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HughRistik comments on Beauty quips, "I'd shut up and multiply!" - Less Wrong
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This was also my understanding of the problem. Are we missing something?
On awakening, I would give:
p(heads) and p(tails) on Monday should be equal (a fair coin was flipped). p(tails) on Monday and p(tails) on Tuesday should also be equal (nothing important changes in the interim).
Even though you knew ahead of time that there was a 50% chance you'd be on the heads path, and a 50% chance you'd be on the tails path, you'd shift those around without probability law justification?
I also think you are not careful with your wording. What does p(heads) on Monday mean? Is it a joint or conditional probability? p(heads | monday) = p(tails | monday), yes, but Beauty can't condition on Monday since she doesn't know the day. If you are talking about joint probabilities, p(heads and monday) does not equal p(tails and monday).
Re: a 50% chance you'd be on the heads path, and a 50% chance you'd be on the tails path.
Those are not the probabilities in advance of the experiment being perfomed. Once the experimental procedure is known the subjective probabilites for Beauty on awakening are 33% for heads and 67% for tails. These probabilities do not change during the experiment - since Beauty learns nothing.
"Once the experimental procedure is known the subjective probabilites for Beauty on awakening are 33% for heads and 67% for tails."
Suppose 50% of the population has some asymptomatic form of cancer. We randomly select someone and do a diagnostic test. If they have cancer (we don't tell them), we wake them up 9 times and ask their credence for cancer (administering amnesia-inducing drug each time). If they don't have cancer, we wake them up once.
The person selected for this experiment knows there is a 50% chance they have cancer. And they decide ahead of time that, upon awakening, they'll be 90% sure they have cancer. And this makes sense to you.
Re: "but Beauty can't condition on Monday since she doesn't know the day."
She could make a bet. You do not have to know what day of the week it is in order to make a bet that it is Monday.
Re: "If you are talking about joint probabilities, p(heads and monday) does not equal p(tails and monday)."
Sure it does - if a fair coin was flipped!
Maybe instead of just saying it's true, you could look at my proof and show me where I made a mistake. I've done that with yours.
I think you already clarified that here.
You interpreted:
"What is your credence now for the proposition that our coin landed heads?"
...as being equivalent a bet along these lines:
"the scenario where at each awakening we offer a bet where she'd lose $1.50 if heads and win $1 if tails, and we tell her that we will only accept whichever bet she made on the final interview."
...which is a tortured interpretation.
The question says "now". I think the correct corresponding wager is for Beauty to make a bet which is judged according to its truth value there and then - not for it to be interpreted later and the payout modified or cancelled as a result of other subsequent events.