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?
The modification you're proposing is analogous to a modification (I think originally due to Michael Titelbaum) called "Technicolor Beauty", in which Beauty sees either a red or a blue piece of paper in her room when she awakens on Monday (determined by a fair coin toss, independently of the "main" coin toss that decides if she's woken once or twice), and on Tuesday (if she's awakened) sees a piece of paper of whichever color she didn't see on Monday. I'll use this example rather than yours because it requires less specification about which coin toss we're talking about. Let "RB" be the hypothesis that the "main" coin toss landed Tails and Red was shown on Monday and Blue was shown on Tuesday. Let BR be the same, except Blue on Monday and Red on Tuesday.
Titelbaum used this to generate the "thirder" (SIA) answer to the problem, but SSA doesn't actually give the same answer, as you suggest it does. Even though Beauty is twice as likely to observe red paper at some point in the experiment, at no point do her conditional probabilities (e.g. for observing red, conditional on Heads or Tails) differ. Briefly: conditional on Heads, she expects to see red with probability 0.5 (because the coin toss was fair). Conditional on Tails, suppose Beauty has her eyes shut while calculating her conditional probabilities for observing red upon opening them, and evenly splits her (conditional) credence between it being Monday and it being Tuesday (SSA requires this). Now, if it's Monday, Beauty's credence in RB and BR is 0.5 for both, so she expects to see red with probability 0.5. Same goes for Tuesday.
It seems I misunderstood what SSA and SIA were. I have corrected this.
For what it's worth, they give roughly the same answers as long as there is a large number of observers that aren't in the experiment. The paper has nothing to do with it.