Some quick thoughts:

On the 10^10 extrapolation - it's interesting to think about how surprised you would be under the different scenarios. Imagine that you wake up, you're asked which way you think the coin came up, and then you're told that it actually came up heads. If you really had 10^10x belief that you were waking up because it had come up tails, you would be absolutely flabbergasted! This would be the most unlikely thing that had every happened to anyone in the world in your lifetime! But would you actually be that surprised? Or would you just think "well, there was a 50% chance that I was just going to wake up once and hear that it was heads." Consider how you would look at someone else who had woken up because the coin came up heads - if this fact absolutely blew their mind, would you think they were being a little bit silly?

On betting - assume you are only going to win or lose the bet once, not once for each time you wake up. Say for the sake of argument that if the coin comes up tails then you will make a bet each time you wake up, but only one of those bets is randomly selected to be the one that "counts". Do you have any strong opinions about how you should bet in this case? You really shouldn't, because you could just as easily enter your bet before you go to sleep and it shouldn't change the outcome. The thirder case for the betting scenario relies on the fact that the reward is greater for correctly guessing tails, not on actually having a strong belief that the coin came up tails.

I came into this thinking that the answer seemed to be YES, but reconsidering the betting scenario in particular has talked me around to thinking NO.

@IsaacCarruthers

On the betting modification that only one of your bets matter, only one of those bets is randomly selected to be the one that "counts", does this modification not effectively remove the distinction that you were woken up at all. If heads you are woken up once, if Tails only one of the times you wake up counts.

I don't think there is any controversy that if you wake up sleeping beauty only once in either senario that her credence should be 50/50.

This goes to whether or not waking up is new information or not. If only one awakening counts then waking up is definitely not new information. If every waking counts but you cannot differentiate the awakenings is it information? If sleepung beauty uses betting to keep score in the post experiment, and the rational sleeping beauty would try to maximize the expected value of the score in the post experiment state.

@ShitakiIntaki Since it's a question of "degree of belief" I'm just trying to align what we believe when we're woken with the expected value of the bet. For instance, before we go to sleep we obviously think that there's a 50/50 chance of the coin coming up heads. However, if we knew that we would get to make a separate bet on each waking, we could pre-commit to betting on tails because the expected value is higher, even though our degree of belief is currently 50%. Since our behavior when we wake is exactly what we would have pre-committed to back when our degree of belief was 50%, it doesn't say anything about our degree of belief on waking. The example above is just my attempt to come up with a situation where, prior to falling asleep, we're ambivalent about which way would pre-commit to betting. You could equally say that, if the coin comes up tails, each of your bets counts for half of the value.
I think the key point here is that, on waking, you will not do anything differently from what you would have written down ahead of time prior to falling asleep. This means that your expectations about the state of the world should be the same as they were before you fell asleep.

@IsaacCarruthers

your expectations about the state of the world [when awoken] should be the same as they were before you fell asleep.

This is true if being awoken is not "new" information upon which you would/could update your belief.

Naively if you have to treat the bets differently upon any given awakening to get the answer you expect, then it feels like there is some sort of information that awakening must impart to the system.

@ShitakiIntaki right but I'm saying you don't treat bets differently on any given awakening. Your actions on awakening are exactly what you would have planned to do ahead of time. For any betting setup, you could write down exact, unconditional instructions for yourself on how to bet prior to falling asleep, and then simply follow those instructions when you wake up. This means that awakening does not impart any new information.

@IsaacCarruthers there is a dependency on timing though, not of when you draft your prescriptive betting strategy, but of when you place the bet.

Everyone agrees if you place your a bet before the experiment starts, your credence must be 50/50.

The camps differ on what your credence should be, and how you should bet, during the experiment at any given awakening.

If every awakening is treated the same and therefor has equal weight in the betting model, then bets placed during an awakening will have a credence that moves away from the pre-experiment 50/50 credence.

@ShitakiIntaki I don't see how what you're saying implies that the timing matters. "If every awakening has equal weight in the betting model" is equivalent to saying that the payoff for betting tails is 2x the payoff for betting heads. If I told you that you could bet on a coin flip, but that if the coin came up tails we would automatically double the size of the bet, of course you would bet on tails. Your policy is exactly the same whether you go through the sleeping/waking rigamarole or not, which implies that your credence is not changed by the sleeping/waking process. All that's changed is the payoffs for different policies.

@IsaacCarruthers

Ah I am envisioning a the same wager to payout ratio for each bet at the time they bet is placed, were you either net lose 1 or gain 1.

So a bet before the experiment is 50/50,

A bet during the awakening is at the same wager to pay ratio in either case but in one case you are placing a wager twice as many times. You don't know the results of the flip yet so you are still in a superpostion of it could be heads or it could be tails.

@ShitakiIntaki yes but the number of times you take the bet is correlated with the outcome. Like I said above, we could set up a bet where I flip a coin, and if it's heads you win or lose $1, but if it's tails you win or lose $2. Obviously it's optimal to pick tails, because when you win you win bigger, and when you lose you lose smaller. This is true even though you know that the coin is 50/50. It's the same in the sleeping beauty problem: you pick tails not because you think the coin is more likely to be tails, but because the correlation between the coin-flip and your effective bet size means that tails is the optimal choice even though you're still 50/50 on the outcome of the coin-flip itself.

@IsaacCarruthers @ShitakiIntaki Another thought experiment that relates to this discussion is, when you wake up, what day do you think it is? Is it Monday or Tuesday? And suppose you are woken up 10^10 times on Tuesday if the coin is Tails. Would you think it's overwhelmingly likely that it's Tuesday, since the vast majority of awakenings (even in expectation) happen on Tuesday? Then you will think it's overwhelmingly likely that the coin is Tails. Or do you think there's only a slightly less than 50% chance that it is Tuesday? Because Tails is 50% and, if the flip is Tails, then of course on any awakening it's overwhelmingly likely that it is Tuesday. Personally, I think that if I woke up, given that I realize that I am awake, I will think that it's overwhelmingly likely that it is Tuesday. (I'm a Thirder)

@DavidPennock if you really take that seriously, like I said in my first post, you should find it earth-shatteringly amazing if it turns out to be Monday, and I just... don't think I would? I think I'd wake up, find out it was Monday, and just shrug and say "well, there was a 50/50 shot of that happening." Do you really think you would be absolutely mindblown if you woke up as what you're saying is the 1/10^10 version of yourself?

@IsaacCarruthers

the number of times you take the bet is correlated with the outcome

Is more or less the crux of the

When betting odds and credences come apart: more worries for Dutch book arguments,

Analysis 66.2, April 2006, pp. 119-27. Darren Bradley and Hannes Leitgeb

https://www.jstor.org/stable/25597713

Bradley and Leitgeb conclude:

If there is a normative link between beliefs and bets, then the corresponding norm should include the proviso: 'Other things being equal (risk-neutral, utility linear with money, . . . , and: it is not that case that the agent is unaware of the fact that the size or the existence of the bet is correlated with the outcome of the event that the bet is on) ...'.

Although, I find the not unaware phrasing a bit awkward and think that maybe they could have phrased the additional proviso "and: the agent is assured that the size or the existence of the bet is not correlated with the out come of the event that the bet is on"

I do struggle with Bradley & Leitgeb's set up with Hallucinated bets to try to prevent awakenings/bets from imparting any information to Sleeping Beauty upon which there could be a rationalization to update her beliefs mid-experiment.

I find it very attractive to believe there to be a normative link between beliefs and bets, and that if you are subject to a dutch book attack that perhaps your beliefs are not correct.

It has been a while since I read it, but I seem to recall that I liked

Synchronic Bayesian updating and the generalized Sleeping Beauty Problem

Analysis 67.1, January 2007, pp. 50-59., Terry Horgan

https://www.jstor.org/stable/25597773

Horgan treats non-Bayesian thirder arguments with suspicion and asserts that

The correct way to reason, in both the original and the generalized versions of the problem, is to invoke synchronic Bayesian updating on newly acquired indexical information.

@IsaacCarruthers Yes, that is a good question and a good thought experiment. I guess I really would be mind blown if it was Monday. And if it was Monday I would be very convinced the flip was Heads. But I will think about this more.

@IsaacCarruthers Maybe it will be helpful to come up with some things that we can all agree on. I feel that we can all agree on the following three statements:

1. If you woke up and someone told you that the flip was tails and the day was Monday, your mind would be blown.

2. If you woke up and someone told you that the flip was heads and the day was Monday, you would think, "ok, there was a 50% chance of that".

3. If you woke up and someone told you that the flip was tails and the day was Tuesday, you would think, "ok, there was a 49.9999999999% chance of that".

Do you agree with these 3 statements? Then how would you feel if you woke up and someone told you only that the day was Monday? Or only that the flip was Heads?

@DavidPennock Here we're in the setup where on tails, you're woken once on Monday and 10^10 times on Tuesday? If so, I disagree with 1. This feels like saying "there are 57600 waking seconds in a day, so on first waking up you should be very surprised to be experiencing the first of those seconds, and not any of the other ones." I'd also change the number in 3 to 50% for the same reason.

@IsaacCarruthers

"there are 57600 waking seconds in a day, so on first waking up you should be very surprised to be experiencing the first of those seconds, and not any of the other ones."

Is this not the case?
Anecdotally I feel like people are often surprised to observe particular times or odometer readings, specifically because they are specific times or odometer readings, and the sampling of times when you are actually observing is more or less random. Oh, wow, my odometer reads 123,456 or the time is exactly 11:11:11 AM.

Observing an arbitrary second is should be no surprise but if you assign a particular second a unique attribute before hand and then observe that unique attribute after a single sampling it is surprising. Like what are the odds you observe a time represented entirely using a single digit in your waking hours? depends on when you are awake but of the total 86400 seconds there are 12 seconds where that happens?

• 1:11:11 [am/pm]

• 2:22:22 [am/pm]

• 3:33:33 [am/pm]

• 4:44:44 [am/pm]

• 5:55:55 [am/pm]

• 11:11:11 [am/pm]

Or only 7 seconds if you are using military time and you lose all the pm occurrences in exchange for the 0:00:00 midnight representation.

@DavidPennock It seems like you are just asking if @IsaacCarruthers is Bayesian? and perhaps based on their response they are not Bayesian.

I assume that Bayesian halfers would assign the odds

• P(Heads & 1st Day| Awake)=1/2

• P(Tails & 1st Day of X | Awake) = 1/(2x)

• P(Tails & 2nd Day of X | Awake) = 1/(2x)

And from there

• P(1st Day & Heads | Awake) = x/(x+1)

• P(1st Day of X & Tails | Awake) = 1/(x+1)

• P(Tails | 2nd Day of X & Awake) = 1

• P(1st Day | Awake) = (x+1)/(2x)

• P(Not 1st Day | Awake) = (x-1)/(2x)

• P(1st Day | Heads & Awake) = 1

So I would have assumed that "If you woke up and someone told you that the flip was tails and the day was Monday" it should seem to be improbable.
Tails and the 2nd Day of X is still some what improbable but tails and Not the 1st day seems like that should be approaching 1/2

@ShitakiIntaki It seems like you're talking about the surprise of looking at the clock and noticing that it happened to be a "special" second at the moment you happened to look. You certainly wouldn't be surprised if you set an alarm to go off at some special second, and then it did in fact go off. What I'm asking is more along the lines of: when you first wake up, are you surprised to find yourself in that first moment of the day as opposed to any other? This is something that, ahead of time, you knew you were going to experience at some point. That doesn't mean that when you experience it you need to be surprised, even though it makes up a very small proportion of your total experiences.

I'll have to get back to you on how I would frame this problem within my bayesianity.

@IsaacCarruthers Yes I roughly agree with @ShitakiIntaki . The thought experiment is less like being surprised it's the first of 57600 seconds, and more like: you are magically teleported to one of 57600 seconds: how surprised would you be if it happened to be the first one? When Sleeping Beauty wakes up, she doesn't know which reality she is in: she has no idea. If someone asked her to guess if it was Monday and the flip was Tails, she would say "probably not -- extremely unlikely". Then if someone told her that indeed it was Monday and the flip was tails, she would be extremely surprised. I admit it's less obvious that she would be surprised to learn only that the flip was Heads. I have to keep thinking about it.

@DavidPennock one thing I keep circling around is that she's guaranteed to experience a wake-up on Monday; this is why I think it's more like waking up in the first second of the day than being randomly transported to the first second of the day. This is an experience that she's guaranteed to have, so why be surprised that she's having it now as opposed to some other time? I also think this is where some kind of mistake is sneaking into the attempts at bayesian calculation (cc. @ShitakiIntaki): in some sense the actual bayes factor for waking up on Monday is simply 1.0: it was evidence that was guaranteed to be observed whether the coin came up heads or tails. The fact that after observing this evidence we are then caused to forget it obviously makes it somewhat hard to reason about, and I fully admit that I can't yet properly sketch out why and when we should treat the bayes factor this way, but this seems like the best way to line up the bayesian calculations with other basic results such as betting strategies.

@IsaacCarruthers I see what you're saying and yes from an English language definition, sleeping beauty won't be "surprised", but from a probability theory definition, I think she would be surprised. I don't think that it matters that she is guaranteed to wake up on Monday since, when she does wake up, she doesn't know which day it is. For example, imagine flipping over all 52 cards in a deck on cards one at a time. You know that eventually you are guaranteed to turn over the Ace of Spades. So you won't be "surprised" when you do finally flip it over. But on any given flip, you will think there is a low probability of flipping the Ace of Spades, you would bet against it turning up, and you will be surprised (in probability terms) to see it.

Interestingly, I think that my thought experiment is leading to a halfer argument even though I was adamantly a thirder before. I have to think more about it.

@ShitakiIntaki It does not change how many times you are woken up.

It does show that you are not sure that you being woken up.

@IsaacCarruthers @ShitakiIntaki I realized that the following experiment is completely identical to sleeping beauty with 52 awakenings on Tuesday, and easier to reason about (for me):

If the coin flip is heads, sleeping beauty is woken up once with an Ace of Spades card face down on a table in the room.

If the coin flip is tails, sleeping beauty is woken up 52 times with one card face down on a table in the room; each awakening uses a different card in a deck of cards.

The day of the week that the awakenings happen doesn't matter: the Ace of Spades plays the exact same role as "Monday" in the original experiment.

Now when a halfer wakes up, if she flips over the card and sees an Ace of Spades, she will think, "the coin flip was almost surely heads". That's because either the flip was heads, which has a 50% chance of happening, or the flip was tails AND the card was the Ace of Spades, which has a 1/104 probability or less than 1% chance of happening. The coin flip being heads is 52 times more likely than tails.

When a thirder wakes up, if she flips over the card and sees an Ace of Spades, she will think, "the coin flip is 50/50 heads or tails". That's because in exactly one awakening the coin flip is heads and the card is Ace of Spades and in exactly one awakening the coin flip is tails and the card is Ace of Spades. Since sleeping beauty has no idea which awakening she is in, these two awakenings are equally likely.

A thirder would reason that the very fact that she is sitting in a room after being woken up and seeing a card face down on the table in front of her is evidence that the coin flip was tails. When the coin flip is tails, she will wake up 52 times more than when the coin flip is heads, so she will have 52 times as many experiences like the one she is in. So since she is having this experience, it's likely that the coin flip was tails.

Now that I replaced the day of week with a playing card, I feel that the halfer argument makes more sense. Why should the heads-Ace-of-Spades awakening have the same probability as the tails-Ace-of-Spades awakening? I used to be an adamant thirder. I used to think that, of course, if sleeping beauty were told it was Monday, the coin flip would still be 50/50. Now I can finally see the halfer argument. But I can also see that this seems like a fundamentally a philosophical question with no exact true answer.

@DavidPennock

when a halfer wakes up, if she flips over the card and sees an Ace of Spades, she will think, "the coin flip was almost surely heads". That's because either the flip was heads, which has a 50% chance of happening, or the flip was tails AND the card was the Ace of Spades, which has a 1/104 probability or less than 1% chance of happening. The coin flip being heads is 52 times more likely than tails.

I apologize my earlier post had erroneous conditionals.

The P(tails|ace of spades) should still be 50%

Whereas P(ace of spades|tails) would be 1/52.

Once you learn it is the ace of spades you would update your beliefes conditionalized on that information rather than continue to use the unconditionalized beliefes.

Conditional upon observing the Ace of spades though, I think everyone (halfers and thirders) agrees that the odds of heads vs tails is 50:50

When you are awake and the card is still face down is where halfers and thirders diverge.

@ShitakiIntaki @IsaacCarruthers Thanks but actually I don't think that's right.

A Halfer reasons just like the experimenter. She thinks that P(tails)=1/2 but that P(tails|AceOfSpades) = 1/53. That's because P(tails|AceOfSpades) = P(AoS|tails)P(tails)/P(AoS) = 1/52*(1/2)/(1*(1/2)+1/52*(1/2)) = 1/104*(104/53) = 1/53.

A Thirder thinks that P(tails) = 52/53 but that P(tails|AoS) = 1/2.

Everyone agrees that each tail awakening has the same probability. Everyone agrees that the total probability of all awakenings is 1. That's already two equations for 3 unknowns. The only difference is:

A Halfer believes that the probability of the one heads awakening is the same as the total probability of all tail awakenings.

A Thirder believes that the probability of the one heads awakening is the same as the probability of one single tail awakening.

After factoring in that fundamental disagreement, both the Halfer and the Thirder are Bayesian.

So I think it really does come down to a philosophical question. What is the probability of the one heads awakening according to SB?

Imagine SB knew that she would die before waking up on Wednesday. Then she might reason that, "the very fact that I am alive and experiencing an awakening is evidence that the coin flip was tails". This is the crux of the Thirder argument.

Ok look: the only argument I am willing to give at this point correct or not is that the Perron-Frobenius eigenvector of the matrix representing the Markov chain that represents this problem comes out to all 0.333s and if you take that as the steady state probability distribution over states (which it by definition is) you see that 1/3 is correct. But regardless of that the question here is „they are asked their degree of belief for the coin having come up heads.“ and if you don’t trust me and run a simulation or some other calculation, you will find that the belief that uniquely minimizes the surprisal you get from the observations is 1/3. and that is what the question is about so unambiguously that is what this should resolve to.