Alternative Possibilities are one of the key requirements for the freedom component of free will, critically needed for libertarian free will. For discussion on the difference between multiple possibilities and the one realized actuality, go here.
Alternative Possibilities have been part of the problem of free will at least from the time of Thomas Hobbes, who denied anyone ever "could have done otherwise".
Harry Frankfurt famously defined what he called "The Principle of Alternate Possibilities" or PAP.
"a person is morally responsible for what he has done only if he could have done otherwise.”
Frankfurt developed sophisticated arguments (thought experiments) to disprove this principle using what might be called Frankfurt's Demon.
Frankfurt asks us to imagine an agent who can control the minds of others, or a demon inside one's mind that can intervene in our decisions. Considering the absurd nature of his hypothetical and counterfactual intervener, the recent philosophical literature is surprisingly full of articles with "Frankfurt-type cases" supporting Frankfurt, and logical counterexamples to his attack on the principle of alternate possibilities. This work is based on a logical fallacy
Frankfurt's basic claim is as follows:
"The principle of alternate possibilities is false. A person may well be morally responsible for what he has done even though he could not have done otherwise. The principle's plausibility is an illusion, which can be made to vanish by bringing the relevant moral phenomena into sharper focus."
Libertarians like Robert Kane, David Widerker, and Carl Ginet have mounted attacks on Frankfurt-type examples, in defense of free will. Their basic criticism is that in an indeterministic world Frankfurt's demon cannot know in advance what an agent will do. As Widerker put it, there is no "prior sign" of the agent's de-liberate choice. This is the epistemic Kane-Widerker Objection to Frankfurt-style cases. In information theoretic and ontological terms, the information about the agent's choice does not yet exist in an indeterministic universe. To block an agent's decision the intervening demon would have to act in advance, without actually know if blocking is necessary. As pointed out first by Kane in 1985, Frankfurt cases "beg the question" by assuming determinism is true (in which case the demon could predict the agent's choice). John Martin Fischer refers to these situations as "flickers of freedom." Determinism would eliminate the agent's control and destroy the presumed "responsibility" of the agent for the choice, despite no available alternative possibilities. This is the ontological Information Objection. Here is a discussion of the problem, from Kane's A Contemporary Introduction to Free Will, 2005, (p.87)
5. The Indeterminist World Objection While the "flicker of freedom" strategy will not suffice to refute Frankfurt, it does lead to a third objection that is more powerful. This third objection is one that has been developed by several philosophers, including myself, David Widerker, Carl Ginet, and Keith Wyma.5 We might call it the Indeterministic World Objection. I discuss this objection in my book Free Will and Values. Following is a summary of this discussion:Suppose Jones's choice is undetermined up to the moment when it occurs, as many incompatibilists and libertarians require of a free choice. Then a Frankfurt controller, such as Black, would face a problem in attempting to control Jones's choice. For if it is undetermined up to the moment when he chooses whether Jones will choose A or B, then the controller Black cannot know before Jones actually chooses what Jones is going to do. Black may wait until Jones actually chooses in order to see what Jones is going to do. But then it will be too late for Black to intervene. Jones will be responsible for the choice in that case, since Black stayed out of it. But Jones will also have had alternative possibilities, since Jones's choice of A or B was undetermined and therefore it could have gone either way. Suppose, by contrast, Black wants to ensure that Jones will make the choice Black wants (choice A). Then Black cannot stay out of it until Jones chooses. He must instead act in advance to bring it about that Jones chooses A. In that case, Jones will indeed have no alternative possibilities, but neither will Jones be responsible for the outcome. Black will be responsible since Black will have intervened in order to bring it about that Jones would choose as Black wanted.In other words, if free choices are undetermined, as incompatibilists require, a Frankfurt controller like Black cannot control them without actually intervening and making the agent choose as the controller wants. If the controller stays out of it, the agent will be responsible but will also have had alternative possibilities because the choice was undetermined. If the controller does intervene, by contrast, the agent will not have alternative possibilities but will also not be responsible (the controller will be). So responsibility and alternative possibilities go together after all, and PAP would remain true—moral responsibility requires alternative possibilities—when free choices are not determined.6 If this objection is correct, it would show that Frankfurt-type examples will not work in an indeterministic world in which some choices or actions are undetermined. In such a world, as David Widerker has put it, there will not always be a reliable prior sign telling the controller in advance what agents are going to do.7 Only in a world in which all of our free actions arc determined can the controller always be certain in advance how the agent is going to act. This means that, if you are a compatibilist, who believes free will could exist in a determined world, you might be convinced by Frankfurt-type examples that moral responsibility does not require alternative possibilities. But if you are an incompatibilist or libertarian, who believes that some of our morally responsible acts must be undetermined you need not be convinced by Frankfurt-type examples that moral responsibility does not require alternative possibilities.
Alternative Possibilities Are NOT ProbabilitiesOne of the major errors in thinking about alternative possibilities is to assume that they are the direct cause of action. This leads many philosophers to make the oversimplified assumption that if there are two possibilities, for example, that they are equally probable, or perhaps one has thirty percent chance of leading to action, the other seventy percent. Alternative possibilities are simply that - possibilities. They only lead to action following an act of determination by the will that the action is in accord with the agent's character and values. And the will is adequately determined. Most philosophers who use the standard argument against free will in their work assume that chance alternative possibilities will show up as random behavior. Here is an example from leading libertarian incompatibilist Peter van Inwagen. He imagines a God who can "replay" exactly the same circumstnces to demonstrate the random willings.
Now let us suppose that God a thousand times caused the universe to revert to exactly the state it was in at t1 (and let us suppose that we are somehow suitably placed, metaphysically speaking, to observe the whole sequence of "replays"). What would have happened? What should we expect to observe? Well, again, we can't say what would have happened, but we can say what would probably have happened: sometimes Alice would have lied and sometimes she would have told the truth. As the number of "replays" increases, we observers shall — almost certainly — observe the ratio of the outcome "truth" to the outcome "lie" settling down to, converging on, some value. We may, for example, observe that, after a fairly large number of replays, Alice lies in thirty percent of the replays and tells the truth in seventy percent of them—and that the figures 'thirty percent' and 'seventy percent' become more and more accurate as the number of replays increases. But let us imagine the simplest case: we observe that Alice tells the truth in about half the replays and lies in about half the replays. If, after one hundred replays, Alice has told the truth fifty-three times and has lied forty-eight times, we'd begin strongly to suspect that the figures after a thousand replays would look something like this: Alice has told the truth four hundred and ninety-three times and has lied five hundred and eight times. Let us suppose that these are indeed the figures after a thousand  replays. Is it not true that as we watch the number of replays increase we shall become convinced that what will happen in the next replay is a matter of chance.In our two-stage model of free will, if Alice is a generally honest person, her character will ensure that she rarely lies even if lying frequently "comes to mind" as one of her alternative possibilities. Another leading libertarian incompatibilist, Robert Kane, makes this mistake when he introduces his "probability bubbles" model of free will.