I’m to lazy to work out how, but it seems very easy to work out an infinite series of bets to give her such that she’ll get 0 with certainty no matter what number of times she is woken up if this is the case. And if the experiential really does never end there still never comes a time when she gets to enjoy the money.
Actually, you’re right about the infinite series of bets. Let N be the number of times Sleeping Beauty is to be woken up. Suppose (edit: on each day she wakes up) Sleeping Beauty is offered the following bets:
$10 if N=1, or $1 otherwise.
$10 if N=2, or $0.50 otherwise.
$10 if N=3, or $0.25 otherwise.
$10 if N=4, or $0.125 otherwise.
And so on.
In each individual bet, the second option has an infinite expectation, while the first has a finite expectation. However, if Sleeping Beauty accepts all the first options, she gets $10 every day she wakes up, for a total of $10N; if Sleeping Beauty accepts all the second options, she gets less than $2 every day she wakes up, for a total of $2N. Even though both options yield infinite expected money, this is still clearly inferior.
I suspect though, that this is a problem with the infinite nature of the experiment, not with Sleeping Beauty’s betting preferences.
That’s not what I meant. I meant… ugh I’m really tired right now and can’t think straight.
maybe:
Pot starts at 1$, each iteration she bets the pot against adding one dollar to it if N is greater than the number of iterations so far, with if needed the extra rule that if she gets woken up an infinite number of time she really gets infinite $.
To sleep deprived to check if the math actually works out like I think it does.
I don’t believe the first point, but I’m not entirely certain you’re wrong, so if you think you have such a construction, I’d like to see it.
As for your second point, the number of times that Sleeping Beauty wakes up is always finite, so no matter what, the experiment does end. It’s just that, due to the heavy tail of the distribution, the expected value is infinite (see also: St. Petersburg paradox). Of course, we would have to adjust rewards for inflation; also, the optimal strategy changes if the universe (or Sleeping Beauty) has a finite lifespan. So there’s a few implementation problems here, yes.
I’m to lazy to work out how, but it seems very easy to work out an infinite series of bets to give her such that she’ll get 0 with certainty no matter what number of times she is woken up if this is the case. And if the experiential really does never end there still never comes a time when she gets to enjoy the money.
Actually, you’re right about the infinite series of bets. Let N be the number of times Sleeping Beauty is to be woken up. Suppose (edit: on each day she wakes up) Sleeping Beauty is offered the following bets:
$10 if N=1, or $1 otherwise.
$10 if N=2, or $0.50 otherwise.
$10 if N=3, or $0.25 otherwise.
$10 if N=4, or $0.125 otherwise.
And so on.
In each individual bet, the second option has an infinite expectation, while the first has a finite expectation. However, if Sleeping Beauty accepts all the first options, she gets $10 every day she wakes up, for a total of $10N; if Sleeping Beauty accepts all the second options, she gets less than $2 every day she wakes up, for a total of $2N. Even though both options yield infinite expected money, this is still clearly inferior.
I suspect though, that this is a problem with the infinite nature of the experiment, not with Sleeping Beauty’s betting preferences.
That’s not what I meant. I meant… ugh I’m really tired right now and can’t think straight.
maybe:
Pot starts at 1$, each iteration she bets the pot against adding one dollar to it if N is greater than the number of iterations so far, with if needed the extra rule that if she gets woken up an infinite number of time she really gets infinite $.
To sleep deprived to check if the math actually works out like I think it does.
I don’t believe the first point, but I’m not entirely certain you’re wrong, so if you think you have such a construction, I’d like to see it.
As for your second point, the number of times that Sleeping Beauty wakes up is always finite, so no matter what, the experiment does end. It’s just that, due to the heavy tail of the distribution, the expected value is infinite (see also: St. Petersburg paradox). Of course, we would have to adjust rewards for inflation; also, the optimal strategy changes if the universe (or Sleeping Beauty) has a finite lifespan. So there’s a few implementation problems here, yes.