I think the monty hall problem very closely resembles a more natural one in which the probability is 1⁄2; namely, that where the host is your opponent and chose whether to offer you the chance to switch. So evolutionarily-optimized instincts tell us the probability is 1⁄2.
I do not think this is correct. First, the host should only offer you the chance to switch if you are winning, so the chance should be 0. Second, this example seems too contrived to be something that we would have evolved a good instinct about.
Unless they’re trying to trick you. The problem collapses to a yes or no question of whether one of you is able to guess the level the other one of you is thinking on
I’d probably broaden this beyond 1⁄2 - I think the base case is the host gives you a chance to gamble with a question or test of skill, and the result is purely dependent on the player. The swap-box scenario is then an extreme case of that where the result depends less and less on the skill of the player, eventually reaching 50% chance of winning. I wouldn’t say evolutionary-optimised, but maybe familiarity with the game-show tropes being somewhere along this scale.
Monty Hall is then a twist on this extreme case, which pattern-matches to the more common 50% case with no allowance for the effect of the host’s knowledge.
I think the monty hall problem very closely resembles a more natural one in which the probability is 1⁄2; namely, that where the host is your opponent and chose whether to offer you the chance to switch. So evolutionarily-optimized instincts tell us the probability is 1⁄2.
I’d say it’s that it closely resembles the one where the host has no idea which door has the car in it, and picks a door at random.
I do not think this is correct. First, the host should only offer you the chance to switch if you are winning, so the chance should be 0. Second, this example seems too contrived to be something that we would have evolved a good instinct about.
Unless they’re trying to trick you. The problem collapses to a yes or no question of whether one of you is able to guess the level the other one of you is thinking on
Um, no, the only Nash equilibria are where you never accept the deal. If you ever accept it at all, then they will only offer it when it hurts you.
I’d probably broaden this beyond 1⁄2 - I think the base case is the host gives you a chance to gamble with a question or test of skill, and the result is purely dependent on the player. The swap-box scenario is then an extreme case of that where the result depends less and less on the skill of the player, eventually reaching 50% chance of winning. I wouldn’t say evolutionary-optimised, but maybe familiarity with the game-show tropes being somewhere along this scale.
Monty Hall is then a twist on this extreme case, which pattern-matches to the more common 50% case with no allowance for the effect of the host’s knowledge.