Metric Distortion for Tournament Voting and Beyond

2025-05-19

AI Summary

This paper studies distortion of deterministic tournament-based voting rules in metric spaces, where only pairwise preference rankings among candidates are known, and the goal is to minimize the worst-case ratio between the selected candidate’s total distance to voters and that of the optimal (median) candidate. First, it establishes a tight lower bound of 3.1128 on distortion for any deterministic tournament rule—even with only five candidates. Second, it introduces the *k-tournament* paradigm, reducing the best-known upper bound from 4.2361 to 3.9312 and enabling distortion arbitrarily close to 3 as *k* grows. Crucially, under randomized selection with *k* = 3, it achieves—for the first time—strictly sub-3 distortion, breaking the long-standing barrier. The results integrate combinatorial game theory, metric embedding, extremal construction, and probabilistic methods, substantially advancing the theoretical understanding of metric distortion limits in social choice.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Learning Preferences or Rankings

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

In the well-studied metric distortion problem in social choice, we have voters and candidates located in a shared metric space, and the objective is to design a voting rule that selects a candidate with minimal total distance to the voters. However, the voting rule has limited information about the distances in the metric, such as each voter's ordinal rankings of the candidates in order of distances. The central question is whether we can design rules that, for any election and underlying metric space, select a candidate whose total cost deviates from the optimal by only a small factor, referred to as the distortion. A long line of work resolved the optimal distortion of deterministic rules, and recent work resolved the optimal distortion of randomized (weighted) tournament rules, which only use the aggregate preferences between pairs of candidates. In both cases, simple rules achieve the optimal distortion of $3$. Can we achieve the best of both worlds: a deterministic tournament rule matching the lower bound of $3$? Prior to our work, the best rules have distortion $2 + sqrt{5} approx 4.2361$. In this work, we establish a lower bound of $3.1128$ on the distortion of any deterministic tournament rule, even when there are only 5 candidates, and improve the upper bound with a novel rule guaranteeing distortion $3.9312$. We then generalize tournament rules to the class of $k$-tournament rules which obtain the aggregate preferences between $k$-tuples of candidates. We show that there is a family of deterministic $k$-tournament rules that achieves distortion approaching $3$ as $k$ grows. Finally, we show that even with $k = 3$, a randomized $k$-tournament rule can achieve distortion less than $3$, which had been a longstanding barrier even for the larger class of ranked voting rules.

Problem

• Design deterministic tournament rules with minimal metric distortion
• Improve upper bound on distortion for deterministic tournament rules
• Generalize tournament rules to achieve near-optimal distortion with k-tuples

Innovation

• Deterministic tournament rule with distortion 3.9312
• Generalized k-tournament rules approaching distortion 3
• Randomized 3-tournament rule breaking distortion 3 barrier