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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.
So, I wanted to give you an opportunity to suggest your own name and conceptual meaning for the number that Alice puts into the computer in Variant 2 before I gave my own take on it. I think at this point, you've had an opportunity and have not taken it, so here goes.
I cannot think of anything that is appropriate to call this number (the number that Alice puts into the computer in Variant 2) other than some form of "the probability that Bob observes an outcome". Alice has to be reasoning about Bob's observation function when she computes this number. It's pretty obvious, because if we keep everything else the same, but fiddle with Bob's observation function, then we see a corresponding change in Alice's computation and the resulting value of this number (and no other change in Alice's reasoning or behavior, as evidenced by her own bets).
As such, if I can put it in a pointed way, why would one think that Alice is smart enough and capable of distinguishing between "the probability that Bob observes an outcome" and "the probability of the coin flip, itself"... but is too stupid to distinguish between "the probability that I, Alice, observe an outcome" and "the probability of the coin flip, itself"?
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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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