A common problem in development economics is that a poor country will be rich in some concentrated resource, like diamonds or oil: a dictator will take control of the resource and use it to fund his own military, which in turn keeps the populace from revolting. The basic problem is that when a valuable resource becomes concentrated, a relatively small group can control it and use it to gain leverage elsewhere. (This strategy, by the way, is central to the plot of Frank Herbert’s Dune.)
I’m wondering if it would make sense to think about the police’s monopoly on violence in an analogous way. Once upon a time, if some kind of violence was going down in your village, you and your friends had to find a way to deal with it because there was no formal police or guards coming to help you. Before the rise of modern technology and state apparatus, there was simply no way for most people to outsource force. In contemporary America, on the other hand, the police are always there: maybe not when you’d like or as fast as you’d like, but there’s basically nowhere within our borders that police or military can’t reach within minutes (or, in extremely rural areas, an hour or two).
I suggest that the rise of police has led to the decline of citizens’ ability to use force and negotiate violence on their own, just as the rise of sewing machines led to a decline in sewing-ability among layfolk. I also suggest that over past century or so, we’ve seen an ever-increasing centralization between police agencies, so that today it’s taken for granted that a rural North Dakota Sheriff will–nay, must–coordinate with and defer to national police if they tell him to.
So police have a monopoly on force (both because citizens have forgotten how to use force, and because non-police use of force is generally punished) and police are highly centralized. This seems to fit the criteria of a concentrated resource which I described above.