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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.
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.
So are you saying that if Player 2 gets to move he will believe that Player 1 picked C?
Yes.
Then please answer the question, "if Player 2 gets to move what should he believe Player 1 has picked?"
Until you can answer this question you can not solve the game. If it is not possible to answer the question, then the game can not be solved. I know that you want to say "Not picking A would prove Player 1 is irrational" but you can't claim this until you tell me what Player 2 would do if he got to move, and you can't answer this last question until you tell me what Player 2 would believe Player 1 had done if Player 1 does not pick A.
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.
By definition of rationality in game theory, Player 1 will maximize his expected payoff and so need to have some belief as to the probabilities. If you can't figure out a way of estimating these probabilities the game has no solution in classical game theory land.
Well, as i said I'm not familiar with the mathematics or rules of game theory so the game may well be unsolvable in a mathematical sense. However, it still seems to me that Player 1 choosing A is the only rational choice. Having thought about it some more I would state my reasoning as follows. For Player 1, there is NO POSSIBLE way for him to maximize utility by selecting B in a non-iterated game, it cannot ever be a rational choice, and you have stated the player is rational. Choosing C can conceivably result in greater utility, so it can't be immediately discarded as a rational choice. If Player 2 finds himself with a move against a rational player, then the only possible choice that player could have made is C, so a rational Player 2 must choose X. Both players, being rational can see this, and so Player 1 cannot possibly choose anything other than A without being irrational. Unless you can justify some scenario in which a rational player can maximize utility by choosing B, then neither player can consider that as a rational option.
Because for Player 1 to increase his payoff over picking A, the only option he can choose is C, based on an accurate prediction via some process of reasoning that player 2 will pick X, thereby making a false prediction about Player 1's behaviour. You have stated both players are rational, so I will assume they have equal powers of reason, in which case if it is possible for Player 2 to make a false prediction based on their powers of reason then Player 1 must be equally capable of making a wrong prediction, meaning that Player 1 should avoid the uncertainty and always go for the guaranteed payoff.
To formulate this mathematically you would need to determine the probability of making a false prediction and factor that into the odds, which I regret is beyond my ability.
But why is not picking A "impossible-in-reality"? You can not answer until you tell me what Player 2's beliefs would be if A was not picked.
Because for Player 1 to increase his payoff over picking A, the only option he can choose is C, based on an accurate prediction via some process of reasoning that player 2 will pick X, thereby making a false prediction about Player 1's behaviour. You have stated both players are rational, so I will assume they have equal powers of reason, in which case if it is possible for Player 2 to make a false prediction based on their powers of reason then Player 1 must be equally capable of making a wrong prediction, meaning that Player 1 should avoid the uncertainty and always go for the guaranteed payoff.
To know that A gives you a higher payoff than C you have to know what Player 2 would do if he got to move, but since Player 2 expects to never move how do you figure this out?
Right that makes sense, but wouldn't Player 1 simply realize that making an accurate forecast of player 2's actions is functionally impossible, and still go with the certain payout of A?
In response to
A simple game that has no solution
I'm not overly familiar with game theory, so forgive me if I'm making some elementary mistake, but surely the only possible outcome is Player 1 always picking A. Either other option is essentially Player 1 choosing a smaller or no payoff, which would violate the stated condition of both players being rational. A nonsensical game doesn't have to make sense.
I did reply to his point. He spoke about nutrition science. That field has it's own problems that psychologists don't have to deal with. It's a bad example if he wanted to make the point you think he wanted to make.
Well fair enough. His use of nutrition science as an example was probably poorly chosen.
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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.