Well, if your decision theory tells you you ought to be the sort of person who would pay up in a Counterfactual Mugging, because that gets you positive utility, then you could end up in with Omega coming and saying "I would have given you a million dollars if your decision theory said not to pay out in a counterfactual mugging, but since you would, you don't get anything."
When you know nothing about Omega, I don't think there's any positive expected utility in choosing to be the sort of person who would have positive expected utility in a Counterfactual Mugging scenario, because you have no reason to suspect it's more likely than the inverted scenario where being that sort of person will get you negative utility. The probability distribution is flat, so the utilities cancel out.
Say Omega comes to you with a Counterfactual Mugging on Day 1. On Day 0, would you want to be the sort of person who pays out in a Counterfactual Mugging? No, because the probabilities of it being useful or harmful cancel out. On Day 1, when given the dilemma, do you want to be the sort of person who pays out in a Counterfactual Mugging? No, because now it only costs you money and you get nothing out of it.
So there's no point in time where deciding "I should be the sort of person who pays out in a Counterfactual Mugging" has positive expected utility.
Reasoning this way means, of course, that you don't get the money in a situation where Omega would only pay you if it predicted you would pay up, but you do get the money in situations where Omega pays out only if you wouldn't pay out. The latter possibility seems less salient from the "before you do that-" standpoint of a person contemplating a Counterfactual Mugging, but there's no reason to assign it a lower probability before the fact. The best you can do is choose according to whatever has the highest expected utility at any given time.
Omega could also come and tell me "I decided that I would steal all your money if you hit the S key on your keyboard between 10:00-11:00 am on a Sunday, and you just did," but I don't let this influence my typing habits. You don't want to alter your decision theories or general behavior in advance of specific events that are no more probable than their inversions.
So there's no point in time where deciding "I should be the sort of person who pays out in a Counterfactual Mugging" has positive expected utility.
Sure, I agree.
What I'm suggesting is that "I should be the sort of person who does the thing that has positive expected utility" causes me to pay out in a Counterfactual Mugging, and causes me to not pay out in a Counterfactual Antimugging, without requiring any prophecy. And that as far as I know, this is representative of the locally bandied-around solutions to decision-theory problems....
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