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.

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. — Michael

That's because the "rational person" that you are picturing is thinking about the Halfer ratio expressed in the second proposition as the definition of their credence. And it would indeed be a non-sequitur to argue from the truth of (1) that (2) is false. But I am not doing that.

Since (1), just like (2), merely is a definition of SB's credence, as such, it is neither rational not irrational to believe in them. What it is that Halfers or Thirders go on to do or infer on the basis of such definitions can be qualified as being rational or irrational. But, given any payout structure, both (1) and (2), in addition to being logically compatible, recommend the exact same betting strategies in all cases.

I think your intuition that the credence expressed by (1) is irrational in the extreme case comes from you being focussed on the prior unlikelihood of 1-runs to the expense of considering the compensating effect of there being so many 1-awakenings in one single 1-run.

Imagine a lottery where 1,000,000 tickets are printed, numbered from 1 to 1,000,000. A fair coin it tossed and if the result is Heads, one random ticket is designated to wins $1.000.000, or else all tickets win $1. After the draw, the ticket validation process has two steps: one revealing if the ticket is a winner and the second one revealing the amount. Suppose you are a member of a team of 1,000,000 friends who all bought a ticket. You all bring your tickets together to the store and the clerk validates them all. You then ask the clerk to hand a winning ticket (irrespective of how many there are) to its initial buyer. A Halfer would reasonably infer that since no information was gained regarding the coin toss result, the winning amount is equally likely to be $1 or $1.000.000. I think this is the epistemic perspective you are focused on, where the winning/awakening event is being considered merely as part of a run.

Consider the different epistemic perspective of a ticket buyer who independently validates their own ticket (first validation step only). You find out that your ticket is a winner. Are the chances of your prize being $1,000,000 still 1/2? Obviously not. One million out of two-million winning tickets sold in such a lottery win $1 while only one of them wins $1.000.000, and your winning ticket just is one such tickets out of 1,000,001.

In a scenario similar to one @JeffJo suggested, the 1.000.000 tickets could be distributed to Sleeping Beauty, with one of them being assigned to her on each one of 1.000.000 consecutive days. On each day, she is awakened and enabled to find out if her ticket is winning. If not, she is put back to sleep. If it is winning, she is asked about her credence in Heads (or equivalently, that the ticket she currently holds is worth $1.000.000). A Thirder would argue quite sensibly that her credence is 1/1.000.001 since she is now in the exact same epistemic situation as a buyer of a single ticket who has just validated it as a winning ticket.