The randomness argument insists that if our actions are not determined by our character, then they are a matter of chance and hence not free. This is the most powerful argument against libertarianism. I want to think about whether the argument presents a challenge to the compatibilism. (Compatibilists think freedom is compatible with determinism; libertarians think that (a) freedom is incompatible with determinism, and (b) there is freedom. I shall assume that freedom and responsibility go together, though a more careful examination will require dropping the assumption.)
The compatibilist had better have something to say about how on her preferred view of action the actions do not come out as "a matter of chance". As a warmup, observe that for the purposes of evaluating whether something is "a matter of chance" in our ordinary intuitive sense—which is presumably the sense at play in the argument—the question isn't whether the event is determined or not. We think of coin flips as a matter of chance, and we would think this even if it turned out that the outcomes of coin flips are determined by the precise values of the initial conditions, so a slightly different angle or velocity would produce a different outcome, and we do not have precise enough control over the angle or velocity. If our actions came from a deterministic coin flip in the head, they would be no less objectionably random than if they came from an indeterministic coin flip.
Now, add this observation. People sometimes make seriously responsible decisions where their desires are very close in strength. For instance, no one would get off on a murder charge just because her desire to do the right thing was almost as strong as her desire to inherit the money from her uncle. But when desires are sufficiently close in strength, then which desire is the stronger is "a matter of chance" in exactly the same sense in which a deterministic coin flip is "a matter of chance". What sense is that? It's the sense that when something depends on very fine differences in conditions, too fine for us to discern or control, it's "a matter of chance." Whether one is seriously responsible for murder or not responsible for murder at all shouldn't depend on such tiny differences in desires: if it depends on the fact that one's desire to commit the murder has strength 27.4837 while one's desire not to commit the murder has strength 27.4836, that's just as chance as a truly indeterministic choice would be.
I think three families of answers are available to the compatibilist.
Answer 1: Insist that as the strengths of desires get closer and closer, the degree of freedom and responsibility decreases to zero.
Objection: It doesn't! While perhaps we want to especially criticize the character of the person whose desire for murder was by far the stronger desire, the person for whom the desires were closely balanced, and who agonized over the decision is still seriously responsible, and not merely vanishingly responsible.
Answer 2: Go for Markosian's agent causal compatibilism.
Objection: I have some problems with Markosian's story, but once one goes for agent causation to respond here, one should allow the libertarian the courtesy of relying on agent causation to respond to the randomness argument as well. But if one does that, one loses the strongest argument against libertarianism.
Answer 3: The person whose desire for murder has strength 27.4837 and whose desire not to commit the murder has strength 27.4838 is still a pretty bad person, even if she does not commit murder. Thus, our moral evaluation of the character of the person who in fact commits the murder because her desire strengths are 27.4837 and 27.4836 for and against respectively and the person who does not commit the murder because her desire strengths are 27.4837 and 27.4838 should be very similar, all other things being equal. The latter person does not commit the murder, but that's just her luck, and it is not a very different kind of luck from the luck of a person who does not murder her uncle because he happens to die before she gets around to trying to kill him. In all of these cases, ordinarily, we are dealing with persons with a corrupt moral character, persons who can be appropriately criticized for that character. Largely for pragmatic reasons, however, we do not punish the person who does not commit the murder, whether because her uncle dies on his own or because her desire for murder is slightly the weaker.
Objection: I do not think this fits with how we ordinarily think about the situations. We think there is something significantly praiseworthy when someone with very closely balanced desires for good and evil does the right thing. Yet the difference between the significantly praiseworthy and the significantly blameworthy action shouldn't be "a matter of chance".
Here, the compatibilist pressing Answer 3 might bite the bullet and say that while for pragmatic reasons we praise the person with such closely balanced desires, perhaps in the hope that the balance is shifting in a good direction and our praise will push it further there, she is a really bad person. But I still doubt this: Is she morally really about as blameworthy as an actual murderer? It still seems that the blameworthiness in the action is objectionably a matter of chance, unless one takes Answer 2 as the way out.
In any case, let's explore this compatibilist answer some more. Even if the bullet has been bitten, it's only going to have any hope of working in cases where the person is responsible for her character. For if her character is not something she's responsible for, it might as well be a matter of chance (and indeed may be a matter of chance circumstances and genetics). In particular, it won't work for the first responsible choices a person makes. The story has to be a story of bootstrapping: one has a very low level of responsibility in the initial choices, but then as one's choices build up a character in concert with them, the level of responsibility rises.
I doubt this this bootstrapping. If the choice itself does not inject any new responsibility, as I think on a deterministic story it does not since it is simply an outcome of earlier character, then one is no more responsible for the up-built character than for the initial character. But that is a different and independent objection to compatibilism, I think.
In sum, I think the randomness argument isn't so much an argument against libertarianism as an argument against the possibility of freedom: it affects compatibilists and libertarians alike. Oh, and I think it doesn't work, either because of agent causation or because it fails to notice how well indeterministic explanation can be made to work.