Ok, let’s skew the odds a little, and have the coin have 4⁄7 probability of being heads (SSA agrees). The SIA probabilities are now 4⁄10 of being heads. You run the simulations 700 times, getting (on average) 400 experiments with only Monday awakening, and 300 with awakenings on both Monday and Tuesday.
You then ask the sleeping beauties to guess what the coin was. Suppose they guess tails, following SIA odds.
We can then ask: how often did Sleeping Beauty guess right? Well, there were 300x2=600 copies that guessed right, and 400 that guessed wrong, as in your example. SIA is the way to go.
But now suppose the question is: in how many simulations did Sleeping Beauty guess right? Well, she guessed right only in 300 simulations, and guessed wrong in 400. So for this criteria, SSA is the way to go.
OK, so the difference is in how you count: SB instances (skewed toward tails) vs simulation instances. Now, when would the latter matter? For example, if a correct guess of day+coin would let the lucky SB stay awake, the SIA is clearly better.
In what scenario would choosing the SSA let the poor girl be less doomed?
P.S. I have calculated the probabilities for skewed odds, and if the probability of heads is p:
Monday (heads): p, Monday (tails): (1-p)/2, Tuesday: (1-p)/2 for SSA
Monday (heads): p/(2-p), Monday (tails): (1-p)/(2-p), Tuesday: (1-p)/(2-p) for SIA
OK, so the difference is in how you count: SB instances (skewed toward tails) vs simulation instances. Now, when would the latter matter?
If a correct guess of coin would mean that the SB was reawakened on the next Sunday and left to go on with her life. Here guessing right twice is of no help, and you should follow SSA odds. But these kind of situations are dealt with in more details in my next two posts.
If a correct guess of coin would mean that the SB was reawakened on the next Sunday and left to go on with her life. Here guessing right twice is of no help
I thought about it, but she has two chances to guess in case of tails, skewing the odds toward the SIA even heavier, if a single correct guess is enough. Unless she has to guess right twice in a row, which is rather artificial.
It appears that here is a small window of probabilities (1/3<p<1/2 for heads) where the two models can be distinguished, but I have not put in enough time to formulate the corresponding setup clearly enough. Hopefully your subsequent posts will make it clearer.
I’m assuming Sleeping Beauty has no access to a random process, so she will guess the same on both occasions. So the two guesses are of no help to her.
Monday (heads): 0.4, Monday (tails): 0.3, Tuesday: 0.3 for SSA
Monday (heads): 0.25, Monday (tails): 0.375, Tuesday: 0.375 for SIA
Thus the SSA SB would always guess Monday (heads) and the SIA SB would guess either Monday (tails) or Tuesday to maximize her odds. Suppose she always picks Tuesday. In 1000 simulations there are 400 heads and 600 tails. 400 SSA SBs survive vs 600 SIA SBs, so the SIA is the way to go.
You’re using the SIA way of counting (considering each agent in tails as separate), and getting an SIA-favouring result.
An SSA way of counting would be that you have to guess what day and coin flip it was, and your chances of surviving is the average number of times you guessed right. Guessing Tuesday(tails) or Monday(tails) would give you a 50-50 chance of surviving in the tails world, since one of the versions of you will get it wrong. Guessing Monday(heads) would give you a certainty of surviving in the heads world (since there is only one of you). 400 SSA SB survive versus 300 SIA SB.
OK, I understand the SSA setup now, though it does look a little contrived to me. I guess I need to read your arxiv paper in more detail to see when this is reasonable. Thanks.
Ok, let’s skew the odds a little, and have the coin have 4⁄7 probability of being heads (SSA agrees). The SIA probabilities are now 4⁄10 of being heads. You run the simulations 700 times, getting (on average) 400 experiments with only Monday awakening, and 300 with awakenings on both Monday and Tuesday.
You then ask the sleeping beauties to guess what the coin was. Suppose they guess tails, following SIA odds.
We can then ask: how often did Sleeping Beauty guess right? Well, there were 300x2=600 copies that guessed right, and 400 that guessed wrong, as in your example. SIA is the way to go.
But now suppose the question is: in how many simulations did Sleeping Beauty guess right? Well, she guessed right only in 300 simulations, and guessed wrong in 400. So for this criteria, SSA is the way to go.
OK, so the difference is in how you count: SB instances (skewed toward tails) vs simulation instances. Now, when would the latter matter? For example, if a correct guess of day+coin would let the lucky SB stay awake, the SIA is clearly better.
In what scenario would choosing the SSA let the poor girl be less doomed?
P.S. I have calculated the probabilities for skewed odds, and if the probability of heads is p:
Monday (heads): p, Monday (tails): (1-p)/2, Tuesday: (1-p)/2 for SSA
Monday (heads): p/(2-p), Monday (tails): (1-p)/(2-p), Tuesday: (1-p)/(2-p) for SIA
Hope this matches your calculations.
If a correct guess of coin would mean that the SB was reawakened on the next Sunday and left to go on with her life. Here guessing right twice is of no help, and you should follow SSA odds. But these kind of situations are dealt with in more details in my next two posts.
Yep.
I thought about it, but she has two chances to guess in case of tails, skewing the odds toward the SIA even heavier, if a single correct guess is enough. Unless she has to guess right twice in a row, which is rather artificial.
It appears that here is a small window of probabilities (1/3<p<1/2 for heads) where the two models can be distinguished, but I have not put in enough time to formulate the corresponding setup clearly enough. Hopefully your subsequent posts will make it clearer.
I’m assuming Sleeping Beauty has no access to a random process, so she will guess the same on both occasions. So the two guesses are of no help to her.
Let’s consider p(head)=2/5. Then the odds are:
Monday (heads): 0.4, Monday (tails): 0.3, Tuesday: 0.3 for SSA
Monday (heads): 0.25, Monday (tails): 0.375, Tuesday: 0.375 for SIA
Thus the SSA SB would always guess Monday (heads) and the SIA SB would guess either Monday (tails) or Tuesday to maximize her odds. Suppose she always picks Tuesday. In 1000 simulations there are 400 heads and 600 tails. 400 SSA SBs survive vs 600 SIA SBs, so the SIA is the way to go.
What am I missing?
Note what we’re doing in these situations: we’re determining the ‘right’ answer, without having to use the anthropic probabilities at all.
You’re using the SIA way of counting (considering each agent in tails as separate), and getting an SIA-favouring result.
An SSA way of counting would be that you have to guess what day and coin flip it was, and your chances of surviving is the average number of times you guessed right. Guessing Tuesday(tails) or Monday(tails) would give you a 50-50 chance of surviving in the tails world, since one of the versions of you will get it wrong. Guessing Monday(heads) would give you a certainty of surviving in the heads world (since there is only one of you). 400 SSA SB survive versus 300 SIA SB.
OK, I understand the SSA setup now, though it does look a little contrived to me. I guess I need to read your arxiv paper in more detail to see when this is reasonable. Thanks.