An extension of May's Theorem to three alternatives: axiomatizing Minimax voting

2023-12-21

arXiv.org

AI Summary

May’s Theorem, characterizing majority rule for two alternatives, lacks a direct generalization to settings with three or more candidates, leaving a gap in the axiomatic foundations of social choice for multi-option elections. Method: We introduce novel axioms—specifically, immunity to vote splitting and avoidance of the strong no-show paradox—alongside standard requirements (anonymity, neutrality, and Pareto efficiency), yielding the first concise and complete axiomatization for the three-alternative case. Contribution/Results: We prove that, over three alternatives, a social choice function satisfies this axiom set if and only if it coincides with the Minimax (Simpson) rule. This establishes the unique axiomatic characterization of Minimax in the three-candidate setting and reveals, for the first time, its essential distinction from other prominent rules such as Borda and Copeland in multi-option frameworks. Our work extends the axiomatic frontier of social choice theory and provides a rigorous foundation for designing small-scale electoral mechanisms.

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.

Problem

• Extending May's Theorem to three alternatives
• Axiomatizing Minimax voting with additional criteria
• Mitigating spoiler effects and avoiding strong no-show paradox

Innovation

• Extends May's Theorem to three alternatives
• Introduces axioms to mitigate spoiler effects
• Uniquely determines Minimax voting method