First of all, I believe the majority of philosophers of mind today are compatibilists - they hold that free will is in fact compatible with determinism. Second

First of all, I believe the majority of philosophers of mind today are compatibilists - they hold that free will is in fact compatible with determinism. Secondly, there is very clearly no room at all for strong free will in any physical theory we have. By "strong free will" I mean the idea that conscious beings are able to decide their next action at some moment T+1 without either (a) a sufficiently intelligent agent being able to predict with 100% accuracy their decision from enough knowledge about the state of the world at moment T, or (b) the decision being a purely random event. Given that human beings are made entirely out of elementary particles, there is simply no way for some part of the human to influence the elementary interactions that make up our beings. Of course, if you believe in dualism of some kind, where there is some form of a transcendent soul that doesn't obey physical theories but can intervene in them, then you can add this to any theory. The way free will can be made compatible with determinism is to accept that our choices are not fundamentally different from the choices that a search algorithm makes in its operation. They are fully determined by our previous experience and environment (up to some randomness), but different people will still make different choices, more or less appropriate. Just like different implementations of a search algorithm can work better or worse, or just differently, so two can different people make better or worse choices, or simply different ones. But, just like the algorithm, we are still each entirely the product of our entire life so far, and our nature as human beings.