Six Candidates Suffice to Win a Voter Majority

TL;DR

This paper proves that for any number of candidates and voters, there always exists a small committee of size 6 that is preferred over any other candidate by a majority, extending previous results on Condorcet winning sets.

Contribution

It establishes the existence of Condorcet winning sets of size 6 universally, and provides a general condition relating committee size and voter preference fractions.

Findings

Condorcet winning sets of size 6 always exist.

A general condition relates committee size to voter preference fractions.

First nontrivial results for all committee sizes greater than or equal to 2.

Abstract

A cornerstone of social choice theory is Condorcet's paradox which says that in an election where $n$ voters rank $m$ candidates it is possible that, no matter which candidate is declared the winner, a majority of voters would have preferred an alternative candidate. Instead, can we always choose a small committee of winning candidates that is preferred to any alternative candidate by a majority of voters? Elkind, Lang, and Saffidine raised this question and called such a committee a Condorcet winning set. They showed that winning sets of size $2$ may not exist, but sets of size logarithmic in the number of candidates always do. In this work, we show that Condorcet winning sets of size $6$ always exist, regardless of the number of candidates or the number of voters. More generally, we show that if $\frac{\alpha}{1 - \ln \alpha} \geq \frac{2}{k + 1}$, then there always exists a committee of size $k$ such that less than an $\alpha$ fraction of the voters prefer an alternate candidate. These are the first nontrivial positive results that apply for all $k \geq 2$. Our proof uses the probabilistic method and the minimax theorem, inspired by recent work on approximately stable committee selection. We construct a distribution over committees that performs sufficiently well (when compared against any candidate on any small subset of the voters) so that this distribution must contain a committee with the desired property in its support.