In a famous paper (“The impossibility of a Paretian liberal”, 1970), Amartya Sen proved a mathematical theorem that shows how the requirements of even rather basic libertarianism (what he at the time called “liberalism”), the idea that in certain spheres we have absolute authority over our own personal decisions, can be incompatible with Pareto-efficiency, the requirement that nobody could be made better off without someone else being worse off. Both requirements seem intuitively plausible, but in certain cases, they simply cannot both apply.
Sen's theorem is valid, so my concern here is not to refute it. Rather, my goal is twofold: First, to offer an example that I think captures the situation better than the one Sen uses, and second, to provide an intuitively plausible constraint on libertarianism which allows it to be reconciled with Pareto-efficiency.
Sen's example is as follows, and I think you'll see why it's less than ideal. He imagines two individuals, A, a Bohemian, and B, a prude, trying to decide who (if anyone) should use pornography. B would prefer that nobody use porn; A would prefer that everyone use porn. On libertarianism, it's A's business whether A uses porn, and likewise it's B's business whether B uses porn. So far, so good.
But in order to make the paradox work, Sen has to assign two more requirements, namely: That A prefers that B be the only one who use porn (rather than A), and similarly that B concurs on this issue. This is, well, weird. It implies that it's more important for A to force his love of porn upon B than it is for A to even get the chance to use porn in the first place. It furthermore implies that B would rather use porn himself than let A use it. Both of these value systems seem inherently anti-libertarian, and so it's not all that surprising that the result turns out that libertarianism doesn't satisfy people's preferences. In a very real sense, we seem to have a preference against freedom itself. Without these requirements, there either is no Pareto-superior alternative, or else the Pareto-superior alternative is the libertarian alternative, namely that A uses porn and B doesn't.
I have an example that is mathematically isomorphic but seems in real life a lot more justifiable. Suppose now that A is an environmentalist, and B is an oil company executive. The question is this: Who will be subject to pollution regulations?
Obviously A prefers that everyone be subject to regulation rather than nobody—that's what makes him an environmentalist. But he also has his own self-interest to consider, and if you're not going to obey the regulations, he'd rather not either. In fact, strictly speaking he prefers that you be the only one obeying (so that his water is clean but he can still burn firewood).
Conversely, B of course prefers not to be regulated. He also prefers that A be regulated, because there are some small ways that burning firewood could potentially cut into his oil business (e.g. by cutting demand for heating oil). His worst-case scenario is that he gets regulated but A doesn't (which is of course A's best-case scenario).
If the regulation is a purely opt-in matter (as libertarianism would demand), then nobody will obey regulations, which is strictly worse for everyone than the case where everyone is regulated.
You'll note that this is isomorphic to the Prisoner's Dilemma, as in fact Sen's original scenario is as well. (It's harder to see in that case, because the “mutual cooperation” scenario is the one where B uses porn and A doesn't—which just seems bizarre.) Also, the items don't correspond in the most obvious way: A using porn is not equivalent to A being regulated, but instead to nobody being regulated. B using porn is not equivalent to B being regulated, but rather to everyone being regulated. Everyone using porn corresponds to B being regulated, and no one using porn corresponds to A being regulated. I blame this lack of correspondence on Sen's poor choice of example.
There is however one advantage of Sen's example: It seems to naturally coincide with what we ordinarily think of as the space of individual liberty. What right do I have to decide whether you use porn? It seems fundamentally tyrannical for me to assert this kind of authority, whatever my preferences.
Environmental regulations, on the other hand, don't naturally coincide to this space of personal freedoms; it doesn't make much sense to say I have the individual liberty to pollute the air you breathe.
What makes this difference? It seems quite obvious, at least in hindsight: My pollution affects you; my porn does not. In economic terms, pollution carries obvious externalities that pornography does not. (You could argue otherwise, I suppose; but if you could convince me that porn carries heavy externalities, I would also be convinced that it doesn't belong in the realm of personal liberty!)
Sen's paradox crucially depends upon these externalities. Without it, we end up asserting some sort of “mere preference”, in which A just happens to prefer that B use porn, and has no actual reason for doing so (it certainly won't affect A in any way). This is why the characters in Sen's original example seem like such busybodies; they are poking their noses into matters which ought not to concern them.
To put it another way, Sen's paradox can only be formalized in terms of utility in the presence of externalities. Without externalities, A must be indifferent to what B does, and B must be indifferent to what A does, and so the libertarian solution is the Pareto-superior (and utility-maximizing) one.
Indeed, a utilitarian (as increasingly I'm coming to think of myself) has a very clear quantitative breaking point for deciding when to apply the libertarian rule and when not to: Do the externalities exceed the internalities? Does the harm to A of B's actions exceed the benefit to B (or vice-versa)? If so, then libertarianism will fail Pareto-efficiency. If not, then libertarianism works just fine.
Hence, we have the standard caveat that even libertarians will generally apply: “I am free to act however I wish, so long as in doing so I do not harm others or infringe upon their liberty.” If the libertarian principle is formulated in this way, then Sen's paradox simply can't arise, and the result is guaranteed to be Pareto-efficient—indeed, utility-maximizing.
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