Sen and the Paretian liberal paradox

In a six-page article Sen (1970a) introduces a notion of individual rights within the Arrovian framework of social choice. These six pages had a fundamental importance on the development of studies on non-welfaristic aspects of normative economics.

Sen introduced two conditions of liberalism, or individual freedom, the second one being a weakening of the first that turns out to be sufficient to get the result.

Condition L (liberalism) For each individual i ∈ N, there are two social states a_i and b_i such that we get a_i >_s b_i whenever a_i >_i b_i and b_i >_s a_i whenever b_i >_i a_i.

The second condition states that at least two individuals enjoy liberalism as defined above.

Condition ML (minimal liberalism) There exist two individuals i and j, two social states a and b for i and two social states c and d for j such that a >_s b whenever a >_i b, b >_s a whenever b >_i a, c >_s d whenever c >_j d and d >_s c whenever d >_j c.

Sen’s impossibility theorem If there are at least two individuals and two social states, there is no social decision function satisfying conditions U, P and ML.

We can note that there is no need of the finiteness of N and of condition I and that the result already holds for two social states. To understand the power of this result - which, incidentally, has no real meaning in a voting context since this would mean that two individuals have a partial dictatorship power - one has to consider that social states are descriptions of the states of the world as detailed as one wishes with possible personal elements.

There are two individuals i and j. The social states a and b are identical except that in a individual i eats legs of lamb with garlic and in b without garlic. Individual i is a garlic addict and, accordingly, strongly prefers a to b. The social states c and d are also identi­cal except that in c individual j puts some Guerlain’s “L’Instant Magic” perfume before going to sleep and, in d, does not. Individual j has a passion for “L’Instant Magic” and so strongly prefers c to d. Now imagine that j is i’s wife and that she hates garlic as much as her husband hates perfume in general and “L’Instant Magic” in particular. On this basis, let us suppose that the two individuals’ preferences are the following: d >_i a >_i b >_i c and b >_j c >_j d >_j a.

Since there is nothing more personal than culinary tastes or tastes related to smells, the social states a and b perfectly fit J.S. Mill’s notion of personal sphere regarding individ­ual i, and likewise for individual j, concerning states c and d (Mill 1859). This illustration exemplifies the difficulty one can encounter with this notion of personal sphere in the presence of what the economists call externalities. We will assume that our society is only composed of i and j. It is in fact very easy to consider a general profile π with appropriate preferences for the other individuals. By Condition P, since both individuals prefer d to a, we have d >_s a, and since they both prefer b to c, b >_s c. Now, since i prefers a to b, by condition ML, a >_s b. Since individual j prefers c to d, then by condition ML, c >_s d. We have accordingly a cycle: a >_s b, b >_s c, c >_s d, d >_s a.

A major by-product of Sen’s paper is the tremendous development of the freedom of choice literature (see, for instance, Dowding and van Hees 2009).