Thanks for presenting my take on Sleeping Beauty. Your generalization beyond my assumption that Beauty’s observations on Monday/Tuesday are independent and low-probability are interesting.
I’m not as dismissive as you are of betting arguments. You’re right, of course, that a betting argument for something having some probability could be disputed by someone who doesn’t accept your ideas of decision theory. But since typically lots of people will agree with with your ideas of decision theory, it may be persuasive to some.
Now, I have to admit that I have personally failed in this respect, responding to this post by Lubos Motl:
I think I failed to get Lubos to even consider my argument, but it might be of interest to people here. Here’s an except from my comment there:
--- (start excerpt)
...modify the usual problem so that Beauty can send a message to her brother recommending how to bet (on the coin having landed H or T), which will be received only after Tuesday. If she sends two messages, only the second one will be received. Let’s suppose that it’s known to everyone from the beginning that there will be exactly two bets on offer—one where one wins $90 on T and loses $100 on H, and the other where one wins $90 on H and loses $100 on T. Should Beauty recommend to her brother one, or the other, or neither of these bets? (Let’s assume that her brother is very likely to follow her recommendation, whatever it is.)
I think we agree that Beauty should recommend that her brother take neither of these bets, which both have a negative expected payoff if H and T are equally likely.
However, I think that this is the conclusion Beauty will reach by following the THIRDER logic. If she follows the HALFER logic, she will do the wrong thing.
If Beauty is a thirder, she thinks upon awakening that there are three possibilities, with the following probabilities:
a) It’s Monday, and the coin landed heads (probability 1⁄3).
b) It’s Monday, and the coin landed tails (probability 1⁄3).
c) It’s Tuesday, and the coin landed tails (probability 1⁄3).
Now, when deciding what action to take, Beauty should consider only those possibilities in which her action actually makes a difference. That eliminates (b), since the message sent in that circumstance will be wiped out by the message that will be sent on Tuesday. The remaining possibilities are (a) and (c), which are equally likely. So the recommendation should be based on H and T being equally likely, which means it should recommend taking neither bet.
In contrast, if Beauty is a halfer, she sees the three possibilities as having the following probabilites:
a) It’s Monday, and the coin landed heads (probability 1⁄2).
b) It’s Monday, and the coin landed tails (probability 1⁄4).
c) It’s Tuesday, and the coin landed tails (probability 1⁄4).
Again, (b) should be ignored, since it makes no difference what Beauty does in that circumstance. That leaves (a) and (c), with (a) having twice the probability as (c). So Beauty—if she is a halfer—should recommend taking the bet that pays $90 on H and loses $100 on T, since it has a positive expected return. But of course this is the wrong answer.
--- (end excerpt)
By the way, what did you think of my “Sailor’s Child” version of the problem? I personally thought that it should put the whole issue to rest in a definitive fashion, but then, that’s a typical view of people making philosophical arguments that don’t in fact end the debate...
I don’t understand how the halfer makes the wrong bet. If we are talking about probabilities, then a), b) and c) all have a probability of 1⁄2 of occurring. If we want the probabilities to sum to 1, then we need to do the following:
If heads occurs, Monday always “counts”
If tails occurs, we need to flip a second coin to determine if Monday or Tuesday “counts”.
So b) is “It’s Monday, and the coin landed tails and Monday counts”
And c) is “It’s Tuesday, and the coin landed tails and Tuesday counts”
So b) and c) are exclusive, so c) doesn’t override b).
On the other hand, if b) and c) aren’t exclusive, then then are both 0.5 instead. So b) being ignored wouldn’t matter as c) would suffice by itself.
The only way we get the wrong answer is if b) and c) overlap and are not 0.5. This makes no sense for the halfer model.
I agree that the Sailor’s Child is the correct translation of Sleeping Beauty into a situation with no copies (Do you remember Psy-Kosh’s non-anthropic problem?), but I think some people might even deny that any such translation exists.
Don’t worry about not being able to convince Lubos Motl. His prior for being correct is way too high and impedes his ability to consider dissenting views seriously.
1. Traditional CDT (causal decision theory) breaks down in unusual situations. The standard example is the Newcomb Problem, and various alternatives have been proposed, such as Functional Decision Theory. The Sleeping Beauty problem presents another highly unusual situation that should make one wary of betting arguments.
2. There is disagreement as to how to apply decision theory to the SB problem. The usual thirder betting argument assumes that SB fails to realize that she is going to both make the same decision and get the same outcome on Monday and Tuesday. It has been argued that accounting for these facts means that SB should instead compute her expected utility for accepting the bet as
Pr(H)⋅payoff(H)+2Pr(T)payoff(T).
3. Your own results show that the standard betting argument gets the wrong answer of 1⁄3, when the correct answer is 1/(3−p(x)). At best, the standard betting argument gets close to the right answer; but if Beauty is sensorily impoverished, or has just awakened, then p(x) can be sufficiently large that the answer deviates substantially from 1/3.
BTW, I was a solid halfer until I read your paper. It was the first and only explanation I’ve ever seen of how Beauty’s state of information after awakening on Monday/Tuesday differs from her state of information on Sunday night in a way that affects the probability of Heads.
With regards to your “Sailor’s Child” problem:
It was not immediately obvious to me that this is equivalent to the SB problem. I had to think about it for some time, and I think there are some differences. One is, again the different answers of 1/3 versus 1/(3−p(x)). I’ve concluded that the SC problem is equivalent to a variant of the SB problem where (1) we’ve guaranteed that Beauty cannot experience the same thing on both Monday and Tuesday, and (2) there is a second coin toss that determines whether Beauty is awakened on Monday or on Tuesday in the case that the first coin toss comes up Heads.
In any event, it was the calculation based on Beauty’s new information upon awakening that I found convincing. I tried to disprove it, and couldn’t.
Thanks for presenting my take on Sleeping Beauty. Your generalization beyond my assumption that Beauty’s observations on Monday/Tuesday are independent and low-probability are interesting.
I’m not as dismissive as you are of betting arguments. You’re right, of course, that a betting argument for something having some probability could be disputed by someone who doesn’t accept your ideas of decision theory. But since typically lots of people will agree with with your ideas of decision theory, it may be persuasive to some.
Now, I have to admit that I have personally failed in this respect, responding to this post by Lubos Motl:
https://motls.blogspot.ca/2015/08/sleeping-beauty-betting-assisting.html
I think I failed to get Lubos to even consider my argument, but it might be of interest to people here. Here’s an except from my comment there:
--- (start excerpt)
...modify the usual problem so that Beauty can send a message to her brother recommending how to bet (on the coin having landed H or T), which will be received only after Tuesday. If she sends two messages, only the second one will be received. Let’s suppose that it’s known to everyone from the beginning that there will be exactly two bets on offer—one where one wins $90 on T and loses $100 on H, and the other where one wins $90 on H and loses $100 on T. Should Beauty recommend to her brother one, or the other, or neither of these bets? (Let’s assume that her brother is very likely to follow her recommendation, whatever it is.)
I think we agree that Beauty should recommend that her brother take neither of these bets, which both have a negative expected payoff if H and T are equally likely.
However, I think that this is the conclusion Beauty will reach by following the THIRDER logic. If she follows the HALFER logic, she will do the wrong thing.
If Beauty is a thirder, she thinks upon awakening that there are three possibilities, with the following probabilities:
a) It’s Monday, and the coin landed heads (probability 1⁄3).
b) It’s Monday, and the coin landed tails (probability 1⁄3).
c) It’s Tuesday, and the coin landed tails (probability 1⁄3).
Now, when deciding what action to take, Beauty should consider only those possibilities in which her action actually makes a difference. That eliminates (b), since the message sent in that circumstance will be wiped out by the message that will be sent on Tuesday. The remaining possibilities are (a) and (c), which are equally likely. So the recommendation should be based on H and T being equally likely, which means it should recommend taking neither bet.
In contrast, if Beauty is a halfer, she sees the three possibilities as having the following probabilites:
a) It’s Monday, and the coin landed heads (probability 1⁄2).
b) It’s Monday, and the coin landed tails (probability 1⁄4).
c) It’s Tuesday, and the coin landed tails (probability 1⁄4).
Again, (b) should be ignored, since it makes no difference what Beauty does in that circumstance. That leaves (a) and (c), with (a) having twice the probability as (c). So Beauty—if she is a halfer—should recommend taking the bet that pays $90 on H and loses $100 on T, since it has a positive expected return. But of course this is the wrong answer.
--- (end excerpt)
By the way, what did you think of my “Sailor’s Child” version of the problem? I personally thought that it should put the whole issue to rest in a definitive fashion, but then, that’s a typical view of people making philosophical arguments that don’t in fact end the debate...
Finally, note that a partially revised version of my paper is available via http://www.cs.utoronto.ca/~radford/anth.abstract.html
I don’t understand how the halfer makes the wrong bet. If we are talking about probabilities, then a), b) and c) all have a probability of 1⁄2 of occurring. If we want the probabilities to sum to 1, then we need to do the following:
If heads occurs, Monday always “counts”
If tails occurs, we need to flip a second coin to determine if Monday or Tuesday “counts”.
So b) is “It’s Monday, and the coin landed tails and Monday counts”
And c) is “It’s Tuesday, and the coin landed tails and Tuesday counts”
So b) and c) are exclusive, so c) doesn’t override b).
On the other hand, if b) and c) aren’t exclusive, then then are both 0.5 instead. So b) being ignored wouldn’t matter as c) would suffice by itself.
The only way we get the wrong answer is if b) and c) overlap and are not 0.5. This makes no sense for the halfer model.
I agree that the Sailor’s Child is the correct translation of Sleeping Beauty into a situation with no copies (Do you remember Psy-Kosh’s non-anthropic problem?), but I think some people might even deny that any such translation exists.
Don’t worry about not being able to convince Lubos Motl. His prior for being correct is way too high and impedes his ability to consider dissenting views seriously.
In regards to betting arguments:
1. Traditional CDT (causal decision theory) breaks down in unusual situations. The standard example is the Newcomb Problem, and various alternatives have been proposed, such as Functional Decision Theory. The Sleeping Beauty problem presents another highly unusual situation that should make one wary of betting arguments.
2. There is disagreement as to how to apply decision theory to the SB problem. The usual thirder betting argument assumes that SB fails to realize that she is going to both make the same decision and get the same outcome on Monday and Tuesday. It has been argued that accounting for these facts means that SB should instead compute her expected utility for accepting the bet as
3. Your own results show that the standard betting argument gets the wrong answer of 1⁄3, when the correct answer is 1/(3−p(x)). At best, the standard betting argument gets close to the right answer; but if Beauty is sensorily impoverished, or has just awakened, then p(x) can be sufficiently large that the answer deviates substantially from 1/3.
BTW, I was a solid halfer until I read your paper. It was the first and only explanation I’ve ever seen of how Beauty’s state of information after awakening on Monday/Tuesday differs from her state of information on Sunday night in a way that affects the probability of Heads.
With regards to your “Sailor’s Child” problem:
It was not immediately obvious to me that this is equivalent to the SB problem. I had to think about it for some time, and I think there are some differences. One is, again the different answers of 1/3 versus 1/(3−p(x)). I’ve concluded that the SC problem is equivalent to a variant of the SB problem where (1) we’ve guaranteed that Beauty cannot experience the same thing on both Monday and Tuesday, and (2) there is a second coin toss that determines whether Beauty is awakened on Monday or on Tuesday in the case that the first coin toss comes up Heads.
In any event, it was the calculation based on Beauty’s new information upon awakening that I found convincing. I tried to disprove it, and couldn’t.