Sorry - if you think the likes of Blackwater or Sandline care about that, you are mistaken. They don't - they really do work for bastards. In Iraq, they really did kill innocent people on behalf of bastards. An inheritance dispute won't trouble them for a nanosecond.
Why don't we take a step back here? I admire the way you put your head above the parapet and made this thread. You have to accept a system that leaves 55% of families depending on the morals of contract gunmen is not good enough. There is no point in spitting out an instant answer which is morally weak.
Why not take the time to think of an actual fix ?
Blackwater and the like are run by military men... by definition psychopaths - created, if not born. I don't expect them to care about humans.
You're being dishonest with yourself if you truly believe that 55% of people (all those who die intestate) would wholly be at the mercy of "the morals of contract gunmen"... you forget that arbitration is for disputes, not defense agencies. You presented a rare case, where one son of three (how many of those 55% had three children?) refused arbitration (how many of those that did have 3 children had enough shit to fight over?) because he believed in primogeniture (how common is this belief?). In that rare case, then his case might be said to be at the mercy of the protection company's morals.
But even then... Out of pure greed, the protection company should refuse his contract. How much is one contract worth, if they lose all their others? Worse, he knows that he can refuse payment, and then just refuse arbitration again. They know that too. It's in their rational best interest not to deal with him.
I think what myrkul is proposing is an Ancap society actually populated with Ancaps. Democracy hinges upon the morality of our gunmen as well.
Yes, I rather thought that was assumed. I guess that proves the old witticism...