Paretian liberals?

Mathematical formalism seems particularly apt to cause misunderstanding. One example is the so-called Sen's "paretian liberal impossibility theorem".

I will describe the theorem with a short and simple example. Suppose that there are two people, a puritan and a libertine; and one only copy of a certain obscene book (for example, "Das Kapital" by Karl Marx). This book can be read by only one person, situations "P" (the puritan reads the book) or "L" (the libertine reads it), or destroyed, situation "N". So, we have two persons and three choices.

The puritan has this set of preferences: N > P > L. This means that he prefers the book to be destroyed rather than reading the book, but he prefers to read the book rather than leaving it to the libertine. The libertine has this set of preferences: P > L > N. This means that the libertine wants the puritan to read the book, otherwise he prefers to read it by himself, while his worst case is that the book is destroyed.

Following Sen, we are going to define liberalism so that every individual has "the last word" at least on one problem: the so called "minimal liberalism" condition is that at least for one choice between two alternatives, the individual choice coincides with the social choice.

Suppose the choice to be between L and N, and leave the "social choice" to the libertine: he will prefer L to N. On the other hand, suppose the choice to be left to the puritan to be between N and P: he will prefer N to P.

These two conditions on the social choice imply, in order for social preferences to be transitive, that L > N > P. That is, L > P. The "problem" is that L is Pareto inefficient with respect to P. In fact, both the libertine and the puritan prefer P to L, for the initial hypotheses. Minimal liberalism is Pareto-inefficient.

However we define rights, there will always be one set of individual preferences which will give rise to the "paretian liberal impossibility paradox". Inefficiency arises for a particular definition of minimal rights, but even if we change the content of these rights, we can always change preferences in order to achieve the paradox...

I'm not an economist and I may get it wrong, but I think the reason is that if we have three choices, A, B and C, and impose the condition for which individual X is decisive between A and B, and individual Y is decisive between B an C (this formulation is general: A, B and C can be chosen or permutated freely); then we can obtain the paradox by choosing X preferences to be C > A > B and Y preferences to be B > C > A. I'm not a mathematician, and I hold that this is enough to show the theorem.

X will force social preferences to be A > B; Y will force social preferences to be B > C; thus A > B > C; thus A > C, even though they both prefer C to A. This gives rise to Pareto-inefficiency.

Notwithstanding these mathematical doubts, the real question is: "What have all this technicalities to do with (classical) liberalism?". My answer is: nothing.

Turning back to our example (the so-called "Lady Chatterley paradox"... the guy who created the paradox didn't despise Marx enough...), we have that, in a liberal society, the book will be owned either by the puritan or by the libertine. In the former case, the puritan will choose, in the latter case the choice will be up to the libertine. There is no "at least one minimal right for each person" condition in this case, because minimal liberalism, as defined by Sen, has nothing to do with liberalism. Liberalism is about legitimate control over resources, that is, property rights.

Even more important, if the puritan owns the book, he has the right to choose between N and P: he has no right, in a liberal society, to force the libertine to read the book; on the other hand, if the libertine owns the book, his choice is between N and L, for the same reason. Ok, I'm criticizing the particular example and not the general theorem. Anyway, this forces us to think about another problem: the "unrestricted domain clause" is a totalitarian hypothesis, and has nothing to do with liberalism at all (it is clear that I use the term "liberalism" as equivalent to "libertarianism", and I don't care a damn about contemporary "liberals", such as Rawls).

There is a third possibility, that the book is res nullius (it has no owner), but it is irrelevant: until it has no owner, there is not a problem; as soon as it gets one, we are back to the "either the libertine or the puritan own the book" case).

The Sen paradox does not prove the inefficiency of classical liberalism, but rather Professor Sen's misunderstanding of what liberalism really is.