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?
Alright, now it's time for my comment about why saying "I'd like to use the SSA" (or, for that matter, "I'd like to use the SIA") is misguided.
Suppose every time Beauty wakes up, she is asked to guess whether the coin landed Heads or Tails. She receives $3 for correctly saying Heads, and $2 for correctly saying Tails.
The SIA says Pr[Heads] = 1/3 and Pr[Tails] = 2/3, so saying Heads has an expected value of $1, and Tails an expected value of $1.33. On the other hand, the SSA says Pr[Heads]=Pr[Tails]=1/2, so saying Heads is expected to win $1.50, while saying Tails only wins $1.
These indicate different correct actions, and clearly only one of them can be right. Which one? Well, suppose Beauty decides to guess Heads. Then she wins $3 when Heads comes up. On the other hand if Beauty decides to guess Tails, she wins $4. So the SIA gives the "correct probability" in this case.
On the other hand, suppose the rewards are different. Now, suppose Beauty receives money on Wednesday -- $3 if she ever correctly said Heads, and $2 if she ever correctly said Tails. In this case, the optimal strategy for Beauty is to act as though Pr[Heads]=Pr[Tails]=1/2, as suggested by the SSA.
Of course, proponents of either assumption, assuming a working knowledge of probability, are going to make the correct guess in both cases, if they know how the game works; suddenly there is no more disagreement. I therefore argue that the things that these assumptions call Pr[Heads] and Pr[Tails] are not the same things. The SIA calculates the probability that the current instance of Beauty is waking up in a Heads-world or a Tails-world. The SSA calculates the probability that some instance of Beauty will wake up in a Heads-world or a Tails-world.
The way I phrase it makes them sound more different than they are, because this latter event is also the event that every instance of Beauty will wake up in a Heads-world or a Tails-world. Since it's certain that the current instance of Beauty wakes up in the same world that every instance of Beauty wakes up in, it's unclear why these probabilities are different.
This ambiguity disappears once you start talking some hand-wavy notion of probability that feels like it's perfectly okay to disagree about, and fix a concrete situation in which you need the correct probability in order to win, as illustrated in the example above.
(One final comment: by using payoffs of $2 and $3, I am technically only determining whether the probability in question is above or below 2/5. Since this separates 1/2 and 1/3, it is all that is necessary here, but in principle you could also use log-based payoffs to make Beauty give an actual probability as an answer.)
Are we dealing with the optimal strategy for her to decide on before-hand, or the one she should decide on mid-experiment?
She may have evidence in the middle of the experiment that she didn't before, as such, the optimal choice may be different. It's similar to Parfit's Hitch-hiker.