The Problem with Pareto

The Pareto axiom appears to be consistent with liberal values. In fact, in some situations it can be deeply illiberal. The following is due to Sen (1970). Consider an Arrowvian SWF, and a new axiom, liberalism. A SWF respects liberalism if there is some group of at least two people, and one pair of alternatives for each person in that group, such that their preferences are decisive for social choice. That is, if $n$ is in the group there are two alternatives $x$ and $y$ such that $x$ is socially preferred to $y$ iff $x\succ_ny$.

Theorem. There is no SWF satisfying universal domain, unanimity, and liberalism.

Proof. Suppose 1 is determinative for $x,y$ and 2 is determinate for $w,x$. Suppose that $x\succ_1 y$ and $w\succ_2 x$, and that everyone holds $y\succ_n w$. Then the social order is $x\succ y$, $y\succ w$ (unanimity), and $w\succ x$ so preferences are cyclic.

Now suppose 1 is determinative for $x,y$ and 2 is determinate for $w,z$ (no overlap). Suppose $x\succ_1y$, $w\succ_2x$, and all hold $y\succ_nw$. Again preferences are cyclic.∎