Tight Bounds On the Distortion of Randomized and Deterministic Distributed Voting

2025-09-21

AI Summary

This paper studies metric distortion in distributed voting: $n$ voters are partitioned into $k$ groups; each group selects a local representative, and a final winner is chosen either from the set of representatives or from the full candidate set. We systematically characterize—both upper and lower bounds—the distortion of deterministic and randomized mechanisms under four cost objectives: $avgavg$, $avgmax$, $maxavg$, and $maxmax$, yielding near-complete theoretical characterization. Our analysis integrates probabilistic mechanism design, combinatorial optimization, and game theory, with a focus on the two-stage voting structure. Key contributions include: (i) improving the deterministic upper bounds for $avgmax$ and $maxmax$ from 11 and 5 to 7 and 3, respectively; (ii) establishing tight distortion bounds of 3 for $maxavg$ and $maxmax$ under two-stage randomization; and (iii) achieving randomized distortion arbitrarily close to the tight bound of 3 for both $avgavg$ and $avgmax$.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We study metric distortion in distributed voting, where $n$ voters are partitioned into $k$ groups, each selecting a local representative, and a final winner is chosen from these representatives (or from the entire set of candidates). This setting models systems like U.S. presidential elections, where state-level decisions determine the national outcome. We focus on four cost objectives from citep{anshelevich2022distortion}: $avgavg$, $avgmax$, $maxavg$, and $maxmax$. We present improved distortion bounds for both deterministic and randomized mechanisms, offering a near-complete characterization of distortion in this model. For deterministic mechanisms, we reduce the upper bound for $avgmax$ from $11$ to $7$, establish a tight lower bound of $5$ for $maxavg$ (improving on $2+sqrt{5}$), and tighten the upper bound for $maxmax$ from $5$ to $3$. For randomized mechanisms, we consider two settings: (i) only the second stage is randomized, and (ii) both stages may be randomized. In case (i), we prove tight bounds: $5!-!2/k$ for $avgavg$, $3$ for $avgmax$ and $maxmax$, and $5$ for $maxavg$. In case (ii), we show tight bounds of $3$ for $maxavg$ and $maxmax$, and nearly tight bounds for $avgavg$ and $avgmax$ within $[3!-!2/n, 3!-!2/(kn^*)]$ and $[3!-!2/n, 3]$, respectively, where $n^*$ denotes the largest group size.

Problem

• Analyzing metric distortion in distributed voting systems with partitioned voters
• Establishing improved distortion bounds for deterministic and randomized mechanisms
• Characterizing cost objectives for election systems like presidential elections

Innovation

• Improved deterministic bounds for distributed voting distortion
• Tight randomized bounds for two-stage voting mechanisms
• Near-complete characterization of four cost objectives