Why Anarcho-Capitalism cannot work.

My Position
As an Objectivist, I agree with the conclusion that a Free Market Capitalist system is the only moral social system because it is the only one compatible to the individualistic nature of our species. We are social because it is to the advantage of the individual, not because it is hardwired in our brain. Humans are contractual animals, not social animals.
In order to live together in a society of individual minds, rules of conduct must be devised and enforced in order to protect the individual's rights: life, liberty and property. Clear objective rules established by a central body have to be enforceable by the use of retaliatory force. This central body, which we can call the state, has it's authority because the individuals within the geographic area in question have delegated to it the individual's sovereignty in the use of retaliatory force. This consent of the governed can be withdrawn by a vigilant citizenry in case the government oversteps its bounds.

The Anarcho-Capitalist (AnCap) Position.
I will refrain from expressing what I think AnCaps think, so I will quote from Wikipedia.

Anarcho-capitalists believe that in the absence of statute (law by decree or legislation), society would improve itself through the discipline of the free market (or what its proponents describe as a "voluntary society").^[6][7] In an anarcho-capitalist society, law enforcement, courts, and all other security services would be operated by privately funded competitors rather than centrally through compulsory taxation. Money, along with all other goods and services, would be privately and competitively provided in an open market. Therefore, personal and economic activities under anarcho-capitalism would be regulated by victim-based dispute resolution organizations under tort and contract law, rather than by statute through centrally determined punishment under political monopolies.^[8]

 My Objections to Anarcho-Capitalism.

1. The use of competing Dispute Resolution Organizations would be a very efficient and cheap alternative to the court system. The problem arises when two of these organizations cannot agree on a resolution. At this point, society needs a Arbiter of Last Recourse. Without one, two competing organizations within one geographic area could end up in a less than amicable conflict, if their clients or the organization itself refuses to settle.
2. The only way to avoid that an irreconcilable conflict degenerates into outright violence is to have a population that is highly rational and self-interested enough to see that violating other people's rights is not in their best interest.Unfortunately, in most if not all societies, many people are driven by their whims and not their reason. Many are motivated more by status than by profit. Some would rather be the Chief of a jungle tribe, then be a butler in Monaco.Many are just bullies trying to get other people to do things their way while others crave to be relieved of personal responsibility.
3. The only way to make sure a society is fully, or mostly, populated by rational citizens is to be able to choose who is admitted to the geographic area in question, as well as having the ability to expel them in a credible way. Here again the need arises for a central choosing authority whose powers emanate from the consent of the governed..
4. Taxation is the forceful confiscation of property by the state. But it is not the only way to fund a government that is just big enough to operate the courts, police and military necessary to defend individual rights. I won't go into this, but you can find some insights into no-taxation funding here and here.

The Non-Aggression Principle (NAP) and the Uses of Force.
The NAP is the idea that No One has the right to initiate force against Anyone. This idea was first proposed by Ayn Rand and is part of her philosophy, Objectivism. The difference, though, between the AnCap and the Objectivist view of it starts in that the former views the NAP as an axiom, while the latter considers it a logical consequence of the morality of individual rights.
The use of force can be classified in three different categories:

1. Initiated or Aggressive Force: The unprovoked use of violence or threat of violence against a person. It is this one to which no one has the right to use.
2. Self Defense: Using force to stop an occurring or imminent aggression. Every individual has a right, if not the moral obligation, to defend against aggression from others, including the state.
3. Retaliation: The use of force to punish right violating aggressors, recover any stolen property and the enforcement of contracts. This use of force, in the Objectivist view, should be delegated to a government constrained by rational and objective laws. It is up to the citizens in the geographic area of influence to keep vigilant against any deviation in the use of this force from the objective rules.

AnCaps reject the idea that retaliatory force should be delegated to a single entity as it would impose a single set of norms and it would eventually lead to abuse. In my opinion, they neglect their right as individuals to defend themselves, with force if necessary, from an aggressive use of force by the state; and they neglect their responsibility as citizens to vocal vigilance against "mission creep".

My Opinion on the Role of Government

A limited government has a legitimate and necessary role in a free society. It must be seen as the Arbiter of Last Recourse in disputes between individuals because of breach of contract, negligence, and outright aggression. Any form of private arbitration, binding or not, between parties must somehow be enforced.The prosecution of criminals should be left to an objective entity, and not to the possibility of gang warfare.

The role of the citizen in a free society is to be vigilant of any "mission creep" by the government and be ready to act, with force if necessary, to stop any abuse by the government. The benevolence of those in control of government must not be taken for granted. They must never be allowed to consider themselves above the law or those they govern.They must always feel the eyes of the citizens upon them.

The lack of a central government did not work well for Europe after the fall of the Roman Empire, nor has it worked well within Somalia.Even within a city, where the police does not make it's presence known, gangs rise up to take control of the streets. And when two gangs try to occupy the same territory, war breaks out. All power vacuums will be filled.