Of all the comments in this block, byrnema's seems the most on-track, having the most ingredients of the solution, in my view. A few points:
I prefer to suppose that Omega has a powerful, detailed model of the local world, or whatever parts of the universe are ultimately factors in Joe's decision. It isn't just the contents of Joe's brain. Omega's track record is strong evidence that his model takes enough into account.
I do not see any backwards-in-time causality in this problem at all. That Joe's state causes both Omega's prediction and Joe's choice is not the same as the choice causing the prediction.
In fact, that's what seems wrong to me about most of the other comments right here. People keep talking about the choice causing something, but the problem says nothing about this at all. Joe's choice doesn't need to cause anything. Instead, Joe's choice and Omega's (prediction->money-hiding) have common causes.
The way I see it, the sleight-of-hand in this problem occurs when we ask what Joe "should" do. I think focusing on Joe's choice leads people to imagine that the choice is free in the sense of being unconnected to Omega's prediction (since the prediction has already happened). But it is not unconnected, because our choices are not un-caused. Neither are they connected backwards-in-time. Omega's actions and Joe's choice are connected because they share common causes.
EDIT: To make this a bit more concrete: Make this a question of what you "should" do if you meet Omega someday. Consider that your decision might be highly influenced by all the musings on the blog, or on Eliezer's or another poster's arguments. If these arguments convince you that you should one-box, then they also cause Omega to predict that you'll one-box. If these arguments fail to convince you, then that circumstance also causes Omega to predict you will two-box.
You've got to resist thinking of the machinery of human decision-making as primary or transcendent. See Thou Art Physics.
(This is the first, and most newcomer-accessible, post in a planned sequence.)
Newcomb's Problem:
Joe walks out onto the square. As he walks, a majestic being flies by Joe's head with a box labeled "brain scanner", drops two boxes on the ground, and departs the scene. A passerby, known to be trustworthy, comes over and explains...
If Joe aims to get the most money, should Joe take one box or two?
What are we asking when we ask what Joe "should" do? It is common to cash out "should" claims as counterfactuals: "If Joe were to one-box, he would make more money". This method of translating "should" questions does seem to capture something of what we mean: we do seem to be asking how much money Joe can expect to make "if he one-boxes" vs. "if he two-boxes". The trouble with this translation, however, is that it is not clear what world "if Joe were to one-box" should refer to -- and, therefore, it is not clear how much money we should say Joe would make, "if he were to one-box". After all, Joe is a deterministic physical system; his current state (together with the state of his future self's past light-cone) fully determines what Joe's future action will be. There is no Physically Irreducible Moment of Choice, where this same Joe, with his own exact actual past, "can" go one way or the other.
To restate the situation more clearly: let us suppose that this Joe, standing here, is poised to two-box. In order to determine how much money Joe "would have made if he had one-boxed", let us say that we imagine reaching in, with a magical sort of world-surgery, and altering the world so that Joe one-boxes instead. We then watch to see how much money Joe receives, in this surgically altered world.
The question before us, then, is what sort of magical world-surgery to execute, before we watch to see how much money Joe "would have made if he had one-boxed". And the difficulty in Newcomb’s problem is that there is not one but two obvious world-surgeries to consider. First, we might surgically reach in, after Omega's departure, and alter Joe's box-taking only -- leaving Omega's prediction about Joe untouched. Under this sort of world-surgery, Joe will do better by two-boxing:
Expected value ( Joe's earnings if he two-boxes | some unchanged probability distribution on Omega's prediction ) >
Expected value ( Joe's earnings if he one-boxes | the same unchanged probability distribution on Omega's prediction ).
Second, we might surgically reach in, after Omega's departure, and simultaneously alter both Joe's box-taking and Omega's prediction concerning Joe's box-taking. (Equivalently, we might reach in before Omega's departure, and surgically alter the insides of Joe brain -- and, thereby, alter both Joe's behavior and Omega's prediction of Joe's behavior.) Under this sort of world-surgery, Joe will do better by one-boxing:
Expected value ( Joe's earnings if he one-boxes | Omega predicts Joe accurately) >
Expected value ( Joe's earnings if he two-boxes | Omega predicts Joe accurately).
The point: Newcomb's problem -- the problem of what Joe "should" do, to earn most money -- is the problem which type of world-surgery best cashes out the question "Should Joe take one box or two?". Disagreement about Newcomb's problem is disagreement about what sort of world-surgery we should consider, when we try to figure out what action Joe should take.