Steel Anarchism

The first thing to understand that the only true anarchism is a flavour of anarcho-capitalism. This is anarchy fully steelmanned, logicked forward until it could be logicked no more. (This is philosophical anarchism, not bomb-things anarchism. Philosophical anarchism is recognizing each dire ape as responsible to themselves and only themselves, and that each should be held responsible for their actions and only their actions.)  I'm going simply call it steel anarchy. (You're supposed to make fun of it. Any other name would be just as ridiculous but trying to hide it.)

The key feature of anarchism is Exit. Because it is illegitimate to actively do anything to someone else without their consent, all anarchic institutions must either be opt-in or opt-out. As a result, if you don't like the rules of any particular anarchy (syndicalism, communism, whatever) you can opt out and they're not allowed to force you back in. Empirically speaking most will opt for private property, because they want their stuff to stay their stuff. Or: markets are what you get when you leave humans alone. See the bazaar. In any case, an anarcho-capitalism has no problem with subsidiary communes or whatever. If they can survive, good on them.
There is only one kind of real anarchy: steel anarchy.

Given these few premises, most of the modern world (eventually) falls out of the logic. Fact is, mayors and police and courts and municipal utilities serve needs that citizens have, and an anarchy is going to have to serve the same needs. Given the IQ of the population isn't going to markedly increase upon switching to anarchy, the solutions are going to look much the same and employ much the same kind of people.

Everything will be the same. Except radically different. More on this in a bit. First, how does anarchy maintain itself?

Anarchy is obviously not particularly natural. While it has arisen in limited historical contexts, it usually doesn't.

Every regime has required a political formula. For kings it was divine right. For democracies it is mob right. However, in all cases these formulae have been lies. Obviously the king wasn't voted in by God. Democracies don't have the consent of the people; if they did they wouldn't have to harp on having the consent, they would just ask citizens to sign normal contracts, and they would willingly do so. When these lies are exposed, the regime collapses; moral legitimacy is critical to regime survival. However, all coercive regimes are inherently illegitimate, (proof in draft stage) so their formulae must be lies.
The true formula is consent, and thus Exit. Once a critical mass of humanity understands this truth, their nation will convert to an anarchy. Any attempted conquerors will have an uphill fight, lies against truth, to convince the people of their right to rule. It will almost certainly be too expensive, and they will fail. It is similarly difficult to dislodge mob right as a political formula. Even North Korea holds elections. Difficult, that is, until it fails naturally due to being a lie.

But what was that about everything being radically different, yet the same?
The point of Exit is to invite Gnon to punish you more swiftly and harshly. The longer Gnon's wrath abides, the more terrible it is.

What terrifies me is that not only do I not think America deserves Mitt Romney, I don't even think America deserves Barack Obama.  After all, a couple of centuries of diligent looking-after has run us up quite a tab with God.  A tab that will be paid or punished.  What terrifies me is that while I see no collective interest in paying the tab, it doesn't seem to me that the punishment has even begun to begin.

Because their customers can Exit at will (more or less) institutions will be directly funded by their beneficiaries. Anyone they might parasitize will immediately Exit. For the same Exit reason, they will have discipline imposed on them; if they do not serve, they will starve. As a result, while (for example) courts will still exist and deal with much the same problems as now, how they deal with them can be radically different.

Let's examine courts in a little detail. Even such bright lights as Nick Szabo don't fully understand the anarchist version of them, so one should not be surprised to to find them difficult to grasp.

Szabo rightly brings up the topic of judicial arbitrage. When I commit a crime, I pick a court likely to find me innocent. My victim, naturally, picks a court likely to find me guilty. Seems worse than having no court. How does anarchy solve this?
By not attempting this system at all. A court will no more accept a case post-facto than a life insurance company will accept an application post-mortem.
In the modern world, your local democracy promises (vaguely) to uphold rule of law, and lets you accuse people and haul them into court. It then does whatever it sees fit and you just have to suck it up. If it doesn't feel like letting you haul someone off for whatever reason, you just have to suck it up.

In an anarchy, you sign up with a security insurance provider. In this agreement, you will have certain obligations, such as turning over evidence, appearing in court, and most importantly, abiding by the court's rulings. In return, it promises to protect your property, indemnify you against loss, and persecute anyone who perpetrates against you. In general, the court under which a perp is tried is determined by who the victim pays to provide court service. If you don't like their laws, don't victimize their patrons.
(You can, if you want, try to see to your own security. Good luck with that. More on this in a bit.)
In other words pretty much what it's supposed to do now, except you sign a contract agreeing to let them do it.
However, that signing step is what creates a radical difference. If you feel the court is not successfully protecting your property or is burdening you beyond what it is benefiting you, you can simply sign on with a different one. As a result, courts will only have effective and efficient laws. (Not 40,000 and growing.) Most likely it will have about three laws: battery, vandalism, and fraud. That is, protecting person, property, and integrity. However, it can be tricky to decide who is in fact aggressing on whom. Punishment and discrimination between small differences will generally follow the English Common Law precedent, being the most advanced body of judgement in the world, though Xeer will eventually catch up.

(Objection for competing courts: courts will want to cooperate, mostly. While they'll have somewhat varying procedures, thus creating desire for regulatory arbitrage, if OneCourt refuses to deal with AlphaCourt, effectively patrons of One can't trade with patrons of Alpha, because AlphaCourt patrons know they can't enforce contracts with them. There would be a lot of shooting instead, as per below. Similarly, rather than calling the cops on a OneCourt thief, they will just shoot, since the cops can't do anything special. That's not a win for OneCourt patrons.)

Finally, if there really is a better solution than courts, it will be impossible to stop it from being used. Groping toward the edge of the box for an example, what if courts guaranteed contracts instead of people? Law a la carte?

It is also possible to provide your own security, much as you can grow your own food. However, it's a tricky proposition. The main function of the court is to hold a set of rules about who the aggressor is. By signing the contract with the security firm, I agree to abide by their judgement of such. I do this because even honest men disagree...and most aren't even honest.

If someone slurs my good name and I shoot them, am I defending my good name or unconscionably escalating a petty conflict? If someone trespasses beyond my fence and I can't shoot them before they get off again, is chasing them necessary to keep the boundary's integrity, or simply another trespass? If someone flagrantly trespasses and I shoot them, did I shoot them for trespassing or did I invite them onto my land, then shoot them in the back? That is, do I owe weregild?

For personal security, I have to do a lot of shooting. I'll meet two kinds of people; those that will honour their agreements, and those that won't sit still for me to apply sanctions. The first I never need to shoot, the latter I might as well shoot now. Knowing this, they might well try to shoot me first...
By contrast, using a guarantor, more agreements become possible. Anyone breaking their sworn word has to evade two vengeful parties instead of just one; this allows contracts that can't be personally overseen. Delivery is a simple example; if I have to personally oversee a delivery, I might as well make the delivery myself. Reputation can help, but why trust only reputation when I can trust the combination of reputation and technology?

However, there are certain bits of personal security the firms will likely endorse. If a probable Ebola patient approaches me, they will likely be fine with me shooting them, as it constitutes a threat to my life, and they have to pay weregild to my dependents if I die. If my neighbour brings home a lion, they probably want to go check it out themselves - it is possible to secure it properly after all - but they're not going to mind if I shoot a loose lion. Similarly they're going to raise premiums on my neighbour if they don't flatly refuse to insure him as long as they have a lion. Consider what happens if they lose their insurance; I shoot my neighbour, I shoot their lion...and nobody is left to object to what happened. It may be moral. Or not. However, anyone seeking vengeance is going up alone against me and my insurer. (For this reason I think it should be considered attempted murder to publish lists of uninsured.) We hire the insurer precisely because without one, morality is not upheld, as it isn't in the modern world.

Second example: mayors-cum-barons.
Since cities will no longer be some special kind of property, supposedly owned by everyone, they will likely be consolidated under one individual who buys out everyone else. Cities will have CEOs, that is, still have mayors, though calling things by their right names they're barons. They don't merely administer the city; they have property right (not only moral, but secured by some insurance firm) in the land and buildings of the city.
Mayors will more or less make the same kinds of decisions they do now. They will make bylaws about parking spaces and decided who to contract out municipal water supplies to.
The difference is the citizens will have to actively agree to let them make these decisions. I'm not sure exactly how this will shake out; social engineering is easy, but not that easy. Perhaps barons will require visitors to agree to local by-laws before they enter the city. Perhaps they'll be tolerant, allow metics, but not give them full citizenship unless they agree to the baron's rule.

(As always, you can form the contract to almost exactly simulate modern 'democratic' cities if you want. We'll just see what Gnon thinks about that...)

Since they're selected for economic competence, frugality, providence, et cetera, anarchic barons will generally be competent rulers, or at least know enough to hire a competent steward. (As in, basically the opposite of being elected for being the best liar.)

However, some barons will make mistakes. In these cases they will lose citizens. (I'm assuming nobody will be dumb enough to opt into a contract that forbids them from emigrating the city.) As a result, the less-mistaken barons will gain citizens. Bad cities will either reform or be gradually, peacefully leached of power. This is one of the non-perfect aspects of anarchism, however, as leaving a city is not cheap, and cannot be made cheap without subsidy. It is difficult to take your friends and family with you. It is impossible to take your geographical familiarity and history with you. Barons will be able to charge rent (neo property taxes) in proportion to how culturally invested their citizens are. However, this gives the baron an incentive to (unironically) culturally enrich their city, so it's not all bad.

Finally, ancap is neocameralist patchwork. Given free Exit, cities will balkanize. You get patchwork for free. As for the neocameralism...

It doesn’t care about competency and order and peace and stability as such.

Anarchy frees humans to pursue whatever they want. Humans care about competency and order and peace and stability. Ancap, unlike Hadley Bennett, does not presume to tell humans they care about these things more than any particular thing else. If they do so care, they will patronize cities and barons that provide those things at the expense of other things. However, Gnon has an opinion on such barons. I don't know what it is, and neither does Bennett. Other barons may have value hierarchies that Gnon likes better. Humans may prefer cities that sacrifice order and stability for those things. Anarchy proposes to let Gnon tell us which is which.