One thing I read Eliezer as saying, in Dissolving the Question, is that the phenomenology of free will is more interesting than the metaphysics:

Your homework assignment is to write a stack trace of the internal algorithms of the human mind as they produce the intuitions that power the whole damn philosophical argument.

(This comment is not a full answer to that "homework assignment.")

In other words, it is a fact that humans do reasonably reliably possess the intuition, "I have free will." We do have that intuition; our having it is something to be explained. And it is a fact that when we examine the processes that we are made of — physics — we do not (contra Penrose) see anywhere for free will to sneak in. Brains use the same atoms that billiard balls and computers do.

(I don't know if you are a coder. A "stack trace" is a snapshot of what is going on, at a particular moment, at every level of abstraction in a computer program. Stack traces are often seen when a program crashes, to let the programmer follow the trail of the code bug or bad data that let to the crash. We might obtain a stack trace of consciousness through introspective techniques such as meditation — I'm not there yet via that particular method, but I think I can follow the arguments.)

This is fine, if that's the answer. But it's not a compatibilist answer.

I take Eliezer to be (heavily) influenced by Daniel Dennett. Dennett in Elbow Room, Freedom Evolves, etc. holds that what we want out of "free will" is that we create choices to influence the future; that we can take reasoned steps to avoid predicted bad outcomes; that we could have done otherwise if we thought or believed differently. This is just as incompatible with indeterminism (wherein our seeming choices are the results of quantum indeterminacy in the neurons (Penrose) as well as with a sort of greedy mechanical determinism where our choices are produced by our bodies without conscious reflection. I take Dennett as implying that our choices are produced by our bodies, but conscious reflection is our name for the mechanism by which they are produced.

(As Eliezer points out in the Anti-Zombie posts, consciousness does have an effect on the world, notably that we talk about consciousness: we discuss our thoughts, plans, fears, dreams.)

Eliezer diagrams this in Thou Art Physics: the determinist claims that the world's future is caused by physics operating on the past, and not by me making choices. But to the computationally minded materialist, "me" is the name of the place within physics where my choices are calculated, and "me" certainly does have quite a bit of control over the future.

I am not convinced that a materialist determinist like Sam Harris or Democritus would be convinced. The fact that I draw a line around some part of physics and call it "me" doesn't mean I control what goes on in that boundary, after all.

(Computation is vital here, because computation and (selective) correlation are never free. In order for a computation to take place, it has to take place somewhere. In order for some outputs (say, my movement towards the cookie jar) to correlate with some inputs (my visual signals about the cookie jar), that correlation has to be processed somewhere. Plot my worldline, and I am in orbit around the cookie jar with some very complex equation modeling my path; but where that equation is actually computed in order to guide my feet, is inside my brain.)

But the reason that determinism worries freshman philosophy students and novice LWers is that it seems to imply fatalism — that the choices we make don't matter, because the universe is scripted in advance. This compatibilist view, though, seems to say that the choices we make do matter, because they are part of how the universe calculates what the future will bring.

Fatalism says we can't change the future, so we may as well just sit on the couch playing video games. Compatibilism says that we are the means of changing the future.