I've seen that argument before. It fails, because I'll simply hire a private protection agency that does not have an agreement with your private protection agency to go to an arbitrator when the two agencies find themselves representing opposing clients.
It doesn't fail. You're assuming that one such agency would exist. In reality, violence is prohibitively expensive. No agency would stay in business for too long if it constantly engages into conflicts on behalf of its customers. Prices for its services would go up, while demand would go down. Also, savvy competitors would team up and contribute to destroying this nasty agency as well, because it hurts their peaceful business.