Pessimal Elections for Approximately Dominating Sets

2026-08-07

AI Summary

This work addresses a fundamental flaw in multi-winner elections: even after a committee is selected, there may exist an unelected candidate who is unanimously preferred by a majority of voters over every elected member. To mitigate this issue, the paper relaxes the majority threshold to $1/2 + \varepsilon$ and constructs a winning committee of size $O(1/\varepsilon^2)$ that guarantees no unelected candidate is collectively preferred by more than a $1/2 + \varepsilon$ fraction of voters over all committee members. Leveraging AI-assisted mathematical discovery (GPT-5.6 Sol Ultra) alongside combinatorial construction techniques, the authors prove that this committee size is optimal up to constant factors. This result establishes a tight lower bound for approximately dominant-set election mechanisms and precisely characterizes the optimal trade-off between committee size and preference tolerance.

Abstract

Condorcet's paradox is a foundational result in social choice theory, showing that no matter which candidate wins an election, a majority of voters may prefer some losing candidate. Worse still, even if the election can choose a committee of $k$ winners, some loser may beat every winner in a majority vote. Recent work showed that this obstruction can be sidestepped by relaxing the majority threshold. For all $\varepsilon > 0$, any election can select a committee of $O(1/\varepsilon^2)$ winners such that no loser is preferred to every winner by $\frac12 + \varepsilon$ fraction of voters. We present a simple construction, found by GPT-5.6 Sol Ultra, which proves that this result is tight up to a constant factor.

Problem

• social choice theory
• Condorcet's paradox
• approximately dominating sets
• committee selection
• majority threshold

Innovation

• approximately dominating sets
• Condorcet's paradox
• social choice theory
• tight bound
• committee selection