Why then would it violate free will if Omega could predict your accuracy in this different situation (one where he’s also able to predict the effects of him telling you) to a similar precision?
Because that’s pretty much our intuitive definition of free will; that it is not possible for someone to predict your actions, announce it publicly, and still be correct. If you disagree, we are disagreeing about the intuitive definition of “free will” that most people carry around in their heads. At least admit that most people would be unsurprised if a person predicted that they would (e.g.) brush their teeth in the morning (without telling them in advance that it had predicted that), versus predicting that they would knock a vase over, and then as a result of that prediction, the vase actually getting knocked over.
Then take my bet situation. I announce your attendance, and cut you in with a $25 stake in attendance. I don’t think it would be unusual to find someone who would indeed appear 99.99% of the time—does that mean that person has no free will?
People are highly, though not perfectly, predictable under a large number of situations. Revealing knowledge about the prediction complicates things by adding feedback to the system, but there are lots of cases where it still doesn’t change matters much (or even increases predictability). There are obviously some situations where this doesn’t happen, but for Newcombe’s paradox, all that is needed is a predictor for the particular situation described, not any general situation. (In fact Newcombe’s paradox is equally broken by a similar revelation of knowledge. If Omega were to reveal its prediction before the boxes are chosen, a person determined to do the opposite of that prediction opens it up to a simple Epimenides paradox.)
Because that’s pretty much our intuitive definition of free will; that it is not possible for someone to predict your actions, announce it publicly, and still be correct. If you disagree, we are disagreeing about the intuitive definition of “free will” that most people carry around in their heads. At least admit that most people would be unsurprised if a person predicted that they would (e.g.) brush their teeth in the morning (without telling them in advance that it had predicted that), versus predicting that they would knock a vase over, and then as a result of that prediction, the vase actually getting knocked over.
Then take my bet situation. I announce your attendance, and cut you in with a $25 stake in attendance. I don’t think it would be unusual to find someone who would indeed appear 99.99% of the time—does that mean that person has no free will?
People are highly, though not perfectly, predictable under a large number of situations. Revealing knowledge about the prediction complicates things by adding feedback to the system, but there are lots of cases where it still doesn’t change matters much (or even increases predictability). There are obviously some situations where this doesn’t happen, but for Newcombe’s paradox, all that is needed is a predictor for the particular situation described, not any general situation. (In fact Newcombe’s paradox is equally broken by a similar revelation of knowledge. If Omega were to reveal its prediction before the boxes are chosen, a person determined to do the opposite of that prediction opens it up to a simple Epimenides paradox.)