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 trillion 3 possible observers, three of which wake up here. Of them, one woke up in a universe with heads, and the other in a universe with tails. The probability of being the one with heads is 1/3.
SSA: Sleeping beauty has an even prior, so the odds ratio of heads to tails is 1:1. She then wakes up in this experiment. If the coin landed on heads, there's a 1/1 trillion 1 chance of this. If the coin landed on tails, there's a 2/1 trillion 2 chance of this. This is an odds ratio of 500,000,000,001:1,000,000,000,001 for heads. Multiplying this by 1:1 yields 500,000,000,001:1,000,000,000,001. The total probability of heads is 1/3 + 2*10^-13.
Your problem seems to be updating on the fact that she's in the experiment without taking into account that this is about twice as likely if the coin landed on heads.
...huh. You have a point. I'll have to think about this for a bit, but it seems right, and if this is what you've been trying to get at this whole time I think everyone may have misunderstood you.
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?