If Player 2 gets to move, then the only possible choice for a rational Player 1 to have made is to pick C, because B cannot possibly maximize Player 1′s utility. The probability for a rational Player 1 to pick B is always 0, so the probability of picking C has to be 1. For Player 1,there is no rational reason to ever pick B, and picking C means that a rational Player 2 will always pick X, negating Player 1′s utility. So a rational Player 1 must pick A.
You seem to be treating the sub-problem, “what would Player 2 believe if he got a move” as if it is separate from and uninformed by Player 1′s original choice. Assuming Player 1 is a utility-maximizer and Player 2 knows this, Player 2 immediately knows that if he gets a move, then Player 1 believed he could get greater utility from either option B or C than he could get from option A. As option B can never offer greater utility than option A, a rational Player 1 could never have selected it in preference to A. But of course that only leaves C as a possibility for Player 1 to have selected and Player 2 will select X and deny any utility to Player 1. So neither option B nor C can ever produce more utility than option A if both players are rational.
Exactly, but B is never a preferable option over A, so the only rational option is for Player 1 to have chosen A in the first place, so any circumstance in which Player 2 has a move necessitates an irrational Player 1. The probability of Player 1 ever selecting B to maximize utility is always 0.
If Player 2 gets to move, then the only possible choice for a rational Player 1 to have made is to pick C, because B cannot possibly maximize Player 1′s utility. The probability for a rational Player 1 to pick B is always 0, so the probability of picking C has to be 1. For Player 1,there is no rational reason to ever pick B, and picking C means that a rational Player 2 will always pick X, negating Player 1′s utility. So a rational Player 1 must pick A.
So are you saying that if Player 2 gets to move he will believe that Player 1 picked C?
Yes.
But this does not make sense because then player 1 will know that player 2 will play X, so Player 1 would have been better off playing A or B over C.
You seem to be treating the sub-problem, “what would Player 2 believe if he got a move” as if it is separate from and uninformed by Player 1′s original choice. Assuming Player 1 is a utility-maximizer and Player 2 knows this, Player 2 immediately knows that if he gets a move, then Player 1 believed he could get greater utility from either option B or C than he could get from option A. As option B can never offer greater utility than option A, a rational Player 1 could never have selected it in preference to A. But of course that only leaves C as a possibility for Player 1 to have selected and Player 2 will select X and deny any utility to Player 1. So neither option B nor C can ever produce more utility than option A if both players are rational.
Exactly, but B is never a preferable option over A, so the only rational option is for Player 1 to have chosen A in the first place, so any circumstance in which Player 2 has a move necessitates an irrational Player 1. The probability of Player 1 ever selecting B to maximize utility is always 0.