There is another way of interpreting the classical dilemma regarding freedom of the will, in terms of possible worlds. I think that this interpretation is worth pursuing:
If determinism is true (of @, the actual world), then @ is the only member of the set of physically possible worlds; there is, at no point, a "branching" of events in the history of @. Without any branching of any sort, we cannot be said to have choices. (I'm aware that this looks suspiciously like PAP, but I'm hoping that distinction between PAP and this model are somewhat evident.)
If indeterminism is true, then there are branching points where there shouldn't be (including our actions); our "choices" would then appear to be random. (I'm thinking of an example used by Robert Kane about the woman deliberating between vacationing in Colorado or Hawaii.)
I think that compatibilism cannot work simply because the existence of "branching points" is vital to free will. My question is, what attempts have been made by compatibilists to counter this model (other than disregarding it as irrelevant)? I know that there are libertarian accounts of countering their issue. Also, from what I've seen, compatibilists are naturalists/physicalists/materialists. Is this always the case? If it is, then the model above certainly offers no way out, I think.