Permutation-invariant social welfare orders are anonymous

We consider two widely discussed impartiality criteria for social welfare relations: Permutation Invariance, which says that every permutation of the population induces an automorphism of the relation, and Anonymity, which says that every permutation of the population induces a permutation that maps every social welfare distribution to one that is equally good. We show that these criteria are equivalent for social welfare orders (i.e., complete social welfare relations). The new result is that Permutation Invariance entails Anonymity for infinite populations. This is not a curiosity, as there are principled reasons (to do with upholding Pareto criteria) that have led many authors to reject Anonymity in favor of Permutation Invariance specifically in the case of infinite populations. A corollary of the present result is that such approaches are incompatible with the use of social welfare orders.