Imagine a centipede game is being played by perfectly rational agents. As has been well-established, the first player reasons using backward induction, and also reasons that their opponent will use backward induction, and therefore to maximise their own utility they are forced to end the game immediately, taking 1 util for themselves and none for their opponent, even when it was possible that they receive (about) 100 utils each had they co-operated.
Here's the question. If the first player didn't end the game immediately, the second player would have good evidence that the first player plans to co-operate for some time to come. If the first player made the co-operative decision on the first turn, then they will on most of the rest since they are identical scenarios. Therefore the second player ought also to co-operate. Even if the other player decides to end the game prematurely in the future for no discernible reason, you still make more than you would by playing an ultra-defensive strategy. However there still is some kind of paradox looming, as the end-game will still be in the minds of the players. I haven't worked this one out yet.
Centipede could be one of those rare games that turns up in game theory where it pains the player to be rational, just as Lewis thought the Newcomb's game paid off those less rational 1-boxers. One problem with this game is about knowledge. In the centipede game, your opponent knows that you are "rational", therefore you cannot choose to play irrationally because it is a given fact already how you will play. Hence decision theory of this sort becomes ironically indecisive and determined. We need to use the concept of common belief rather than common knowledge. In Centipede, if you play the co-operative move you also falsify the other player's belief, which will cause them to play differently. Not irrationally. It just happens that their behaviour is identical to the behaviour of an irrational agent, but not their thought process. This is a phenomenon that is observed very frequently in the world of professional poker.
Suppose CBR, common belief in rationality, and suppose that Backwards Induction is essential for rationality. My earliest instinct then is that, given both players have CBR, if the first chooses to play co-operatively, the second player no longer has CBR, i.e. he comes to believe that the first player cannot be using backwards-induction and is therefore irrational. However, this cannot be the case. There are players who do use backwards induction but who still co-operate because they expect or suspect their opponent of being able to sympathise with them. So when the first player co-operates, essentially the other player still cannot tell whether the first is using BI or not. Like in poker, when an opponent makes a play you can only tell that they are not an intermediate, they could still either be an expert or a beginner.
