Your new formulation is much better. Now I can identify the pain point.
The probability of my waking up this time having been a waking-event following a tails coinflip is 2⁄3.
I think I unconditionally agree with this one (I’m not certain, though).
The probability of the coinflip having come up tails is 1⁄2.
This is when I get confused. See, if you ask the question before drugging SB, it feels obvious that she should answer “1/2”. As you say, she gains no information by merely waking up, because she knew she would in advance. Yet she should still bet 2:1 odds, whether it’s money or log-odds. In other words, how on Earth can the subjective probability be different from the correct betting odds?!
Currently, I see only two ways of solving this apparent contradiction. Either estimating 2:1 odds from the beginning, or admitting that waking up actually provided information. Both look crazy, and I can’t find any third alternative.
(Note that we assume she will be made to bet at each wake up no matter what. For instance, if she knows she only have to bet Monday, then she wakes up and is told to bet, she gains information that tells her “1/2 probability, 1⁄2 betting odds”. Same thing if she only know she will bet once.)
how on Earth can the subjective probability be different from the correct betting odds?!
Because the number of bets she makes will be different in one outcome than the other. it’s exactly like the bookie software bug example I gave. Normally you don’t need to think about this, but when you begin manipulating the multiplicity of the bettors, you do.
Let’s take it to extremes to clarify what the real dependencies are. Instead of waking Bea 2 times, we wake her 1000 times in the event of a tails flip (I didn’t say 3^^^3 so we wouldn’t get boggled by logistics).
Now, how surprised should she be in the event of a heads flip? Astonished? Not that astonished? Equanimous? I’m going with Equanimous.
Your new formulation is much better. Now I can identify the pain point.
I think I unconditionally agree with this one (I’m not certain, though).
This is when I get confused. See, if you ask the question before drugging SB, it feels obvious that she should answer “1/2”. As you say, she gains no information by merely waking up, because she knew she would in advance. Yet she should still bet 2:1 odds, whether it’s money or log-odds. In other words, how on Earth can the subjective probability be different from the correct betting odds?!
Currently, I see only two ways of solving this apparent contradiction. Either estimating 2:1 odds from the beginning, or admitting that waking up actually provided information. Both look crazy, and I can’t find any third alternative.
(Note that we assume she will be made to bet at each wake up no matter what. For instance, if she knows she only have to bet Monday, then she wakes up and is told to bet, she gains information that tells her “1/2 probability, 1⁄2 betting odds”. Same thing if she only know she will bet once.)
Because the number of bets she makes will be different in one outcome than the other. it’s exactly like the bookie software bug example I gave. Normally you don’t need to think about this, but when you begin manipulating the multiplicity of the bettors, you do.
Let’s take it to extremes to clarify what the real dependencies are. Instead of waking Bea 2 times, we wake her 1000 times in the event of a tails flip (I didn’t say 3^^^3 so we wouldn’t get boggled by logistics).
Now, how surprised should she be in the event of a heads flip? Astonished? Not that astonished? Equanimous? I’m going with Equanimous.