Which is why I had included the proviso that the (rare) opportunities be proportional to the number of days the hostage is held captive. Under those conditions, they carry no information to the hostage. — Pierre-Normand

Can you give actual numbers? Because that determines the answer. If there's a 90% opportunity to escape on day 1 in safehouse #1 but a 1% opportunity to escape on each day in safehouse #2 then P(Dice roll 6|opportunity to escape) < P(Dice roll 1-5|opportunity to escape).

Or if you set the %s up right then it can work out as P(Dice roll 6|opportunity to escape) = P(Dice roll 1-5|opportunity to escape), and so it's equally likely.

It's quite straightforward that P(Dice roll 6|opportunity to escape) > P(Dice roll 1-5|opportunity to escape) — Michael

Indeed, which is basically the 'thirder' solution (in this case, the 5/11er solution).

Indeed, which is basically the 'thirder' solution (in this case, the 5/11er solution). — Pierre-Normand

Yes, but it's only P(Dice roll 6|opportunity to escape) > P(Dice roll 1-5|opportunity to escape) because of the opportunity to escape (and where the %s are set up a certain way). Without that opportunity to escape it's P(Dice roll 1-5|no opportunity to escape) > P(Dice roll 6|no opportunity to escape), which is comparable to the Sleeping Beauty problem.

Likewise, enabling Sleeping Beauty to bet on H on each awakening provides no information to her, provided only the payouts are delivered after the experiment is over. — Pierre-Normand

You may have missed my edit above:

I don't understand the connection between Sleeping Beauty's credence that the coin landed heads and the tracked frequency of heads-awakenings. It's a non sequitur to claim that because tails-awakenings are twice as frequent over repeated experiments then a coin toss having landed tails is twice as likely in any given experiment.

Sleeping Beauty is being asked "in this current, one-off experiment, what is the probability that the coin I tossed on Sunday evening landed heads?".

She's not being asked to guess if it's heads or tails and then being rewarded for each successful guess.

Her choice of guess in the latter has nothing to do with what her answer would be to the former.

If I were Sleeping Beauty I would answer "1/2" and guess tails.

Can you give actual numbers? — Michael

Suppose there is a 0.01% chance to find an opportunity to escape on any given day held captive regardless of that day being the only one or one among six in a kidnapping event. Finding such opportunities doesn't yield any updating of credence. That would actually yield a very small updating since the fact that you didn't yet escape reduces a little the chances that you are not experiencing your first day of captivity. But this is quite immaterial. We can suppose that you always are recaptured but that your goal nevertheless is to maximize your number of stints (while warding off the crocodiles or lions). In Sleeping Beauty's case, opportunities to bet could be 100%, and provided the payoffs are paid after the experiment is over, that wouldn't impact her credence either.

Suppose there is a 0.01% chance to find an opportunity to escape on any given day held captive regardless of that day being the only one or one among six in a kidnapping event. Finding such opportunities doesn't yield any updating of credence. — Pierre-Normand

It does.

Dice roll 1-5 (safehouse #1): day 1, 0.01% opportunity to escape
Dice roll 6 (safehouse #2): day 1, 0.01% opportunity to escape; day 2, 0.01% opportunity to escape; day 3, 0.01% opportunity to escape; day 4, 0.01% opportunity to escape; day 5, 0.01% opportunity to escape; day 6, 0.01% opportunity to escape

If I'm doing my maths right, the probability of being offered an opportunity to escape from safehouse #2 is 0.058, so you're almost 6 times more likely to be offered the opportunity to escape from safehouse #2, so upon being offered the opportunity to escape, you can condition on this and determine that P(Dice roll 6|opportunity to escape) > P(Dice roll 1-5|opportunity to escape).

Conversely, upon not being offered the opportunity to escape, you can condition on this and determine that P(Dice roll 6|no opportunity to escape) < P(Dice roll 1-5|no opportunity to escape).

And it would be strange to say that P(Dice roll 6|no opportunity to escape) < P(Dice roll 1-5|no opportunity to escape) where there is a 0.01% chance of an opportunity to escape each day but then to say P(Dice roll 6) > P(Dice roll 1-5) where there is a 0% chance of an opportunity to escape each day.

It does. — Michael

I don't understand your argument at all. Finding an escape opportunity with probability P is a constant feature of the prisoner's environment. Sleeping Beauty's waking days (of the prisoner's captivity days) aren't being distinguished by one another in point of betting or escape opportunities. Suppose there always is a tulip on Sleeping Beauty's nightstand. Would her seeing the tulip enable her to change her credence in H? What if there is a 1% chance that the tulip is red on any given awakening day? Would that make any difference?

What if there is a 1% chance that the tulip is red on any given awakening day? Would that make any difference? — Pierre-Normand

Yes. The probability of it being red on a waking day if heads is 1%. The probability of it being red on a waking day if tails is 1 - 0.99^2 = 1.99%.

The probability of it being red on a waking day if tails is greater than the probability of it being red on a waking day if heads, therefore if it's red on a waking day then it is more likely tails and if it's not red on a waking day then it is more likely heads.

This might be clearer if we say that the probability of it being red on a waking day if heads is 1.5%. Sleeping Beauty should still reason that it is more likely tails. But if we increase it further to 2% then she should reason that it is more likely heads.

Introducing these additional probabilistic events upon which to condition her credence changes the answer and so doesn't accurately represent the problem without them.

Yes. The probability of it being red if heads is 1%. The probability of it being red if tails and Monday is 1%. The probability of it being red if tails and Tuesday is 1%. The probability of it being red if tails is 1 - 0.99^2 = 0.199.

The probability of it being red if tails is greater than the probability of it being red if heads, therefore if it's red then it is more likely tails. — Michael

I believe you meant 0.0199.

That represents the probability of the tulip being red at least once across the two days of awakening in the event that the coin lands tails, correct? While it's true that Sleeping Beauty, upon observing the red tulip, can deduce the truth of this disjunctive proposition, the two events are probabilistically independent. Thus, although it's more probable that at least one of these events will occur if there are two opportunities for it to occur within a single experiment run rather than just one, the observation of one such event doesn't entail that it's more probable that you're in an experiment run wherein two such opportunities occur.

Consider this analogy: you're entering a movie theater where there's an even chance of a double feature being shown (according to a coin flip). There's a one percent chance that any given film will feature Rex Harrison. Suppose you see Harrison featured in the first film. Does that increase your credence that there will be a subsequent feature?

If you think about it, Lewis's notion—that Sleeping Beauty can conclude from knowing it's Monday that a future coin toss is more likely to yield heads with a 2/3 probability—is already rather puzzling. It would be even more extraordinary to believe that any random observation made by Sleeping Beauty could allow her to update her credence about an unrelated coin flip.

Consider this analogy: you're entering a movie theater where there's an even chance of a double feature being shown. There's a one percent chance that any given film will feature Rex Harrison. Suppose you see Harrison featured in the first film. Does that increase your credence that there will be a subsequent feature? — Pierre-Normand

No, but if I walk in not knowing if it’s the first or second film then my credence favours it being part of a double feature.

If you think about it, Lewis's notion—that Sleeping Beauty can conclude from knowing it's Monday that a future coin toss is more likely to yield heads with a 2/3 probability—is already rather puzzling. — Pierre-Normand

I agree, which is why I offered the example of taking balls from a bag which is a double halter interpretation of the problem.

Sorry, misunderstand the movie example. It’s a different answer if I only get to walk into one film, which would be comparable to Sleeping Beauty only waking on Monday (or Tuesday) if tails.

In such a case I can’t say it’s more likely a double feature, and thirders would say Sleeping Beauty can’t say it’s more likely tails.

No, but if I walk in not knowing if it’s the first or second then my credence favours it being part of a double feature. — Michael

Yes, me too. But that's just because two out of three showings are parts of double features. Seeing Rex Harrison being featured doesn't change my credence. It remains 2/3.

Sorry, misunderstand the movie example. It’s a different answer if I only get to walk into one film, which would be comparable to Sleeping Beauty only waking on Monday (or Tuesday) if tails. — Michael

The main point is that seeing Rex Harrison being featured (while knowing that 1% of the movies randomly being shown in this theater feature him) doesn't impact your credence in this movie being part of a double feature.

The main point is that seeing Rex Harrison being featured (while knowing that 1% of the movies randomly being shown in this theater feature him) doesn't impact your credence in this movie being part of a double feature. — Pierre-Normand

That's only because I walk into one film. If I'm given amnesia and walk into the second film (if there is a second film) then it affects my credence.

It's exactly like my scenario with the coin toss and prizes. If heads then the car is the possible prize, otherwise the motorbike is the possible prize. If a car then a single coin toss determines if I get it (if heads), if a motorbike then two coin tosses determine if I get it (one head is enough to win).

If I'm told that I've won a prize then my credence favours that it is a motorbike. If I'm told that I didn't win a prize then my credence favours that the first coin toss was heads.

Knowing that I've won (or not won) a prize is the same as (not) seeing a red tulip or Harrison. It conditions my credence of that first coin flip.

That's only because I walk into one film. If I'm given amnesia and walk into the second film (if there is a second film) then it affects my credence.

It's exactly like my scenario with the coin toss and prizes. If heads then the car is the possible prize, otherwise the motorbike is the possible prize. If a car then a single coin toss determines if I get it (if heads), if a motorbike then two coin tosses determine if I get it (one head is enough to win). — Michael

In your scenario, the nature of the prize is conditioned on the coin toss results. In the film example, the number of features (and the coin toss result that determined this number) is probabilistically independent of the probability that Rex Harrison will be featured. And likewise with Sleeping Beauty and the betting opportunities, or the prisoner and their escape opportunities.

In your scenario, the nature of the prize is conditioned on the coin toss results. — Pierre-Normand

Then forget the nature of the prize.

If I know that I’ve won a prize my credence favours the first coin toss having landed tails, otherwise it favours heads. If I see that the tulip is red my credence favours the coin toss having landed tails, otherwise it favours heads.

Then forget the nature of prize. If I know that I’ve won a prize my credence is that the first coin toss landed tails. — Michael

In your prize scenario, knowing that you've won a prize indeed gives you new information, and this can rightly affect your credences. However, in the scenarios with Sleeping Beauty and the prisoner, merely being presented with an opportunity to bet or escape does not give them any new information about the outcome of the coin toss (or throw of the die). They must decide how to take advantage of this opportunity (by choosing to carry the torch or the plank, or choosing what safehouse address to communicate to the police) before gaining any knowledge about the success of the attempt. The offering of the opportunities carry no information and provide no ground for updating credences.

However, in the scenarios with Sleeping Beauty and the prisoner, merely being presented with an opportunity to bet or escape does not give them any new information about the outcome of the coin toss (or throw of the die). They must decide how to take advantage of this opportunity (by choosing to carry the torch or the plank, or choosing what safehouse address to communicate to the police) before gaining any knowledge about the success of the attempt. The offering of the opportunities carry no information and provide no ground for updating credences. — Pierre-Normand

She can and should use known priors to condition her credence, and one such prior is that she is more likely to win a prize/have the opportunity to escape if tails/a dice roll of 6. As such, if she wins a prize or has the opportunity to escape she should condition on this and her credence should favour tails/a dice roll of 6, otherwise she should condition on not winning a prize or having the opportunity to escape and her credence should favour heads/a dice roll of 1-5.

And if she's guaranteed the opportunity to escape each day, or she never has the opportunity to escape each day, then her credence should favour a dice roll of 1-5.

But honestly, all this talk of successes is irrelevant anyway. As I said before, these are two different things:

1. Sleeping Beauty's credence that the coin tossed on Sunday for the current, one-off, experiment landed heads
2. Sleeping Beauty's most profitable strategy for guessing if being asked to guess on heads or tails over multiple games

It's simply a non sequitur to argue that if "a guess of 'tails' wins 2/3 times" is the answer to the second then "1/3" is the answer to the first.

She can and should use known priors to condition her credence, and one such prior is that she is more likely to win a prize/have the opportunity to escape if tails/a dice roll of 6. As such, if she wins a prize or has the opportunity to escape she should condition on this and her credence should favour tails/a dice roll of 6, otherwise her she should condition on not winning a prize or having the opportunity to escape and her credence should favour heads//a dice roll of 1-5. — Michael

She can't condition her credence on H by winning a prize if she only learns about her victory after the experiment ends. The same applies to her opportunity to bet — this fact is probabilistically independent of H and thus doesn't allow for an update in credence. Would you still argue that your belief that the movie you're watching is part of a double feature increases the moment you see Rex Harrison appearing on screen? Or that it increases even further when you notice he isn't wearing a top hat, or when he is?

And if she's guaranteed the opportunity to escape each day her credence should favour a dice roll of 1-5.

So, she should be carrying a plank and end up being eaten by lions on 6 out of 11 escape attempts?

But honestly, all this talk of successes is irrelevant anyway. As I said before, these are two different things:

1. Sleeping Beauty's credence that the coin tossed for the current, one-off, experiment landed heads
2. Sleeping Beauty's most profitable strategy for guessing if being asked to guess on heads or tails over multiple games

It's simply a non sequitur to argue that if "tails" is the answer to the second then "1/3" is the answer to the first.

The manner in which (1) is stated suggest that Sleeping Beauty is referring to the wide centered possible world spanning the whole experiment run. In that case, her credence in H should be 1/2.

The second one makes it rational for her to rely on her credence regarding narrow centered possible worlds spanning single awakening episodes. There indeed isn't any entailment from the suitability of one framing of the question from (1) to (2) or vice versa. The two sentences concern themselves with different questions.

By the way, I've just finished reading Groisman's The End of Sleeping Beauty’s Nightmare. Although I found it somewhat glib, it's undoubtedly the most readable paper on this topic that I've encountered so far. Groisman makes some very compelling points that are directly relevant to the discussion we are presently having. However, I prefer Laureano Luna's approach, which elaborates further on Groisman's work and brings more nuance.

So, she should be carrying a plank and end up being eaten by lions on 6 out of 11 escape attempts? — Pierre-Normand

5 out of every 6 victims escape. I count by participants, not by escape attempts. I think it's more reasonable.

The manner in which (1) is stated suggest that Sleeping Beauty is referring to the wide centered possible world spanning the whole experiment run. In that case, her credence in H should be 1/2.

The second one makes it rational for her to rely on her credence regarding narrow centered possible worlds spanning single awakening episodes. There indeed isn't any entailment from the suitability of one framing of the question from (1) to (2) or vice versa. The two sentences concern themselves with different questions. — Pierre-Normand

And that first question is the premise of the problem. Sleeping Beauty is asked her credence that the current experiment's coin toss landed heads. That's it. She's not being asked to consider the most profitable betting strategy for multiple games, which even Lewis (I assume) would agree is to bet on tails.

5 out of every 6 victims escape. I count by participants, not by escape attempts. I think it's more reasonable. — Michael

Since on my approach probabilities track frequencies, even if there is just one kidnapping event, the hostage's chances of survival are 5 in 11 whenever an escape attempt occurs.

Since on my approach probabilities track frequencies, even if there is just one kidnapping event, the hostage's chances of survival are 5 in 11 whenever an escape attempt occurs. — Pierre-Normand

So then there are two different ways to reason with nothing to prove that one or the other is the "right" way?

And that first question is the premise of the problem. Sleeping Beauty is asked her credence that the coin landed heads. That's it. She's not being asked to consider the most profitable betting strategy for multiple games. — Michael

The state "the coin landed heads" is ambiguous with respect to the span of the centered possible world that it might be referring to. I think you would benefit from reading Groisman.

So then there are two different ways to reason with nothing to prove that one or the other is the "right" way? — Michael

That's correct since events that happen in the world don't come flagged with sign posts that say: "the current event begins here" and "the current event terminates here." How credences in the probabilities of events are assessed depend on the way those events are individuated and this can be dictated by pragmatic considerations.

I think you would benefit from reading Groisman. — Pierre-Normand

I've just read it. Seems to be saying what I said here:

There are two ways to reason:

1. $2\over3$ of all interviews are 100 heads in a row interviews, therefore this is most likely a 100 heads in a row interview
2. $1\over2^{100}$ of all participants are 100 heads in a row participants, therefore I am most likely not a 100 heads in a row participant

I would say that both are true...

That's correct since events that happen in the world don't come flagged with sign posts that say: "the current event begins here" and "the current event terminates here." How credences in the probabilities of events are assessed depend on the way those events are individuated and this can be dictated by pragmatic considerations. — Pierre-Normand

I don't think this is right. The experiment is only being conducted once and Sleeping Beauty is asked "what is your credence that the coin I tossed on Sunday landed heads?" I think there's only one appropriate way to reason, which above you seemed to accept gives an answer of $1\over2$.

But if you want to insist that it's ambiguous, I'll just fall back to that previous application of Bayes' theorem (despite ChatGPT's objection):

P(Heads | Mon or Tue) = P(Mon or Tue | Heads) * P(Heads) / P(Mon or Tue)
P(Heads | Mon or Tue) = 1 * 1/2 / 1
P(Heads | Mon or Tue) = 1/2

As Elga says, she learns nothing new (and Lewis would agree); she only recognises that her temporal location (of which all she knows is it's either Monday or Tuesday) is relevant. But contrary to Elga's reasoning (which I believe I showed to be faulty) this doesn't change her credence.

It might just be as simple as this.

The issue of optimum betting strategies over multiple games is irrelevant to the problem. There's nothing to debate on that; it's obviously better to bet on tails. I'm sure Lewis and every other halfer would agree.

But as this debate has gone on long enough and I don't think I have the energy to continue it much more, I'm happy to just say that both $1\over2$ and $1\over3$ are correct answers to distinct but equally valid interpretations of the question.

So thanks for the enlightening discussion.

But as this debate has gone on long enough and I don't think I have the energy to continue it much more, I'm happy to just say that both 1/2 and 1/3 are correct answers to distinct but equally valid interpretations of the question. — Michael

I'm still mulling it over despite myself, but I think there's something to this.

Rather than getting back into the nitty-gritty, I'm thinking about the stuff I posted a while back, the possible self slices and all that.

It could be the two natural positions represent slightly different attitudes toward the self.

Halfers define participants as ending up having one interview, or ending up with two. It's a cumulative view of the self: I am my life story, and the story draws a connected arc passing through circumstances; what matters most is that storyline, and situations are only things I (robust, impervious) encounter along the way.

Thirders define participants by what they're experiencing at the moment, and seem less confident that what matters most is the unique historical self for whom this is merely an incident in their story; thirders seem to feel that the situation in part defines them, they are just the someone this is happening too -- anything could happen to anyone or not happen to them, and that's who you are at that moment.

It's hard to state each side clearly, but you could also say, roughly, that for a halfer the question is only: what happened? how did the coin land? For thirders, the question is: who am I? what's happening to me right now that defines who I am? (This is pretty clear in @JeffJo's thirder analysis, posted here on the forum and also here on a site dedicated to the sleeping beauty problem, in which the whole point is that participants don't know which sort of participant they are, and recognize that roles could be shuffled amongst them.)

The who am I? question doesn't really seem to be on the halfer radar at all, thus my suggesting that halfers view the self as more robust in the face of circumstances.

Maybe one more analogy: thirders might be more likely to buy the results Zimbardo claimed for the Stanford Prison Experiment. --- That's more than a bridge too far for me, but might be the clearest way to put the distinction.

I wonder if the variation I considered here highlights the difference in perspective, with the question forcing you to consider it from the halfer perspective:

There are two Sleeping Beauties; one will be woken on Monday and one on both Monday and Tuesday, determined by a coin toss.

What is their credence that they have been or will be woken twice?

We can even consider it with just one Sleeping Beauty, as with the traditional problem.

Does it make sense for her credence to be that she's twice as likely to be woken twice simply because she's woken twice if she is woken twice compared to once if woken once? I don't think it does. The very question requires that you think of both wakings as being part of the same outcome, with your credence in that outcome being what is questioned.

But given that you're woken twice iff the coin landed tails, it stands to reason that the answer to "what is your credence that you have been or will be woken twice" is the same as the answer to "what is your credence that the coin landed tails".

Yeah there's some similarity to Bostrum's thing. In Stanford terms, you could say that thirders are identifying the self with the role they are playing at the moment, and it's a matter of chance that they are playing one role at a given moment rather than another.

Self and repetition are linked in Nietzsche's puzzling doctrine that truly to will something is to will its eternal recurrence. The Good Place is also oddly sleeping-beauty adjacent:

The Monty Hall "problem" has an answer: it's just an illustration of a tempting but fallacious way of reasoning about probability. Two Envelopes rises to the level of paradox because a clearly and provably wrong answer can be arrived at by reasoning, the flaw in which is so difficult to determine that there is no professional consensus on what the flaw is.

I'm not sure SB is the same sort of thing: which is the natural, obvious, tempting mistake and which the correction? If you haven't been around these sorts of puzzles, maybe "a fair coin is 50-50, period" is the obvious answer, and SB is one of the only scenarios bizarre enough to undermine that confidence; if you've been around these sorts of puzzles, carving the space into quarters and conditioning to get one third might be the obvious thing to do, but that's to be tricked into misreading the source of randomness here. Because the analysis of the 1200-sophomore study I posted is so straightforward, but getting standard SB to align with it is so difficult, I have even wondered if SB doesn't undermine the whole idea of "subjective probability". A more optimistic take would be that SB is unsolvable, and shows you what background assumptions are necessary for probabilistic reasoning to work -- memory, continuity of self, objective verification, something in here.

Rather than getting back into the nitty-gritty, I'm thinking about the stuff I posted a while back, the possible self slices and all that. — Srap Tasmaner

I think this observation is very astute. As I am developing a pragmatist conception of credence, I am trying to move away from the idea that one's credence in the state H is entirely determined by the specification of the ways in which one can come to be in that state. The Sleeping Beauty problem completely specifies how (and in which circumstances) Sleeping Beauty can come to be in state H, however one wants to define what counts as "being in state H". But it fails to specify how one might want to individuate such states (or 'self slices') so as to determine the relevant frequency or probability ratios that we ordinarily express as credences.

Whenever Sleeping Beauty is awakening during a H experimental run, she is awakening during a H day, and vice versa. But there isn't a one to one relationship between the two sets of 'events' (i.e. between the two sets of centered possible worlds) once they have been individuated.

The relationship is one to one between types but not between tokens, we might say.

In the Sleeping Beauty problem the equivalence relation is:

H-run <--> H&Monday (H-day)
T-run <--> T&Monday or T&Tuesday (T-day)

The ratio considered by the thirders is the ratio of H-days per day long centered possible worlds (or individual awakenings). It is indeed 1/3. The ratio considered by the halfers is the ratio of H-runs per run-long centered possible worlds. It is indeed 1/2. The credences expressed by thirders and by halfers therefore have different topics.

I would say that neither the halfers nor the thirders position concerns credence. Plus the concept of "credence" is a minefield of unresolved philosophical difficulty, which is usually overlooked by advocates of probabilism.

The halver's position is based on the common-sense postulate that the physical properties and propensities of objects remain unchanged when their states are conditioned upon irrelevant information. Given acceptance of that postulate plus the premise of a fair coin, why shouldn't the halver insist that they are talking about reality, as opposed to their mere mental state?

The thirder's position is indeed a ratio of possible words, but there is scant evidence to support the idea that credences are accurately represented by taking ratios over possible worlds.

For example, if one were to ask the average person to express their credences regarding the outcome of a two horse race that they know absolutely nothing about, they will simply say "I don't know who will win" and refrain from assigning any odds, equal or otherwise. They will also tend to accept bets in which they have knowledge that the physical probabilities are 50/50 over bets that they are totally ignorant about.

Fundamentally, it is difficult to even know what the object of a credence or belief is, which relates to an overlooked problem of epistemology: If every belief is fully explained by it's natural causes, then how are false beliefs possible and where does the error reside? In the mind of the speaker who expresses their beliefs, or in the mind of the listener who interprets the speaker?

Whatever SB says when interrogated upon awakening, if she is assumed to have amnesia then how are her interrogators supposed to interpret her "credence estimate"? Why should they consider her uttered "credence estimate" as being qualitatively different from yawning?