We can discard one sense of anarchism as vacuous: everyone always, by definition, actually does what she thinks is in her own best individual self-interest, all things considered, including considering how other people will probably act (perhaps coercively) in their own interests.
Taken to the logical extreme of its roots in a non-vacuous sense, anarchism would entail absolutely no coercion whatsoever. The only acceptable means of influencing another person's behavior would be to threaten to or actually withhold active cooperation. Killing or injuring someone in self-defense is obviously coercion. Locking your front door or physically hanging onto your wallet is coercive: you are using physical force to prevent a person from doing what he wants to do.
We could achieve the logical extreme of anarchism if it were physically impossible to coerce another person. Extreme anarchism is also possible if no one ever wants to coerce another person.
It's instructive to read the science fiction of Greg Egan*. In many of his novels and short stories, he envisions a future where coercion is literally impossible. In these stories everyone exists as programs running on very fast, very large computers; it's impossible for one person (program) to coercively influence another. But it's instructive that he has to go "outside" the system, to the operating system of these giant computers, to make coercion impossible. But someone has to write and maintain these operating systems, and there's nothing to prevent one person from creating a new computer with an operating system that does allow him to coerce its inhabitants. Fundamentally, he assumes that no one will want to coerce others.
*I don't mean at all to put Mr. Egan down. He's a brilliant, imaginative writer, one of the best writing today (or ever), and his work has been a source of tremendous inspiration for my own ideas.
But how do we make sure that no one wants to coerce others? Wanting to coerce others, the subjective desire to coerce others, is not materially self-selecting, in the same sense that not wanting to eat food is self-selecting for modern humans. To eliminate the subjective desire to coerce others, you have to either impose artificial selection, or completely eliminate uncorrelated heritable variation in subjective desire. The latter seems sterile — there's no growth without uncorrelated variation — the former is obviously coercive.
So the logical extreme of anarchism seems physically or logically impossible.
Can we adjust the definition of anarchism away from the logical extreme? How far can we adjust the definition before the label itself becomes ridiculous or intellectually dishonest?
We could simply say that anarchism is the absence of punishment. Whatever coercion we apply to each other, we never apply coercion that's intentionally designed to cause subjective harm to a person. We put people in prison to hurt them; we maintain threat of prison to (at best) discourage people from hurting each other; but we're still actively hurting some people. (The notion of prisons as "correctional" facilities goes beyond a polite fiction to an obvious lie.)
I myself would definitely approve of a system without punishment, and such a system seems plausibly realistic. The problem with this definition is that you could still have a ruled/ruling class political-economic system without punishment. Even if the ruling class doesn't punish the ruled class, the presence of a ruling class would seem to make this definition too weak to be called specifically anarchism.
Anarchism might be defined as simply the absence of a ruling class. Again, perhaps desirable and feasible, but the absence of a ruling class doesn't sufficiently distinguish anarchism from democracy (by its roots "rule of the people"); this definition still seems too weak.
We might further restrict the definition of anarchism to a democracy without any violent coercion; we could still lock our doors and hold onto our wallets. This definition seems like the most we can adjust the definition without losing the intellectual validity of the label of anarchism. But, however, this definition begins to pose severe practical difficulties.
Again, we can turn to science fiction for an example of an anarchist democracy in Eric Frank Russell's novella, And Then There Were None. It's notable that Russell has to posit several conditions to make this society seem superficially plausible. The society is isolated: it's enormously expensive (according to the story) to send just a few hundred soldiers to the Gands' world; a full-scale invasion is impossible. The society is homogeneous, and each individual is inculcated to be fanatically resistant to even the hint of coercion, as well as fanatically idiosyncratic. The society is small, and there appears to be no long-range transportation: someone who egregiously abuses the lack of violent coercion (except in immediate self-defense) will be coercively frozen out and starved. The world is underpopulated, and there is no competition for physical resources. Most importantly, the world has inherited technology from Earth. There is no need to create large-scale productive enterprises; all technological production appears to be at a small enough scale for easy voluntary cooperation.
Russell gives us a nice package, but he ignores the fundamental power of dialectical materialism and evolution. There will be variation in individual attitudes, caused by genetics and the variability of childhood indoctrination. People who egregiously abuse the system are selected against, but that's about it; there is scope for variation that's not selected against.
People can subtly abuse the system, continually pushing the limits of how much the system can "push back". A person might, for example, always give back "low value" obs for higher value obs. A person who successfully does so without getting frozen out will have at least a small material advantage over his neighbors, and this propensity can be transmitted to others.
People can also become more conformist. In a system where the good will of one's neighbors is of paramount importance, someone who's not quite so idiosyncratic as his neighbors can have an advantage, and it's hard to disapprove of someone who's not being idiosyncratic enough. One can easily imagine a group of rebellious children all dressing in uniforms just to piss off their idiosyncratic parents! Childish, to be sure, but sometimes childish attitudes become adult beliefs.
It's also possible for people to become submissive. It's hard to retain fanaticism without actual oppression. The founders of the society might be fanatically resistant to coercion, but this very fanaticism will ensure that people don't generally try to coerce others. At first, then, a person a little more submissive won't even be coerced; he can get a "free ride" on his neighbors' resistance to coercion. It's also plausible that even a slightly dominant-submissive relationship might have a local, immediate economic advantage over equalitarian relationships. Even when the communists were fighting the fascists in Spain, communist soldiers were still shot if they didn't follow their officers' orders.
It's impossible to say how Russell's society would evolve even if it could exist, but it's clear that it would evolve away from his democratic anarchism into something very different.
There's also the severe problem of protection for unpopular minorities under any form of democratic anarchism. Either the democratic mechanisms used to influence others actually work, or they do not work. If they don't work, it's hard to see how you can have any form of mutual cooperation where individual betrayal confers an immediate advantage. If they do work, then sooner or later, the majority will get a bug up their ass about some otherwise inoffensive minority: blacks, Jews, women, religious dissenters, redheads, left-handers, people whose favorite color is blue, whatever. It takes a minority with disproportionate power to protect another minority, but a minority with disproportionate power is the very definition of a ruling class.
I have no doubt that we do not today live in the best kind of society that's practically feasible. There is enormous scope for improvement, improvement that's possible with the level of intelligence, philosophy and ethical enlightenment we've already achieved. But we have to achieve this improvement by using rational thought, not Utopian magical thinking*.
*Nor by slandering as a fascist anyone who refuses to adopt one's magical thinking and historical revisionism.