Van Zuylen et al. introduced the notion of a popular ranking in a voting context, where each voter submits a strictly-ordered list 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 \emph{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 by van Zuylen et al. We also point out strong connections to the famous open problem of finding a Kemeny consensus with 3 voters.
翻译:Van Zuylen et al. 介绍了在投票背景下民众排名的概念,即每个选民提交严格排序的所有候选人名单。在以下意义上,受欢迎的排名$pi$至少与任何其他排名$$$gma$一样好:如果我们把美元比作美元,那么,至少一半选民总是不太喜欢~$pi$。选民是否偏爱排名中的排名,根据Kendall距离来计算。更传统的流行定义 -- -- 适用于民众匹配,这是计算社会选择中公认的主题 -- -- 更为严格,因为它要求至少一半的选民在$pi$和$\sgma$之间不无差别,而不是选择~$\pi$。在本文中,我们从两种环境中都得出结构和算法结果,同时改进van Zuylen等人的结果。我们还指出,在与与3名选民达成Kemeny共识的著名公开问题之间有着密切的联系。