Actually I take it back. I think that what I would do depends on what I know of how Omega functions (exactly what evidence lead me to believe that he was good at predicting this).
Omega #1: (and I think this one is the most plausible) You are given a multiple choice personality test (not knowing what’s about to happen). You are then told that you are in a Newcomb situation and that Omega’s prediction is based on your test answers (maybe they’ll even show you Omega’s code after the test is over). Here I’ll two-box. If I am punished I am not being punished for my decision to two-box, I am being punished for my test answers, and in reality am probably being punished for having personality traits that correlate well with being a two-boxer. I can rationally regret having the wrong personality traits.
Omega #2: You are sent through the Newcomb dilemma, given an amnesia pill and then sent through for real. Omega’s prediction is whatever you did the first time (this is similar to the simulation case). If I know this is going on, I clearly one-box because I don’t know whether this is the first time through or the second time through.
Omega #3: Omega makes his prediction by observing me and using a time machine. Clearly I one-box.
Omega #4: It is inscribed in the laws of physics somewhere the Omega cannot make a prediction that comes out wrong. Clearly I one-box.
But I think that the problem as stated is ill posed since I don’t know what my probability distribution over Omegas should be (given that it depends a lot on exactly what evidence convinces me that Omega is actually a good predictor).
1) I would one-box. Here’s where I think the standard two-boxer argument breaks down. It’s the idea of making a decision. The two-boxer idea is that once the boxes have been fixed the course of action that makes the most money is taking both boxes. Unless there is reverse causality going on here, I don’t think that anyone disputes this. If at that moment you could make a choice totally independently of everything leading up to that point you would two-box. Unfortunately, the very existence of Omega implies that such a feat is impossible.
2) A mildly silly argument for one-boxing: Omega plausibly makes his decision by running a simulation of you. If you are the real copy, it might be best to two-box, but if you are the simulation then one-boxing earns real-you $1000000. Since you can’t distinguish whether this is real-you or simulation-you, you should one-box.
3) Would it change things for people if instead of $1000000 vs $1000 it were $1001 vs $1000? Where is the line drawn?
4) Eliezer: just curious about how you deal with paradoxes about infinity in your utility function. If for each n, on day n you are offered to sacrifice one unit of utility that day to gain one unit of utility on day 2n and one unit on day 2n+1 what do you do? Each time you do it you seem to gain a unit of utility, but if you do it every day you end up worse than you started.