Teaching Arrow's Impossibility Theorem

I regularly teach undergrads about Arrow’s impossibility theorem. In previous years, I’ve simply presented a statement of the theorem and provided a proof in the optional readings. Arrow’s proof is rather complicated; and while there are several simpler presentations of the proof, they are still too complicated for me to cover with philosophy undergraduates.

Preparing for class this year, I realized that, if Arrow’s theorem is slightly weakened, we can give a proof that is much easier to follow—the kind of proof I’m comfortable presenting to undergraduate philosophy majors. The point of the post today is to present that proof.

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The Impossibility of a Paretian Liberal

A foundational assumption in welfare economics is that, if everyone prefers A to B, then A is better than B. This normative claim is called the ‘weak Pareto principle’. At a first glance, this principle can appear unimpeachable. At the very least, it appears to be a sensible principle for policy makers to adopt.

Many of us are also committed to the principle that there are some choices that should be left up to the individual. Even if everyone else prefers that my nose be pierced, if I prefer it unpierced, then it is better for my nose to remain unpierced. A state-of-affairs in which I am compelled to pierce my nose, against my wishes, is worse than a state-of-affairs in which I get to make up my own mind about whether my nose is pierced. And that’s so no matter how many other people would prefer to see me with a pierced nose. Call this committment ‘minimal liberalism’.

In 1970, Amartya Sen produced an amazing result which seems to show that minimal liberalism and the weak Pareto principle are inconsistent with one another. Today I want to rehearse Sen’s result, and introduce an objection to Sen’s way of formalizing ‘minimal liberalism’ first made by Allan Gibbard. I think that Gibbard’s objection teaches us that Sen formulated liberalism incorrectly. However, I’ll conclude by showing that a better formulation of liberalism, one that avoids Gibbard’s objection, is also inconsistent with the weak Pareto principle.

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