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Minarchism is a political philosophy advocating the reduction of government functions to a minimum scope, providing people with only bare-bones services such as law and security. It's considered a form of libertarianism, and since it finds some useful role for the state, it's incompatible with anarcho-capitalism (Unless, of course, you consider the Friedmanesque voucher a bridge between anarcho-capitalism and minarchism), another major form of libertarianism. Ayn Rand's views could be described as minarchist.
Minarchists advocate for a "night-watchman state" that is not responsible for the education, health care, employment or transportation of its citizens, neither makes it any use of natural resources in its territory. All of this is instead held privately or publicly, but is never susceptible to any interference of the state, its law or its representatives. Minarchy is, of course, different from anarchy, since the latter term means a complete absence of a government with all services, including even law and security, done or exercised by people themselves.
The following two sections describe how a person supporting and another opposing minarchism would argue.
Even the Austrian Economists do not advocate for this. The free market fails utterly when dealing with Public Goods, that is, goods that are non-rivalrous — my consumption of the good doesn't diminish your ability to consume it — and non-excludable — you can't stop me from consuming it. For example, there are only so many fish in a location, and Alice can't stop Bob from fishing there, so Alice and Bob catch too many fish and next year there are practically no fish left. Alice and Bob could make some sort of agreement, but nothing prevents Charlie, Denise and Emily from still fishing. So everyone is poorer than had there been some sort of limit on over-fishing.
Next comes Externalities, which are detriments (or benefits) that the producer of the good or service doesn't have to deal with. An example of this is pollution. If a firm can lower their average costs in order to make significantly more profit by dumping their waste in the local river, so long as the owners don't drink from said river, it will probably do so to maximise its profit margins. This generally harms the local population far more than the company gains from not properly disposing its waste, so overall it's a net loss for society. The third party effects arising from production of this good are greater than that of the private benefits incurred.
Things like education, safety regulations, roads, research subsidies, and so forth often add to the economy more than they cost. Welfare and subsidised housing are often far more effective at reducing crime — for cheaper — than extra police alone. What Minarchists have a hard time understanding is that their income depends on things like roads and communication systems being available, that having access to adequate healthcare increases worker productivity, and a whole slew of other things. Even if you built your business out of gumption and bootstraps alone, virtually all of your customers and employees relied on something provided via government, and without those customers you wouldn't have that business to be taxed in the first place. It's far better to pay 50% tax on $200,000 than pay 5% on $20,000.
Without international bodies imposing rules on corporations (such as the Institute of Electrical and Electronics Engineers) there also wouldn't be any universal standards and regulations to which all corporations should abide, meaning that literally every single corporate entity has to expand or cooperate with another one to form any sort of coherent universal standard and even then you'd be lucky if every corporation in the world agreed by it. While that may not seem all that shocking at first, this means that there will be situations where you can't phone some of your friends because their telephones don't know how to register your telephone signal or that you can't do your job abroad without re-learning how to do it because the machines they work with have notably different architecture from the ones you are used to work with. In the end it ends up complicating the life for corporations and people alike.
Response to criticism
Proponents for minarchism  argue that the assignment of property rights would solve both the issue of negative externalities and the issue of the conservation of common (i.e 'non-rivalrous) goods. Let us return to the scenarios outlined in the previous section, we can see that by awarding either party ownership of the lake will reduce both the rate of pollution, and extirpate any chance of over-fishing. If we give property rights over the lake to Bob, he will have an active, personal interest in preserving the number of fish in the lake. He would likely begin issuing fishing permits, and or an upper-limit on the fish which may be caught by fishermen using his property. These measures will speedily eliminate the risk of over-fishing .
The issue of pollution would be solved in a similar manner. If we award ownership of the relevant lake to the polluter, the polluter will either recognize that using the lake (read; their property) as a rubbish-dump is not a profitable utilization of the asset, and will instead move to conserve the asset to preserve its value and put it to some alternative use. Alternatively, even if the polluter determines that the lake is best used as a refuse-pit, victims of the pollution can launch legal action (public courts of law which exist to defend human and proprietary rights being-- of course-- characteristic of minarchism) on the grounds that the pollution is harming their property (i.e., their bodies). Now forced to pay compensation to the individuals who have suffered property damage (i.e, bodily harm) as a result of the polluter's use of the lake, the polluter will be compelled-- by reason of simple profit and loss-- to either limit their pollution to a rate that does not harm anyone's body, or transition to another method of waste disposal; whichever option is cheaper will prevail. This bears repeating: if the benefit of dumping is offset by the cost of the compensation payments completely, the polluter will-- of its own volition-- seek other methods of waste management.
The assertions that the free market would be unable to provide services like education, regulation, and infrastructure have been long debunked. Although hard to imagine for those who have grown dependent on government, all these services would be provided more freely and more efficiently under a free market. The essential reason for the private market's superiority-- with respect to the provision of goods and services-- comes down to the private sector being beholden to profit-and-loss considerations, proprietary interest, and competition. 
Corporate regulation comes freely and naturally by its nature in a free market as well. This occurs all across the world, and readily occurs even in jurisdictions already laden with regulation. Professional associations create standards and give "seals of approval" to businesses that adhere to their guidelines, and in turn the consumer is protected from predatory practices by the people who understand the business the most. To suggest that corporations will not self-regulate is akin to saying that school children will simply not cooperate unless ordered to by their school-master-- a clearly specious assertion.
Minarchy has also been advocated by some non-anarchist libertarian socialists and other left-libertarians. Minarcho-socialism is also an ideology that combines the principles of minarchy with socialism.
- See, Robert Nozick, Tibor Machan, Roderick T. Long, etc.
- Karpoff, Jonathan M., The Tragedy of ‘The Tragedy of the Commons’ – Hardin vs. the Property Rights Theorists (July 5, 2021). Forthcoming, The Journal of Law and Economics, European Corporate Governance Institute – Finance Working Paper No. 750/2021, Available at SSRN: https://ssrn.com/abstract=3531981 or http://dx.doi.org/10.2139/ssrn.3531981
- Nozick, Robert (2013). Anarchy, State and Utopia (Reprint ed.). ISBN 978-0465051007.
- Bei Zhang (2013) Market-based solutions: An appropriate approach to resolve environmental problems, Chinese Journal of Population Resources and Environment, 11:1, 87-91, DOI: 10.1080/10042857.2013.777526
- Larson, Paul. (2009). Public vs. Private sector perspectives on supply chain management. Journal of Public Procurement. 9. 222-247. 10.1108/JOPP-09-02-2009-B003.
- Curioni, A., Voinov, P., Allritz, M., Wolf, T., Call, J., & Knoblich, G. (2022). Human adults prefer to cooperate even when it is costly. Proceedings. Biological sciences, 289(1973), 20220128. https://doi.org/10.1098/rspb.2022.0128
- See the Wikipedia article on Demanding the Impossible.
- Minarcho-Socialism, polcompball.miraheze.org.