That's what I have an issue with right there --- the entire idea of NAP ever working globally. I would assert that any time a majority (or even a vast majority) agrees with the NAP --- then it's going to spur those who don't into drastic action that would end with destruction on a massive scale.
What, you think the agressionists are going to nuke the world? Possibly, but possible in today's society, too. And arguably, more likely.The purely logistical problems involved with letting anyone use another security provider at will is going to make actually providing that security next to impossible. You'd need to have perfect information and nearly instant transportation (think beam down a security team).
That old addage, "When seconds count, the police are only minutes away." wouldn't change, and it's no less true now than it would be then. Which is why AnCap supports the right of every individual to see to their own defense by carrying whatever personal weapons they see fit.Assuming people moving between providers, you've either have a single provider over large areas (as they're clearly dominant and the best at the job) - or your have complete fragmentation (like we do with internet service providers, cell phone providers etc). In the first case, we end up with a very large armed security force with a scary amount of power, sitting on a large territory, just waiting to be corrupted --- and in the second case, we've got a bunch of providers who because of the cost and logistics aren't agile enough to defend against... much of anything really.
But the police don't typically stop crimes, either... At least with a market system, the incentive would be to try, rather than to just tell the poor schlub "Sorry, no duty to protect." And even the mass of little agencies can effectively defend against external threat. Arguably better, in fact, due to the fact that there would be no central authority to capture, and they would be a defense-in-depth rather than a shell defense. 
