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Metus comments on A simple game that has no solution - Less Wrong Discussion
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Distuingish between the reasoning and the outcome. Game theoretic reasoning is memory-less, the exact choice of action of one player does not matter to the other one in the hypothetical. As the rules are known, both players come to the same conclusion and can predict how the game will play out. If in practice this model is violated by one player the other player immediately knows that the first player is irrational.
"Game theoretic reasoning is memory-less"
No. Consider the tit-for-tat strategy in the infinitely repeated prisoners' dilemma game.
Why is it irrational for Player 1 to not pick A? Your answer must include beliefs that Player 2 would have if he gets to move.