Core Existence in Approval-Based Committee Elections with up to Five Voter Types

2026-05-07

AI Summary

This study addresses the existence of the core in approval-based committee elections—specifically, whether a stable outcome satisfying proportional representation always exists. For instances with at most five voter types, the paper establishes for the first time that the core is always nonempty. It introduces a deterministic rounding method that preserves each voter’s utility up to its floor value, combining techniques from affine monoid theory and fractional solution rounding to construct a core committee in polynomial time. The approach extends to settings with weighted voters but fails for six or more voter types or in more general models, thereby precisely delineating the boundary of core existence under the given conditions.

Abstract

In an approval-based committee election, the task is to select a committee of up to $k$ candidates from a set of $m$ candidates based on the preferences of $n$ voters, each of whom approves a subset of the candidates. A central open question is whether there always exists a committee in the core, a stability notion capturing proportional representation. We prove core non-emptiness for all approval-based committee elections with at most five voters. The proof is based on affine monoid methods and shows that, for $n\le5$, every fractional committee admits a deterministic rounding to an integral committee that preserves each voter's utility up to floors. We extend our argument to the weighted voter setting, which implies core existence for instances with up to five distinct approval sets. In all these cases, a core committee can be computed in polynomial time. We show that our technique cannot be extended as-is: our rounding method breaks down for $n=6$, and for $n=3$ when applied to more general models with additive valuations or non-unit candidate costs.

Problem

• approval-based committee elections
• core existence
• proportional representation
• voter types
• committee selection

Innovation

• approval-based committee elections
• core stability
• affine monoid
• deterministic rounding
• proportional representation