I've reviewed again the argument in Section 2 of Elga's paper (https://www.princeton.edu/~adame/papers/sleeping/sleeping.pdf) and it seems valid to me. It doesn't have the same simplicity of some other arguments for 1/3, though, such as the betting arguments, and my argument in https://glizen.com/radfordneal/anth.abstract.html that the probability of anyone having your detailed experience on awakening is twice are large if you are woken twice.
There are many good arguments for 1/3. The arguments for 1/2 are all bad, and defending them forces people into contorted positions such as that there is no actual probability in this situation, or the probability depends on what class of people you see yourself as belonging to, or as seen in a comment below, that 1/2 and 1/3 are both valid answers. (Regarding the later, Beauty can decide to break down the door of the room, go out, and ask someone what day it is. There is nothing ambiguous about asking what the probability is that she will find out that it is Monday.)
For a contrary position, see https://philpapers.org/archive/BURSBR-2.pdf (which I just found by a search, and haven't read yet), which however does acknowledge that 1/3 is the favoured position.
I stand by my "valid answers for different questions" position, but it's pretty difficult to swallow "The arguments for 1/2 are all bad", without being clear that you're excluding the simplest of questions: if nothing changes from the setup, what is Beauty's expected experience of being told on Wednesday what the results were?
The chance of a fair coin coming up tails is 1/2. Beauty has no evidence on waking up to alter that, as she'd experience that either way.
There are OTHER questions, and betting-odds considerations that lead to 1/3 being ab...
I can't tell you whether it is considered to be sound. In my opinion, it is. But I do know that the issues causing the controversy to continue for 23 years are not a part of the actual problem. And so are unnecessary.
If anyone thinks that sounds odd, they should go back and read the question that Elga posed. It does not mention Monday, Tuesday, or that a Tails-only waking follows a mandatory waking. Those were elements he introduced into the problem for his thirder solution.
In the problem as posed, the subject (I'll call her SB, even though in the posed problem it is "you") is woken once, or twice, based on the outcome of a fair coin flip (Heads=once, Tails=twice). Elga enacted that description with a mandatory waking on Monday, and an optional one on Tuesday. This way, could create two partial solutions by revealing two different bits of information that removed one of the three possibilities. The unfortunate side effect of this was that the conditions surrounding Monday and Tuesday are different; thirders require Tuesday to be part of a different outcome, and halfers insist it is the same outcome as Monday+Tails.
And that is what is unnecessary. Instead of that Monday/Tuesday schedule, simply flip two coins (call them C1 and C2) after SB is first put to sleep. Then:
Now turn coin C2 over to show its other face, and repeat these steps.
When SB is woken, she knows that: (A) She is in step 2 of a pass thru these six steps. (B) In step 1, there were four equally-likely states for the two coins: HH, HT, TH, and TT. (C) Since she is awake, the state HH is eliminated but the other three remain equally likely. (D) In only one of those states is C1 showing Heads. So she can confidently state that her credence is 1/3.
The difference between this, and how Elga enacted the problem he posed, is that here there is no ambiguity about how she arrived at her current, awake situation.
I originally upvoted your answer as it presented an interesting version of the SB which I didn't see before.
However, it has similar problems to the original and it's even more convoluted, you may notice that there is no need to throw the second coin other than to confuse everyone. You can just put it tails and then turn over to get the exact same results.
HH, HT, TH and TT are not four elementary outcomes of the experiment, as there is causal connection between them, even if such formulation makes it less obvious and harder to talk about it.
I assume you mean from https://www.princeton.edu/~adame/papers/sleeping/sleeping.pdf ? The one about "if you knew it was Monday it would be 0.5, surely adding Tuesday as an option causes an update," followed by a standard claim that symmetry implies 1/3?
Both parts are independently correct. But I think there's probably not enough step-by-step talk about symmetry for this to serve the social role of a proof for confused people. I've found it's important, when explaining anthropics, to really focus on "centered possible worlds" (or hypotheses that reproduce your experiences, to frame it more like Solomonoff induction), and why a question like "what is the universe like outside my head?" can be answered by considering hypotheses anew when you have different memories.
I think a link or summary of the "proof", and description of the parts of it that you're wondering about, would go a long way toward making this a useful question.
I couldn't find a concise summary of exactly which argument you're asking about, so I'll just say that all probability is in the agent's map - it's a prediction of future experiences. There may or may not be an underlying "reality", but even if there is, there's no access to any outside observer.
So 1/2 or 1/3 are valid answers to different questions.
I've spend sometime looking into the issue and I'm quite confident that this proof isn't sound. Is it already a known fact?