traitorously.
The centipede game is more complicated, but in the formulation I typically see (two pots, one k * the size of the other, two players, choose to pass and double the pot sizes or take one pot and give the other player the other pot, at most N steps) the only rational strategy is for the players to pass the pot between each other until the number of steps remaining is small enough that the small pot can no longer exceed the large pot's current value and then pass it between them an even number of times. The 'winner' and 'loser' are decided at the game's start by N and k; the game then is about maximizing everyone's profit. The typical argument against this strategy is that at the $n$th step, the choosing player has no incentive not to take the pot, but it ignores that for a rational player to take the pot, taking the pot must be the rational choice, and if it were rational to take the pot now it would have been rational to take the pot at the first opportunity, and if it were rational to take the pot at the first opportunity then they would have taken the pot at the first opportunity, and gotten a much smaller pot, which is clearly not the rational choice. The incentive against taking the pot before the point at which the small pot can no longer reach the size of the large pot is that taking it sooner guarantees a smaller payout than the above strategy. After that point taking the pot out of turn would invalidate the earlier incentive, and so cannot be the rational choice.
Edit in response to Shane's comment, since it seems like this will come up a lot.
It seems like my argument regarding the prisoner's dilemma is being misconstrued. I will restate it here using hopefully more precise language.
The game is set up so that there are two, rational players, each of whom choose between A (cooperation) or B (betrayal). If both players choose A they each get a large reward. If both choose B they each get a small reward. If one chooses A and the other chooses B, the one choosing A gets no reward and the one choosing B gets a very large reward. Both players have complete knowledge of the game, and can take as long as they'd like to choose between A and B, though they cannot communicate with each other.
The game is completely symmetrical, and there is no way to distinguish between the players, and the players know this. Therefore, each player independently can reason that, if there is a better choice for themselves between A and B, then that same choice is also better for the other player to make. Therefore, if it is better for them to choose B, then it is also better for the other player to choose B, resulting in the small reward for each of them. However, if it is instead better to choose A, then it is also better for the other player to choose A, resulting in the large reward for each of them. Since both players getting a large reward is better for both players than both players getting a small reward, it is better to choose A, and therefore both players choose A.