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

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 seven voters. The proof is based on affine monoid methods and shows that, for $n\le 5$, every fractional committee admits a deterministic rounding to an integral committee that preserves each voter's utility up to floors. This no longer applies for larger $n$. However, for $n\in \{6,7\}$, we show that a Lindahl equilibrium can be adapted and rounded to obtain a core committee. For $n\le 5$, we further provide a polynomial-time algorithm for computing a committee in the core. Our arguments work for the weighted voter setting, which implies core existence for instances with up to seven distinct approval sets. We conclude by providing examples where our methods fail for more general models with additive valuations, non-unit candidate costs, or the related Droop core.

50 pages. Strengthens existence result to apply up to n=7 (v2) instead of n=5 (v1)