What if we modify the problem slightly, and ask sleeping beauty for her credence that it’s Monday? That is, her credence that “it” “is” Monday. If the coin came up heads, there is only Monday, but if it came up tails, there is a Monday observer and a Tuesday observer. AA/SHA reasons purely from the perspective of possible worlds, and says that Monday is consistent with observations, as is Tuesday, and refuses to speculate further on which “observer” among possible observers she “is”. Again, given an actual decision problem with an actual payoff structure, AA/SHA will quickly reach the correct decision, even while refusing to assign probabilities “between observers”.
As much as I agree that the term “observers” causes needless confusion, I don’t think you adequately defend the position above.
Let’s say Sleeping Beauty acts under the following reasoning: If you count all the instances where she is woken up, both in the possible universe where the coin came up heads and where it came up tails, there are two instances where she woke up on monday, and one instance where she woke up on tuesday. This yields 2⁄3 probability that it is a monday if she is woken up, and she should therefore accept 2-1 odds against that it is monday. Say that the testers agree to pay her 1 dollar if it is monday and she pays them 2 dollars if it is tuesday whenever she wakes up. If the testers then run the experiment 1000 times, the coin will come up heads 500 times, in which case they pay 1 dollar each time, and tails 500 times, in which case they first pay 1 dollar and then receive 2 dollars the next day. In total they pay 1000 dollars and receive 1000 dollars on average, making the bet entirely fair if you base it on 2⁄3 odds that it is monday. So if we use your logic of basing the decision on bets, it works out if you assume 2⁄3 odds that it is monday.
Yes, I also had this concern when I read that paragraph.
If Sleeping Beauty were paid $1 for a correct guess as to which day it is, she should probably always answer “today is Monday”.
If the coin came up heads, she is correct. If the coin came up tails, she will be correct on Monday and wrong on Tuesday. So this guarantees that she earns $1 every time the experiment runs.
If she instead always answers “today is Tuesday”, then if the coin came up heads, she earns nothing; if the coin came up tails, she earns nothing on Monday and $1 on Tuesday. So this strategy has an expected revenue of $0.50.
So is it fair to “say that Monday is consistent with observations, as is Tuesday, and refuse to speculate further”? I think that’s yielding too much ground.
This particular case isn’t a philosophical question about getting information from knowledge of being an observer among all sets of observers. It’s a purely pragmatic analysis that says you’ll always be woken up on a Monday but not always on a Tuesday, so Monday awakenings are simply more probable. That’s the structure of the experiment, and has nothing to do with the anthropic principle.
An analogous experiment that removes the observer is: The experimenter has a bag with only a white ball and a green ball in it. He draws once from the bag randomly, but if he drew the white ball, he draws again. Anyone can tell you that he’ll produce a green ball every round, but a white ball only half the time. If it’s done behind a screen and you’re asked “Was a green ball drawn this round?”, the smart bet is yes.
As much as I agree that the term “observers” causes needless confusion, I don’t think you adequately defend the position above.
Let’s say Sleeping Beauty acts under the following reasoning: If you count all the instances where she is woken up, both in the possible universe where the coin came up heads and where it came up tails, there are two instances where she woke up on monday, and one instance where she woke up on tuesday. This yields 2⁄3 probability that it is a monday if she is woken up, and she should therefore accept 2-1 odds against that it is monday. Say that the testers agree to pay her 1 dollar if it is monday and she pays them 2 dollars if it is tuesday whenever she wakes up. If the testers then run the experiment 1000 times, the coin will come up heads 500 times, in which case they pay 1 dollar each time, and tails 500 times, in which case they first pay 1 dollar and then receive 2 dollars the next day. In total they pay 1000 dollars and receive 1000 dollars on average, making the bet entirely fair if you base it on 2⁄3 odds that it is monday. So if we use your logic of basing the decision on bets, it works out if you assume 2⁄3 odds that it is monday.
Yes, I also had this concern when I read that paragraph.
If Sleeping Beauty were paid $1 for a correct guess as to which day it is, she should probably always answer “today is Monday”.
If the coin came up heads, she is correct. If the coin came up tails, she will be correct on Monday and wrong on Tuesday. So this guarantees that she earns $1 every time the experiment runs.
If she instead always answers “today is Tuesday”, then if the coin came up heads, she earns nothing; if the coin came up tails, she earns nothing on Monday and $1 on Tuesday. So this strategy has an expected revenue of $0.50.
So is it fair to “say that Monday is consistent with observations, as is Tuesday, and refuse to speculate further”? I think that’s yielding too much ground.
This particular case isn’t a philosophical question about getting information from knowledge of being an observer among all sets of observers. It’s a purely pragmatic analysis that says you’ll always be woken up on a Monday but not always on a Tuesday, so Monday awakenings are simply more probable. That’s the structure of the experiment, and has nothing to do with the anthropic principle.
An analogous experiment that removes the observer is: The experimenter has a bag with only a white ball and a green ball in it. He draws once from the bag randomly, but if he drew the white ball, he draws again. Anyone can tell you that he’ll produce a green ball every round, but a white ball only half the time. If it’s done behind a screen and you’re asked “Was a green ball drawn this round?”, the smart bet is yes.