Condorcet-Consistent Choice Among Three Candidates

TL;DR

The paper investigates Condorcet extensions for three candidates, focusing on reinforcement and no-show paradoxes. It shows reinforcement is unavoidable for any Condorcet extension with at least eight voters, while certain refinements of maximin avoid reinforcement up to seven voters; it also proves that only homogeneous Condorcet extensions immune to the no-show paradox are refinements of maximin. The authors provide axiomatic characterizations for maximin and two refinements, leximin and Nanson's rule, linking their properties to optimism-based participation and continuity. Overall, maximin-based rules emerge as particularly robust for three-candidate elections, offering principled guidance for real-world voting design within this narrow but important setting.

Abstract

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.

Figures (4)

• Figure 1: An example of a preference profile over $N = \{1, \dots, 5\}$ and its margin graph and ordinal margin graph. The ordinal margin graph encodes the information that $m_{a,b} > m_{b,c} = m_{c, a} > 0$.
• Figure 2: Hasse diagram of three-candidate Condorcet extensions where lower rules refine higher ones, and rules at the same node are identical.
• Figure 3: A proof sketch of \ref{['thm:maximin-refinement']}. In each panel, arrows from $x$ to $y$ indicate that $x$ has a non-negative majority margin against $y$: bold arrows represent the largest margins, normal arrows smaller but strictly positive margins, and dashed arrows the smallest margins, which may be zero. Between panels, the arrow from $A$ to $B$ explains why the choice of a candidate in panel $A$ would contradict the choices established in panel $B$.
• Figure 4: Fraction of anonymous profiles in which SCFs return more than one winner, computed using Ehrhart theory BGS15a and the Normaliz package BIR+15a.

Theorems & Definitions (27)

• Theorem 1
• proof
• Definition 1: Reinforcement
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Definition 2: Resolute participation, Moul88b