Conditional on the fact that the boxes have already been filled, two-boxing has a $1000 higher expected value. If I know only one box is filled, I should take both. If I know both boxes are filled, I should take both. If I know I’m in one of those situations but not sure of which it is, I should still take both.
You are focusing too much on the “already have been filled”, as if the particular time of your particular decision is relevant. But if your decision isn’t random (and yours isn’t), then any individual decision is dependent on the decision algorithm you follow—and can be calculated in exactly the same manner, regardless of time. Therefore in a sense your decision has been made BEFORE the filling of the boxes, and can affect their contents.
You may consider it easier to wrap your head around this if you think of the boxes being filled according to what result the decision theory you currently have would return in the situation, instead of what decision you’ll make in the future. That helps keep in mind that causality still travels only one direction, but that a good predictor simply knows the decision you’ll make before you make it and can act accordingly.
You are focusing too much on the “already have been filled”, as if the particular time of your particular decision is relevant. But if your decision isn’t random (and yours isn’t), then any individual decision is dependent on the decision algorithm you follow—and can be calculated in exactly the same manner, regardless of time. Therefore in a sense your decision has been made BEFORE the filling of the boxes, and can affect their contents.
You may consider it easier to wrap your head around this if you think of the boxes being filled according to what result the decision theory you currently have would return in the situation, instead of what decision you’ll make in the future. That helps keep in mind that causality still travels only one direction, but that a good predictor simply knows the decision you’ll make before you make it and can act accordingly.