Okay. Read this line carefully. I'm taking it straight from Wikipedia and you linked to the Wikipedia articles, so you must have read it. "All other things equal, an observer should reason as if they are randomly selected from the set of all actually existent observers (past, present and future) in their reference class."
In this case, conditioned on the two coinflips being TH, the observers in the reference class are all the observers in the TH world, but after waking up in the experiment, we know that only two possible observers remain: Beauty waking up on Monday, and Beauty waking up on Tuesday. We have no information to select one of these over the other and so each is 50% likely. Therefore the probability of being shown heads, given that the two coinflips are TH, is 50% by the SSA.
The difference between SSA and SIA is that in SSA we first randomly pick one of the possible worlds, and then pick one of the possible observers in that world. In SIA, we randomly pick one of the possible observers in all worlds.
Edit: to emphasize -- it is irrelevant how many other observers there are besides Sleeping Beauty. Once Beauty wakes up and looks around, she knows that she is Sleeping Beauty as opposed to, say, the person running the experiment. However, she has no additional information on which instance of Sleeping Beauty she is, which is what the thought experiment is all about.
We could postulate some additional number of observers that wake up in the same situation for completely different reasons -- say, someone else is running a simulation of lots of people waking up. In that case, the probabilities of 1/3 and 1/2 are conditional probabilities -- conditional on Sleeping Beauty actually being part of this experiment. In the original formulation of the problem, we do not postulate these additional observers waking up -- because their existence is independent of the coin flips, including them or not does not differentiate between the SIA and the SSA, so we either don't think about them or we deal with the conditional probabilities.
What I'm arguing against is apparently neither SIA or SSA. I made a mistake. Are we arguing about what I originally intended to, or about my statement that they both predict 1/3?
My intent was to argue the, if you only update on the existence of an observer, rather than anything about how unlikely it is to be them, the probability will work out the same.
If you would like to discuss why SIA and SSA give the same result:
For simplicity, we'll assume 1 trillion observation days outside the experiment.
SIA: Sleeping beauty wakes up in this experiment. There are 2...
leeping Beauty is put to sleep on Sunday. If the coin lands on heads, she is awakened only on Monday. If it lands on tails, she is awaken on Monday and Tuesday, and has her memory erased between them. Each time she is awoken, she is asked how likely it is the coin landed on tails.
According to the one theory, she would figure it's twice as likely to be her if the coin landed on tails, so it's now twice as likely to be tales. According to another, she would figure that the world she's in isn't eliminated by heads or tails, so it's equally likely. I'd like to use the second possibility, and add a simple modification:
The coin is tossed a second time. She's shown the result of this toss on Monday, and the opposite on Tuesday (if she's awake for it). She wakes up, and believes that there are four equally probable results: HH, HT, TH, and TT. She then is shown heads. This will happen at some point unless the coin has the result HT. In that case, she is only woken once, and is shown tails. She now spreads the probability between the remaining three outcomes: HH, TH, and TT. She is asked how likely it is that the coin landed on heads. She gives 1/3. Thanks to this modification, she got the same answer as if she had used SIA.
Now suppose that, instead of being told the result of second coin toss, she had some other observation. Perhaps she observed how tired she was when she woke up, or how long it took to open her eyes, or something else. In any case, if it's an unlikely observation, it probably won't happen twice, so she's about twice as likely to make it if she wakes up twice.
Edit: SIA and SSA don't seem to be what I thought they were. In both cases, you get approximately 1/3. As far as I can figure, the reason Wikipedia states that you get 1/2 with SIA is that it uses sleeping beauty during the course of this experiment as the entire reference class (rather than all existent observers). I've seen someone use this logic before (they only updated on the existence of such an observer). Does anyone know what it's called?