It will be in their interest not to fight each other. They'll try to agree. But agreeing can be hard. People try to talk things out and come to an agreement a lot, and sometimes it works, but sometimes it doesn't. So, some disagreements will be resolved without a big hassle. But there will be some others that won't get resolved so easily. The security forces will either need to peacefully co-exist, side by side, *while they disagree about some things*, or there will be violence.
One way to try to solve this problem is not to have multiple security forces. If there's only one game in town, then there can't be any disagreements! Problem avoided. Right?
No. People will still disagree. The difference is that people who disagree with the only game in town have no good options. They'll be forced to monetarily fund a security force that is operating in a way they disagree with, rather than one they approve of more. That's bad! It's not nearly as bad as frequent gun fights in the streets would be, but it is quite bad.
Anarcho-capitalism wants to get rid of that badness. It wants disagreements not to be suppressed. Not expressed violently either. We need something else: a system that is responsive to people's opinions, and non-violent. Allowing multiple security forces, and having people choose which to subscribe to, has the "responsive to people's opinions" part covered much better than having only one security force that everyone subscribes to whatever they think of it. It harnesses the power of the market, so that security forces that please customers prosper, and ones that are not responsive to customers fail. Of course the market doesn't do that perfectly, but it does it better than anything else.
That leaves the violence issue. How will violence be averted?
Let's think about how violence is averted today. Here is one story of how it works: People do disagree with the Government about all sorts of things, like whether marijuana should be treated as similar to cocaine, or not. But they aren't going to take out a gun and do something violent about it. They would be up against overwhelming force. There are disagreements within the Government too. But for any given thing, when it comes down to it, there is always *one law* about who gets their way, and overwhelming force backing up that law. Sometimes the answer is complex, and involves multiple people and even voting, but there is always one unambiguous outcome backed by overwhelming force.
So the general idea is: for any given issue, there is some ultimate authority, backed by overwhelming force. That is how we avert violence. No one who disagrees wants to fight over it when they'll just lose really badly.
If there were a dozen security forces, and none had overwhelming force, there would sometimes be situations where two forces disagree, and taking into account their allies, they are roughly evenly matched. Close enough the outcome is in doubt. So will they fight it out? How do we make damn sure they don't?
First of all, pretend for a moment *you* are in charge of a security force. Would you want to fight it out? Would that be a temptation? Or would you bend over backwards to avoid it? I know I would want to avoid it.
Now let's consider again the story of how we avert violence today. Is it really because there is one clear law everyone follows? No, it can't be, because sometimes the law is ambiguous. Sometimes we have situations the law makers didn't foresee. Our real system involves people making judgment calls, and it's pretty adaptable. Now, those people making judgment calls sometimes disagree. Think of the 2000 election where people disagreed about counting votes. That could have been a serious problem! It could have turned violent! The country was split fairly evenly about which President they wanted. But it did not turn violent. Why? I think the most important factor, the thing that reliably averts violence, is that just like you and me, the people making those decisions did not want violence. It was more important to them to avoid violence than to become President, or get their way. They were willing to bend over backwards to avoid violence. Everyone involved on both sides was so averse to violence that it never came close to violence. I think the fact that people hate violence is a much larger factor than the threat that if they fought they would lose badly. Americans care more about what is right than which side is more powerful.
Why will security forces by run by people who are any less averse to violence than average American is today? In fact they will be run by people who are even more averse to violence, and better at avoiding it, than the average American today. If I have the choice between subscribing to a security force that has that kind of leader, and one that doesn't, it's obvious to me which to choose. And it would be obvious to most people.
Alright, now let's suppose the security forces are run by decent people who are eager to avoid violence. Not just because it'll save them money, but also because of morality. But still they don't agree. What will they do?
I'm sure there are multiple possible good solutions to this problem. Here is one: They will bid money.