If one party or the other doesn't abide by the terms of their contract, there can never be a peaceful resolution, governing body or no. It is impossible to eliminate the need to use force. The only question is who is exercising that force and by what authority. In the case of arbitration in an anarchic society, the arbitrator’s private defence agency has the (fire)power to enforce contracts, and their authority comes from the contract itself.
And then a private entity can buy the arbitrators, their private defense agencies, and set up a local zone with its own set of laws favorable to the private entity. There would be no motivation to compete because certain entities, "corporations," would become so big as to buy out any form of competition, or simply turn the laws in their favor. There are no checks on different agencies, and these arbitrators would duplicate services and be much more liable to buy-out than even our current lobbying system in the United States.
Let me use another example. Suppose there was a man with no family and no friends who was shot and killed by another man with money and friends, in other words, better known (in a good way). The murderer could simply pay off an arbitrator to write of the crime...the arbitrator doesn't have to worry about people striking against their agency because of their decision because the murderer is known and the victim was a nobody. That is
not justice, and like it or not, that is what will become of a system like this. I can think of countless more examples of how this system will devolve into stagnation.
Nothing so primitive. Really, the whole system is not so different from the current system of courts, police, etc, except that these functions are privatised. I think we're having a misunderstanding over the meaning of the word "government". By "government", I mean an institution with a monopoly on legislation, regulation, justice, economic policy, and so forth. I don't mean that these functions are unnecessary or not useful, just that they should be provided by private organisations which are competing for voluntarily-paid fees. Such organisations could be described as "governing bodies", though the lack of monopoly power and enforced taxation makes them completely unlike a "government" in the usual sense of the word.
The difference between a well-run government and a corporate monopoly is that a well-run government has a system of checks to protect the rights of the people as well as the legitimacy of the law, while a private monopoly's primary motivation is profit. There are certain things that are run better privatized, but a government is needed to oversee all of these.
We need a government to stop individuals from selling their own private property for profit? 
Is that seriously the point you're making? ...
No, the point is this is a company acting to remove competition in a way other than providing a better product or lower prices. This is directly contrary to your claim.
What claim? I never said big companies couldn't use their capital to put up barriers to entry, I said if they did and they raise the prices or reduce the quality of their products, competitors will overcome whatever barriers are put up in search of their own profits. A monopolised industry is a profitable one, and profitable industries attract competitors.
Your theories are just that...theoretical. In that "axiom," you give no time frame for how long that would even take. It could be years and years before a competitor could even materialize, and by then, a company can become so big as to completely crush competition, making the barrier to entry extremely high. Take our current government in the United States for example. Suppose we all aren't happy with a certain law. We don't just stop paying our taxes; we vote the person who made the law out of office and vote another representative who will work to repeal the law (so long as it is not infringing on the US Constitution, and even then with enough votes, we can amend it). If we all stopped paying our taxes and ignored the law, our current system of government would collapse. Here is a true axiom:
Not everyone is rational. With this axiom in mind, anarchy is one of the worst types of societies. There is the slightest chance that it would work in very small communities or in a Star Trek or transhuman society, but in our present day and for much of the future, it simply cannot work. It is inefficient, unstable, relies on both rational and irrational conjectures with equal weight, and puts justice second to purchasing power.
Funny, you admit that their IS government regulation, and yet you point to this as an example of it working without government regulation. The "(for the most part)" is extremely telling of the weakness of your position. Because what it says is that the government is there to step in as needed to rectify things and keep the free market running smoothly. Like I said, the mere threat of bringing in the law can do a lot to keep people operating honestly. So you haven't at all demonstrated your case.
You'd need an example where there is NO government involvement and NO threat of possible government involvement. I'm not sure if a real world example of this actually exists in the modern age in anything but very small scale because such systems break down. (By "government" I am including any governing body.)
I freely admit that no anarchic society has ever existed since prehistoric times except briefly during wars, and this discussion is entirely hypothetical. But the non-existence of such a society at present does not preclude the possibility of an anarchy emerging in the future, and it is possible to speculate as to how this society would function (or not) and what benefits and flaws it might have over other societies.
Every historical case of an anarchy breaking down happened because the existing government was suddenly removed by force, without warning, and with nothing to replace it. This always happened with no intention of introducing anarchy as a stable system, and no modern anarchist would condone, let alone recommend, dismantling a government in this manner. (The right way to go about introducing anarchy is to gradually replace each branch of the government with a private system that performs an equivalent function, but to my knowledge this has never even been attempted.)
When you translate governmental services like justice and lawmaking to private entities, they would lose their moral sentiment. Once again, I believe certain things can be privatized, but the government as a whole cannot. In introducing an anarchic system, it will simply devolve into a poorly run monopoly that was intended to be avoided. The consumer chooses what type of society they want to live in. That's why revolutions occur, and that's why new systems of government are erected. Anarchism, even when it is gradually introduced with a plan, will devolve because of the larger influence of irrationality and anti-morals through people's direct purchasing power to destroy rational and "moral" systems of government in times of increased emotion or short-term loss of faith in private entities.
