On the Distortion of Multi-winner Election Using Single-Candidate Ballots

TL;DR

This paper analyzes the distortion bounds in multi-winner elections with single-candidate ballots, revealing how the ratio of maximum to minimum distances among candidates affects the efficiency of voting mechanisms.

Contribution

It establishes new bounds on the distortion for both truthful and non-truthful mechanisms in multi-winner elections based on metric space properties.

Findings

Distortion is linear in the ratio σ of candidate distances.

All mechanisms have a distortion greater than 1 + (w-1)/(w+1)(σ-1).

The SNTV mechanism's distortion is at most 1 + 2σ.

Abstract

In this paper, we study the distortion bounds for voting mechanisms in multi-winner elections in general metric spaces. Our study pertains to the case in which each voter only reports her favorite candidate amongst $m$ possible choices. Given that candidates' locations are undisclosed to the mechanism, the mechanism has to form a $w-$winner committee based solely on the number of votes received by candidates. We establish distortion bounds for both truthful and non-truthful mechanisms. Our research highlights the significance of the $\sigma$ parameter, which represents the ratio between maximum and minimum distances among all candidate pairs. We show that the distortion is linear in $\sigma$. First, we demonstrate that all mechanisms possess a distortion greater than $1+\frac{w-1}{w+1}(\sigma-1)$. To give an upper bound, we study the Single Non-Transferable Vote (SNTV) mechanism, whose distortion is at most $1+2\sigma$. Second, we retrieve the upper bounds for strategyproof mechanisms. In particular, we infer an upper bound by examining the Random Sequential Dictator mechanism that achieves a distortion less than $1+4\sigma$ when $w=2$.