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Eponymuse comments on Counterfactual Mugging - Less Wrong
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The only mechanisms I know of by which Omega can accurately predict me without introducing paradoxes is by running something like a simulation, as others have suggested. But I really, truly, only care about the universe I happen to know about, and for the life of me, I can't figure out why I should care about any other. So even if the universe I perceive really is just simulated so that Omega can figure out what I would do in this situation, I don't understand why I should care about "my" utility in some other universe. So, two box, keep my $100.
Edit: I should add that my not caring about other universes is conditional on my having no reason to believe they exist.
Ah. But under mild assumptions about how Omega's simulation works, I can expect that with some probability p bounded away from zero, I am in a simulation. So with probability at least p, there is another universe I care about, and I can increase utility there.
So, I guess I do pay $100, but only because my utility function values the utility of others. I remain unconvinced that paying is winning for someone with a different utility function.