Rationalizations of Condorcet-Consistent Rules via Distances of Hamming Type

TL;DR

This paper explores how Condorcet-consistent voting rules can be rationalized through distance measures, introducing new insights into Young's and Maximin rules and revealing computational complexity challenges.

Contribution

It introduces novel distance rationalizations for Young's and Maximin rules, clarifies misconceptions about their relation to Condorcet consensus, and identifies computational hardness.

Findings

Young's rule is not rationalized by Hamming distance and Condorcet class.

A new, previously unstudied rule is introduced with similar properties to Young's rule.

Winner determination for the new rule is computationally hard.

Abstract

The main idea of the {\em distance rationalizability} approach to view the voters' preferences as an imperfect approximation to some kind of consensus is deeply rooted in social choice literature. It allows one to define ("rationalize") voting rules via a consensus class of elections and a distance: a candidate is said to be an election winner if she is ranked first in one of the nearest (with respect to the given distance) consensus elections. It is known that many classic voting rules can be distance rationalized. In this paper, we provide new results on distance rationalizability of several Condorcet-consistent voting rules. In particular, we distance rationalize Young's rule and Maximin rule using distances similar to the Hamming distance. We show that the claim that Young's rule can be rationalized by the Condorcet consensus class and the Hamming distance is incorrect; in fact, these consensus class and distance yield a new rule which has not been studied before. We prove that, similarly to Young's rule, this new rule has a computationally hard winner determination problem.