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twanvl comments on Notes on logical priors from the MIRI workshop - Less Wrong
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A priori I wouldn't trust Omega to be fair. I only know that he doesn't lie. If Omega also said that he chose the logical statement in some fair way, then that would assure me the logical coin is identical to a normal coin. He can do this either using real uncertainty, like rolling a die to pick from a set of statements where half of them are true. Or he could use logical uncertainty himself, by not calculating the digit of pi before deciding to make the bet, and having a prior that assigns 50% probability to either outcome.
For what it's worth, the post assumes that Omega decides to participate in the game unconditionally, its code doesn't have a branch saying it should play only if such-and-such conditions are met. I'm not sure if that answers your question.