I think that the most "intuitive" illustration of the concept is in the form of gamebooks. Consider:
You enter the tavern, eyeing the other customers warily. You pick a secluded spot, away from as much of the garish music and gibbering of the inebriated guests as possible. Eventually, a waitress approaches you. She seems amiable enough, and asks you if you would like a drink.
- You ask for a shot of vodka. Go to page x.
- You ask for a glass of mead. Go to page y.
- You say nothing, but you smile and shrug to indicate you don't intend to drink anything right now. Go to page z.
Your "experience" of this narrative structure would be same initial conditions, different possible outcomes. That the kind of disjunctive imperative!! at issue would ever only be resolved "at random" might also be your "experience," and so you might reject the whole sense of the "situation" as a (transcendental) illusion. However, if free will is supposed to be neither predeterminate nor random, and if there is a difference between chance and randomness too, then it is not clear (to me) that strong free will (as in the strong ability to do otherwise) is impossible: maybe it's as different from chance and randomness as those are different from determinism, and as similar to those as it is to determinism: it's its own thing, it's a unique concept in the same family, irreducible in the limit.
I wanted to quote something from John L. Austin (I believe was the guy), from the SEP article on him in which they go over his (positive) treatment of this strong "ability to do otherwise" concept. But the SEP is not working on my laptop for some reason right now, so... This question of mine on this SE contains the desired quotes, though.
Now, consider that mathematics by now either has no absolute foundations, or the best theory of its foundations is a category-theoretic amalgamation with multiversal set theory. If entire mathematical universes, upon which the mathematics of probability would in turn be grounded, are themselves objects of unlimited intellectual will, formed and mediated and traversed by whatever
abstract interiority Hamkins and his colleagues are adverting to, then it is not normal game theory, neither probability theory, that covers the mathematics of free will. (An abnormal game theory, or games on the multiverse of sets, could be in play, but here we verge on the debate between semiotic formalists and ante rem realists, among other things.)
Instead, it is the mathematics of the set-theoretic multiverse that is itself the closest thing possible to the mathematics of (strong) free will. One might turn this into a transcendental argument for such will: if strong free will did not exist, we would not know what we know about the set-theoretic multiverse. Since we do know such things, we apparently must have such will available to us. QED
Lastly, though, I don't know that I've ever seen it asked whether some of our actions might be determined, some random/chancy, and others the result of free will. Kant said that free will proper doesn't even directly manifest in the empirical world; it is intelligibly attributed to the meaning of the world from a transcendental source. At any rate, I am not an aggressive enough person to indelibly condemn anyone I see doing something I believe is wrong, neither to overly praise those who I see doing what I believe is right. The question of moral luck weighs on my mind a lot, seeing as it seems like it's due to moral luck that I didn't join the conspiracy cult that is trying to destroy my homeland. They tempted me with drugs, sex, money, weapons, and political power, and I'm pretty surprised I didn't fail the test. Schizophrenia-spectrum issues, of all things, shielded me, because according to my pre-existent delusions in this case, according to my own conspiracy theories, that conspiracy cult is the incarnation of a (metaphorically) demonic evil, and so I resisted them (and continue to resist them) with all my might. Was this an example of my free will? Did my schizotypal character predetermine my resistance? Did my mental health problems lead to a random disavowal of the cult? I don't quite know.
So I won't pretend to have "solved the problem of free will." I will add in this, though:
Consider the following imperatives:
- Do A or B.
- Do ~A or do B.
- Do ~A or do ~B.
If you never have a pure free choice, then none of these imperatives should be intelligible. So if they are intelligible to you, then you have some "sense" of the strong "ability to do otherwise." But note also that (3) is not, in the limit, entirely meaningful: a choice between two negations is an empty choice.
!!In another post on this site, I asked about "disjunctive imperatives," and one respondent claimed I was misusing the word disjunctive, since this supposedly is only "meant" to apply to assertoric functions. However, in his deontic writing, Immanuel Kant spoke of hypothetical and categorical imperatives, translating those concepts from his theoretical to his practical realm. "Originally" (in the first Critique), categorical and hypothetical logical determination is also grouped with (you guessed it!) disjunctive logical functionality. If someone doesn't understand the notion of a disjunctive imperative, that's on them.