First of all, the “Free Will Problem” is a stupid name. I mean, do I have to think there is such a thing as a “will” in order to believe in free will? Or “freedom,” beyond that of being able to go to the mall if I want to or not if don’t?
So, I’d rather talk about the “agency problem.” The trouble is, there isn’t one. A “problem” suggests that something is wrong, and in this case nothing is. Everyone thinks people have agency. Of course, not everyone says that. There are some philosophers (and some scientists trying to do philosophy in their free time) who say that we don’t. But, they are lying. Or if they aren’t, they have no idea what they are saying or what would follow, if they actually meant it. It reminds me of Chesterton’s observation that “the determinist … finds that he cannot say ‘if you please’ to the housemaid.” The fact is that if you’ve ever given someone advice, asked someone for something, told someone to do something, praised or blamed someone for doing something, or engaged in any number of other garden-variety activities with people, then you believe they have agency. So why would you say there isn’t any? Just to publish papers on Free Will? Please, don’t. There’s only so much more bullshit anyone should have to take, and we’re already swimming in it.
Determinism can be taken as a kind of skepticism, and with skepticism there is a smart, interesting way of doing it as well as a silly, boring one. With regard to skepticism about the external world it seems that thankfully – blissfully – a good number of people have found their way to the smart way of doing it. The proposition that the world doesn’t really exist or that there really aren’t other people or other such stuff aren’t positions that anyone holds but a way of posing a challenge, the purpose of which is to help us think more carefully about the reasons we give for the things we believe and how those reasons function. And if we really learn something from the exercise, we might come to understand that regardless of the inquiry, there will always be things which are not possible to doubt, because they have to “stand fast” (as Wittgenstein described it) if the particular line of inquiry is to proceed in the first place. If I am engaged in an inquiry with another person about our solar system, for example, I could doubt his claims regarding the number of moons that orbit Saturn and demand that he provide reasons for why I should believe him. But I could not doubt that there are planets or moons and demand reasons for thinking there are. Or, rather, I could, but I would be taken as a “half-wit,” as Wittgenstein (admittedly, somewhat untactfully) put it. That there are planets and moons must stand fast, if we are to have conversations about and engage in inquiry with regard to our solar system, which includes planets and moons. To demand reasons for that belief, in this context, is to misunderstand what reasons are and why one gives them. 
With regard to skepticism about agency, however – alas – many if not most people seem stuck on the silly, boring side of things, for the view that there is no agency is taken as an actual position (with a name and everything), and those who think that people do genuinely act are expected to provide reasons for thinking so. I’m not suggesting that no one has found their way to the smart, interesting side of the question. There are still some desultory Wittgensteinians and other assorted wise-people wandering around the philosophical landscape trying to remind everyone that one could try to understand the relationship between reasons and actions, in terms other than those currently fashionable in what passes today for the philosophy of mind, but they are few and far between, and have about as much effect as ghosts whispering things in empty rooms.
I blame science for this to some degree or more precisely, the ham-fisted importing of hard-scientific notions into our discussion of human activity and affairs, something for which both scientists and philosophers are to blame. That there is no agency involved in billiard balls bouncing off of one another and ending up in various holes (or not) – because their movements are governed by strict causal laws – makes perfect sense, given our conception of the things involved. But importing these notions into our effort to understand human conduct – treating our actions as nothing more than movements and the relationship between the states of mind responsible for them as strictly causal, in the same sense – involves a category error. Billiard balls and their movements are mathematically quantifiable magnitudes. People and their actions are not. The latter, unlike the former, are only completely describable in what philosophers call “intentional terms,” meaning that one has not fully characterized either people or their actions, until one has included how they represent things or how they are represented. This is not true of billiard balls and their motions, which is why one can explain their various goings-on entirely in the language of classical mechanics, while one cannot do the same for people and the things we do.
Going to the mall or passing someone the mustard are actions that people engage in, which may involve the movements of various body parts but are not reducible to them, and because any accounting for them is steeped in intentionality, they presuppose agency, which simply is the doing of things for various reasons. Asking, “Why did you go to the mall?” or “Why did you pass him the mustard” are not equivalent to asking “Why did the nerves and muscles in your arms and legs and the rest do such and such?” and the causality involved when one cites one’s reasons for doing the various things in question, like “because I wanted to go and hang out with my friends” or “because I know he loves mustard on his hot dogs,” is not equivalent to – or anything like, really – the accounts one gives of the billiard balls’ movements or of the movements of one’s arms and legs or other body parts. Returning to Wittgenstein’s remarks on skepticism, if we are speaking of actions and our reasons for engaging in them, that people have agency must stand fast and is not something about which it makes any sense to demand reasons for believing. I can doubt your reason for going to the mall or even that you went there at all. But what I can’t doubt is that there is such a thing as malls or people or reasons people go to malls – and I mean people going to them, not peoples’ body parts moving through them.
Now, beyond Wittgenstein’s “half-wits,” there are also genuinely crazy people – or those pretending to be crazy – who will want to say, “so much the worse for actions and reasons and agency then” and suggest that motor movements and their mechanical (and other hard-scientific) causes are all there “really” is. The truly crazy ones, like B.F. Skinner, may even be somewhat consistent in their stand and go on to say things like “and worse still for responsibility, reward, desert, and all the other things that presuppose actions and reasons and agency,” as he did in his gloriously crazy book, with its suitably crazy title, Beyond Freedom and Dignity.  But being more consistent doesn’t make any of it any less silly. It’s not just that I’m sure that when Skinner was alive he asked people to pass him the mustard or that he blamed people when they cheated him or went to the store, but that the behaviorist’s supposedly purely mechanistic accounts never could be made to work, without intentionality and agency somehow sneaking back in. Sure, it’s likely that if you deprive someone of food or water long enough, he’ll eat or drink something if offered it, but if he’s on a hunger strike and believes he shouldn’t, he might not. And if you lower the mall prices to where every item is a penny, it is likely to be flooded with customers, but if it is revealed that the mall is infested with rats, those same shoppers may go somewhere else. So, while one may find the nice, neat, cause and effect chains determinists like so much if one focuses attention on peoples’ pharynxes or esophagi or major muscle groups, once you start talking about people and drinking and hunger striking and shopping or not shopping in a particular place, you’ve left that framework behind and the neat causality and determinism with it. You are in a world full of – and defined by – agency.
Earlier, I mentioned the philosophy of mind, and it really is the sorry state of that once fascinating, productive discipline that is to blame here. Maybe it’s just physics envy, to which philosophers have always been susceptible, or maybe it’s the fact that somehow they just didn’t notice the word ‘social’ in front of ‘social science’ – of which psychology is one – but philosophers working in the philosophy of mind have largely adopted the billiard-ball model of causal explanation in their accounts of human action, and not just middlebrow philosophers, but the brightest lights in the discipline. In this area, however, the stuff they are doing is no better than it was in Descartes’ day, something that poor souls like Peter Hacker, who really is doing the Lord’s work, have been trying to point out, now for decades, to little avail. 
For these people, heads are like boxes at the end of one’s neck, in which we find things called “mental states” that by way of the nerves cause other things that go on in the rest of the body – including the mouth – called behaviors. (They tend not to use ‘actions’ as the connotations are too difficult to ignore and risk upsetting the “we’re doing science” cart.) Funnily – sadly – this isn’t very far from Descartes’ good old lever-and-pulley metaphor, in which the nerves act like bell-ropes and human actions are conceived of in a way that suggests we are like organic marionettes. That mental states are thought of today in terms of neurochemicals and nerves as transmitters doesn’t make things better, but actually worse, because it has sent a whole crowd of otherwise very smart people chasing down one rabbit-hole after another trying to figure out how a bunch of neurochemicals can represent something or have qualitative characteristics. They can’t of course – such characteristics only arise from a point of view – and so the holes always turn up empty. To explain peoples’ actions is as much to interpret them as it is to do anything of the sort one does when accounting for the movements of billiard balls or muscles and limbs, by which I mean it is to render them intelligible, in the sense of understanding them from the actor’s own point of view.
But, what about these Harry Frankfurt-style thought experiments? A person wants to vote Democrat, and he does. A mad scientist has set up a failsafe, such that if the person decides to vote Republican instead, it will make him vote Democrat. The case is supposed to be significant, because in contemporary philosophical discussions on Free Will, the determinist bogeyman has been largely understood though the idea that if people act freely, they could have acted otherwise, and if Determinism is true, then people cannot act other than the way they acted. In this version of the case, then, nothing could have happened other than the person voting Democrat (because of the failsafe), but our “intuition” is that the person still acted freely and is properly praised or blamed for what he did, and this is supposed to show that Free Will and Determinism can go together.
It’s a bizarre example. For one thing, it isn’t true that nothing could have happened other than the person voting Democrat – he might have decided to vote Republican and succeeded, the machine having short-circuited and electrocuted the Mad Scientist instead – but like children and their toys, philosophers will have their thought experiments. The weirder thing is that it doesn’t address any real challenge to the question of agency, and that’s because in the example – and the discussion in which the example is raised – agency and more specifically, who and what agents and actions are, is misunderstood.
The person acted – and thus, has agency – if he voted. If he, the person, engaged in the act of voting. And according to the Frankfurt case he did, as the Mad Scientist’s failsafe never went off. Had the counterfactual situation obtained, however, and the Mad Scientist’s failsafe gone off, it’s not as if he still would have voted, but against his will. Rather, though his hands and fingers would have engaged in various motor movements and certain buttons would have been depressed as a result (causal chain!), he would not have been the person who voted. Rather, the Mad Scientist would have voted, using the person’s body as a proxy. How do we know this? Well, for one thing, you could not interpret the person’s bodily movements in light of his reasons – could not see the action from his point of view – as his intentions were to vote Republican. The bodily movements are only interpretable as reflecting the Mad Scientist’s reasons and are thus only intelligible as actions, from his point of view. And for another, if the whole sordid affair was discovered by the local authorities, the Mad Scientist would be indicted for fraudulent voting, which indicates that as far as the law is concerned, it was he who voted, not the brain-and-body hijacked person.
The whole question of the relationship between agency and whether I could have done otherwise is also misunderstood in the case and the discussions in which such cases are deployed. That I could have done otherwise is a consequence of my agency, not a condition for it. Alternative possibilities are not like pre-existing roads or a selection of cupcakes, where my agency is defined in terms of my being able to pick one or another, without interference. Rather, they exist as a result of the fact that when one acts, one does so for one reason or another, and there are many reasons why one might do something. That’s just the kind of things actions are.
 This is Wittgenstein’s account of the reason why relative to any particular inquiry, some propositions will be necessary and not subject to demands for warrant. As for the “half-wit” line:
§257. If someone said to me that he doubted whether he had a body I should take him to be a halfwit. But I shouldn’t know what it would mean to try to convince him that he had one. And if I had said something, and that had removed his doubt, I should not know how or why.
Ludwig Wittgenstein, On Certainty, tr. by Denis Paul and G.E.M. Anscombe (1969). The book is a compilation of Wittgenstein’s extended reflections on G.E. Moore’s essay, “A Defence of Common Sense.”
 B.F. Skinner, Beyond Freedom and Dignity (1971).
 I’ve posted these videos so many times that I should receive royalties. They should be required watching, before anyone is permitted to work in the philosophy of mind.
https://www.youtube.com/watch?v=LZx93eov5i4&t (Hacker begins at 1:00.00)