One charge leveled at market anarchism is that it cannot solve the problem of public goods. After all, who would "pony up" for a good such as national defense if others can free ride off of his purchase? Government is necessary, the argument holds, to compel people to contribute so no one gets cheated and defense from foreign invasions can exist.
Let us take the extreme case then. Assume there is a city that can be protected from all attacks by a giant shield if only X dollars can be raised to pay for it. Without it, the city is defenseless. A democratic government can solve this without a problem: just levy some tax until X dollars are in a public fund and buy the shield. Without a government, it just won't happen, because no one would pitch in, knowing there would be free riders. Or so we're told.
But even in this extreme case, several factors are left out. First, while the democratic government can guarantee the shield is built, it will be more expensive than if a single person (for the sake of argument) built it. This is because the person would pocket the full value of any reductions in price he can find. The government, on the other hand, accountable to only disinterested voters must struggle to produce incentives for its researchers to find ways to cut costs. At best, it can provide them with a tiny fraction of the reduction in price they can find, greatly reducing their incentives to search for one. It can thus be reasonably predicted that a privately built shield would be substantially cheaper (cost some amount Y which is less than X) than a publicly built shield.
Why on earth would a person pay to build such a thing, though? Let us look at his incentives. Assume that he too hates free riders and for that reason has class envy. He also has the incentive that he, his family, his friends, and all of his property in the city is protected from foreign aggressors if he builds the shield – an amazing offer. Assume further that he is very rich, in fact, the richest person in the city as measured by physical assets that he cannot hide outside the city. If the cost of the shield is less than the value of the property he stands to lose, and he remains at the top of the pecking order after paying, he is likely to build it since he keeps both his property and his status. (Keep in mind that this would not be possible for a business to pay the cost because this puts it at a competitive disadvantage, though of course a business could be paid to put up the shield.)
This arrangement could be possible if we keep in mind, as argued above, that the cost of the shield is substantially cheaper when built by one person. Also consider how many public works (such as libraries) are built by the rich when their family, friends, property, and possibly life are not at stake. Even better, the rich person is unlikely to fully own all assets he wishes to protect; he may be only a share owner in some of them. This rich person then brings in many contributors since everyone who is an investor in one of his assets is effectively contributing to the incentive for him to build the shield.
However, it is not even necessary that just one person build the shield. There will always exist some number of people n that is capable of keeping an agreement. n will be less than the total population but almost certainly greater than one, assuming some civilization exists. The argument from above then generalizes: n people are likely to pool Y dollars to build a shield, with each person contributing an amount proportional to the wealth in the city he wishes to defend, if they are the n richest people, their wealth ranking stays the same, and the poorest of n is still richer than the (n+1)th richest person. Now an even greater number of people is contributing to defense because every investor in one of the collective assets of the n richest is a contributor.
This still does not completely solve the free rider problem: lots of people could still be free riding. However, let us take a moment to compare how big the free rider problem is in a democracy. To do so we must look at how the democracy raises X dollars. It will tax each person a fraction of their visible wealth. The fraction can itself vary with the visible wealth, being progressive, regressive, or proportional. Assuming wealth is distributed unevenly, the tax voted in to raise X is likely to be progressive. This is because more than 50 percent must vote for the tax, and people will not vote for the tax if they are taxed more than they stand to lose. It is entirely possible that some could pay nothing, since a politician can buy votes this way by appealing to class envy. In fact, as some have already noted, the tendency in a democracy is for an increasingly large portion of the population to be net tax consumers living off the wealth of an increasingly small portion of the rich. It is thus clear that a democracy has free riders as well.
With this fact in mind, one observation can be made: it is ambiguous which system has a greater free rider problem. In market anarchy, it's possible that almost everyone could be saving for retirement (or be a dependent of such a person) and have a broad portfolio that includes investments in the n's assets, meaning almost everyone contributes to the city's defense. On the other hand, it's possible that in a democracy, huge swaths of people pay nothing for the city's defense.
In conclusion, even in the extreme case of a shield that covers everyone, there are many dynamics at play in the concept of public goods. The question should not be, "What does market anarchism do about free riders?" but "How many free riders are acceptable, and at what cost should they be minimized?" And if such an extreme case can handle the public goods problem, one should have little doubt about less extreme cases.
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Special thanks to Jason Ditz for his input.