Although the Pareto principle is a very intuitive property, it has been observed that many results in social choice theory do not require it as an assumption. For example, R. B. Wilson [J. Econ. Theory 5, 478- 486 (1972)] proved an impossibility theorem concerning the aggregation of individual preferences without the Pareto principle. Since Wilson’s theorem implies as a corollary the Arrow impossibility theorem when the Pareto principle is also assumed, it has been viewed by some as more fundamental than Arrow’s theorem. However, most proofs of Wilson’s theorem (in particular a recent ultrafilter approach proof by D. E. Campbell [J. Econ. Theory 50, No. 2, 414-423 (1990; Zbl 0705.90003)] are so close to known proofs of Arrow’s theorem that one might conjecture that these two results are essentially the same.
We will verify that this conjecture is indeed true. We actually will prove a more general proposition: under proper domain restrictions of preference profiles, a social welfare function that satisfies independence of irrelevant alternatives and non-imposition, two basic conditions in Arrow’s original treatment, is either null, or Pareto optimal, or anti-Pareto optimal. The domain restrictions under which this proposition holds include some condition weaker than the well-known free- triple condition for Arrow’s theorem. Hence, this proposition implies that the Pareto principle in Arrow’s theorem is virtually redundant. It also implies that Wilson’s theorem is in fact a corollary of Arrow’s theorem, or perhaps more appropriately, a refined version of the original Arrow’s theorem.