Easy solution for the Sleeping Beauty problem: instead of merely asking her her subjective probability, we can ask her to bet. The question now becomes “at what odds would you be willing to bet?”. So here are the possibilities:
Heads. There will be one bet, Monday.
Tails. There will be two bets, Monday, and Tuesday.
Heads or tails comes up with equal probability (0.5). But when it comes up Tails, the stakes double (because she will bet twice). So, what will generate the correct bets is the assumption that Tails will subjectively come up 2⁄3 of the time.
I know it looks cheap, because it doesn’t answer the question “But what really is the subjective probability?”. I don’t know, but I’ll find a way to make the correct decision anyway.
By asking, “At what odds would you be willing to bet?”, you’ve skewed the payout matrix, not the probabilities—even subjectively. If she offers the bet at 2:1 odds, it’s so that when her future/past twin makes the same bet, it corrects the payout matrix. She adjusts in this way because the probability is 1⁄2.
It’s just like if an online bookie discovers a bug in their software so that when someone takes the first option in a bet, and if they win, they get paid twice. He needs to lower the payout on option 1 by a factor of 2 (on the backend, at least—no need to embarrass everyone by mentioning it on the front end).
Sleeping Beauty can consistently say, “The probability of my waking up this time having been a waking-event following a tails coinflip is 2⁄3. The probability of the coinflip having come up tails is 1⁄2. On either of these, if you demand to bet on the issue, I’m offering 2:1 odds.”
Consistently? Sorry, I can’t even parse the sentence that follows. Trying to understand it:
“this event”
Could you mean “the fact that I just woke up from drugged induced sleep”? But this event is not correlated with the coin flip to begin with. (Whether it ends up head or tail, you will wake up seemingly for the first time.)
“The probability of the coinflip having come up tails is 1⁄2.”
Whose probability?
Also, how my solution could lead Sleeping Beauty to be Dutch-booked? Could you provide an example, please?
SB is required, on sunday, to lay odds on the coin flip; the coin will be shown to her on wednesday, and the outcome judged. She is given the opportunity to change her mind about the odds she’s laying at any point during the experiment before it’s over. Should she change her odds? No.
About Dutch-booking—You must have gotten in there before I rewrote it, which I did before you finished posting. I realized I may have been misusing the term. Does the version up now make sense? Oh, heck, I’ll rewrite it again to make it even clearer.
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.
I know it looks cheap, because it doesn’t answer the question “But what really is the subjective probability?”
I don’t think using betting is cheap at all, if you want to answer questions about decision-making. But I still wanted to answer the question about probability :D
Easy solution for the Sleeping Beauty problem: instead of merely asking her her subjective probability, we can ask her to bet. The question now becomes “at what odds would you be willing to bet?”. So here are the possibilities:
Heads. There will be one bet, Monday.
Tails. There will be two bets, Monday, and Tuesday.
Heads or tails comes up with equal probability (0.5). But when it comes up Tails, the stakes double (because she will bet twice). So, what will generate the correct bets is the assumption that Tails will subjectively come up 2⁄3 of the time.
I know it looks cheap, because it doesn’t answer the question “But what really is the subjective probability?”. I don’t know, but I’ll find a way to make the correct decision anyway.
By asking, “At what odds would you be willing to bet?”, you’ve skewed the payout matrix, not the probabilities—even subjectively. If she offers the bet at 2:1 odds, it’s so that when her future/past twin makes the same bet, it corrects the payout matrix. She adjusts in this way because the probability is 1⁄2.
It’s just like if an online bookie discovers a bug in their software so that when someone takes the first option in a bet, and if they win, they get paid twice. He needs to lower the payout on option 1 by a factor of 2 (on the backend, at least—no need to embarrass everyone by mentioning it on the front end).
Sleeping Beauty can consistently say, “The probability of my waking up this time having been a waking-event following a tails coinflip is 2⁄3. The probability of the coinflip having come up tails is 1⁄2. On either of these, if you demand to bet on the issue, I’m offering 2:1 odds.”
Consistently? Sorry, I can’t even parse the sentence that follows. Trying to understand it:
Could you mean “the fact that I just woke up from drugged induced sleep”? But this event is not correlated with the coin flip to begin with. (Whether it ends up head or tail, you will wake up seemingly for the first time.)
Whose probability?
Also, how my solution could lead Sleeping Beauty to be Dutch-booked? Could you provide an example, please?
My hunch is that any solution other than yours allows her to be Dutch-booked...
Not after correction for payout matrix, as described...
Hers, right then, as she says it.
Let’s go ahead and draw a clearer distinction.
SB is required, on sunday, to lay odds on the coin flip; the coin will be shown to her on wednesday, and the outcome judged. She is given the opportunity to change her mind about the odds she’s laying at any point during the experiment before it’s over. Should she change her odds? No.
About Dutch-booking—You must have gotten in there before I rewrote it, which I did before you finished posting. I realized I may have been misusing the term. Does the version up now make sense? Oh, heck, I’ll rewrite it again to make it even clearer.
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.
I don’t think using betting is cheap at all, if you want to answer questions about decision-making. But I still wanted to answer the question about probability :D