Frankfurt counterexamples to the Principle of Alternate Possibilities (PAP) have worried libertarians. However, they should have also worried compatibilists. Traditionally, compatibilists have accepted PAP, but given it a counterfactual spin (see my previous post). Suppose Jones freely chooses to push button A. On the standard Humean analysis, this implies that were Jones to have chosen not to push A, he would not have pushed A. But a fairly crude Frankfurt case will provide a counterexample to this. Imagine Black stands by with his neuroscope and has a firm plan that if he sees Jones choosing not to push A, he will make Jones push A. Then it is true that were Jones to have chosen not to push A, he would still have pushed A.
Hence, Frankfurt cases are also counterexamples to the Humean version of PAP, and indeed are better counterexamples to it than to the libertarian PAP (standard Frankfurt cases are known to beg the question against many libertarians, since they require a sign that is nomically connected with the action in a way that many will not accept).
PAP is very plausible. So it is an important task not just for the incompatibilist but also for the compatibilist to find a version of it that survives Frankfurt counterexamples. Here is my hypothesis: A plausible PAP that survives Frankfurt counterexamples will still be sufficient for incompatibilist arguments once one plugs in a non-Humean analysis of "could have". If this is right, and if I am right that everybody needs a PAP, then Frankfurt examples do not in the end weaken the incompatibilist's case—just as before Frankfurt examples the question was whether the Humean analysis of "could have done" was right, so too this is the question after Frankfurt examples, once one correctly formulates PAP.
My own preferred PAP is flickery and fits well with the above remark: If x freely does A, then x could have failed to freely do A. Actually, it may be that the libertarian is in a better position than the compatibilist when it comes to formulating a PAP. For flickery PAPs like the above don't fit well with the Humean analysis of "could have done". Would the Humean want to say that "x could have failed to freely do A" means "were x not to have willed to do A, then x would not have freely done A"? But that's just a tautology and does no justice to the intuitions behind PAP.
In summary: Everybody who believes in free will—compatibilist or incompatibilist—needs PAP. Frankfurt examples affect both the compatibilist and the incompatibilist. It is a bit easier for the incompatibilist to find a replacement for PAP that survives the examples, but the replacement-finding task is one that both compatibilists and incompatibilists need to engage in. But the real question, as before Frankfurt, is how to understand "could have done" conditions.