What is going to be done with these numbers? If Sleeping Beauty is to gamble her money, she should accept the same betting odds as a thirder. If she has to decide which coinflip result kills her, she should be ambivalent like a halfer.
Halfer makes sense if you pre-commit to a single answer before the coin-flip, but not if you are making the decisions independently after each wake-up event. If you say heads, you have a 50% chance of surviving when asked on Monday, and a 0% chance of surviving when asked on Tuesday. If you say tails, you have a 50% chance of surviving Monday and a 100% chance of surviving Tuesday.
I’ve updated my comment. You are correct as long as you pre-commit to a single answer beforehand, not if you are making the decision after waking up. The only reason pre-committing to heads works, though, is because it completely removes the Tuesday interview from the experiment. She will no longer be awoken on Tuesday, even if the result is tails. So, this doesn’t really seem to be in the spirit of the experiment in my opinion. I suppose the same pre-commit logic holds if you say the correct response gets (1/coin-side-wake-up-count) * value per response though.
The disagreement is how to factorise expected utility function into probability and utility, not which bets to make. This disagreement is still tangible, because the way you define your functions have meaningfull consequences for your mathematical reasoning.
I mean I think the “gamble her money” interpretation is just a different question. It doesn’t feel to me like a different notion of what probability means, but just betting on a fair coin but with asymmetric payoffs.
The second question feels closer to actually an accurate interpretation of what probability means.
Probability is not some vaguely defined similarity cluster like “sound”. It’s a mathematical function that has specific properties. Not all of them are solely about betting.
We can dissolve the semantic disagreement between halfers and thirders and figure out that they are talking about two different functions p and p’ with subtly different properties while producing the same betting odds.
This in itself, however, doesn’t resolve the actual question: which of these functions fits the strict mathematical notion of probability for the Sleeping Beauty experiment and which doesn’t. This question has an answer.
What is going to be done with these numbers? If Sleeping Beauty is to gamble her money, she should accept the same betting odds as a thirder. If she has to decide which coinflip result kills her, she should be ambivalent like a halfer.
Halfer makes sense if you pre-commit to a single answer before the coin-flip, but not if you are making the decisions independently after each wake-up event. If you say heads, you have a 50% chance of surviving when asked on Monday, and a 0% chance of surviving when asked on Tuesday. If you say tails, you have a 50% chance of surviving Monday and a 100% chance of surviving Tuesday.
If you say heads every time, half of all futures contain you; likewise with tails.
I’ve updated my comment. You are correct as long as you pre-commit to a single answer beforehand, not if you are making the decision after waking up. The only reason pre-committing to heads works, though, is because it completely removes the Tuesday interview from the experiment. She will no longer be awoken on Tuesday, even if the result is tails. So, this doesn’t really seem to be in the spirit of the experiment in my opinion. I suppose the same pre-commit logic holds if you say the correct response gets (1/coin-side-wake-up-count) * value per response though.
Betting argument are tangential here.
https://www.lesswrong.com/posts/cvCQgFFmELuyord7a/beauty-and-the-bets
The disagreement is how to factorise expected utility function into probability and utility, not which bets to make. This disagreement is still tangible, because the way you define your functions have meaningfull consequences for your mathematical reasoning.
I mean I think the “gamble her money” interpretation is just a different question. It doesn’t feel to me like a different notion of what probability means, but just betting on a fair coin but with asymmetric payoffs.
The second question feels closer to actually an accurate interpretation of what probability means.
https://www.lesswrong.com/posts/Mc6QcrsbH5NRXbCRX/dissolving-the-question
Probability is not some vaguely defined similarity cluster like “sound”. It’s a mathematical function that has specific properties. Not all of them are solely about betting.
We can dissolve the semantic disagreement between halfers and thirders and figure out that they are talking about two different functions p and p’ with subtly different properties while producing the same betting odds.
This in itself, however, doesn’t resolve the actual question: which of these functions fits the strict mathematical notion of probability for the Sleeping Beauty experiment and which doesn’t. This question has an answer.