Is Violence a Natural Monopoly?

A long time ago, I saw a very heated argument in a comments thread on Hulu (yes really) between an anarcho-capitalist and a sane person. It went something like this, though truthfully I'm probably steel-manning the Ancap.

Ancap: We shouldn't have a state; the market will provide.
Sane: What about roads? Courts? Protecting your private property?
Ancap: The market will provide! Private companies will build the roads, and I'll pay a usage fee! Private arbitrators will resolve disputes like in 9th century Iceland or pre-colonial Somalia!
Sane: And your private property? Who will protect your stuff in this brave new world?
Ancap: I'll protect it myself or contract out someone to do this!

I think about that exchange a lot, and every time I want to reach through my screen, grab the Ancap by his (these guys are almost always male) shoulders and shout:

What could you possibly offer a mercenary that's worth more to him than just shooting you and taking all of your stuff right now? A series of payments over time? Congratulations motherfucker, you just invented the state! But not even a good state, with rights or democratic accountability or anything, but the most shitty, primitive kind of state, a mere stationary bandit. You've just reset political development back to square negative fucking one! The blood of millions shed over ten millennia to get to where we are now, boiling away in this stateless vacuum! All for what! For some macho bullshit, some myth of total self-sufficiency, or have you just not thought this through?

...deep breath...

And from that my mind turns to a more intellectual question: Is (legitimate) violence a natural monopoly?

Max Weber defined a state as being the institution with the monopoly on the legitimate use of force in some geographic area. But supposing an Anarcho-capitalist society existed, and there was a competitive market for legitimate violence. Would such a market remain competitive, or would a monopoly on legitimate violence (i.e. a state) emerge?

Having defined "state", let me define "legitimate violence". By "legitimate violence" I mean violence to protect property or enforce a contract or agreement.

Now my assumptions:
Suppose there were n mercenary firms,  numbering from 1 to n, with 1 being the most powerful and n being the least powerful.

Now if I contract with a firm, it can assuredly protect me against less powerful firms, but it cannot secure me at all against more powerful firms.

A firm will not destroy itself resisting a more powerful firm - i.e. it is costless for stronger firms to overcome weaker ones

The unprotected lose everything - even a lone individual cannot resist the least powerful mercenary. Thus all people would hire someone

But who would hire firm n, which cannot protect against any other? No-one, so firm n exits the market. But now firm n-1 is the weakest, and so its services are therefore worthless, and so it too must exit the market. Thus by induction, only 1 firm can exist in equilibrium. Thus a state emerges naturally from (capitalist) anarchy. Intuitively, the thinking might be summarized as "why would you hire the second strongest mercenary?"

This begs some questions.

First, if a single state is emergent from anarchy, how is that one universal state hasn't emerged from the present international anarchy? My simplified model must be omitting some important details - and indeed it its.

By assumption, distance to irrelevant in my model; all the mercenaries contest the same geography. A more "realistic" model would have a spatial component, weakening firms the further out they are from their "base". I suspect the the equilibrium there would have multiple firms spread out, like our present international anarchy.

My model also assumes the mercs have fixed strength relative to one another, although given the strong presumably have the greatest ability to earn money from their "customers", this seems to enhance a centralizing equilibrium.

And of course i ignores the role of prices in consumer choice.  I have avoided this so as to avoid solving equations to maximize utility, etc. which a reader might find tedious or intimidating. It won't make a difference though - as long as the leading firm asks for less than everything, their "customers" will pay it before losing everything.

Second, I assumed that overcoming weaker rivals is costless. If it isn't costless, then its possible for the profits gained from a new "customer" to be insufficient to justify the costs of acquiring them, and a firm could stay independent.

At this point, we've away from economics and markets and into international relations, so I'll bring it back into my comfort zone by asking the next obvious question: suppose there was a competitive market for "legitimate violence" - why wouldn't the firms just combine into a monopoly to maximize their profits? Such a thing could not be prevented in Anarcho-Capitalistan. And such a firm (state, now, I suppose) could maintain such a monopoly by murdering or co-opting anyone who threatens it. State monopolies are after all a time honored tradition.