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LimberLarry comments on A simple game that has no solution - Less Wrong Discussion
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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.
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?
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.
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.
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.