An extension of May's Theorem to three alternatives: axiomatizing Minimax voting
Wesley H. Holliday, Eric Pacuit
TL;DR
The paper provides an axiomatic extension of May's Theorem to three alternatives by introducing positive involvement and (near) immunity to spoilers alongside homogeneity and block preservation. It proves that any voting method on profiles with two or three alternatives satisfying anonymity, neutrality, and weak positive responsiveness, plus the additional axioms, must refine the Minimax rule, thereby selecting among Minimax winners in three-alternative elections (modulo tie-breaking). The authors extend the analysis to independence of axioms, discuss implications for four or more alternatives, and relate the results to Condorcet-consistent methods such as Kemeny, Ranked Pairs, and Schulze, arguing that Minimax serves as a principled canonical solution in the three-alternative domain. They also consider refinements like Minimax MB and discuss future directions for axiomatic characterizations beyond three alternatives, including normative considerations for top-three elections.
Abstract
May's Theorem [K. O. May, Econometrica 20 (1952) 680-684] characterizes majority voting on two alternatives as the unique preferential voting method satisfying several simple axioms. Here we show that by adding some desirable axioms to May's axioms, we can uniquely determine how to vote on three alternatives (setting aside tiebreaking). In particular, we add two axioms stating that the voting method should mitigate spoiler effects and avoid the so-called strong no show paradox. We prove a theorem stating that any preferential voting method satisfying our enlarged set of axioms, which includes some weak homogeneity and preservation axioms, must choose from among the Minimax winners in all three-alternative elections. When applied to more than three alternatives, our axioms also distinguish Minimax from other known voting methods that coincide with or refine Minimax for three alternatives.