A voting rule is a Condorcet extension if it returns a candidate that beats every other candidate in pairwise majority comparisons whenever one exists. Condorcet extensions have faced criticism due to their susceptibility to variable-electorate paradoxes, especially the reinforcement paradox (Young and Levenglick, 1978) and the no-show paradox (Moulin, 1988). In this paper, we investigate the susceptibility of Condorcet extensions to these paradoxes for the case of exactly three candidates. For the reinforcement paradox, we establish that it must occur for every Condorcet extension when there are at least eight voters and demonstrate that certain refinements of maximin, a voting rule originally proposed by Condorcet (1785), are immune to this paradox when there are at most seven voters. For the no-show paradox, we prove that the only homogeneous Condorcet extensions immune to it are refinements of maximin. We also provide axiomatic characterizations of maximin and two of its refinements, Nanson's rule and leximin, highlighting their suitability for three-candidate elections.

翻译：若一个投票规则在存在孔多塞胜者（即在成对多数比较中击败所有其他候选人的候选人）时总能返回该胜者，则称其为孔多塞扩展规则。孔多塞扩展规则因其易受可变选民规模悖论的影响而受到批评，尤其是强化悖论（Young and Levenglick, 1978）和不参与悖论（Moulin, 1988）。本文研究了在恰好有三名候选人的情况下，孔多塞扩展规则对这些悖论的易感性。针对强化悖论，我们证明当选民数至少为八时，该悖论必然发生在每一个孔多塞扩展规则上；同时，我们证明了当选民数不超过七时，由孔多塞（1785）最初提出的投票规则——极大极小规则（maximin）的某些改进版本对该悖论具有免疫性。针对不参与悖论，我们证明了唯一对其免疫的齐次孔多塞扩展规则是极大极小规则的改进版本。我们还给出了极大极小规则及其两种改进版本——南森规则（Nanson's rule）和字典序极小规则（leximin）的公理化刻画，突显了它们在三候选人选举中的适用性。