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arXiv · 2501.18304

The Core of Approval-Based Committee Elections with Few Seats

Dominik Peters
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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.

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Dominik Peters. 2025-01-30. The Core of Approval-Based Committee Elections with Few Seats. https://arxiv.org/abs/2501.18304

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