Distortion of Metric Voting with Bounded Randomness

TL;DR

This work shows that breaking the distortion barrier of $3$ in metric voting is possible with strictly bounded randomness. By developing and combining RepApx variants of Maximal Lotteries and Stable Lotteries within the biased-metric framework, the authors construct a rule that samples from a constant-size candidate list and achieves distortion $\le 3 - \varepsilon$. The analysis pivots on robust distortion bounds for Maximal Lotteries, near-3 distortion for RepApx Maximal Lotteries, and controlled distortion for RepApx Stable Lotteries, all tied together via a mixing strategy that works under both consistent and inconsistent biased metrics. The results yield a polynomial-time method to identify suitable constant-size multisets and imply practical implications for fixed-size winner lists in committee selection, with broader connections to multi-winner settings and interpretability concerns.

Abstract

We study the design of voting rules in the metric distortion framework. It is known that any deterministic rule suffers distortion of at least $3$, and that randomized rules can achieve distortion strictly less than $3$, often at the cost of reduced transparency and interpretability. In this work, we explore the trade-off between these paradigms by asking whether it is possible to break the distortion barrier of $3$ using only "bounded" randomness. We answer in the affirmative by presenting a voting rule that (1) achieves distortion of at most $3 - \varepsilon$ for some absolute constant $\varepsilon > 0$, and (2) selects a winner uniformly at random from a deterministically identified list of constant size. Our analysis builds on new structural results for the distortion and approximation of Maximal Lotteries and Stable Lotteries.

Figures (5)

• Figure 1: $\ell(D, t)$ and $r(t)$.
• Figure 2: Partition of the interval $[0, 1]$ induced by the preference order $d \succ_v c \succ_v b \succ_v a$. The multiset is $S = \left\{a, a, b, c, c, c\right\}$. To compute $\mathop{\mathrm{\mathbf{Pr}}}\limits\limits_{}{\left[c \succ_v S\right]}$, we sample $X_1, \dots, X_6$ and $Z$ independently from the corresponding subintervals of $[0, 1]$. In this example, we have $\mathop{\mathrm{\mathbf{Pr}}}\limits\limits_{}{\left[c \succ_v S\right]} = \mathop{\mathrm{\mathbf{Pr}}}\limits\limits_{}{\left[\max\left\{X_1, \dots, X_6\right\} < Z\right]} = 1/4.$
• Figure 3: Pictorial illustration of $\ell(D_\mathrm{ML}, t)$ v.s. $r(t)$, and $\ell(D_{\varepsilon^2\text{-}\mathrm{ML}}, t)$ v.s. $r(t) + 2\varepsilon$
• Figure 4: Pictorial illustration of $\ell(D, t)$ and $r(t) + 2\varepsilon$
• Figure 5: Pictorial illustration of $f_1(p)$, $f_2(p)$, and $f_3(p)$.

Theorems & Definitions (74)

• proof : Proof of \ref{['fac:set2dis']}
• Lemma 2.2: see, e.g., DBLP:journals/jacm/CharikarRWW24
• Definition 2.3: Pseudometric Space
• Definition 2.4: Metric Distortion
• Definition 2.5: Biased Metrics
• proof : Proof of \ref{['fac:d_to_s_1']}
• proof : Proof of \ref{['fac:d_to_s_2']}
• Theorem 2.8: DBLP:journals/jacm/CharikarRWW24
• proof : Proof of \ref{['thm:biased_metric']}
• Corollary 2.9: DBLP:conf/soda/CharikarR22DBLP:journals/jacm/CharikarRWW24