While don't necessarily disagree with the issues of free will Strawson raises, I would be interested to see how he responds to the work of Benjamin Libet. In short, Libet's work leads one to conclude that free will isn't so much the will to do a thing, but rather the ability to intervene and stop oneself from doing a thing. If you're propelled to action by some irrational process, free will, in Libet's estimation, allows one to prevent oneself from conducting that action.
For instance, if seeing something throws you into a rage, something that may be a natural reaction to something rage-inducing, your free will doesn't propel you to action, but rather the irrational response of being flooded by the right combination of neurotransmitters. However, before your go on your rampage, the rational side intervenes, preventing you from some epic stupidity.