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PhilGoetz comments on Exterminating life is rational - Less Wrong
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So does your answer change once you've drawn 10 cards and are still alive?
No, if my guess is correct, then some time before I'm offered the 11th card, Omega will say "I can't double your utility again" or equivalently, "There is no prize I can offer you such that you'd prefer a .5 probability of it to keeping what you have."