The part to note is that almost all of this is a red herring. Its irrelevant if she remembers or not. Its just word play to get us out of the raw math. The odds are still the same.

Flip heads, 1 result
Flip tails, 2 results

Put the pile of results as total possible outcomes. You have 3 possible outcomes. In two of the outcomes, tails was flipped. Put it in a hat and draw one. You have a 2/3rd chance that its a tails outcome.

To be clear, it is a 50/50 shot as to whether heads or tails is picked. Meaning that both are equally like to occur. But since we have more outcomes on tails, and we're looking at the probability of what already happened based on outcomes, not prediction of what will happen, its a 2/3rds chance for tails. — Philosophim

The issue with her remembering or not is that if, as part of the protocol, she could remember her Monday awakening when the coin landed tails and she is being awakened again on Tuesday, she would be able to deduce that the coin landed Tails with certainty and, when she couldn't remember it, she could deduce with certainty that "today" is Monday (and that the probability of Tails is 1/2). That would be a different problem, and no problem at all.

Your argument in favor of the Thirder credence that the coin landed Tails (2/3) relies on labeling the awakening episodes "the outcomes". But what is it that prevents Halfers from labelling the experimental runs "the outcomes" instead? Your ball picking analogy also has been produced by Berry Groisman to illustrate this ambiguity in his The end of Sleeping Beauty's Nightmare paper (although I don't fully agree with his conclusions).

The issue with her remembering or not is that if, as part of the protocol, she could remember her Monday awakening when the coin landed tails and she is being awakened again on Tuesday, she would be able to deduce that the coin landed Tails with certainty and, when she couldn't remember it, she could deduce with certainty that "today" is Monday — Pierre-Normand

Correct. My point was that its just used as a word problem way of saying, "We have 3 outcomes we reach into a hat and pull from"

Your argument in favor of the Thirder credence that the coin landed Tails (2/3) relies on labeling the awakening episodes "the outcomes". But what is it that prevents Halfers from labelling the experimental runs "the outcomes" instead? — Pierre-Normand

Because there are two different outcomes. One with one day, and one with two days. If you pick any day and have no clue if its a day that resulted from a heads or tails outcome, its a 2/3rds chance its the tails outcome. The heads and tails is also irrelevant. The math is, "Its as equally likely that we could have a series of one day or two day back to back in this week. If you pick a day and you don't know the outcome or the day, what's the odds its a tails day vs a heads day?"

The odds of whether its head or tails is irrelevant since they are the same and can be effectively removed from the problem.

Correct. My point was that its just used as a word problem way of saying, "We have 3 outcomes we reach into a hat and pull from" — Philosophim

You are using the word "outcome" ambiguously and inconsistently. In your previous post you had stated that "You have 3 possible outcomes. In two of the outcomes, tails was flipped."

And now you are saying that:

Because there are two different outcomes. One with one day, and one with two days. If you pick any day and have no clue if its a day that resulted from a heads or tails outcome, its a 2/3rds chance its the tails outcome. The heads and tails is also irrelevant. The math is, "Its as equally likely that we could have a series of one day or two day back to back in this week. If you pick a day and you don't know the outcome or the day, what's the odds its a tails day vs a heads day?"

The odds of whether its head or tails is irrelevant since they are the same and can be effectively removed from the problem.

So, now you are back to treating experimental runs rather than awakening episodes as the "outcomes". This sort of ambiguity indeed is the root cause of the misunderstanding that befalls Halfers and Thirders in their dispute.

When Sleeping Beauty is being asked, on one particular awakening occasion, what her credence is that the coin landed Tails, she must ponder over what the odds are that the epistemic situation she currently is in (given the information available to her) is such that the coin landed Tails when she is in that situation. In other words, she takes herself to be experiencing one among a range of possible and undistinguishable (from her current point of view) events (or "outcomes") such that a proportion P of them occur when the coin landed Tails, in the long run. All of this leaves it undefined what the events or "outcomes" are that we're talking about.

Thirders interpret those outcomes as awakening episodes and Halfers interpret them as experimental runs. Their expressed credences, 2/3 and 1/2 respectively, therefore are answers to different questions (or to the same question differently disambiguated, if you will).

Thirder Sleeping Beauty expects, reasonably enough, that in the long run awakening episodes like the one she is currently experiencing will turn out to have occurred when the coin had landed Tails two thirds of the times.

Halfer Sleeping Beauty expects, equally reasonably, that, in the long run, experimental runs like the one she is currently experiencing (regardless of how many more times she already was or will be awakened during that run) will turn out to have occurred when the coin had landed Tails one half of the times.

Credences implicitly are about ratios. Halfers and Thirders disagree about the denominator that is meant to figure in the relevant ratio.

So, now you are back to treating experimental runs rather than awakening runs as the "outcomes". This sort of ambiguity indeed is the root cause of the misunderstanding that befalls Halfers and Thirders in their dispute. — Pierre-Normand

I'm not seeing the ambiguity here, but maybe I'm not communicating clearly. There are two outcomes based on context.

1 Outcome of the coin
2. Outcome from the result of the coin

The outcome from the coin is always 50/50. Because of this, you can ignore the outcome and simply say, "Its equally likely that the result of one coin flip will happen as the other result"

When we look at all the outcomes from the result of the coin, we see that if its tails, there are two days, and only 1 day if heads. So if you don't know which day it is, (previous days can be removed as we 'forget') then it is a 1/3 chance that your day is a heads, and 2/3rd chance that your day is a tails. Is it equally likely that a heads or a tails was flipped in isolation from the days? Yes. But since we have different outcomes from the flip itself, we have to take them all in consideration and realize that if either side is equally likely to occur, it is a much better odds to say that its heads because the consequence of it being heads makes it twice as likely you awake two days.

One more way to think about it is you have a hat that you could either reach in one time, or two times. Each is equally likely to occur. You forget how many times you've reached into the hat. Is it more likely that you're on an outcome where you reach in two times, or one time? Obviously its two times.

I'm not seeing the ambiguity here, but maybe I'm not communicating clearly. There are two outcomes based on context. — Philosophim

I assume what you now mean to say is that there are two possible ways to think of the "outcomes" based on context. Well, sure, that's pretty much what I have been saying. But I'm also arguing that the original Sleeping Beauty problem fails to furnish the relevant context.

If we think of the experimental runs following the coin toss as the "R-outcomes" (an equal amount of T-runs and H-runs are expected) and the awakening episodes as the "A-outcomes" (twice as many T-awakenings as H-awakenings are expected), then we've resolved part of the ambiguity. But Sleeping Beauty isn't being asked about specific kinds of outcomes explicitly. Rather she is being asked about her credence regarding the current state of the coin. She can reason that the current state of the coin is Tails if and only if she is currently experiencing a T-awakening and hence that the current state of the coin is twice as likely to be Tails than it is to be Heads. But she can also reason that the current state of the coin is Tails if and only if she is currently experiencing a T-run and hence that the current state of the coin is equally as likely to be Tails than it is to be Heads.

Another way to state the Halfer interpretation that makes it intuitive is to suppose Sleeping Beauty will be given a bag of Twizzlers (T-candy) at the end of the experiment if the coin landed Tails and a bag of Hershey's Kisses (H-candy) if it landed Heads. The fact that she's awakened twice rather than once when she's scheduled to receive Twizzlers doesn't make it more likely that she will receive them at the end of the run. Hence her credence remains 1/2 that she will receive Twizzlers. This is consistent with her credence being 2/3 that her current awakening episode (A-outcome) is twice as likely to be one that puts her on a path towards getting Twizzlers. But since the Twizzlers reward is an outcome that is being realized if and only if she is currently experiencing a T-awakening, she can sensibly reason that the odds of that are 1/2 also.

The key to understanding the consistency between the two apparently contradictory credences regarding the very same coin toss result is to realize that the two T-awakening outcomes occur in the same timeline and hence them being more frequent than H-awakenings doesn't increase the relative frequency of the Twizzlers rewards (or of her having experienced a T-run, regardless of how many times she was awakened in this run).

But Sleeping Beauty isn't being asked about specific kinds of outcomes explicitly. Rather she is being asked about her credence regarding the current state of the coin. She can reason that the current state of the coin is Tails if and only if she is currently experiencing a T-awakening and hence that the current state of the coin is twice as likely to be Tails than it is to be Heads. But she can also reason that the current state of the coin is Tails if and only if she is currently experiencing a T-run and hence that the current state of the coin is equally as likely to be Tails than it is to be Heads. — Pierre-Normand

She can reason that its equally likely that the result of the coin flip is 50/50, but that doesn't mean its likely that the day she is awake is 50/50. Lets flip it on its head and note how the likelihood that she would be wrong.

If she always guesses heads, she's wrong twice if its tails. If she always guesses tails, she's only wrong once. Thus, she is twice as likely to be wrong if she guesses heads on any particular day woken up, and twice as likely to guess correctly if she guesses tails. If the total odds of guessing correctly were 50/50, then she would have an equal chance of guessing correctly. She does not.

We can see this by extending the days out. Lets say that if its heads, she's woken up one day. If its tails she's woken up 10 days. Again, its better odds to guess tails, despite the outcome of the coin being equal as mentioned above.

As I've been noting, because the probable outcome of the coin is the same, you can effectively remove the probable outcome of the coin from the equation. r = r can be divided out. Lets change up the odds of the coin flip, and it changes the our guess.

Going back to the original setup, lets Say that its twice as likely that the coin lands on heads. Now the probability of the coin becomes essential to the outcome of the follow up outcomes.

So its two times more likely that its heads, but only one day. 1 times as likely for two days. This appears to equate to 50/50 when considering what day is randomly picked out of the outcomes.

Now, there is absolutely zero doubt that in the original case it is not a 50/50 outcome. However, I do doubt if it ends up being 2/3rds. There may be some multiplicative statistical shenanigans needed here that I'm not aware of that is being disguised by making the heads/tails outcome equal and eliminating it as a consideration. Perhaps the imaginary case I made above would be more revealing of any special case. Thus I may be wrong that 2/3rds is the answer, but there is no question that 1/2 is wrong.

She can reason that its equally likely that the result of the coin flip is 50/50, but that doesn't mean its likely that the day she is awake is 50/50. — Philosophim

Sure, but the former precisely is what she is being asked. She is being asked what her credence about the coin will be on that occasion, and not what the proportion of such occasions are that are T-occasions. One can argue than she is being asked this implicitly, but in that case it's still open to interpretation what those "occasions" are meant to be, as we've already discussed.

Lets flip it on its head and note how the likelihood that she would be wrong.

If she always guesses heads, she's wrong twice if its tails. If she always guesses tails, she's only wrong once. Thus, she is twice as likely to be wrong if she guesses heads on any particular day woken up, and twice as likely to guess correctly if she guesses tails. If the total odds of guessing correctly were 50/50, then she would have an equal chance of guessing correctly. She does not.

That's right, and this is a good argument favoring the Thirder position but it relies on explicitly introducing
a scoring procedure that scores each occasion that she has to express her credence: once for each awakening episode. If you would rather score those statements only once per run, regardless of how many times she is being asked about her credence in that run, then she would be right half the times. This also makes sense if you view all of the separate awakening episodes occurring during a single Tails run to be part of the same "outcome" (as you've indeed yourself favored doing earlier).

Sure, but the former precisely is what she is being asked. She is being asked what her credence about the coin will be on that occasion, and not what the proportion of such occasions are that are T-occasions. — Pierre-Normand

I would say she is being asked what the odds are of it being a day in which a T side vs a H side coins is flipped. If she's only being asked what the percent chance of the coin ended up being at, the answer is always 50/50. The odds of the coin flip result don't change whether its 1 or 1,000,000 days. What changes is from the result of that coin flip, and that is the pertinent data that is important to get an accurate answer.

This is very similar to the old Monty Hall problem. You know the three doors, make a guess, then you get to make another guess do you stay or change?

On the first guess, its always a 1/3 shot of getting the door wrong. But it can also be seen as a 2/3 chance of getting the door wrong. When given another chance, you simply look at your first set of odds and realize you were more likely than not wrong, so you change your answer. The result matches the odds.

Same with the situation here. Run this experiment 100 times and have the person guess heads 50 times, then tails 50 times. The person who guesses tails every time 50 times will be right 2/3rds of the time more than the first. Since outcomes ultimately determine if we are correct in our odds, we can be confident that 1/2 odds is incorrect.

By the way, very nice discussion! I appreciate your insight and challenging me to view things I might not have considered.

I would say she is being asked what the odds are of it being a day in which a T side vs a H side coins is flipped. — Philosophim

I was talking about what she is being asked, literally, in the original formulation of the problem discussed in the OP. From Wikipedia:

This has become the canonical form of the problem: [...] During the interview Sleeping Beauty is asked: "What is your credence now for the proposition that the coin landed heads?"

Your reading is a sensible interpretation of this question, but it isn't the only one.

If she's only being asked what the percent chance of the coin ended up being at, the answer is always 50/50. The odds of the coin flip result don't change whether its 1 or 1,000,000 days. What changes is from the result of that coin flip, and that is the pertinent data that is important to get an accurate answer.

Yes, quite, but now with the phrase "the result of that coin flip" you still seem to be gesturing to the "occasions" being generated by those coin flip results. And how is it that those events are meant to be individuated (one occasion per run or one per awakening) is what's at issue.

This is very similar to the old Monty Hall problem. You know the three doors, make a guess, then you get to make another guess do you stay or change?

On the first guess, its always a 1/3 shot of getting the door wrong(sic). But it can also be seen as a 2/3 chance of getting the door wrong. When given another chance, you simply look at your first set of odds and realize you were more likely than not wrong, so you change your answer. The result matches the odds.

That's not quite how the Monty Hall problem is setup so I'm unsure about the analogy your intending to make. The contestant has a 1/3 chance of getting the price-hiding door right on their first try. But the game host then reveals among the two other doors one that they (i.e. the host) know to be hiding a goat. The contestant then is being offered an opportunity to switch their initial choice for the other unopened door. The correct reasoning is that since contestants that switch their choices will win the prize on each occasion where their first choice was wrong, they have a 2/3 change of winning the prize if they switch their choice.

Same with the situation here. Run this experiment 100 times and have the person guess heads 50 times, then tails 50 times. The person who guesses tails every time 50 times will be right 2/3rds of the time more than the first. Since outcomes ultimately determine if we are correct in our odds, we can be confident that 1/2 odds is incorrect.

Yes, the person who guesses that the coin landed Tails will turn out to have made a correct guess two thirds of the times on average, thereby matching the Thirder credence. But the Halfer argues that this is just because they were able to make more guesses during T-runs. Under the Thirder interpretation of the meaning of SB's credence, though, her being afforded more opportunities to express her credence during T-runs doesn't make those runs more likely to happen, and this is what to them matter.

By the way, very nice discussion! I appreciate your insight and challenging me to view things I might not have considered.

Cheers!

On the occasion of an awakening, what is Sleeping Beauty's expectation that when the experiment is over ... — Pierre-Normand

This is what invalidates your variation. She is asked during the experiment, not before or after. Nobody contests what her answer should be before or after. And you have not justified why her answer inside the experiment should be the same as outside.

The issue with her remembering or not is that if, as part of the protocol, she could remember her Monday awakening when the coin landed tails and she is being awakened again on Tuesday, she would be able to deduce that the coin landed Tails with certainty and, when she couldn't remember it, she could deduce with certainty that "today" is Monday (and that the probability of Tails is 1/2). That would be a different problem, and no problem at all. — Pierre-Normand

Gee, if she is given a different set of information (knowing that it is not Tuesday, after Heads is information, as is all of these) produces different conditional probabilities. Possibly certainties. And?

Here's one more attempt. It's really the same thing that you keep dodging by changing the timing of the question, and claiming that I have "vallid thirder logic" while ignoring that it proves the halfer logic to be inconsistent.

• Get three opaque note cards.
• On one side of different cards, write "Monday and Heads," "Monday and Tails," and "Tuesday and Tails.
• Turn the cards over, shuffle them around, and write "A," "B," and "C" on the opposite sides.
• Before waking SB on the day(s) she is to be woken, put the appropriate card in the table in her room, with the letter side face up.

Let's say she sees the letter "B." She knows, as a Mathematical fact, that there was a 1/3 probability that "B" was assigned to the card with "Heads" written on the other side. And a 2/3 chance for "Tails."

By halfer logic, while her credence that "Heads" is written on the "B" card must be 1/3, her credence that the coin landed on Heads is 1/2. This is a contradiction - these two statements represent the same path to her current state of knowledge, regardless of what day it is.

Your justification for considering the halfer logic is that there may be a different path that _includes_ her her current state of knowledge and another that has a different day written on the card. My reason for rejecting that justification outright, besides the contradiction that it produces, is that this other day is not a path of the the path she sees.

On the occasion of an awakening, what is Sleeping Beauty's expectation that when the experiment is over ...
— Pierre-Normand

This is what invalidates your variation. She is asked during the experiment, not before or after. Nobody contests what her answer should be before or after. And you have not justified why her answer inside the experiment should be the same as outside. — JeffJo

It looks like you didn't parse correctly the sentence fragment that you quoted. It is indeed on the occasion of an awakening (as I said) that she is being asked about her credence regarding the coin, not later. I did make reference in the question to the end-off-run verification conditions of the credence statement that SB is asked to express. The reason this reference is made (to the future verification conditions) is to disambiguate the sense of the question, in accordance with the Halfer interpretation in this case. But the question still is about her credence "now".

Compare: (1) "What are the chances, now, that your lottery ticket is the winning ticket?" and (2) "What are the chances, now, that your lottery ticket has the number that will be drawn as the winning number?" It's the exact same question and the odds are the same (one in a million, say). The second question merely makes explicit what "winning" means.

Here's one more attempt. It's really the same thing that you keep dodging by changing the timing of the question, and claiming that I have "vallid thirder logic" while ignoring that it proves the halfer logic to be inconsistent.

Get three opaque note cards.
On one side of different cards, write "Monday and Heads," "Monday and Tails," and "Tuesday and Tails.
Turn the cards over, shuffle them around, and write "A," "B," and "C" on the opposite sides.
Before waking SB on the day(s) she is to be woken, put the appropriate card in the table in her room, with the letter side face up.

Let's say she sees the letter "B." She knows, as a Mathematical fact, that there was a 1/3 probability that "B" was assigned to the card with "Heads" written on the other side. And a 2/3 chance for "Tails."

By halfer logic, while her credence that "Heads" is written on the "B" card must be 1/3, her credence that the coin landed on Heads is 1/2. This is a contradiction - these two statements represent the same path to her current state of knowledge, regardless of what day it is.

The Halfer logic is to reason that although T-runs, unlike H-runs, are such that SB will be presented with a card two different times (one on Monday and one on Tuesday), nevertheless, on each occasion where the experiment is performed and she finds herself being involved in an individual experimental run, the likelihood that this run is a T-run is 1/2. That's because in the long run there are as many T-runs as there are H-runs. By that logic she can also reason that, on an particular occasion where she is being awakened, the chances that she is holding the "Monday and Heads" card is 1/2. The fact that she now finds out her card to be labelled "B" doesn't alter those odds since the labelling procedure is probabilistically independent from the coin toss result and hence this result conveys no information to her regarding the coin toss result.

By the way, for the same reason, it would not convey any information to her either from a Thirder perspective. Her credence that the coin landed Tails would remain 2/3 before and after she saw the "B" label on her card.

Remember: SB isn't betting on the card (neither is she betting on the current awakening episode). She's betting on the current coin toss outcome. How those outcomes must be considered to map to her possible ways to experience them (in separate awakenings or separate runs) is a matter of interpretation that isn't spelled out in the original SB problem. It's true, though, that under the Thirder interpretation, betting on the wakening episodes or betting on the cards is equivalent. But that's just because the cards and the awakening episodes are mapped one-to-one.

The reason this reference is made (to the future verification conditions) is to disambiguate the sense of the question, ... — Pierre-Normand

Perhaps you didn't parse correctly. There is no ambiguity. If she is asked to project her state of knowledge on Wednesday, or to recall it from Sunday, of course the answer is 1/2.

Where, exactly, do you think projection or recollection is implied or stated? You are forcing this issue into a place where it does not belong, in order to justify saying that 1/2 is a possible answer.

Remember: SB isn't betting on the card (neither is she betting on the current awakening episode). She's betting on the current coin toss outcome.

I keep looking at the problem, and I can't find a reference to betting anywhere. The reason I don't like using betting is because anybody can re-define how and when the bet is made and/or credited, in order to justify the answer they like. One is correct, and one is wrong.

They ask her one question after each time she awakens, however: What is the probability that the coin shows heads.

Do you see the words "landed when it was flipped" here? No? How about "will be showing after the experiment ends"? Still no? Then stop inserting them. What is the probability that the coin is showing "Heads"? This is in the present tense.

So, if a bet were to exist, and assuming she uses the same reasoning each time? She risks her $1 during the interview, and is credited her winnings then also. If she bets $1 on Heads with 2:1 odds, she gains $2 if the coin landed Heads, and loses 2*$1 if it landed on Tails. If she bets on Tails with 1:2 odds, she loses $1 if the coin landed Heads, and gains 2*$0.50=$1 if it landed Tails.

But if she bets $1 on Heads with 1:1 odds, she gains $1 if the coin landed Heads, and loses 2*$1=$2 if it landed on Tails. If she bets on Tails with 1:1 odds, she loses $1 if the coin landed Heads, and gains 2*$1=$2 if it landed Tails.

The answer, to the question that was asked and not what you want it to be, is 1/3.

Her Thirder-credence would then be pragmatically relevant to selecting the destination most likely to afford her a sunny trip. — Pierre-Normand

Again, there's not much sense in this so-called "pragmatically relevant" credence. Even before being put to sleep – and even before the die is rolled – I know both that the die is most likely to not land on a 6 and that betting that it did will offer the greater expected return in the long run. So after waking up I can – and will – continue to know that the die most likely did not land on a 6 and that betting that it did will offer the greater expected return in the long run, and so I will bet against my credence.

With respect to "pragmatic relevance", Thirder reasoning is unnecessary, so if there's any sense in it it must be somewhere else.

Under the Thirder interpretation, all three of those biconditionally related "experienced" events are actual on average 2/3 of the times that SB is experiencing a typical awakening episode. — Pierre-Normand

My argument is that a rational person should not – and would not – reason this way when considering their credence, and this is most obvious when I am woken up 2¹⁰¹ times if the coin lands heads 100 times in a row (or once if it doesn't).

It is true that if this experiment were to be repeated 2¹⁰¹ times then we could expect $2\over3$ of all awakenings to occur after the coin landed heads every time, but it's also irrelevant. The experiment is only performed once. I strongly believe that it is irrational for one's credence to consider this long term average; a rational person, after waking up and knowing that the experiment is only performed once, will only consider the sheer improbability of the coin landing heads every time. Their credence remains ${1\over{2^{100}}}$. There is no ambiguity in the question or the answer.

Thirder reasoning only has its place, if it has a place at all, if both a) the experiment is repeated 2¹⁰¹ times and b) Sleeping Beauty is also made to forget between experiments. It matters that the problem does not stipulate these two conditions.

Perhaps you didn't parse correctly. There is no ambiguity. If she is asked to project her state of knowledge on Wednesday, or to recall it from Sunday, of course the answer is 1/2. — JeffJo

I was explicitly referring to her state of knowledge at the time when the interview occurs. There is no projection of this state into the future. Likewise, when you buy a lottery ticket and express your credence that it is the winning ticket as one in one million, say, what you mean is that there is a one in one million chance that, when the winning number will be drawn (or will be revealed, if it has already been drawn), your ticket will be the winner. You're not projecting your state of knowledge into the future. You're merely stating the conditions of verification regarding what your present state of knowledge (i.e. your credence) is about.

I keep looking at the problem, and I can't find a reference to betting anywhere. The reason I don't like using betting is because anybody can re-define how and when the bet is made and/or credited, in order to justify the answer they like. One is correct, and one is wrong.

Establishing a betting protocol with a well defined payout structure enable SB to put her money where her mouth is, and also to clarifies what it is that her stated credence is about. It highlights the tension between saying that you have a high credence that some outcome is true but that you wouldn't bet on it. Once you acknowledge that it is rational to make an even money bet on an outcome that you believe to be more likely to occur (or to be actual) than not, then the specification of the payout structure help clarify what the stated credence is about (i.e. what it is exactly that you take to be most likely to occur, or to be actual). This indeed goes beyond the original statement of the problem, but since it is precisely my contention that the original statement is ambiguous, it's a useful way to highlight the ambiguity.

So, if a bet were to exist, and assuming she uses the same reasoning each time? She risks her $1 during the interview, and is credited her winnings then also. If she bets $1 on Heads with 2:1 odds, she gains $2 if the coin landed Heads, and loses 2*$1 if it landed on Tails. If she bets on Tails with 1:2 odds, she loses $1 if the coin landed Heads, and gains 2*$0.50=$1 if it landed Tails.

But if she bets $1 on Heads with 1:1 odds, she gains $1 if the coin landed Heads, and loses 2*$1=$2 if it landed on Tails. If she bets on Tails with 1:1 odds, she loses $1 if the coin landed Heads, and gains 2*$1=$2 if it landed Tails.

The answer, to the question that was asked and not what you want it to be, is 1/3.

Indeed, such a payout structure clarifies what the bettor means when they express a 2/3 credence that the the coin landed tails. They mean that the epistemic situations that they find themselves in when awakened are T-awakenings (as opposed to H-awakenings) two thirds of the time. A different payout structure that rewards the bettor only once when they place a winning bets during an experimental run clarifies what the bettor means when they express a 1/2 credence that the the coin landed tails. They mean that the epistemic situations that they find themselves in when awakened are T-runs (as opposed to H-runs) one half of the times. As @Michael correctly argues in defence of a Halfer interpretation, merely being afforded more opportunities to bet on a given ticket (outcome) doesn't make it any more likely that this ticket is the winning ticket (or that the outcome is actual).

Again, there's not much sense in this so-called "pragmatically relevant" credence. Even before being put to sleep – and even before the die is rolled – I know both that the die is most likely to not land on a 6 and that betting that it did will offer the greater expected return in the long run. So after waking up I can – and will – continue to know that the die most likely did not land on a 6 and that betting that it did will offer the greater expected return in the long run, and so I will bet against my credence.

With respect to "pragmatic relevance", Thirder reasoning is unnecessary, so if there's any sense in it it must be somewhere else. — Michael

It is indeed somewhere else. Look at the payout structure that @JeffJo proposed in their previous post. Relative to this alternative payout structure, your own Halfer reasoning is unnecessary.

My argument is that a rational person should not – and would not – reason this way when considering their credence, and this is most obvious when I am woken up 2^101 times if the coin lands heads 100 times in a row (or once if it doesn't).

It is true that if this experiment were to be repeated 2^101 times then we could expect 2/3 of all awakenings to occur after the coin landed heads every time, but it's also irrelevant.

It's only irrelevant to the determination of your credence about the experimental run that you are experiencing (regarding what proportion of such runs are T-runs). Regarding the determination of your credence about the specific awakening episode that you are experiencing, though, it's rather the fact that T-runs and H-runs are equally frequent that is irrelevant. Taking the case to such wild extremes, though, makes your intuition about the comparative utility of betting on such unlikely outcomes (i.e. H-awakenings) relative to the utility of betting on the likeliest outcome (T-awakenings) play into your intuition about the rational credence. (Why would anyone risk a virtually guaranteed and useful $1 for an infinitesimal chance of winning a bazillion dollars that one wouldn't even be able to stash away in a Sun-sized vault?) But that just a psychological fact. Using more sensible win/loss ratios of 2/3 vs 1/3, or 6/11 vs 5/11 in the die case, doesn't reveal anything odd about the Thirder interpretation of her credence, or about her betting behavior.

[/quote]Thirder reasoning only has its place, if it has a place at all, if both a) the experiment is repeated 2^101 times and b) Sleeping Beauty is also made to forget between experiments. It matters that the problem does not stipulate these two conditions.[/quote]

The experiment needs no be repeated many times for SB's expression of her 2/3 credence (under the Thirder interpretation of her credence) to make sense, or for her associated betting behavior to be rational. The case of a single experimental run (following a single coin flip) was addressed specifically in my Leonard Shelby Christmas gift case. You can refer to it for the relevant Bayesian updating calculations, but here is another variation that may be more intuitive:

For this year’s annual cocktail party at the Sleeping Beauty Experimental Facility, Leonard Shelby is among the guests. Drinks are being served by two butlers: Alfred and Lurch. Each guest is entitled to three complimentary drinks.

In Leonard's case, as with every other guest, a fair coin is secretly tossed beforehand. If the coin lands Tails, Alfred is assigned to serve him two of his drinks and Lurch one. If it lands Heads, their roles are reversed: Lurch serves two drinks and Alfred one. The guests are informed of this protocol, and Leonard has made a note of it in his memento notepad.

Because of his anterograde amnesia, Leonard cannot keep track of how many drinks he has already received, if any. Nor does he initially recognize either butler by name, but he can read their name tags when they approach him.

A final feature of the protocol is that, at the end of the evening, guests whose coin landed Tails (and thus received two drinks from Alfred and one from Lurch) will be given a bag of Twizzlers ("T-candy"). Those whose coin landed Heads will receive a bag of Hershey’s Kisses ("H-candy").

At any given moment during the party, when Leonard sees a butler bringing him a drink, his credence that this drink is unique (that is, not one of two planned drinks from the same butler) is 1/2, as is his credence that the coin landed Heads. However, upon reading the name tag and discovering that the butler is Alfred, he updates his credence that the coin landed Tails to 2/3, since there are twice as many situations in which the coin landed Tails and Alfred serves him a drink as there are situations where the coin landed Heads and Alfred serves him one. This mirrors the Thirder interpretation in the Sleeping Beauty problem.

That seems straightforward enough until someone asks him, "So, Leonard, do you think you’ll get Twizzlers or Kisses at the end of the night?"

He frowns, checks his notepad, and realizes that by the same reasoning that gave him 2/3 for Tails a moment ago, he ought also to think he's more likely than not to get Twizzlers. But that can't be right. The coin decides both outcomes, doesn’t it?

The trick, of course, is in what Leonard’s belief precisely is about when he thinks about the coin toss "outcome". When he reasons about this drink—the one Alfred is serving him—he’s locating himself among drink-moments. In that frame, a Tails-run simply generates twice as many such moments involving Alfred. But when he wonders what candy he'll get later, he's no longer locating himself in a drink-moment but in an entire "run of the evening": the single history that will end either in Twizzlers or Kisses. And there, each run counts only once, no matter how many times Alfred appeared in it.

Two T-drinks in a Tails-run correspond to just one Twizzlers outcome (in the same timeline), while one H-drink in a Heads-run corresponds to one Kisses outcome. Once you factor that mapping in, the overall odds of Twizzlers or Kisses even out again. (Since experiencing one of two T-drink event doesn't exclude but rather ensures the actuality of the other T-drink event in the same timeline).

So Leonard’s probabilities fit together neatly after all. In the middle of the party, as Alfred hands him a glass, he can think, "This is probably a T-drink". Yet, looking ahead to the end of the night, he can just as honestly write in his notebook, "Chances of T-candy: fifty-fifty."

Your argument in favor of the Thirder credence that the coin landed Tails (2/3) relies on labeling the awakening episodes "the outcomes". — Pierre-Normand

Uh, yeah?

Write "Heads and Monday" on one notecard. Write "Tails and Monday" on another, and "Tails and Tuesday" on a third. Turn them over, and shuffle them. Then write "A," B," and "C" on the other sides.

Pick one. What is the probability that it says "Heads" on the other side? What is the probability that it says "Tails" on the other side? Call me silly, but I'd say 1/3 and 2/3, respectively.

Each morning of the experiment when SB is to be awakened, put the appropriate card on a table in her room, with the letter side up. Hold the interview at that table.

What is the probability that the card, regardless of what letter she sees, says "Heads" on the other side? Or "Tails?" This "outcome" can be defined by the letter she sees. But that does not define what an outcome is, being the description of the experiment's result, in SB's knowledge, is. If she wakes on a different day, that is a different result. Being determined by the same coin flip does not determine that.

Now, did these probabilities change somehow? For which letter(s) do they change? Or are they still 1/3 and 2/3?

Within SB's knowledge, is not the outcome where it says "Heads" the exact same outcome where the coin is showing Heads? And the same with "Tails?" If it says "Tails and Monday," is there not another interview along this same path where it says "Tails and Tuesday?" Does that change the probability that this card says "Tails?" How does that carry over to the one time it would say "Heads?"

Again, halfers are constructing inconsistent logic to support the answer they desire. Not using valid logic to answer the question.

But what is it that prevents Halfers from labelling the experimental runs "the outcomes" instead?

Because it is not both Monday, and Tuesday, when she is asked the question? What else may or may not happen is irrelevant.

That's right, and this is a good argument favoring the Thirder position but it relies on explicitly introducing a scoring procedure that scores each occasion that she has to express her credence: once for each awakening episode. — Pierre-Normand

A "scoring procedure" based on imagined repeats is a way of testing your probabilities, not of defining it. It does not work in the SB problem, as should be painfully obvious, because each side will define the number of trials differently since repeated runs require looking at more than one outcome, and the number changes based on the subject event.