Excellent! You raise an important and valid issue. Some would contend that this inevitably results in rival warlords: the "Pinkertons" versus "Securitas," for example.
However, there is an answer to this dilemma. It's worth pointing out that our respective police forces don't actually want to get into a shooting war over my dog digging in your petunias. There are various ways they can avoid a shooting war, so I don't know which one they would pick. One is for the two to agree upon a third judge, and inform you and me that they will drop us as customers unless we agree to enter arbitration with the third judge. If he says you're in the right, then my agency will drop me as a customer unless I pay for your petunias--which, of course, leaves me defenseless against your agency.
There's a separate consideration also. Each of us going to separate judges is pretty unproductive in the first place, and we knew that already. In most cases, therefore, we'll make stipulations in our original contract as to which judges we would consider acceptable in the event of a dispute. If we have no contractual relationship--for example, we're involved in a random car accident--then we would presumably pick the arbitration agency with the best reputation for fair judgments. An arbitator who always decides for his client will soon have no customers.
The thing is that in the sort of capitalism today, where an individual interacts on a daily basis with such a huge variety of people, transaction costs mount so quickly that a state that can monopolize the policing business will be more efficient than a market of competing police and courts systems. Case in point: What about international business transactions? How would you get a foreign client to pick, let alone trust, one of many private court systems?
"However, there is an answer to this dilemma. It's worth pointing out that our respective police forces don't actually want to get into a shooting war over my dog digging in your petunias. There are various ways they can avoid a shooting war, so I don't know which one they would pick. One is for the two to agree upon a third judge, and inform you and me that they will drop us as customers unless we agree to enter arbitration with the third judge. If he says you're in the right, then my agency will drop me as a customer unless I pay for your petunias--which, of course, leaves me defenseless against your agency."
So compulsion enters it's ugly head. Just exactly how is this different from us 'agreeing' that there is a need for a power capable of impossing it's decisions upon us in matters of law enforcement and the judiciary?
"There's a separate consideration also. Each of us going to separate judges is pretty unproductive in the first place, and we knew that already. In most cases, therefore, we'll make stipulations in our original contract as to which judges we would consider acceptable in the event of a dispute. If we have no contractual relationship--for example, we're involved in a random car accident--then we would presumably pick the arbitration agency with the best reputation for fair judgments. An arbitator who always decides for his client will soon have no customers."
Excellent arguments, in theory and the Libertarian philosophy is great in theory. The problem is the real world with real people and real emotions. My in-laws are in such a property dispute. I suggested they get a surveyor acceptable to both of them. Nope, they each hired their own surveyor and I'm sure you can guess what happened. Now they each have their own lawyers and are headed to court. Just imagine how fun it would be if they could each hire their own judge. BTW, a judge who relies upon individuals to pay his fee, won't have many fees unless he remember who is paying him.