I wish I had seen this topic sooner. This is one of my pet subjects.
I think Alvin Plantinga gave a really clear explanation of libertarian free will in his book, God, Freedom, and Evil. And WLC & J.P. Moreland gave basically the same definition in their book, Philosophical Foundations For a Christian Worldview.
Basically, they said if an act is free, that means there are no antecedent conditions prior to and up to the moment of choice that are sufficient to determine what that choice will be. Absent any Frankfurt type scenarios (which never happen in the real world), if a person does X, they could have done otherwise even if everything prior to and up to the moment of choice had been exactly the same. That includes all of their prior mental states, including their desires, motives, preferences, beliefs, etc. People with libertarian free will have the capacity to be uncaused first causes. They can, through an act of the will, initiate a chain of causes without themselves being caused to do so.
Personally, I don't see how that differs from a random event. We have no more control over a random event than we do over an event that is determined by blind mechanistic causes. So it seems that to have free will of any variety, there must be some sense in which a person has control over their choice.
But what does it mean to have control? Libertarians have answered this in different ways. One way they call agent causation. I remember reading on Alex Pruss' blog one time something like, "The agent causes the free action." This struck me as being problematic for a couple of reasons. One reason is that if an act is caused by anything at all, then it isn't a free act in the libertarian sense. To be free in the libertarian sense is to be free of antecedent causes, especially if those causes are deterministic. Another reason it's problematic is because it just pushes the random problem back one step when we ask, "WHY does the agent cause the free act?" and "Isn't the agent's causal activity itself an act of some sort?" If there is no sufficient cause or reason for why the agent causes the act, then it's still a completely random event. In what sense is the agent in control of its own causal activity?
I'm a compatibilist because it seems to me that an act can only be under one's control to the degree that it follows from a person's own antecedent desires, motives, intentions, inclinations, preferences, etc. The further you divorce an action from a person's antecedent mental states, the less control that person has over the action. And that is true whether the action is causally determined (like a bully grabbing your wrist and making you hit yourself), or the action is completely random (like a muscle spasm or a tick, which I admit isn't random but is the closest analogy I could think of). To be in control of your actions is precisely to act out of your own desire and motives, so the more hand your desires and motives have in bringing about your action, the more control you are exercising in your behavior. It follows that you have the most control over your behavior when your desire and motives determine your actions.
As far as moral accountability goes, I think compatibilism gives a better account of moral accountability than libertarianism does. In libertarianism, antecedent conditions can INFLUENCE your behavior, but if they DETERMINE your behavior, then that removes moral responsibility. But influence comes in degrees. For example, the stronger your desire to do something, the more influence that desire has over your behavior. So imagine a scenario in which your desire was so strong that you could not resist it. In that case, the desire would be sufficient to determine your choice, and you wouldn't be responsible for your choice. But if the desire falls short of being strong enough to determine your behavior, you still have some responsibility. This would seem to indicate that moral responsibility also comes in degrees. The stronger your desire to do X, the closer that desire is to determining that you do X. And since you would be completely innocent if your desire were strong enough to determine that you do X, it would follows that your guilt is somewhat ameliorated the closer your desire is to determining that you do X. So the stronger your desire to do X, the less blamable you can be for doing X (or the less praiseworthy you are for doing X). With this in mind, it would also seem to follow that the weaker your desire to do X, the more responsible you are for your choice. And you are the MOST responsible for your choice when the desire has no influence over your choice at all. Actually, you'd be most responsible for your choice when acting out of complete indifference with no desire at all. That seems odd to me.
So suppose you had a desire to help somebody. Well, with the above reasoning, it would seem to follow that the deeper your desire to help somebody, the closer that desire is to determining your behavior, and therefore, the less praiseworthy you are for helping them. That strikes me as being completely backward. The deeper your desire to do good, the less praiseworthy you would be since you'd be all the more close to having your choice be determined by your desire.
Likewise, the deeper your desire to do evil, the less blameworthy you are since the deeper your desire to do evil, the closer that desire is to determining your behavior. This strikes me as being completely backward.
When I've talked to people in the past about this, they usually deny that culpability comes in degrees. You could have a desire that is 99.99999999999% close to being sufficient to determine your behavior, and as long as it's missing that 0.000000000001%, then you are fully responsible. There's no diminishment to your responsibility at all.
But think about that. If no degree of influence can do anything to diminish your responsibility, then your responsibility can never be diminished even if the degree is 100%. After all zero times any number is still zero. So if no degree of influence can diminish your responsibility, then you cannot be absolve of all responsibility even if the influence is sufficient to determine your behavior.
I think we all have an intuitive awareness that things are exactly backward from the consequences of the libertarian view. Your motives have a lot to do with whether your actions are praiseworthy or blameworthy. If you shove an old lady out of a motive to save her from being hit by a bus, then you're praiseworthy. If you shove the same old lady just as hard out of a motive to hurt her because you hate old ladies, then you're blameworthy. The more you actions are motivated by evil desire, the more evil those actions are and the more blameworthy you are. And the more your actions are motivated by good desires, the more good those actions are and the more praiseworthy you are. You are responsible to the degree that your antecedent desire had a hand in bringing about your actions. The more hand, the more responsible, and the less hand, the less responsible. After all, if they had no hand at all, then your action was an accident, and you can't be responsible for an accident. Since you are most responsible when your desire have the most hand in bringing about your actions, it follows that you have the most responsibility for your actions when they are DETERMINED by your desires.
So I think compatibilism makes more sense of morality than libertarianism does.