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I phrased it a bit flippantly, but I do think that the original question is not ambiguously phrased. We do not say that people who think that there is no advantage to switching doors in the Monty Hall problem are answering a different question than the people who say that there is an advantage to switching. We say they are wrong.
It depends on how it's phrased. If they are given the proper version of the Monty Hall problem, then 1/2 is wrong. But if the problem description is sloppy and underspecified then it's legitimately ambiguous and they ARE answering a different question (The Monty Fall problem) correctly. Half the confusion with the Monty Hall problem is that midwits who are trying to be clever but don't fully understand the logic give an underspecified version of the problem half the time and don't notice, or do it deliberately to invite ambiguity so they get opportunities to smugly correct people.
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Verily, in the Monty Hall problem. There, you actually do have a very very clear moment where information is gained and there is no ambiguity about which question you are being asked. But in this problem, if Alice tells Bob what you seem to want to have her tell him, we would say that she is wrong. We'd even say that she's extra wrong if she said she "updated".
Sorry, my comment was ambiguously phrased. I was referring to the cannonical form of the Sleeping Beauty question from Wikipedia:
This question is not ambiguous. The correct answer is 1/3. If you ran this experiment on people who think the answer is 1/2 you could take their money.
Are you taking Alice's money or Bob's money? From what I can see, they've got a nice system set up that's not letting you take their money, but it's not the case that the only number involved in their system is 1/3.
I think there's a sort of Monty Hall-style switcharoo going on with regards to what Alice puts into the computer. Only what she puts in the computer on Monday matters, so she should put the probability conditional on it being Monday into the computer, but bet her true probability (since she doesn't know whether or not it is Monday) herself.
You're being shifty with your language, though. First, you have an underspecified "probability conditional on it being Monday". I'm being a bit picky with this one, but please fully state "probability of ______" even if you think it's "...conditional on it being Monday". Second, you have that she is supposed to bet her "true probability", but what do you mean by "true" probability? This phrase is not defined. "True probability" of what?
Notice further that there are three variants. She doesn't always put the same number in the computer in each variant. How does that work? What name would you call the number that Alice puts into the computer in Variant 2, for example?
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