On weakly and strongly popular rankings

TL;DR

This paper explores different notions of popular rankings in voting, providing structural and algorithmic insights, and relates these concepts to the open problem of Kemeny consensus with three voters.

Contribution

It introduces new structural and algorithmic results for weakly and strongly popular rankings, improving upon previous work and connecting to the Kemeny consensus problem.

Findings

Derived structural properties of popular rankings.

Developed algorithms for computing popular rankings.

Connected popular rankings to the Kemeny consensus problem.

Abstract

Van Zuylen et al. [35] introduced the notion of a popular ranking in a voting context, where each voter submits a strict ranking of all candidates. A popular ranking $\pi$ of the candidates is at least as good as any other ranking $\sigma$ in the following sense: if we compare $\pi$ to $\sigma$, at least half of all voters will always weakly prefer $\pi$. Whether a voter prefers one ranking to another is calculated based on the Kendall distance. A more traditional definition of popularity -- as applied to popular matchings, a well-established topic in computational social choice -- is stricter, because it requires at least half of the voters who are not indifferent between $\pi$ and $\sigma$ to prefer $\pi$. In this paper, we derive structural and algorithmic results in both settings, also improving upon the results in [35]. We also point out connections to the famous open problem of finding a Kemeny consensus with three voters.