It ultimately depends on how you define probabilities, and it is possible to define them such that the answer is $\frac{1}{2}$.

I personally think that the only "good" definition (I'll specify this more at the end) is that a probability of $\frac{1}{4}$ should occur one in four times in the relevant reference class. I've previously called this view "generalized frequentism", where we use the idea of repeated experiments to define probabilities, but generalizes the notion of "experiment" to subsume all instances of an agent with incomplete information acting in the real world (hence subsuming the definition as subjective confidence). So when you flip a coin, the experiment is not the mathematical coin with two equally likely outcomes, but the situation where you as an agent are flipping a physical coin, which may include a 0.01% probability of landing on the side, or a ${10}^{-15}$ probability of breaking in two halfs mid air or whatever. But the probability for it coming up heads should be about $\frac{1}{2}$ because in about $\frac{1}{2}$ of cases where you as an agent are about to flip a physical coin, you subsequently observe it coming up heads.

There are difficulties here with defining the reference class, but I think they can be adequately addressed, and anyway, those don't matter for the sleeping beauty experiment because there, the reference classes is actually really straight-forward. Among the times that you as an agent are participating in the experiment and are woken up and interviewed (and are called Sleeping Beauty, if you want to include this in the reference class), one third will have the coin heads, so the probability is $\frac{1}{3}$. This is true regardless of whether the experiment is run repeatedly throughout history, or repeatedly because of Many Worlds, or an infinite universe, etc. (And I think the very few cases in which there is genuinely not a repeated experiment are in fact qualitatively difference since now we're talking logical uncertainty rather than probability, and this distinction is how you can answer $\frac{1}{3}$ in Sleeping Beauty without being forced to answer $\frac{1}{1000000}$ on the Presumptuous Philosopher problem.)

So RE this being the only "good" definition, well one thing is that it fits betting odds, but I also suspect that most smart people would eventually converge on an interpretation with these properties if they thought long enough about the nature of probability and implications of having a different definition, though obviously I can't prove this. I'm not aware of any case where I want to define probability differently, anyway.

What is going to be done with these numbers? If Sleeping Beauty is to gamble her money, she should accept the same betting odds as a thirder. If she has to decide which coinflip result kills her, she should be ambivalent like a halfer.

I would frame the question as "What is the probability that you are in heads-space?", not "What is the probability of heads?". The probability of heads is 1/2, but the probability that I am in heads-space, given I've just experiences a wake-up event, is 1/3.

The wake-up event is only equally likely on Monday. On Tuesday, the wake-up event is 0%/100%. We don't know whether it is Tuesday or not, but we know there is some chance of it being Tuesday, because 1/3 of wake-up events happen on Tuesday, and we've just experienced a wake-up event:

P(Monday|wake-up) = 2/3
P(Tuesday|wake-up) = 1/3
P(Heads|Tuesday) = 0/1
P(Heads|Monday) = 1/2
P(Heads|wake-up) = P(Heads|Monday) * P(Monday|wake-up) + P(Heads|Tuesday) * P(Tuesday|wake-up) = 1/3

Thirder here (with acknowledgement that the real answer is to taboo 'probability' and figure out why we actually care)

The subjective indistinguishability of the two Tails wakeups is not a counterargument  - it's part of the basic premise of the problem. If the two wakeups were distinguishable, being a halfer would be the right answer (for the first wakeup).

Your simplified example/analogies really depend on that fact of distinguishability. Since you didn't specify whether or not you have it in your examples, it would change the payoff structure.

I'll also note you are being a little loose with your notion of 'payoff'. You are calculating the payoff for the entire experiment, whereas I define the 'payoff' as being the odds being offered at each wakeup. (since there's no rule saying that Beauty has to bet the same each time!)

To be concise, here's my overall rationale:

Upon each (indistinguishable) wakeup, you are given the following offer:

• If you bet H and win, you get $N$ dollars.
• If you bet T and win, you get 1+$ϵ$ dollars.

If you believe T yields a higher EV, then you have a credence $P\left(T\right)\ge \frac{N}{N+1}$

You get a positive EV for all N up to 2, so $P\left(T\right)=\frac{2}{3}$. Thus you should be a thirder.

Here's a clarifying example where this interpretation becomes more useful than yours:

The experimenter flips a second coin. If the second coin is Heads (H2), then N= 1.50 on Monday and 2.50 on Tuesday. If the second coin is Tails, then the order is reversed.

I'll maximize my EV if I bet T when $N=1.5$, and H when $N=2.5$. Both of these fall cleanly out of 'thirder' logic.

What's the 'halfer' story here? Your earlier logic doesn't allow for separate bets on each awakening.

The question "What is the probability of Heads?" is about the coin, not about your location in time or possible worlds.

This is, I think, the key thing that those smart people disagree with you about.

Suppose Alice and Bob are sitting in different rooms. Alice flips a coin and looks at it - it's Heads. What is the probability that the coin is Tails? Obviously, it's 0% right? That's just a fact about the coin. So I go to Bob in the other room and and ask Bob what's the probability the coin is Tails, and Bob tells me it's 50%, and I say "Wrong, you've failed to know a basic fact about the coin. Since it was already flipped the probability was already either 0% or 100%, and maybe if you didn't know which it was you should just say you can't assign a probability or something."

Now, suppose there are two universes that differ only by the polarization of a photon coming from a distant star, due to hit Earth in a few hours. And I go into the universe where that polarization is left-handed (rather than right-handed), and in that universe the probability that the photon is right-handed is 0% - it's just a fact about the photon. So I go to the copy of Carol that lives in this universe and ask Carol what's the probability the photon has right-handed polarization, and Carol tells me it's 50%, and I say "Wrong, you've failed to know a basic fact about the photon. Since it's already on its way the probability was already either 0% or 100%, and maybe if you don't know which it was you should just say you can't assign a probability or something."

Now, suppose there are two universes that differ outside of the room that Dave is currently in, but are the same within Dave's room. Say, in one universe all the stuff outside the room is arranged is it is today in our universe, while in the other universe all the stuff outside the room is arranged as it was ten years ago. And I go into the universe where all the stuff outside the room is arranged as it was ten years ago, which I will shorthand as it being 2014 (just a fact about calendars, memories, the positions of galaxies, etc.), and ask Dave what's the probability that the year outside is 2024, and Dave tells me it's 50%...

You need to start by clearly understanding that the Sleeping Beauty Problem is almost realistic - it is close to being actually doable. We often forget things. We know of circumstances (eg, head injury) that cause us to forget things. It would not be at all surprising if the amnesia drug needed for the scenario to actually be carried out were discovered tomorrow. So the problem is about a real person. Any answer that starts with "Suppose that Sleeping Beauty is a computer program..." or otherwise tries to divert you away from regarding Sleeping Beauty as a real person is at best answering some other question.

Second, the problem asks what probability of Heads Sleeping Beauty should have on being interviewed after waking. This of course means what probability she should rationally have. This question makes no sense if you think of probabilities as some sort of personal preference, like whether you like chocolate ice cream or not. Probabilities exist in the framework of probability theory and decision theory. Probabilities are supposed to be useful for making decisions. Personal beliefs come into probabilities through prior probabilities, but for this problem, the relevant prior beliefs are supposed to be explicitly stated (eg, the coin is fair). Any answer that says "It depends on how you define probabilities", or "It depends on what reference class you use", or "Probabilities can't be assigned in this problem" is just dodging the question. In real life, you can't just not decide what to do on the basis that it would depend on your reference class or whatever. Real life consists of taking actions, based on probabilities (usually not explicitly considered, of course). You don't have the option of not acting (since no action is itself an action).

Third, in the standard framework of probability and decision theory, your probabilities for different states of the world do not depend on what decisions (if any) you are going to make. The same probabilities can be used for any decision. That is one of the great strengths of the framework - we can form beliefs about the world, and use them for many decisions, rather than having to separately learn how to act on the basis of evidence for each decision context. (Instincts like pulling our hand back from a hot object are this sort of direct evidence->action connection, but such instincts are very limited.) Any answer that says the probabilities depend on what bets you can make is not using probabilities correctly, unless the setup is such that the fact that a bet is offered is actual evidence for Heads versus Tails.

Of course, in the standard presentation, Sleeping Beauty does not make any decisions (other than to report her probability of Heads). But for the problem to be meaningful, we have to assume that Beauty might make a decision for which her probability of Heads is relevant. 

So, now the answer... It's a simple Bayesian problem. On Sunday, Beauty thinks the probability of Heads is 1/2 (ie, 1-to-1 odds), since it's a fair coin. On being woken, Beauty knows that Beauty experiences an awakening in which she has a slight itch in her right big toe, two flies are crawling towards each other on the wall in front of her, a Beatles song is running through her head, the pillow she slept on is half off the bed, the shadow of the sun shining on the shade over the window is changing as the leaves in the tree outside rustle due to a slight breeze, and so forth. Immediately on wakening, she receives numerous sensory inputs. To update her probability of Heads in Bayesian fashion, she should multiply her prior odds of Heads by the ratio of the probability of her sensory experience given Heads to the probability of her experience given Tails.

The chances of receiving any particular set of such sensory inputs on any single wakening is very small. So the probability that Beauty has this particular experience when there are two independent wakening is very close to twice that small probability. The ratio of the probability of experiencing what she knows she is experiencing given Heads to that probability given Tails is therefore 1/2, so she updates her odds in favour of Heads from 1-to-1 to 1-to-2. That is, Heads now has probability 1/3. 

(Not all of Beauty's experiences will be independent between awakenings - eg, the colour of the wallpaper may be the same - but this calculation goes through as long as there are many independent aspects, as will be true for any real person.)

The 1/3 answer works. Other answers, such as 1/2, do not work. One can see this by looking at how probabilities should change and at how decisions (eg, bets) should be made.

For example, suppose that after wakening, Beauty says that her probability of Heads is 1/2. It also happens that, in an inexcusable breach of experimental protocol, the experimenter interviewing her drops her phone in front of Beauty, and the phone display reveals that it is Monday. How should Beauty update her probability of Heads? If the coin landed Heads, it is certain to be Monday. But if the coin landed Tails, there was only a probability 1/2 of it being Monday. So Beauty should multiply her odds of Heads by 2, giving a 2/3 probability of Heads.

But this is clearly wrong. Knowing that it is Monday eliminates any relevance of the whole wakening/forgetting scheme. The probability of Heads is just 1/2, since it's a fair coin. Note that if Beauty had instead thought the probability of Heads was 1/3 before seeing the phone, she would correctly update to a probability of 1/2.

Some Halfers, when confronted with this argument, maintain that Beauty should not update her probability of Heads when seeing the phone, leaving it at 1/2. But as the phone was dropping, before she saw the display, Beauty would certainly not think that it was guaranteed to show that it is Monday (Tuesday would seem possible). So not updating is unreasonable.

We also see that 1/2 does not work in betting scenarios. I'll just mention the simplest of these. Suppose that when Beauty is woken, she is offered a bet in which she will win $12 if the coin landed Heads, and lose $10 if the coin landed Tails. She know that she will always be offered such a bet after being woken, so the offer does not provide any evidence for Heads versus Tails. If she is woken twice, she is given two opportunities to bet, and could take either, both, or neither. Should she take the offered bet?

If Beauty thinks that the probability of Heads is 1/2, she will take such bets, since she thinks that the expected payoff of such a bet is (1/2)*12-(1/2)*10=1. But she shouldn't take these bets, since following the strategy of taking these bets has an expected payoff of (1/2)*12 - (1/2)*2*10 = -4. In contrast, if Beauty thinks the probability of Heads is 1/3, she will think the expected payoff from a bet is (1/3)*12-(2/3)*10=-2.666... and not take it.

Note that Beauty is a real person. She is not a computer program that is guaranteed to make the same decision in all situations where the "relevant" information is the same. It is possible that if the coin lands Tails, and Beauty is woken twice, she will take the bet on one awakening, and refuse the bet on the other awakening. Her decision when woken is for that awakening alone. She makes the right decisions if she correctly applies decision theory based on the probability of Heads being 1/3. She makes the wrong decision if she correctly applies decision theory with the wrong probability of 1/2 for Heads.

She can also make the right decision by incorrectly applying decision theory with an incorrect probability for Heads, but that isn't a good argument for that incorrect probability.

If the experiment instead was constructed such that:

1. If the coin comes up heads, Sleeping Beauty will be awakened and interviewed on Monday only.
2. If the coin comes up tails, Sleeping Beauty's twin sister will be awakened and interviewed on Monday and Sleeping Beauty will be awakened and interviewed on Tuesday.

In this case it is "obvious" that the halfer position is the right choice. So why would it be any different if Sleeping Beauty in the case of tails is awakened on Monday too, since she in this experiment have zero recollection of that event? It does not matter how many other people they have woken up before the day she is woken up, she has NO new information that could update her beliefs. 

Or say that the experiment instead was constructed that she for tails would be woken up and interviewed 999999 days in row, would she then say upon being woken up that the probability that the coin landed heads is 1/1000000?

If you look over all possible worlds, then asking "did the coin come up Heads or Tails" as if there's only one answer is incoherent. If you look over all possible worlds, there's a ~100% chance the coin comes up as Heads in at least one world, and a ~100% chance the coin comes up as Tails in at least one world.

But from the perspective of a particular observer, the question they're trying to answer is a question of indexical uncertainty - out of all the observers in their situation, how many of them are in Heads-worlds, and how many of them are in Tails-worlds? It's true that there are equally as many Heads-worlds as Tails-worlds - but 2/3 of observers are in the latter worlds.

Or to put it another way - suppose you put 10 people in one house, and 20 people in another house. A given person should estimate a 1/3 chance that they're in the first house - and the fact that 1 house is half of 2 houses is completely irrelevant. Why should this reasoning be any different just because we're talking about possible universes rather than houses?

I prefer to just think about utility, rather than probabilities. Then you can have 2 different "incentivized sleeping beauty problems"

• Each time you are awakened, you bet on the coin toss, with $ payout. You get to spend this money on that day or save it for later or whatever
• At the end of the experiment, you are paid money equal to what you would have made betting on your average probability you said when awoken.

In the first case, 1/3 maximizes your money, in the second case 1/2 maximizes it.

To me this implies that in real world analogues to the Sleeping Beauty problem, you need to ask whether your reward is per-awakening or per-world, and answer accordingly

Alternatively I started out confused.

Debating this problem here and with LLMs convinced me that I'm not confused and the thirders are actually just doing epistemological nonsense.

It feels arrogant, but it's not a poor reflection of my epistemic state?

Welcome to the club.

I have read some of the LW posts on the canonical problem here. I won't be linking them due to laziness.

I suppose my posts are among the ones that you are talking about here?