P(“odd”|Odd)*( P(“odd” n Odd)*100 + P(“even” n Odd)*100) + P(“even”|Odd)*( P(“odd” n Odd)*0 + P(“even” n Odd)*0) is the utility for the half of imaginable worlds where Q is odd (all possible worlds if Q is odd).
Consider expected utility [P(“odd” n Odd)*100 + P(“even” n Odd)*100)] from your formula. What event and decision is this the expected utility of? It seems to consider two events, [“odd” n Odd] and [“even” n Odd]. For both of them to get 100 utils, the strategy (decision) you’re considering must be, always answer-odd (since you can only answer in response to indication on the calculators, and here we have both indications and the same answer necessary for success in both events).
But U_replace estimates the expected utility of a different strategy, of strategy where you answer-even on your own “even” branch and also answer-even on the “odd” branch with Omega’s help. So you’re already computing something different.
Then, in the same formula, you have [P(“odd” n Odd)*0 + P(“even” n Odd)*0]. But to get 0 utils in both cases, you have to answer incorrectly in both cases, and since we’re considering Odd, this must be unconditional answer-even. This contradicts the way you did your expected utility calculation in the first terms of the formula (where you were considering the strategy of unconditional answer-odd).
Expected utility is computed for one strategy at a time, and values of expected utility computed separately for each strategy are used to compare the strategies. You seem to be doing something else.
Expected utility is computed for one strategy at a time, and values of expected utility computed separately for each strategy are used to compare the strategies. You seem to be doing something else.
I’m calculating for one strategy, the strategy of “fill in whatever the calculator in the world Omega appeared in showed”, but I have a probability distribution across what that entails (see my other reply). I’m multiplying the utility of picking “odd” with the probability of picking “odd” and the utility of picking “even” with the probability of picking “even”.
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.
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?
Consider expected utility [P(“odd” n Odd)*100 + P(“even” n Odd)*100)] from your formula. What event and decision is this the expected utility of? It seems to consider two events, [“odd” n Odd] and [“even” n Odd]. For both of them to get 100 utils, the strategy (decision) you’re considering must be, always answer-odd (since you can only answer in response to indication on the calculators, and here we have both indications and the same answer necessary for success in both events).
But U_replace estimates the expected utility of a different strategy, of strategy where you answer-even on your own “even” branch and also answer-even on the “odd” branch with Omega’s help. So you’re already computing something different.
Then, in the same formula, you have [P(“odd” n Odd)*0 + P(“even” n Odd)*0]. But to get 0 utils in both cases, you have to answer incorrectly in both cases, and since we’re considering Odd, this must be unconditional answer-even. This contradicts the way you did your expected utility calculation in the first terms of the formula (where you were considering the strategy of unconditional answer-odd).
Expected utility is computed for one strategy at a time, and values of expected utility computed separately for each strategy are used to compare the strategies. You seem to be doing something else.
I’m calculating for one strategy, the strategy of “fill in whatever the calculator in the world Omega appeared in showed”, but I have a probability distribution across what that entails (see my other reply). I’m multiplying the utility of picking “odd” with the probability of picking “odd” and the utility of picking “even” with the probability of picking “even”.
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?