I don't know how to respond to this or Morendil's second comment. I feel like I am missing something obvious to everyone else but when I read explanations I feel like they are talking about a completely unrelated topic.
Things like this:
You seem to be confused about free will. Keep reading the Sequences and you won't be.
Confuse me because as far as I can tell, this has nothing to do with free will. I don't care about free will. I care about what happens when a perfect predictor enters the room.
Is such a thing just completely impossible? I wouldn't have expected the answer to this to be Yes.
If you do know what the prediction is, then the way in which you react to that prediction determines which prediction you'll hear. For example, if I walk up to someone and say, "I'm good at predicting people in simple problems, I'm truthful, and I predict you'll give me $5," they won't give me anything. Since I know this, I won't make that prediction. If people did decide to give me $5 in this sort of situation, I might well go around making such predictions.
Okay, yeah, so restrict yourself only to the situations where people will give you the $5 even though you told them the prediction. This is a good example of my frustration. I feel like your response is completely irrelevant. Experience tells me this is highly unlikely. So what am I missing? Some key component to free will? A bad definition of "perfect predictor"? What?
To me the scenario seems to be as simple as: If Omega predicts X, X will happen. If X wouldn't have happened, Omega wouldn't predict X.
I don't see how including "knowledge of the prediction" into X makes any difference. I don't see how whatever definition of free will you are using makes any difference.
"Go read the Sequences" is fair enough, but I wouldn't mind a hint as to what I am supposed to be looking for. "Free will" doesn't satiate my curiosity. Can you at least tell me why Free Will matters here? Is it something as simple as, "You cannot predict past a free will choice?"
As it is right now, I haven't learned anything other than, "You're wrong."
When a human brain makes a decision, certain computations take place within it and produce the result. Those computations can be perfectly simulated by a sufficiently-more-powerful brain, e.g. Omega. Once Omega has perfectly simulated you for the relevant time, he can make perfect predictions concerning you.
Perfectly simulating any computation requires at least as many resources as the computation itself (1), so AFAICT it's impossible for anything, even Omega, to simulate itself perfectly. So a general "perfect predictor" may be impossible. B...
Omega has appeared to us inside of puzzles, games, and questions. The basic concept behind Omega is that it is (a) a perfect predictor and (b) not malevolent. The practical implications behind these points are that (a) it doesn't make mistakes and (b) you can trust its motives in the sense that it really, honestly doesn't care about you. This bugger is True Neutral and is good at it. And it doesn't lie.
A quick peek at Omega's presence on LessWrong reveals Newcomb's problem and Counterfactual Mugging as the most prominent examples. For those that missed them, other articles include Bead Jars and The Lifespan Dilemma.
Counterfactual Mugging was the most annoying for me, however, because I thought the answer was completely obvious and apparently the answer isn't obvious. Instead of going around in circles with a complicated scenario I decided to find a simpler version that reveals what I consider to my the fundamental confusion about Omega.
Suppose that Omega, as defined above, appears before you and says that it predicted you will give it $5. What do you do? If Omega is a perfect predictor, and it predicted you will give it $5... will you give it $5 dollars?
The answer to this question is probably obvious but I am curious if we all end up with the same obvious answer.
The fundamental problem behind Omega is how to resolve a claim by a perfect predictor that includes a decision you and you alone are responsible for making. This invokes all sorts of assumptions about choice and free-will, but in terms of phrasing the question these assumptions do not matter. I care about how you will act. What action will you take? However you label the source of these actions is your prerogative. The question doesn't care how you got there; it cares about the answer.
My answer is that you will give Omega $5. If you don't, Omega wouldn't have made the prediction. If Omega made the prediction AND you don't give $5 than the definition of Omega is flawed and we have to redefine Omega.
A possible objection to the scenario is that the prediction itself is impossible to make. If Omega is a perfect predictor it follows that it would never make an impossible prediction and the prediction "you will give Omega $5" is impossible. This is invalid, however, as long as you can think of at least one scenario where you have a good reason to give Omega $5. Omega would show up in that scenario and ask for $5.
If this scenario includes a long argument about why you should give it $5, so be it. If it means Omega gives you $10 in return, so be it. But it doesn't matter for the sake of the question. It matters for the answer, but the question doesn't need to include these details because the underlying problem is still the same. Omega made a prediction and now you need to act. All of the excuses and whining and arguing will eventually end with you handing Omega $5. Omega's prediction will have included all of this bickering.
This question is essentially the same as saying, "If you have a good reason to give Omega $5 then you will give Omega $5." It should be a completely uninteresting, obvious question. It holds some implications on other scenarios involving Omega but those are for another time. Those implications should have no bearing on the answer to this question.