In The Virtue of Selfishness, chapter 14 titled "The Nature of Government", objectivist founder Ayn Rand criticizes anarchy. An anarchistic society is one in which there exists no government and no institutionalized coercion. (I prefer the term "voluntarism" over "anarchy" because "anarchy" conjures up images of chaos and violence and that's not what anarchists/voluntarists/libertarians want, but here I will use Rand's terminology for consistency.)

The standard assumption shared by over 99% of all people is that anarchy could never work because you need roads, police, a legal system, education, etc—all the things government currently provides. Objectivists agree with anarchists that one does not, in fact, need a government to build roads or schools. (If you're wondering why, see the libertarian FAQ.) This they have in common with anarchists. But anarchists go one step further: they think there is nothing that could not be provided by private corporations (and that wouldn't be worse in the hands of government), which includes a police force, an army, and a legal system. Objectivists, on the other hand, believe that for these three specific services you need a government. Rand writes:

The proper functions of a government fall into three broad categories, all of them involving the issues of physical force and the protection of men’s rights: the police, to protect men from criminals—the armed services, to protect men from foreign invaders—the law courts, to settle disputes among men according to objective laws.

At this point, one may wonder what the special ingredient is that allows governments to provide these services no other group of people could. Rand says it is this: objectivity. But governments are just made of people, like all other groups. What is it that enables them to be objective and prevents other groups from the same? It is unclear to me.

Objective law is a form of knowledge. And objective knowledge needs no special source. Instead, it requires good explanations: good legal, moral, political, etc explanations. As David Deutsch explains in chapter 1 of his book The Beginning of Infinity, good explanations are objectively different from bad explanations because they are "hard to vary": it's difficult to change a good explanation without it breaking apart or at least losing quality while still accounting for what it purports to account for. That is not true of bad explanations: you can vary parts of them without changing much of what they account for.

So objectivity is attainable in an anarchistic society if that society creates objectively good explanations in general and objectively good legal explanations in particular. Doing so requires not a government, but hard-to-vary explanations. And industries not currently monopolized by the government already create objective knowledge all the time (arguably much more so than government ever has): scientists, engineers, architects, doctors, programmers, etc create objective knowledge daily.

However, the alleged lack of objectivity is not the only problem Rand sees with anarchy. She writes:

[S]ome people are raising the question of whether government as such is evil by nature and whether anarchy is the ideal social system. [...] A recent variant of anarchistic theory [...] is a weird absurdity called “competing governments.”

That isn't full anarchy, it's only governments that would be in an anarchistic relationship with each other in that case (as they are already, by the way), not their subjects—but to make it work, let's imagine that what she means by "government" is a collection of private-security companies and arbitration services. They provide protection and legal-conflict resolution—the things Rand states are only legitimate in the hands of a government. She continues:

[S]uppose Mr. Smith, a customer of [arbitration service] A, suspects that his next-door neighbor, Mr. Jones, a customer of [arbitration service] B, has robbed him; a squad of Police A proceeds to Mr. Jones’ house and is met at the door by a squad of Police B, who declare that they do not accept the validity of Mr. Smith’s complaint and do not recognize the authority of [arbitration service] A. What happens then? You take it from there.

She implies that Mr. Smith and Mr. Jones could never resolve their conflict because they do not have a shared jurisdiction—one legal system that subsumes both arbitration services and by whose rules both must abide. By not stating this explicitly, Rand assumes that this is obvious. That is a mistake. And if she had elaborated more, maybe she would have realized her mistake.

Conflict resolution takes knowledge—knowledge that may need to be created first. In other words, it takes creativity. As such, conflict resolution is no different from the creation of knowledge in response to other problems. Problems, universally, are just conflicts between ideas, legal or otherwise. Legal problems have no special epistemological status, so we can conclude that their solution does not require any special treatment or any special source of knowledge such as a government.

Why does Rand think that Smith and Jones could never resolve their conflict without a shared jurisdiction? Her mistake is an instance of what Karl Popper called "the myth of the framework". He explains it in a lecture you can find here (translation by the Philosophy Overdose Youtube channel):

The myth of the [framework] can be summed up in one sentence as follows: it is impossible to discuss a problem rationally and fruitfully if the participants do not stand on a common ground of basic assumptions, or if they do not at least provisionally accept such a common intellectual [framework] for the sake of discussion.

In other words, it is the notion that unless two parties share some common assumptions, they cannot solve problems together or come to an understanding. It is the idea that separate frameworks cannot talk to each other.

Now, in the quote above, replace "framework" with "legal framework" and "common ground of basic assumptions" with "common legal code" or "shared jurisdiction" and we have Rand's idea that people require a shared jurisdiction to solve legal problems. Rand's mistake is a special case of Popper's myth of the framework.

Popper's first criticism of the myth is that discussions in which participants do share all or most assumptions are often much less fruitful than those in which they do not share many assumptions. People who agree don't have much to discuss. On the contrary, discussions between people with very different viewpoints can be tremendously fruitful, in the sense that they learn something new, extend their horizons, or are "shaken in their beliefs".

Then Popper asks: "Is such a discussion impossible [, as the myth claims]?" He gives an account by Herodotus of two ancient Greek tribes which have diametrically opposed burial rituals: one that eats their dead, and one that does not. The former is asked how much money it would take to get them to stop eating their dead, to which they respond they could not entertain such an awful idea. The latter is asked how much money it would take to get them to start eating the dead, to which they have the same response. To each tribe, the other's ritual is an unspeakable horror. And yet, if these tribes ever met, many of their members might be shaken in their most dearly held beliefs, and they almost certainly would learn something from each other, and maybe even learn to respect the other's rituals and start to cooperate.

Popper concludes that what's needed is not some minimum set of shared assumptions, as the myth alleges, but simply a willingness to discuss and solve problems. This willingness is epitomized in his famous quote:

I may be wrong and you may be right, and by an effort, we may get nearer to the truth.

In identifying the myth of the framework and proposing this replacement, Popper made a universal epistemological discovery. It applies to all kinds of conflicts, including legal ones. This solves the problem Rand deemed insoluble without government. Both arbitration services have an incentive to bridge the gap between their frameworks because they want to be known as companies that can solve this problem. Their livelihood depends on it. People would not hire them if they could not solve this simple problem that would arise often. It's just not the case that these companies would just say, "well shucks, I guess we can't solve that", and that society would soon collapse into a state of chaos. People want to solve problems. No man can solve every problem in his life by himself. He depends, at least in part, on the knowledge of others, so he will want to cooperate. This is what holds society together—not government! It is uncharacteristically pessimistic of Rand, who rightly celebrated reason and creativity, to underestimate people's creativity in the legal context.

How, specifically, could these separate arbitration services solve the problem of having different legal frameworks? Well, how does Rand think countries around the world solve the problem when they want to resolve disputes? They are in a state of anarchy with each other. How does she think citizens and companies cooperating across borders solve this problem today? How does she think telecommunications services enable their customers to call each other despite being on separate networks and despite those companies being competitors? Both within and across borders? Etc. So, one specific solution is for both arbitration services to communicate beforehand and agree on rules and set up an infrastructure. That infrastructure could in turn involve banks agreeing to confiscate their customers' funds should they resist cooperation in a suit. That's one specific idea, but one does not need to propose solutions for the claim that legal problems are soluble absent a leviathan to be valid.

Rand arrived at a mistaken political and legal philosophy because of this epistemological mistake identified by Popper. Epistemology is upstream of politics and law. This underlines the need for the correction of epistemological mistakes to fix mistakes in political philosophy.

It is interesting to think about how an ideal world as Rand envisioned it would evolve. As I said, Rand wants the police force, army, and legal services in the hands of the government only. And she wants all three financed by voluntary taxes. What happens if private corporations realize their creative potential, think they can do a better job than the government, and want to offer innovative security or arbitration services? Is the government going to shut them down violently to enforce its monopoly? It can't, since according to Rand, the government may only employ violence to defend men. So if it cannot go against private corporations that want to compete with it, then it will automatically lose its monopoly in the areas Rand reserves specifically for it. It will become just another provider of services, like any other corporation. It seems that the society Rand envisioned would be unstable and would sooner or later evolve to become a stable, anarchistic society anyway.