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Manfred comments on The Ultimate Sleeping Beauty Problem - Less Wrong Discussion
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Comments (39)
50%. Upon finding that the expected physical situation is undefined (a limit that does not converge), sensible agents should default to using a more limited set of information.
EDIT: All right then, if you downvoters are so smart, what would you bet if you were in sleeping beauty's place?
So if I ask you if you think I can hit immeasurable set A on a dartboard, you'd say 50%. Same with disjoint immeasurable set B. Same with A U B. I now offer to bet with 1:1 odds that you can hit A, can hit B and can't hit A U B. If you hit A, you win the first bet, but lose the second two. If you hit B, you win the second, but lose the other two. If you miss both, you lose all three. No matter what, I get money.
Hm. The simplest way around this is to treat the fact that an immeasurable disjoint set B exists as new information to our agent. E.g. if you just tell me to bet on hitting some immeasurable set A, I'll think the possibilities are just (A) or (not A), and in my state of ignorance will bet at 1:1 odds. But if you then tell me there's some disjoint set B, now the possibilities are (A), (B), (neither). Maximum entropy dictates that I only assign a 1/3 probability to hitting A or B. This handles the dutch book correctly.
If you add knowledge about relationships between a jillion more immeasurable sets, it still produces sensible answers. The biggest trouble I can see is that representing the things we know about relationships between immeasurable sets in this way is tedious.
This is a fair point. Your's is an attempt at a real answer to the problem. Mine and most answers here seem to say something like that the problem is ill-defined, or that the physical situation described by the problem is impossible. But if you were actually Sleeping Beauty waking up with a high prior to trust the information you've been given, what else could you possibly answer?
If you had little reason to trust the information you've been given, the apparent impossibility of your situation would update that belief very strongly.