...is treat security as a consumer service, and then apply normal free market economic analysis - and they apply it correctly. This leads them to conclude that a security monopoly cannot emerge in a free market: in the same way that a widget monopoly cannot emerge in a free market. If a security monopoly cannot emerge, then a state cannot emerge, as a state is a security monopoly.
This argument is based on two premises. The first is that the principles of free market economics, such as allow us to argue against the possibility of monopoly formation in widget markets, are correct. Minarchists obviously can't attack this premise, as they believe in the same economic principles. The second premise is that security is a consumer service, not fundamentally different from any other consumer service. Here is where minarchists try to attack ancapism; they try to argue that security is special somehow; that the normal economic analysis does not apply in the case of security; that, unlike every other market, the security market is prone to monopoly (and therefore the state, and therefore ancapism is impossible).
And, IMO, that objection is absolutely correct, but it has not yet been proven. It has been asserted over and over, and hinted at, but not proven. And the burden of proof is on the critics of ancapism, since they are the one's claiming that there is an exception to the principles of free market economics. In other words, if minarchists can't explain why security is an exception, then logically they must either embrace ancapism or reject free market economics. Specifically, what minarchists need is an argument for why/how a monopoly will emerge in the security market. I don't mean some speculations with a dash of empirical evidence, I mean a logically rigorous argument - and to my knowledge, that does not yet exist.
"Alas! I believe in the virtue of birds. And it only takes a feather for me to die laughing."
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