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Vladimir_Nesov comments on Counterfactual Calculation and Observational Knowledge - Less Wrong
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So that's what happens when you don't describe what strategy you're computing expected utility of in enough detail in advance. By problem statement, the calculator in the world in which Omega showed shows "even".
But even if you expect Omega to appear on either side, this still isn't right. Where's the probability of Omega appearing on either side in your calculation? The event of Omega appearing on one or the other side must enter the model, and it wasn't explicitly referenced in any of your formulas.
But implicitly.
P(Omega_in_Odd_world)=P(Omega_in_Even_world)=0.5, but
P(Omega_in_Odd_world|Odd)= P(Omega_in_Even_world|Even)=1
And since every summand includes a P(Odd n X) or a P(Even n X) everything is already multiplied with P(Even) or P(Odd) as appropriate. In retrospect it would have been a lot clearer if I had factored that out, but I wrote U_not_replace first in the way that seemed most obvious and merely modified that to U_replace so it never occured to me to do that.
Omega visits either the "odd" world or "even" world, not Odd world or Even world. For example, in Odd world it'd still need to decide between "odd" and "even".
That's what multiplying with P("odd"|Odd) etc was about. (the probability that, given Omega appearing in an Odd world it would appear in an "odd" world). I thought I explained that?