On the Distortion of Multi-winner Election Using Single-Candidate Ballots
Gennaro Auricchio, Zeyu Ren, Zihe Wang, Jie Zhang
TL;DR
The paper investigates distortion in multiwinner elections with single-candidate ballots in general metric spaces. It introduces the $\sigma$-ratio $\sigma = d_{\max}/d_{\min}$ and derives both lower and upper distortion bounds for non-strategyproof and strategyproof mechanisms, including a lower bound of $1+\frac{w-1}{w+1}(\sigma-1)$ and an upper bound of $1+2\sigma$ for SNTV, plus a strategyproof bound via Random Sequential Dictator of $<1+4\sigma$ when $w=2$. It also shows that truthful mechanisms are independent of irrelevant candidates, proves that no anonymous deterministic strategyproof mechanism can have finite distortion, and presents Sequential Dictator achieving $2(n-w)\sigma+1$ when anonymity is relaxed. These results clarify how the $\sigma$ parameter governs distortion and suggest directions for extending to higher-ranked ballots and alternative objectives.
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 $σ$ parameter, which represents the ratio between maximum and minimum distances among all candidate pairs. We show that the distortion is linear in $σ$. First, we demonstrate that all mechanisms possess a distortion greater than $1+\frac{w-1}{w+1}(σ-1)$. To give an upper bound, we study the Single Non-Transferable Vote (SNTV) mechanism, whose distortion is at most $1+2σ$. 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 $1+4σ$ when $w=2$.