The key point you’ve missed in your analysis, however, is that Omega is almost always correct in his predictions.
It doesn’t matter how Omega does it—that is a separate problem. You don’t have enough information about his process of prediction to make any rational judgment about it except for the fact that it is a very, very good process. Brain scans, reversed causality, time travel, none of those ideas matter. In the paradox as originally posed, all you have are guesses about how he may have done it, and you would be an utter fool to give higher weight to those guesses than to the fact that Omega is always right.
The if observations (that Omega is always right) disagree with theory (that Omega cannot possibly be right), it is the theory that is wrong, every time.
Thus the rational agent should, in this situation, give extremely low weight to his understanding of the way the universe works, since it is obviously flawed (the existence of a perfect predictor proves this). The question really comes down to 100% chance of getting $1000 plus a nearly 0% chance of getting $1.01 million, vs nearly 100% chance of getting $1 million.
What really blows my mind about making the 2-box choice is that you can significantly reduce Omega’s ability to predict the outcome, and unless you are absolutely desperate for that $1000* the 2-box choice doesn’t become superior until Omega is only roughly 50% accurate (at 50.1% the outcome equalizes). Only then do you expect to get more money, on average, by choosing both boxes.
In other words, if you think Omega is doing anything but flipping a coin to determine the contents of box B, you are better off choosing box B.
*I could see the value of $1000 rising significantly if, for example, a man is holding a gun to your head and will kill you in two minutes if you don’t give him $1000. In this case, any uncertainty of Omega’s abilities are overshadowed by the certainty of the $1000. This inverts if the man with the gun is demanding more than $1000 - making the 2-box choice a non-option.
The key point you’ve missed in your analysis, however, is that Omega is almost always correct in his predictions.
It doesn’t matter how Omega does it—that is a separate problem. You don’t have enough information about his process of prediction to make any rational judgment about it except for the fact that it is a very, very good process. Brain scans, reversed causality, time travel, none of those ideas matter. In the paradox as originally posed, all you have are guesses about how he may have done it, and you would be an utter fool to give higher weight to those guesses than to the fact that Omega is always right.
The if observations (that Omega is always right) disagree with theory (that Omega cannot possibly be right), it is the theory that is wrong, every time.
Thus the rational agent should, in this situation, give extremely low weight to his understanding of the way the universe works, since it is obviously flawed (the existence of a perfect predictor proves this). The question really comes down to 100% chance of getting $1000 plus a nearly 0% chance of getting $1.01 million, vs nearly 100% chance of getting $1 million.
What really blows my mind about making the 2-box choice is that you can significantly reduce Omega’s ability to predict the outcome, and unless you are absolutely desperate for that $1000* the 2-box choice doesn’t become superior until Omega is only roughly 50% accurate (at 50.1% the outcome equalizes). Only then do you expect to get more money, on average, by choosing both boxes.
In other words, if you think Omega is doing anything but flipping a coin to determine the contents of box B, you are better off choosing box B.
*I could see the value of $1000 rising significantly if, for example, a man is holding a gun to your head and will kill you in two minutes if you don’t give him $1000. In this case, any uncertainty of Omega’s abilities are overshadowed by the certainty of the $1000. This inverts if the man with the gun is demanding more than $1000 - making the 2-box choice a non-option.