I think it's pretty obvious that you can't predict someone's decisions if you show him the prediction before he makes his final choice. So let's consider a different flavor of prediction. Suppose every time you make a choice, I can predict the decision, write it down before you do it, and then show it to you afterwards. Neither the infinite recursion argument nor the no fixed point argument work against this type of prediction. If this is actually possible, what would that imply for free will?
If you are an AI, this would be fairly easy to do. I'll just make a copy of you, run your copy until it makes a decision, then use that as the "prediction". But in this case I am not able to predict the decision of the copy, unless I made another copy and ran that copy first. The point is that algorithms have minimal run-time complexities. There are many algorithms which have no faster equivalents. The only way to find out their results is to actually run them. If you came up with an algorithm that can predict someone's decisions with complete accuracy, it would probably have to duplicate that person's thought processes exactly, perhaps not on a microscopic level, but probably on a level that still results in the same conscious experiences. So now there is nothing to rule out that the prediction algorithm itself has free will. Given that the subject of the prediction and the prediction algorithm can't distinguish between themselves from their subjective experiences, they can both identify with the prediction algorithm and consider themselves to have free will. So you can have free will even if someone is able to predict your actions. The more obvious fact that you can't predict your own actions really has less to do with free will, and more with the importance of the lack of logical omniscience in decision theory. Classical decision theory basically contradicts itself by assuming logical omniscience. You already know only one choice is logically possible at any given time in a deterministic universe, and with logical omniscience you know exactly which one is the possible one, so there are no more decisions to be made. But actually logical omniscience is itself logically impossible, because of problems with infinite recursion and lack of fixed points. That's why it's great to see a decision theory that does not assume logical omniscience. So please read that paper (referenced in the first post in this thread) if you haven't already.
