“If player [1] already chose to play either B or C, the game transforms into a game with a simple 2x2 payoff matrix.”
No because Player 2 knows you did not pick A and this might give him insight into what you did pick. So even after Player 1 picks B or C the existence of strategy A might still effect the game because of uncertainty.
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.
“If player [1] already chose to play either B or C, the game transforms into a game with a simple 2x2 payoff matrix.”
No because Player 2 knows you did not pick A and this might give him insight into what you did pick. So even after Player 1 picks B or C the existence of strategy A might still effect the game because of uncertainty.
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.