Sleeping Beauty should believe in a 1⁄3 probability of heads, but this doesn’t mean she should guess “heads” 1⁄3 of the time, and “tails” 2⁄3 of the time. It makes no sense to ever go with the option that has a smaller chance of winning, if the object is to correctly guess the coin toss as often as possible.
I haven’t checked the math, but if I remember my information theory correctly, the proper way to elicit an accurate prediction from Sleeping Beauty is the following: ask her for a credence P that the coin is heads. If the coin is heads, she gets $1000 log(P). If the coin is tails, she gets $1000 log(1-P).
Since 0<P<1, she’s losing money either way, so if that bothers you, pay her some constant amount of money every time you do this to make up for it.
P. S. See “Scoring rule” on Wikipedia for the more general case.
So to elicit honest credences, scale the payoff by the log of the credence. And the whole problem here was how to elicit honest credences from Sleeping Beauty. You just solved my whole problem, thanks!
Also, I think it’s time for me to reread Technical Explanation.
Sleeping Beauty should believe in a 1⁄3 probability of heads, but this doesn’t mean she should guess “heads” 1⁄3 of the time, and “tails” 2⁄3 of the time. It makes no sense to ever go with the option that has a smaller chance of winning, if the object is to correctly guess the coin toss as often as possible.
I haven’t checked the math, but if I remember my information theory correctly, the proper way to elicit an accurate prediction from Sleeping Beauty is the following: ask her for a credence P that the coin is heads. If the coin is heads, she gets $1000 log(P). If the coin is tails, she gets $1000 log(1-P).
Since 0<P<1, she’s losing money either way, so if that bothers you, pay her some constant amount of money every time you do this to make up for it.
P. S. See “Scoring rule” on Wikipedia for the more general case.
So to elicit honest credences, scale the payoff by the log of the credence. And the whole problem here was how to elicit honest credences from Sleeping Beauty. You just solved my whole problem, thanks!
Also, I think it’s time for me to reread Technical Explanation.