Positively responsive collective choice rules and majority rule: a generalization of May’s theorem to many alternatives
May’s theorem (1952) shows that if the set of alternatives contains two members,
an anonymous and neutral collective choice rule is positively responsive if and only
if it is majority rule. We show that if the set of alternatives contains three or more
alternatives only the rule that assigns to every problem its strict Condorcet winner
satisfies the three conditions plus Nash’s version of “independence of irrelevant
alternatives” for the domain of problems that have strict Condorcet winners. We
show also that no rule satisfies the four conditions for domains that are more than
slightly larger.