I’m slowly reading philosopher Robert Nozick’s 1974 book Anarchy, State, and Utopia, which begins with an explanation of why some kind of minimal state would automatically evolve out of a Lockean state of nature: when individuals hired “protective associations” (or private police forces) they’d inevitably evolve toward territorially concentrated monopolies on violence.
And to some extent we see that in the real world. But we also see a trend toward overlapping “protective associations.”
In New Orleans, for example, a trash collection entrepreneur named Sidney Torres (he’s known as “Trashanova” for his boy band Bruce Wayne good looks) has started the French Quarter Task Force to halt gunplay. 
He pays armed rent-a-cops in kick-ass golf carts to respond to citizen complaints entered via an Uber-like app.
This is not an isolated example of overlapping police forces. For example, in this post-9/11 era, Washington DC has a remarkable number of federal security forces. (This may be helping propel the rapid gentrification of Washington.)
Another example of overlapping police departments involve college police forces that patrol off campus, which has suddenly become controversial when a U. of Cincinnati cop shot a black motorist.
For example, I visited Tulane a couple of years after Hurricane Katrina, and the university president explained that the Tulane police department patrolled the streets for a mile beyond the campus borders. (Back then, Tulane was the most functional institution in New Orleans, so also the students had organized an ambulance service to get people from the campus area uptown to the Tulane medical center downtown.)
New Orleans is of course a severely dysfunctional, damaged place. But this kind of double protection is also seen in more average cities, like Chicago. Perhaps the most spectacular example is the U. of Chicago’s fiefdom sprawling far beyond its campus. Its police force has full jurisdiction over 50,000 residents who aren’t students.
The President of the United States long made sure to only live within the U. of C. cops’ jurisdiction so he wouldn’t have to put up solely with wimpy Chicago PD protection, but could also call upon the university’s private police force. Or as Obama explained in Dreams from My Father, after one scary night in Hyde Park in which he racially profiled four black youths as being of malevolent intent:
“As I stand there, I find myself thinking that somewhere down the line both guilt and empathy speak to our own buried sense that an order of some sort is required, not the social order that exists, necessarily, but something more fundamental and more demanding; a sense, further, that one has a stake in this order, a wish that, no matter how fluid this order sometimes appears, it will not drain out of the universe.”
I think this means that Obama realized he was on the side of the cops, not the crooks.
One interesting question in the history of thought is whether the University of Chicago having a privatized police force that patrols the nicer part of the South Side has anything to do with the Chicago School views on privatization.
