Not , but a small element would be running wild , which is enough to make the rest of us feel unsafe , and look for means of protection.
Why would a small element be running wild? Do you not think the protection agencies hired by average people, along with local militias and the armed populace itself, would be capable of stopping them?
This usually means making an alliance with whoever your local Warlord is.
This leads to majority domination of the minorites who are still trying to live independently without protection , which in turn eventually leads back to the OPPOSITE of what the anarchists strived for - TYRANNY.
For your minarchy to work, you require a populace, the overwhelming majority of which understand the principles of liberty, and are willing to remain vigilant to keep the government in check. Do you think a population like this would be unable to stop a local gang?
You suppose they can hold a central, national government in check, which has a monopoly on agressive violence, no competiton, etc, yet they can't stop the local gangster. It's beyond absurd.
You're not comparing apples to apples. Your conception of minarchy exists among a nearly angelic populace, and your conception of "anarchy" exists among a bunch of devils.
A simple understanding of human nature is all it takes to know that absolute anarchy won't work in large societies.
If human nature is as you describe, your minarchy won't work either. If the people are not vigilant, the minarchy will grow out of control. If the "majority" is quite willing to tyranize the "minority", as you suppose, then they will use your minarchy to do so -- and it will be even harder for the minority to escape. Perhaps you should think a bit more carefully, and not let your imagination run wild.
A government is just a protection agency which forces its customers to pay, and refuses to allow them to choose any alternative, or competitor. Why do you think this type of protection agency is necessary? Why do you believe it is preferable? Would not the necessity of aquiring subscriptions voluntarily, rather than by force, hold the agency more accountable to the people?