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timtyler comments on Beauty quips, "I'd shut up and multiply!" - Less Wrong
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The problem posed is, p(heads | Sleeping Beauty is awake). There is no payoff involved. Introducing a payoff only confuses matters. For instance, Roko wrote:
This is true; but that would be the answer to "What is the probability that the coin was heads, given that Sleeping Beauty was woken up at least once after being put to sleep?" That isn't the problem posed. If that were the problem posed, we could eliminate her forgetfulness from the problem statement.
If you agree that the forgetfulness is necessary to the story, then 1/2 is the wrong answer, and 1/3 is the right answer. If you don't agree it's necessary, then its presence suggests that the speaker intended a different semantics than you're using to interpret it.
ADDED: This is depressing. Here we have a collection of people who have studied probability problems and anthropic reasoning and all the relevant issues for years. And we have a question that is, on the scale of questions in the project of preparing for AGI, a small, simple one. It isn't a tricky semantic or philosophical issue; it actually has an answer. And the LW community is doing worse than random at it.
In fact, this isn't the first time. My brief survey of recent posts indicates that the LessWrong community's track record when tackling controversial problems that actually have an answer is random at best.
Re: "Introducing a payoff only confuses matters."
Personally, I think it clarifies things - though at the expense of introducing complication. People disagree over which bet the problem represents. Describing those bets highlights this area of difference.
I see what you mean. But some comments have said, "I can set up a payoff scheme that gives this answer; therefore, this is an equally-valid answer." The correct response is to state the payoff scheme that gives your answer, and then admit your answer is not addressing the problem if you can't find justification for that payoff scheme in the problem statement.
Indeed - that would be bad - and confusing.
It is both bad and confusing that people are defending the idea that this problem is not clearly-stated enough to answer.
I suspect this happens because, people don't like criticising the views of others. They would rather just say 'you are both right' - since then no egos get bruised, and a costly fight is avoided. So, nonsense goes uncriticised, and the innocent come to believe it - because nobody has the guts to knock it down.