The Core of Approval-Based Committee Elections with Few Seats
Dominik Peters
TL;DR
This work investigates core stability in approval-based multi-winner elections, focusing on whether a core-stable committee always exists. It leverages linear-programming analyses of Proportional Approval Voting (PAV) and introduces a recursive PAV rule to patch PAV outcomes when core objections arise, proving core existence for $k \le 8$ and for $m \le 15$ across all $n$. The results show PAV guarantees a core committee for $k \le 7$ and at least one core committee for $k=8$, while a constructive recursive method certifies core existence for up to $m=15$ candidates; these conclusions are supported by extensive LP-based certificates. The Droop quota variant reveals limits: the core remains challenging to guarantee, with the Droop core failing for even small $k$ (e.g., $k=6$), underscoring open questions about core existence in broader settings. Overall, the paper strengthens confidence in PAV for small-seat elections and provides a computable, certificate-backed approach for larger-but-limited candidate sets, while outlining avenues for further exploration of core existence under alternative quota notions.
Abstract
In an approval-based committee election, the goal is to select a committee consisting of $k$ out of $m$ candidates, based on $n$ voters who each approve an arbitrary number of the candidates. The core of such an election consists of all committees that satisfy a certain stability property which implies proportional representation. In particular, committees in the core cannot be "objected to" by a coalition of voters who is underrepresented. The notion of the core was proposed in 2016, but it has remained an open problem whether it is always non-empty. We prove that core committees always exist when $k \le 8$, for any number of candidates $m$ and any number of voters $n$, by showing that the Proportional Approval Voting (PAV) rule due to Thiele [1895] always satisfies the core when $k \le 7$ and always selects at least one committee in the core when $k = 8$. We also develop an artificial rule based on recursive application of PAV, and use it to show that the core is non-empty whenever there are $m \le 15$ candidates, for any committee size $k \le m$ and any number of voters $n$. These results are obtained with the help of computer search using linear programs.