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Desrtopa comments on Timeless Decision Theory: Problems I Can't Solve - Less Wrong
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Fine if you think that Omega would have told me about the previous coin flip consider this:
There are two different supernatural entities who can correctly predict my response to the counterfactual mugging. There's Omega and Z.
Two things could theoretically happen to me:
a) Omega could present me with the counterfactual mugging problem.
b) Z could decide to steal $1000000 from me if and only if I would have given Omega $1000 in the counterfactual mugging.
When I am trying to decide on policy for dealing with counterfactual muggings I should note that my policy will affect my outcome in both situation (a) and (b). The policy of giving Omega money will win me $499500 (expected) in situation (a), but it will lose me $1000000 in situation (b). Unless I have a reason to suspect that (a) is at least twice as likely as (b), I have no reason to prefer the policy of giving Omega money.
The basis of the dilemma is that you know that Omega, who is honest about the dilemmas he presents, exists. You have no evidence that Z exists. You can posit his existence, but it doesn't make the dilemma symmetrical.
But if instead Z exists, shows up on your doorstep and says (in his perfectly trustworthy way) "I will take your money if and only if you would have given money to Omega in the counterfactual mugging", then you have evidence that Z exists but no evidence that Omega does.
The point is that you need to make your policy before either entity shows up. Therefore unless you have evidence now that one is more likely than the other, not paying Omega is the better policy (unless you think of more hypothetical entities).