Part of something is better than all of nothing, and remember, once they've selected "their" firms, those arbitrators take over for them, and are capable of selecting new arbitrators to decide between them. The market incentive is to get the decision made quickly, and with as few layers of that as possible, because each new layer cuts into the pie.
There is a free market in law. Arbitration is free market law. Until you understand that, you're just going to make yourself look more foolish with every statement that you make.
So you think that property rights can be taken off a person on the basis that "Part of something is better than all of nothing." Nice.
I'm not sure where you want to go with this. So far, all you have said is that without a state, there are no property rights. I agree - that's one of the reasons a state is a good thing.