The Core of Approval-Based Committee Elections with Few Seats

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 ≤ 8, for any number of candidates m and any number of voters n, by showing that the Proportional Approval Voting (PAV) rule, proposed by Thiele in 1895, always satisfies the core when k ≤ 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 ≤ 15 candidates, for any committee size k ≤ m and any number of voters n. These results are obtained with the help of computer search using linear programs.