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

A team at the University of Oxford has cracked a central open problem in computational social choice: proving that for approval-based committee elections with up to five voter types, a committee in the core always exists. The core is a strict stability concept that guarantees no group of voters can defect to form a better committee for themselves. Until now, it was unknown whether core committees existed even for small numbers of voter profiles. The authors use affine monoid methods to show that any fractional committee (a convex combination of candidates) can be rounded to an integral committee without dropping any voter's approved candidates below a floor of their utility. This rounding works deterministically for up to five voter types, yielding a polynomial-time algorithm to find a core committee.

The breakthrough comes with clear limitations: the rounding technique breaks down for six voter types, and for three voter types under additive valuations or non-unit candidate costs. Still, the result has immediate implications for designing fair voting systems in small committees—such as university hiring or conference program committees—where diverse stakeholder groups need proportional representation. The paper also extends the result to weighted voters, meaning the number of distinct approval sets (not total voters) is what matters. This work advances both the theory of proportional representation and practical algorithm design for multi-winner elections.