Both answers are valid because the question is ambiguous. Asking "what is your belief" doesn't work because there are multiple ways to interpret that. You can clarify such situations by putting them in a game. Probabilities are about how much you would bet on it.
Game 1: You get $10 every time you guess correctly. Best strategy: Guess tails because if it's correct you get $20. If you guess heads correct you get only $10. If you got $3 for guessing tails correctly and $6 for guessing heads correctly then it would not matter what strategy you used.
Game 2: If you guess correctly then you get $10 by the end of the experiment. Best strategy: it doesn't matter what you guess, if you guess tails correctly you get the $10, and if you guess heads correctly you also get $10.
Thirders think the question is like game 1: every time they guess correctly they win correctness points. Halfers think the question is like game 2: they either guess correctly or they don't; you don't get additional correctness points by guessing right twice.
> Game 2: If you guess correctly then you get $10 by the end of the experiment.
The question is clear you get a reward each time you awaken, not at the end of the game.
The reward is correctness (Humans like being correct = reward, else things like 'should believe' becomes meaningless hence the question becomes meaningless Q.E.D. it's not meaningless and correctness is a reward)
"When you are awakened, to what degree should you believe that the outcome of the coin toss was Heads?"
This fits with my understanding of what's happening here. The SO discussion is a bit unclear as you go down the page, but the initial statement of problem up top clearly claims that you're questioned each time you awaken.
I think you're quite right to point out that being correct is the only fair heuristic for this question. Several of the defenses of a 'halfer' belief amount to "you ought to believe this so that your beliefs obey this type of decision theory". Actively believing something with a (reliably) worse payout to conform to a rule about belief seems like a terrible form of epistemology.
On a related note, are you familiar with Newcomb's Paradox and the LessWrong debates over it? They get to a similar idea of "correctness above all".
Your Game 2 is poorly defined. What does it mean to "guess correctly" when you're asked to guess twice? If you systematically take the first (or the last) answer, then it's equivalent to being awaken only once on Tails. If you take the logical OR of the correctness of two guesses, then the sleeping beauty can win 5/8 of the time by answering randomly, which is better than 1/2 by guessing a fixed answer (the only strategy you analyzed).
Right, but I didn't want to complicate the explanation. Only deterministic strategies are allowed, or require both answers to be correct in order to get a reward.
If the original coin was heads, your chance to win is still 50%.
But if the original was tails, and you only have to guess once correct, your chance to guess wrong twice is only 25%.
So a 50% chance of winning half the time and a 75% chance half the time gives you an overall better chance that the 50% of sticking to one answer because it doesn't matter.
It seems a random guess is the best solution for game 2.
Game 1: You get $10 every time you guess correctly. Best strategy: Guess tails because if it's correct you get $20. If you guess heads correct you get only $10. If you got $3 for guessing tails correctly and $6 for guessing heads correctly then it would not matter what strategy you used.
Game 2: If you guess correctly then you get $10 by the end of the experiment. Best strategy: it doesn't matter what you guess, if you guess tails correctly you get the $10, and if you guess heads correctly you also get $10.
Thirders think the question is like game 1: every time they guess correctly they win correctness points. Halfers think the question is like game 2: they either guess correctly or they don't; you don't get additional correctness points by guessing right twice.