If this was a normal game-theory situation, you wouldn’t easily know what to do—your best move depends on Omega’s move.
I would! If this were a normal game theory situation, I’d simply observe that I’m $1k better off regardless of Omega’s decision if I 2-box, and I’d 2-box with a smile on my face.
Of course, Newcomb’s problem complicates things. When Omega knows what I’ll play, the payoff matrix in the post is no longer accurate, because Omega’s ability to predict my play eliminates the ($1M, 2-box) and ($0, 1-box) moves. Hence the only possible plays are those on the main diagonal, and 1-boxing therefore dominates 2-boxing. (And, to head off the usual distraction at the pass, I’m pretty sure it doesn’t matter if Omega only has a 99.9999999999% chance of correctly guessing my move; it just means you have to give up on the lossy normal form description of the game, and switch to the extensive form.)
I would! If this were a normal game theory situation, I’d simply observe that I’m $1k better off regardless of Omega’s decision if I 2-box, and I’d 2-box with a smile on my face.
Of course, Newcomb’s problem complicates things. When Omega knows what I’ll play, the payoff matrix in the post is no longer accurate, because Omega’s ability to predict my play eliminates the ($1M, 2-box) and ($0, 1-box) moves. Hence the only possible plays are those on the main diagonal, and 1-boxing therefore dominates 2-boxing. (And, to head off the usual distraction at the pass, I’m pretty sure it doesn’t matter if Omega only has a 99.9999999999% chance of correctly guessing my move; it just means you have to give up on the lossy normal form description of the game, and switch to the extensive form.)