well I think you're missing the brilliance of the idea of giving EVERYONE guns, especially the nutters: if you give a nutter a gun, he's not going to be a nutter much longer. He's going to be DEAD because he can't handle himself and the gun properly. There's less reason to kill him off if he isn't being an active pest, so give him a gun as a test. If/when he misbehaves with it, shoot him. You need to make the nutters obvious/out in the open or else you cant kill them all off.
The idea is that if everyone is armed and left to fend for themselves in utter anarchy, the only people left standing alive are going to be civil-minded, cooperative and justice-oriented human beings - everyone else will be fucking dead. Of course some of those justice-minded folk will die killing off all the assholes/nutters and pests, but "You can't make an omlette w/o breaking some eggs" etc.