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Comment by LargeWu | Hacker News Reader

The difference here is that the Monty Hall problem has an explanation that while counterintuitive, is statistically sound. You should always switch, because the probability you picked the correct door is locked in at the time you made the choice between 3 doors. It is 1/3 that you picked correctly, and 2/3 that you picked incorrectly. The counterintuitive part is that if you switch, you are effectively selecting all the doors you did not pick originally. It's your original door, vs the field.

The green hat problem hinges on subjective interpretations of the meaning of both "liar" and the different ways in which the liar's sentence may be false. It may be false because the liar owns many hats, none of which are green. Or they own many hats, only some of which are green. Or they own no hats. These are all reasonable interpretations of how the sentence might be false, and the answers presented are not necessarily mutually exclusive.

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The problem here is that the usual explanation sneaks in multiple rarely stated assumptions.

If Monty knows the door with the prize and is aiming for the game to continue, then you should switch. (This is the usual argument.)

If Monty doesn't know where the prize is, then you learned nothing. (Monty's result was luck, and he can't impart information that he doesn't have.)

If Monty knows where the prize is and wants you to lose, absolutely don't switch. (Monty will only drag the game out as a way to try to make you lose.)

The reasoning behind these statements is completely solid, and there are no hidden assumptions being snuck in.

I think, canonically, Monty always knows where the prize is, and will always eliminate all doors except for one, and will never eliminate a door with the prize, and will always give you a choice to switch. There's no room for Monty's motivation.

Those assumptions are required for the usual explanation. But they are very rarely stated. And so the usual explanation is not logically solid.

You can't just sneak in the assumptions. You have to state them somewhere.

the assumption is told to you in that the game is expected to continue after monty opens a door. if he could open the prize door, the game would describe what happens when he does that

No. That condition holds in all three of the scenarios that I stated. And yet you have 3 different answers. Switching 2/3 win, 1/2 win, and 0/1 win respectively.

You specifically need information that we haven't been given about the counterfactuals. What might Monty have done in other scenarios that we have not yet observed? We don't know. And we're not actually told. That makes that an implicit assumption that wasn't specified.

well, they are part of the original, canonical problem. it's not the original problem's fault if re-tellers omit key information.

They're not. It says he "opens another door, say No. 3, which has a goat". That could mean he deliberately chose a door with a goat or he chose one by some unstated process and it happened to have a goat by chance. It says he "knows what's behind the doors" but that statement means nothing because it doesn't say how he uses that knowledge, if at all. It's full of language ambiguity.

Last time I claimed this was the correct answer, I was linked to the Monty Crawl problem. https://www.probability.ca/jeff/writing/montyfall.pdf

That's fine, but given the parameters of the problem change with the Monty Crawl variant, it's not the same problem, and doesn't invalidate the answer of the base variant.

The problem is that the parameter was unspecified in the original problem. Your answer is equivalent to the shaky answer referred to in the paper. Without knowing that the host selects between 2 goats with 50/50 chance, you cannot give a general answer.

I don't follow, which parameter was unspecified in the original problem?

My last sentence. Or which parameter do you think was modified?

The host might be selecting between two goats, or might be selecting between a goat and a car. Either way, it doesn't matter because we don't get any additional information about whether our original choice was correct. (To clarify, this applies to the original problem, not the Crawl variant, where we either sometimes get definitive information, or sometimes get no information)

Edit: Furthermore, I don't think that author's solution to the Crawl problem is correct. When the host eliminates a door, either you will get information that says you should switch, and you'll win 100% of the time; or you won't get information, and you should still always switch and win 2/3 of the time.

> it doesn't matter because we don't get any additional information about whether our original choice was correct

That's the missing assumption. I would say assuming that people are perfectly random falls into the "standardized test" category.

> you won't get information, and you should still always switch and win 2/3 of the time.

You always get some information, the set of possible results becomes narrower, so saying the probabilities don't change is not sufficient. Not a good idea to discuss the problem in informal language though.

Years ago, the Monty Fall variation was mentioned on a local telnet forum I visited. The "consensus" of the thread was the solution of Monty Fall is the same as that of Monty Hall: switching gives you 2/3 chance to win.

That was when I realized many people just memorize the Monty Hall's solution without understanding it, a.k.a. "standardized tests".

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