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.
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.