Approval Gap of Weighted k-Majority Tournaments

TL;DR

This paper introduces weighted k-majority tournaments to measure approval gaps, establishing bounds and constructions for the maximum approval gap parameter, and analyzing the minimum size of such tournaments for given parameters.

Contribution

It defines weighted k-majority tournaments and the approval gap parameter, providing bounds and constructions, advancing understanding of voter preference aggregation.

Findings

Bounds on approval gap: k/2 ≤ γ_w(T) ≤ 2k-1.

Constructed tournaments with any rational approval gap within bounds.

Analyzed minimum vertices m(q,k) for given approval gap q.

Abstract

A $k$-majority tournament $T$ on a finite set of vertices $V$ is defined by a set of $2k-1$ linear orders on $V$, with an edge $u \to v$ in $T$ if $u>v$ in a majority of the linear orders. We think of the linear orders as voter preferences and the vertices of $T$ as candidates, with an edge $u \to v$ in $T$ if a majority of voters prefer candidate $u$ to candidate $v$. In this paper we introduce weighted $k$-majority tournaments, with each edge $u \to v$ weighted by the number of voters preferring $u$. We define the maximum approval gap $\gamma_w(T)$, a measure by which any dominating set of $T$ beats the next most popular candidate. This parameter is analogous to previous work on the size of minimum dominating sets of (unweighted) $k$-majority tournaments. We prove that $k/2 \leq \gamma_w(T) \leq 2k-1$ for any weighted $k$-majority tournament $T$, and construct tournaments with $\gamma_w(T)=q$ for any rational number $k/2 \leq q \leq 2k-1$. We also consider the minimum number of vertices $m(q,k)$ in a $k$-majority tournament with $\gamma_w(T)=q$.