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steven0461 comments on St. Petersburg Mugging Implies You Have Bounded Utility - Less Wrong
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Why should I not attach a probability of zero to the claim that you are able to grant unbounded utility?
Let GOD(N) be the claim that you are a god with the power to grant utility at least up to 2**N. Let P(GOD(N)) be the probability I assign to this. This is a nonincreasing function of N, since GOD(N+1) implies GOD(N).
If I assign a probability to GOD(N) of 4**(-N), then the mugging fails. Of course, this implies that I have assigned GOD(infinity), the conjunction of GOD(N) over all N, a probability of zero, popularly supposed to be a sin. But while I can appreciate the reason for not assigning zero to ordinary, finite claims about the world, such as the existence of an invisible dragon in your garage, I do not see a reason to avoid this zero.
If extraordinary claims demand extraordinary evidence, what do infinite claims require?
Even if you do assign zero probability, what makes you think that in this specific case zero times infinity should be thought of as zero?
Because otherwise you get mugged.
You don't literally multiply 0 by infinity, of course, you take the limit of (payoff of N)*probability(you actually get that payoff) as N goes to infinity. If that limit blows up, there's something wrong with either your probabilities or your utilities. Bounding the utility is one approach; bounding the probability is another.