I take SB's expected value to be a function of both her credence and of the payout structure. — Pierre-Normand

So then what is this calculation using pounds and pence?

I, however, don't take her credence to be a well defined value in the original Sleeping Beauty problem due to an inherent ambiguity in resolving what kind of "event" is implicitly being made reference to in defining her "current epistemic situation" whenever she awakens. — Pierre-Normand

I disagree that there's any ambiguity.

This is an "$n$-sided die rolled a 1" awakening if and only if this is an "$n$-sided die rolled a 1" experiment.

Therefore, a rational person's credence that this is an "$n$-sided die rolled a 1" awakening must equal their credence that this is an "$n$-sided die rolled a 1" experiment, because anyone who claims both of these isn't rational:

1. I believe it most likely that I am in a "die rolled a 1" awakening
2. I believe it most likely that I am not in a "die rolled a 1" experiment

And anyone who claims both of these isn't rational:

1. If I get to place a new bet each time I wake up then I believe it most likely that the die rolled a 1
2. If I only get to change my original bet each time I wake up then I believe it most likely that the die did not roll a 1

(on this point: what is your credence if you don't know the payout structure or if you don't get to place a bet at all?)

I really just think that there's only one correct answer, regardless of "interpretation" or payout structure (whether or not there even is is one, and whether or not it is known), and that answer is most obvious when we consider the 2¹⁰⁰-sided die; a rational person's credence is $1\over{2^{100}}$, never $2\over3$.

Therefore, a rational person's credence that this is an "n

-sided die rolled a 1" awakening must equal their credence that this is an "n

-sided die rolled a 1" experiment, because anyone who claims both of these isn't rational:

1. I believe it most likely that I am in a "die rolled a 1" awakening
2. I believe it most likely that I am not in a "die rolled a 1" experiment

And anyone who claims both of these isn't rational — Michael

It's precisely due to the usual one-to-one mapping I mentioned in my previous post that believing both those propositions at the same time usually is irrational. It is precisely what makes the Sleeping Beauty problem so interesting (and makes it contested between smart people in the literature to this day) that this usual one-to-one mapping doesn't hold in the proposed scenario. In order to display how the first proposition, as asserted by Thirders is consistent with the second one, as asserted by a Halfers, it is necessary to explicitly disambiguate the two different uses of the word "likely" that they are employing:

(1) I believe that awakening episodes such a the one I am currently experiencing turn out to be (i.e. are expected by me to be) 1-awakenings (i.e. awakening episodes that have been spawned by a die landing on "1") two thirds of the time on typical experimental runs.

(2) I believe that experimental runs such a the one I am currently being awakened into typically turn out not to be (i.e. are expected by me not to be) 1-runs (i.e. runs that have not been triggered by a die landing on "1") (2^(100)-1)/2^100)% of the time.

Even though the biconditional proposition "this is a 1-awakening if and only if this is a 1-run" holds true, the many-to-one mapping of 1-awakenings to 1-runs accounts for the divergence of the two different ratios expressed above.

(1) I believe that awakening episodes such a the one I am currently experiencing turn out to be (i.e. are expected by me to be) 1-awakenings (i.e. awakening episodes that have been spawned by a die landing on "1") two thirds of the time on typical experimental runs. — Pierre-Normand

Which is true.

But I believe the step from this to "therefore, my credence that this is a 1-awakening is $2\over3$" is a non sequitur.

This extreme example is a reductio ad absurdum that shows that given the peculiarities of the Sleeping Beauty experiment it is irrational for one's credence to be determined by the long-term average ratio of awakenings.

So it doesn't matter that (1) is true. Any rational person being subject to this experiment should be almost certain that a) the die didn't roll a 1, and so that b) this is not a 1-awakening.

I don't know if it's quite comparable but it reminds me of the Boltzmann brain paradox. Although the probability of a Boltzmann brain forming is vanishingly small, given sufficient time they would dwarf the number of "real" brains.

Should I then believe that I am most likely a Boltzmann brain?

Although perhaps this is best comparable to the variation where both a) the experiment is repeated 2¹⁰¹ times and b) I am made to forget which experiment I'm on. I think only then could the long-term average ratio of awakenings factor into my credence.