Pixy's theory explains free will Ok to me, its basically a computation, if a multi faceted and integrated one.
The will is a choice or a succession of choices and for this to occur, a self is required to make the choice. The self can also be explained approximately with a computation as well, in terms of performance.
In my view, the self doesn't pre-date the rest of it. Rather, it is the result of the choices and not independent. I know it "feels" like the choice is made by an independent entity, but I'm proposing that is an illusion. And we are already familiar with these types of illusions in the brain.
A couple of examples of how the brain creates a false narrative should suffice. If I touch my toe to my nose, I feel both sensations at once. But there is a nerve transmission delay that should have me feeling from the nose before the toe. If I watch as I touch one toe to another, my eyes should see it happen before I feel it. But my brain corrects the narrative.
This also happens in a suite of similar experiments where our brains lie to us about our perceptions. (Inversion glasses, 3D from 2D, moving colored dots where inbetween states are "filled in" and many others.) I suspect the brain is lying to us and creating a narrative with the feeling of choosing as well.
Here is an abstract of a study showing that decisions are made long before conscious awareness of those decisions: http://www.ncbi.nlm.nih.gov/pubmed/18408715
From wiki:
One significant finding of modern studies is that a person's brain seems to commit to certain decisions before the person becomes aware of having made them. Researchers have found delays of about half a second (discussed in sections below). With contemporary brain scanning technology, other scientists in 2008 were able to predict with 60% accuracy whether subjects would press a button with their left or right hand up to 10 seconds before the subject became aware of having made that choice.
The main article is here: http://en.wikipedia.org/wiki/Neuroscience_of_free_will
If you think about it, one of the brain's jobs is to create a model of the world with good fidelity. It is advantageous to hide from awareness those background activities which do not contribute to the picture. For example, it is more useful to have an "urge to pee" rather than a direct experience of muscles in the bladder expanding, or an "urge to breathe" rather than a direct measure of blood acidity (from CO2 build-up). There are many things I could be thinking about that would overwhelm my conscious analysis -- what's the current oxygenation level of my muscles? Has gravity changed? Which direction are my feet oriented? Are both my eyes pointing in the same direction?
All I'm saying is that, at the level of awareness, there's already been a great deal of editing and processing that creates a narrative, generally true, but specifically false, and part of that narrative is a sense of an independent observer.
My claim is that there is no separate "me" in the event: "me writing this post," but that the "me" is part and parcel of the activity. I'd also point out that when there is no "me," as during a deep and dreamless sleep or anesthesia, there is also no willed activity. The agent and the action are intimately bound together. The two are always paired, even if the activity isn't visible, and nothing but introspection. I think they are the same thing and that choices aren't made by us, but are part of what makes us.
So, in a sense, we are the result of our choices, but the chooser is also created as part of the process and simultaneous with the choice.
Here is how I experience it. First there are options I am aware of. Later, I have made a choice. In-between, I can find no point where the choice is actually happening. I can only catch it after a decision has been made. So too, I can see a finger still and I can see a finger move, but I cannot access the part where the nerve transmission is going from my brain to my finger.
