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mendel comments on Counterfactual Mugging - Less Wrong
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Your version of the story discards the most important ingredient: The fact that when you win the coin toss, you only receive money if you would have paid had you lost.
As for Omega, all we know about it is that somehow it can accurately predict your actions. For the purposes of Counterfactual Mugging we may as well regard Omega as a mindless robot which will burn the money you give to it and then self-destruct immediately after the game. (This makes it impossible to pay because you feel obligated to Omega. In fact, the idea is that you pay up because you feel obligated to your counterfactual self.)