Related papers: Metric geometry for ranking-based voting: Tools fo…
We extend the recently introduced theory of Lovasz-Bregman (LB) divergences (Iyer & Bilmes 2012) in several ways. We show that they represent a distortion between a "score" and an "ordering", thus providing a new view of rank aggregation…
The concept of median/consensus has been widely investigated in order to provide a statistical summary of ranking data, i.e. realizations of a random permutation $\Sigma$ of a finite set, $\{1,\; \ldots,\; n\}$ with $n\geq 1$ say. As it…
This paper introduces a new metric and mean on the set of positive semidefinite matrices of fixed-rank. The proposed metric is derived from a well-chosen Riemannian quotient geometry that generalizes the reductive geometry of the positive…
The central problem in this work is to compute a ranking of a set of elements which is "closest to" a given set of input rankings of the elements. We define "closest to" in an established way as having the minimum sum of Kendall-Tau…
We study how electoral rules shape polarization dynamics when voters and candidates both adapt to repeated election outcomes. We introduce two geometric primitives for comparing rules under this feedback: the \emph{winner radius} $R_t =…
We study the performance of voting mechanisms from a utilitarian standpoint, under the recently introduced framework of metric-distortion, offering new insights along three main lines. First, if $d$ represents the doubling dimension of the…
Polyhedral geometry can be used to quantitatively assess the dependence of rankings on personal preference, and provides a tool for both students and universities to assess US News and World Report rankings.
"Compactness," or the use of shape as a proxy for fairness, has been a long-running theme in the scrutiny of electoral districts; badly-shaped districts are often flagged as examples of the abuse of power known as gerrymandering. The most…
In recent years, in an effort to promote fairness in the election process, a wide variety of techniques and metrics have been proposed to determine whether a map is a partisan gerrymander. The most accessible measures, requiring easily…
We introduce and study isomorphic distances between ordinal elections (with the same numbers of candidates and voters). The main feature of these distances is that they are invariant to renaming the candidates and voters, and two elections…
We consider the asymptotic joint distributions among several families of well-known metrics on $S_n$, the symmetric group. These include the bi-invariant metrics such as the Cayley and Hamming distance, and the left-invariant metrics such…
We compare two link analysis ranking methods of web pages in a site. The first, called Site Rank, is an adaptation of PageRank to the granularity of a web site and the second, called Popularity Rank, is based on the frequencies of user…
We introduce a novel definition for a small set R of k points being "representative" of a larger set in a metric space. Given a set V (e.g., documents or voters) to represent, and a set C of possible representatives, our criterion requires…
Ranking and comparing items is crucial for collecting information about preferences in many areas, from marketing to politics. The Mallows rank model is among the most successful approaches to analyse rank data, but its computational…
Winner selection by majority, in an election between two candidates, is the only rule compatible with democratic principles. Instead, when the candidates are three or more and the voters rank candidates in order of preference, there are no…
Kemeny's rule is one of the most studied and well-known voting schemes with various important applications in computational social choice and biology. Recently, Kemeny's rule was generalized via a set-wise approach by Gilbert et. al. This…
This note outlines three intellectually distinct but not mutually exclusive strategies for measuring partisan gerrymandering: partisan symmetry, efficiency gap, and algorithmic sampling.
Introduced by Mallows as a ranking model in statistics, Mallows permutation model is a class of non-uniform probability distributions on the symmetric group $S_n$. The model depends on a distance metric on $S_n$ and a scale parameter…
We present a unifying framework encompassing many social choice settings. Viewing each social choice setting as voting in a suitable metric space, we consider a general model of social choice over metric spaces, in which---similarly to the…
Rank aggregation seeks a representative permutation for a collection of rankings and plays a central role in areas such as social choice, information retrieval, and computational biology. Two fundamental aggregation tasks are the center and…