Ok, but what if Ann's mom is right 99% of the time about how you would choose when playing her?
I would one-box. I gave the relevant numbers on this in my previous comment; one-boxing has an expected value of $990,000,000 to the expected $10,001,000 if you two-box.
I agree that one-boxers make more money, with the numbers you used, but I don't think that those are the appropriate expected values to consider. 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.
When you're dealing with a problem involving an effective predictor of your own mental processes (it's not necessary for such a predictor to be perfect for this reasoning to become salient, it just makes the problems simpler,) your expectation of what the predictor will do or already have done will be at least partly dependent on what you intend to do yourself. You know that either the opaque box is filled, or it is not, but the probability you assign to the box being filled depends on whether you intend to open it or not.
Let's try a somewhat different scenario. Suppose I have a time machine that allows me to travel back a day in the past. Doing so creates a stable time loop, like the time turners in Harry Potter or HPMoR (on a side note, our current models of relativity suggest that such loops are possible, if very difficult to contrive.) You're angry at me because I've insulted your hypothetical scenario, and are considering hitting me in retaliation. But you happen to know that I retaliate against people who hit me by going back in time and stealing from them, which I always get away with due to having perfect alibis (the police don't believe in my time machine.) You do not know whether I've stolen from you or not, but if I have, it's already happened. You would feel satisfied by hitting me, but it's not worth being stolen from. Do you choose to hit me or not?
Another analogous situation would be that you walk into an exam, and the professor (who is a perfect or near-perfect predictor) announces that he has written down a list of people whom he has predicted will get fewer than half the questions right. If you are on that list, he will add 100 points to your score at the end. The people who get fewer than half of the questions right get higher scores, but you should still try to get questions right on the test... right? If not, does the answer change if the professor posts the list on the board?
If the professor is a perfect predictor, then I would deliberately get most of the problems wrong, thereby all but guaranteeing a score of over 100 points. I would have to be very confident that I would get a score below fifty even if I weren't trying to on purpose before trying to get all the questions right would give me a higher expected score than trying to get most of the questions wrong.
If the professor posts the list on the board, then of course it should affect the answer. If my name isn't on the list, then he's not going to add the 100 points to my test in any case, so my only recourse to maximizing my grade is to try my best on the test. If my name is on the list, then he's already predicted that I'm going to score below 50, so whether he's a perfect predictor or not, I should try to do well so that he's adding 100 points to as high a score as I can manage.
The difference between the scenario where he writes the names on the board and the scenario where he doesn't is that in the former, my expectations of his actions don't vary according to my own, whereas in the latter, they do.
Do you choose to hit me or not?
No, I don't, since you have a time-turner. (To be clear, non-hypothetical-me wouldn't hit non-hypothetical-you either.) I would also one-box if I thought that Omega's predictive power was evidence that it might have a time turner or some other way of affecting the past. I still don't think that's relevant when there's no reverse causality.
Back to Newcomb's problem: Say that brown-haired people almost always one-box, and people with other hair colors almost always two-box. Omega predicts on the basis of hair color: both bo...
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