タイトル：Acyclicity, Anonymity, and Prefilters (Joint work with Walter Bossert)
We analyze the decisiveness structures associated with acyclical collective choice rules. In particular, we examine the consequences of adding anonymity to weak Pareto, thereby complementing earlier results on acyclical social choice. Both finite and countably infinite populations are considered. As established in contributions by Brown and by Banks, acyclical social choice is closely linked to prefilters in the presence of the weak Pareto principle. In the finite-population case, adding anonymity implies that the only decisive set is the set of the entire population if there are at least as many alternatives as individuals. When there are more individuals than alternatives, a different decisiveness structure emerges—namely, collections of decisive sets that are special cases of symmetric prefilters. Moving to infinite populations, we again obtain the entire population as the unique decisive set as a possibility, along with additional options. These consist of a new class of prefilters that we refer to as symmetric free Frechet prefilters. The choice of the term Frechet prefilter is motivated by the observation that they share a defining property with the well-known Frechet filter—namely, that all sets in the requisite collection are such that their complement is finite.