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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2501.18304 |
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| _version_ | 1866909609885696000 |
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| author | Peters, Dominik |
| author_facet | Peters, Dominik |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_18304 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Core of Approval-Based Committee Elections with Few Seats Peters, Dominik Computer Science and Game Theory 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. |
| title | The Core of Approval-Based Committee Elections with Few Seats |
| topic | Computer Science and Game Theory |
| url | https://arxiv.org/abs/2501.18304 |