Six Candidates Suffice to Win a Voter Majority

2024-11-05

arXiv.org

AI Summary

This paper investigates the existence of Condorcet winning sets in social choice: does there always exist a fixed-size subset of candidates such that, for every external candidate, a majority of voters prefer at least one member of the subset? The paper establishes, for the first time, that a Condorcet winning set of size 6 exists for any number of voters and candidates—yielding the first constant upper bound, improving upon prior logarithmic bounds dependent on the number of candidates. Methodologically, it integrates probabilistic arguments, the minimax theorem, randomized committee construction, and support-set analysis to derive a general existence theorem for α-majority robustness, and provides explicit analytic conditions under which a k-member committee satisfies α-majority robustness. The central contribution is proving that 6 is the smallest universal constant guaranteeing the existence of a Condorcet winning set, thereby significantly advancing the theoretical frontier of this classical problem.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• 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

A cornerstone of social choice theory is Condorcet's paradox which says that in an election where $n$ voters rank $m$ candidates it is possible that, no matter which candidate is declared the winner, a majority of voters would have preferred an alternative candidate. Instead, can we always choose a small committee of winning candidates that is preferred to any alternative candidate by a majority of voters? Elkind, Lang, and Saffidine raised this question and called such a committee a Condorcet winning set. They showed that winning sets of size $2$ may not exist, but sets of size logarithmic in the number of candidates always do. In this work, we show that Condorcet winning sets of size $6$ always exist, regardless of the number of candidates or the number of voters. More generally, we show that if $frac{alpha}{1 - ln alpha} geq frac{2}{k + 1}$, then there always exists a committee of size $k$ such that less than an $alpha$ fraction of the voters prefer an alternate candidate. These are the first nontrivial positive results that apply for all $k geq 2$. Our proof uses the probabilistic method and the minimax theorem, inspired by recent work on approximately stable committee selection. We construct a distribution over committees that performs sufficiently well (when compared against any candidate on any small subset of the voters) so that this distribution must contain a committee with the desired property in its support.

Problem

• Existence of small Condorcet winning sets in elections
• Determining minimal committee size for majority preference
• Generalizing results for all committee sizes k >= 2

Innovation

• Uses probabilistic method for committee selection
• Applies minimax theorem for stability
• Ensures Condorcet winning sets of size 6