Do otherwise in the same circumstances
The ability to choose and do otherwise in exactly the same circumstances is one of two criteria considered essential for libertarian free will and for moral responsibility. The other is the existence of alternative possibilities for action.
The Fixed Past and the Laws of Nature 
The idea that a free agent "could have done otherwise" in "exactly the same circumstances" is a key element in the libertarian free will argument.
Because "could have done" is a modal conditional, the argument is subtle. Once an action is completed, it falls into the "fixed past," which most philosophers agree is now unchangeable. So in order to "have done otherwise," something would have to have been different in the moments leading up to the decision and action.
The argument turns on the conditions and timing for reproducing "exactly the same circumstances." Some philosophers have maintained that reproducing the same conditions is impossible, at a minimum because the agent would have a memory of being in the earlier situation. But for the sake of the argument, we can assume exactly the same conditions, to exhibit the freedom to do otherwise at any time, and not the lesser freedom - "to do otherwise if the agent had decided to do otherwise," which requires a different past.
Given the "laws of nature" and the "fixed past" just an instant before a decision, philosophers wonder how a free agent can have any possible alternatives. This is partly because they imagine a timeline for the decision that shrinks the decision process to a single moment.
Collapsing the decision to a single moment between the closed fixed past and the open and ambiguous future makes it difficult to see the free thoughts of the mind followed by the willed and adequately determined action of the body.
This illustration still makes an artificial separation between the creative and partially random generation of alternative possibilities and the deliberative evaluation. These two capabilities of the mind can be going on at the same time. That can be visualized by the occasional decision, time permitting, to go back and think again, when the available alternatives are not good enough to satisfy the demands of the agent's character and values.
This diagram shows how given "exactly the same circumstances" at the end of the "fixed past" interval, the agent may generate alternative possibilities, some of which are generated indeterministically.
If the agent can consciously and clearly recall the alternatives, he or she can correctly say "I could have done otherwise." And even if the process described in the diagram is largely unconscious and exceedingly fast, if in similar past circumstances the agent recalls doing otherwise, she can say the same. For example, Keith Lehrer noted that if there is some one thing she has done often and not done on other times, she can reflect on a specific instance and say "In that instance I could have done otherwise."
Creativity, evolution, and doing otherwise 
If it were impossible to generate new alternative possibilities for thoughts and actions, there would be "nothing new" under the sun. All the human artifacts ever produced would already be implicit in earlier states of the creator.
Note the similarity to biological evolution. All the species would be implicit in the earlier species at any stage of evolution. But we know that there is novelty in evolution, driven in part by chance variations in the gene pool, some of which outreproduce existing species and so are naturally selected.
The ability to do otherwise is needed to create surprising new behaviors in animals and humans.
Arguments of the philosophers 
To appreciate the subtle arguments of philosophers over the centuries on this capability at the heart of the problem of free will and determinism, it is critical to study their exact wording.
Thomas Hobbes 
In the 17th century Thomas Hobbes said the idea that one could ever do otherwise than what one actually had done was a contradiction and nonsense.
"I hold that ordinary definition of a free agent, namely that a free agent is that which, when all things are present which are needful to produce the effect, can nevertheless not produce it, implies a contradiction and is nonsense; being as much as to say the cause may be sufficient, that is necessary, and yet the effect shall not follow." 
John Bramhall 
Hobbes's contemporary John Bramhall debated Hobbes on liberty, denying Hobbes's necessity, and saw no contradiction that one could will or not will in the same circumstances.
"I cannot see this nonsense nor discover this contradiction. For (a) in these words, 'all things needful' or 'all things requisite', the actual determination of the will is not included....As he that has pen and ink and paper, a table, a desk, and leisure, the art of writing, and the free use of his hand, has all things requisite to write if he will; and yet he may forbear if he will....indeed if the will were (as [Hobbes] conceives it is) necessitated extrinsically to every act of willing, if it had no power to forbear willing what it does will, nor to will what it does not will, then if the will were wanting, something requisite to the producing of the effect was wanting. 
David Hume 
David Hume's compatibilist account of liberty and necessity is similar to Hobbes. Freedom to do otherwise was "unintelligible," a word very common in modern debates between compatibilists and libertarians.
"I believe we may assign the three following reasons for the prevalence of the doctrine of liberty, however absurd it may be in one sense, and unintelligible in any other. First, After we have perform'd any action; tho' we confess we were influenc'd by particular views and motives; 'tis difficult for us to persuade ourselves we were govern'd by necessity, and that 'twas utterly impossible for us to have acted otherwise; the idea of necessity seeming to imply something of force, and violence, and constraint, of which we are not sensible. Few are capable of distinguishing betwixt the liberty of spontaneity, as it is call'd in the schools, and the liberty of indifference; betwixt that which is oppos'd to violence, and that which means a negation of necessity and causes." 
G. E. Moore 
G. E. Moore argued in 1912 in his Ethics that one could do otherwise, but only if one had chosen to do otherwise. Many later philosophers found this idea to be empty, since under determinism one could not have so chosen, unless the "fixed past" and/or the "laws of nature" had been different.
"It must be remembered that our theory does not assert that any agent ever could have chosen any other action than the one he actually performed. It only asserts, that, in the case of all voluntary actions, he could have acted differently, if he had chosen: not that he could have made the choice. It does not assert, therefore, that right and wrong depend upon what he could choose. As to this, it makes no assertion at all: it neither affirms nor denies that they do so depend. It only asserts that they do depend upon what he could have done or could do, if he chose. In every case of voluntary action, a man could, if he had so chosen just before, have done at least one other action instead. That was the definition of a voluntary action : and it seems quite certain that many actions are voluntary in this sense."
Moritz Schlick 
In "When Is a Man Responsible?, chapter 6 of his book Ethics, Moritz Schlick followed Moore in the idea that he could have done otherwise had he willed to do otherwise.
"more important than the question of when a man is said to be responsible is that of when he himself feels responsible....This feeling is simply the consciousness of freedom, which is merely the knowledge of having acted of one's own desires. And "one's own desires" are those which have their origin in the regularity of one's character in the given situation, and are not imposed by an external power....The absence of the external power expresses itself in the well-known feeling (usually considered characteristic of the consciousness of freedom) that one could also have acted otherwise. How this indubitable experience ever came to be an argument in favor of indeterminism is incomprehensible to me. It is of course obvious that I should have acted differently had I willed something else; but the feeling never says that I could also have willed something else, even though this is true, if, that is, other motives had been present. And it says even less that under exactly the same inner and outer conditions I could also have willed something else." 
C. A. Campbell 
In 1951, C. A. Campbell attacked Moritz Schlick's claim that free will was a pseudo-problem, since it could not be logically verifiable. He denied Sclick's claim the freedom which moral responsibility implies is no more than 'the absence of compulsion' (Freedom of action is not freedom of will).
"Now it may, of course, be an error thus to assume that a man is not morally responsible for an act, a fit subject for moral praise and blame in respect of it, unless he could have acted otherwise than he did. Or, if this is not an error, it may still be an error to assume that a man could not have acted otherwise than he did, in the sense of the phrase that is crucial for moral responsibility, without there occurring some breach of causal continuity."
From the stand-point of moral praise and blame, he [Schlick] would say — though not necessarily from other stand-points — it is a matter of indifference whether it is by reason of some external constraint or by reason of his own given nature that the man could not help doing what he did. It is quite enough to make moral praise and blame futile that in either case there were no genuine alternatives, no open possibilities, before the man when he acted. He could not have acted otherwise than he did.
Campbell argues that if one ought to have done something, then it implies that we could have done that something otherwise than what we did.
The moral 'ought' implies 'can'. If we say that A morally ought to have done X, we imply that in our opinion, he could have done X. But we assign moral blame to a man only for failing to do what we think he morally ought to have done. Hence if we morally blame A for not having done X, we imply that he could have done X even though in fact he did not. In other words, we imply that A could have acted otherwise than he did. And that means that we imply, as a necessary condition of a man's being morally blameworthy, that he enjoyed a freedom of a kind not compatible with unbroken causal continuity. 
R. E. Hobart 
R. E. Hobart (the pseudonym of Dickinson S. Miller) claimed in 1934 that free will involved the ability to do otherwise. His essay is considered one of the definitive statements of determinism and compatibilism. Hobart's compatibilism followed earlier landmark positions by Thomas Hobbes and David Hume, and refined the 19th-century compatibilist views of John Stuart Mill, Henry Sidgwick, and F. H. Bradley.
Writing about six years after the discovery of quantum indeterminacy, Hobart explicitly does not endorse strict logical or physical determinism, and he explicitly does endorse the existence of alternative possibilities, which can depend on absolute chance.
"I mean free will in the natural and usual sense, in the fullest, the most absolute sense in which for the purposes of the personal and moral life the term is ever employed. I mean it as implying responsibility, merit and demerit, guilt and desert. I mean it as implying, after an act has been performed, that one "could have done otherwise" than one did."
"I am not maintaining that determinism is true; only that it is true in so far as we have free will. That we are free in willing is, broadly speaking, a fact of experience. That broad fact is more assured than any philosophical analysis. It is therefore surer than the deterministic analysis of it, entirely adequate as that in the end appears to be. But it is not here affirmed that there are no small exceptions, no slight undetermined swervings, no ingredient of absolute chance. All that is here said is that such absence of determination, if and so far as it exists, is no gain to freedom, but sheer loss of it; no advantage to the moral life, but blank subtraction from it. "Thus it is true, after the act of will, that I could have willed otherwise. It is most natural to add, "if I had wanted to"; but the addition is not required. The point is the meaning of "could". I could have willed whichever way I pleased. I had the power to will otherwise, there was nothing to prevent my doing so, and I should have done so if I had wanted. If someone says that the wish I actually had prevented my willing otherwise, so that I could not have done it, he is merely making a slip in the use of the word "could".
A. J. Ayer 
A. J. Ayer discussed whether free will requires the ability to do otherwise in his 1954 Philosophical Essays.
"When I am said to have done something of my own free will it is implied that I could have acted otherwise; and it is only when it is believed that I could have acted otherwise that I am held to be morally responsible for what I have done. For a man is not thought to be morally responsible for an action that it was not in his power to avoid. But if human behaviour is entirely governed by causal laws, it is not clear how any action that is done could ever have been avoided. It may be said of the agent that he would have acted otherwise if the causes of his action had been different, but they being what they were, it seems to follow that he was bound to act as he did. Now it is commonly assumed both that men are capable of acting freely, in the sense that is required to make them morally responsible, and that human behaviour is entirely governed by causal laws: and it is the apparent conflict between these two assumptions that gives rise to the philosophical problem of the freedom of the will.
J. J. C. Smart 
J. J. C. Smart tried to refute the idea of free will in his 1961 Mind article "Free Will, Praise, and Blame". He explained that "pure chance" (by which he means quantum indeterminacy) exists to some extent within the universe. But he does not use it to generate alternative possibilities that would permit one to have done otherwise. Instead he looks for multiple senses of "could have done otherwise."
There are some events that even a superhuman calculator could not predict, however precise his knowledge of however wide a region of the universe at some previous time...It is important to distinguish "pure chance" from "chance" or "accident." Things may happen by chance or accident in a purely deterministic universe...Now there is perhaps a sense of "could not have done otherwise" in which whether or not a person could or could not have done otherwise depends on whether or not the universe is deterministic...But it does not follow that if a person could not have done otherwise in this special sense then he could not have done otherwise in any ordinary sense. Taken in any ordinary sense, within some concrete context of daily life, "he could have done otherwise" has no metaphysical implications.
Roderick Chisholm 
Roderick Chisholm, in his 1964 Lindley Lecture "Human Freedom and the Self" (adapted as his essay Freedom and Action, "could have done otherwise if he had chosen otherwise" as a strategem used, among others, by Jonathan Edwards in the early 18th century. Chisholm says it lacks a third step to justify moral responsibility:
(a) He could have done otherwise,
it is argued means no more or less than
(b) If he had chosen otherwise, then he would have done otherwise.
Chisholm maintains we could not have made an inference to (a) from (b) unless we can also assert:
(c) He could have chosen to do otherwise.
Keith Lehrer 
Keith Lehrer thought he could prove that someone who showed he could do something (by doing it) could equally refrain, and therefore establish that he could always do otherwise. Lehrer thought this argument strong enough to constitute an empirical disproof of determinism.
"I now wish to argue that we can know empirically that a person could have done otherwise. A person could have done otherwise if he could have done what he did not do. Moreover, if it is true at the present time that a person can now do what he is not now doing, then, later, it will be true that he could have done something at this time which he did not do. This, of course, follows from the fact that "could" is sometimes merely the past indicative of "can." What I now want to argue is that we do sometimes know empirically that a person can do at a certain time what he is not then doing, and, consequently, that he could have done at that time what he did not then do. Moreover, we can obtain empirical evidence in such a way that our methods will satisfy the most rigorous standards of scientific procedure." 
- David Widerker and Michael McKenna, Moral Responsibility and Alternative Possibilities, Ashgate, 2006
- E. O. Wilson, Consilience
- Thomas Hobbes, Of Liberty and Necessity, 1654, § 32
- John Bramhall, "A Defense of Liberty", § 32
- David Hume, A Treatise of Human Nature, Book II, Part III, Sections I-II, p.407
- John Martin Fischer, "Introduction:Responsibility and Freedom," In J. Fischer, ed., Moral Responsibility. Ithaca, NY: Cornell, pp.60-1
- G. E. Moore, "Ethics", p.12
- Moritz Schlick, "The Pseudo-Problem of Freedom of the Will", Ethics, Ch. VII
- Is Free Will A Pseudo-Problem?, Mind, vol.60, no. 240, 1951, pp.245-61
- R. E. Hobart, "Free Will as Requiring Determination, and Inconceivable Without It," Mind, Vol XLIII, 169 (1934) p.2
- A. J. Ayer, "Freedom and Necessity",
- J. J. C. Smart, "Free-Will, Praise and Blame," Mind, LXX, 279 (1961) pp.294-296
- Roderick Chisholm, "Freedom and Action", in Freedom and Determinism, ed. Keith Lehrer
- Keith Lehrer, "Empirical Disproof of Determinism," in Freedom and Determinism, ed. Keith Lehrer, p.177