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Chasing this tangent a bit -- I have never been happy with the Monty Hall problem as posed. To me the problem is that it is posed as a one-shot question. If yo

Chasing this tangent a bit -- I have never been happy with the Monty Hall problem as posed. To me the problem is that it is posed as a one-shot question. If you were in this actual situation, how do you know that Monty is not deliberately trying to make you lose? He could, for example, have just let you open the first door you picked, revealing the goat. But he chose to ask you to switch, then maybe that is a big hint that you picked the right door the first time? If the game is just "you will pick a door, he will reveal another door, and then you can choose to switch" then clearly the "usual" answer is correct; always switch because the only way you lost is if you guessed correctly the first time (1/3). But if the game is "try to find the car while the host tries to make you lose" then you should never switch. His ideal behavior is that if you pick the door with the goat then he gives you the goat; if you pick the door with the car then he tries to get you to switch.

wruza 2y ago
His ideal adversarial strategy becomes non-trivial when you know about it. It is very likely “just flip a coin to turn it back to 50/50” but may be something statistically sophisticated.
- andrewla 2y ago
  If his desire is for the contestant to lose, then he can't really do better (formally) than winning 2/3 of the time by simply opening the door that they choose. In practice, always opening a goat-door and always asking to switch for a car-door can do slightly better than 2/3 because some contestants, unaware of his strategy and objectives, might choose to switch. If his objective is more subtle -- increasing suspense or entertainment value or getting a kick out of people making a self-destructive choice or just deciding whether he likes a contestant -- then I'm not sure what the metrics are or what an optimal strategy would be in those cases. Given that his motives are opaque and given no history of games upon which to even inductively reason, I don't think you can reach any conclusion about whether switching is preferable. Given the spread of possibilities I would tend to default to 50/50 for switch/no-switch, but I don't have a formal justification for this.

wruza 2y ago

His ideal adversarial strategy becomes non-trivial when you know about it. It is very likely “just flip a coin to turn it back to 50/50” but may be something statistically sophisticated.

andrewla 2y ago
If his desire is for the contestant to lose, then he can't really do better (formally) than winning 2/3 of the time by simply opening the door that they choose. In practice, always opening a goat-door and always asking to switch for a car-door can do slightly better than 2/3 because some contestants, unaware of his strategy and objectives, might choose to switch. If his objective is more subtle -- increasing suspense or entertainment value or getting a kick out of people making a self-destructive choice or just deciding whether he likes a contestant -- then I'm not sure what the metrics are or what an optimal strategy would be in those cases. Given that his motives are opaque and given no history of games upon which to even inductively reason, I don't think you can reach any conclusion about whether switching is preferable. Given the spread of possibilities I would tend to default to 50/50 for switch/no-switch, but I don't have a formal justification for this.