Anarchy would result in a natural aristocracy, which really just means that people who earn favorable reputations through the demonstration of good judgment, upstanding moral conduct, and logical consistency would be naturally sought out as fair and impartial arbitrators of conflict between individuals over the employment of scarce resources toward mutually exclusive ends. Arbitration would very much be an economy of reputation that would often be provided free of charge due to a felt sense of civic duty on the part of arbitrators.
This naturally occurring phenomenon is observable in small communities and families even to this day. In a market unrestricted by violent interventionism, people would be free to choose arbitrators like Tom Woods, Larken Rose, Stefan Molyneux, Marc Stevens, Hans-Hermann Hoppe, Andrew Napolitano, Ron Paul, or any other mutually trusted third party they deem fit.
This means that "authority" would accrue to those who are honest, intelligent and consistent.
I'm okay with that.
And in fact, arbitration is a service that could be provided more efficiently through the app store via a reputation-based platform like Steemit than through a territorial monopolist on ultimate decision making. The benefits of this should be obvious: reduced overhead costs, no lack of impartial third parties to whom you can appeal in the event that you enter into conflict with an arbitrator, vastly improved efficiency and speed, and no arbitrators arbitrating conflicts concerning themselves. Contrast this with the "arbitration" provided by a territorial monopoly on ultimate decision making - it's hugely expensive, there's no appealing its ultimate decisions, it's slow as hell, it always takes a huge cut of the restitution for itself, and it "arbitrates" conflicts that it provokes. And to top it all off, a territorial monopolist on ultimate decision making also holds a territorial monopoly on setting the price of arbitration, meaning the costs are completely divorced from price discovery and supply and demand.
This is kind of a no-brainer. "The state" is the cause of every problem to which it presents itself as the solution. There is no service that couldn't be provided more efficiently and satisfactorily through a market unrestricted by violent interventionism.