{"type":"ScholarlyArticle","title":"The Core of Approval-Based Committee Elections with Few Seats","authors":[{"name":"Dominik Peters","slug":"dominik-peters","given":"Dominik","family":"Peters","affiliation":""}],"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 ≤ 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.","doi":"10.24963/ijcai.2025/447","url":"https://ijcai.org/proceedings/2025/447","pdf":"https://ijcai.org/proceedings/2025/0447.pdf","markdown":"https://ijcai.org/proceedings/2025/447.md","bibtex":"https://ijcai.org/proceedings/2025/bibtex/447","volume":{"slug":"2025","title":"Proceedings of the Thirty-Fourth International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2025","url":"https://ijcai.org/proceedings/2025/","year":2025,"location":"Montreal, Canada","dates":"16-22 August 2025","editors":"James Kwok","isbn":"978-1-956792-06-5","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2025"},"section":"Game Theory and Economic Paradigms","track":"Main Track","number":447,"pages":{"first":4014,"last":4022},"keywords":["Game Theory and Economic Paradigms: GTEP: Computational social choice"],"videos":[],"references":[],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}