{"type":"ScholarlyArticle","title":"Proportionality in Approval-Based Elections With a Variable Number of Winners","authors":[{"name":"Rupert Freeman","slug":"rupert-freeman","given":"Rupert","family":"Freeman","affiliation":"Microsoft Research New York City"},{"name":"Anson Kahng","slug":"anson-kahng","given":"Anson","family":"Kahng","affiliation":"Carnegie Mellon University"},{"name":"David M. Pennock","slug":"david-m-pennock","given":"David M.","family":"Pennock","affiliation":"Rutgers University"}],"abstract":"We study proportionality in approval-based multiwinner elections with a variable number of winners, where both the size and identity of the winning committee are informed by voters' opinions. While proportionality has been studied in multiwinner elections with a fixed number of winners, it has not been considered in the variable number of winners setting. The measure of proportionality we consider is average satisfaction (AS), which intuitively measures the number of agreements on average between sufficiently large and cohesive groups of voters and the output of the voting rule. First, we show an upper bound on AS that any deterministic rule can provide, and that straightforward adaptations of deterministic rules from the fixed number of winners setting do not achieve better than a 1/2 approximation to AS even for large numbers of candidates. We then prove that a natural randomized rule achieves a 29/32 approximation to AS.","doi":"10.24963/ijcai.2020/19","url":"https://ijcai.org/proceedings/2020/19","pdf":"https://ijcai.org/proceedings/2020/0019.pdf","markdown":"https://ijcai.org/proceedings/2020/19.md","bibtex":"https://ijcai.org/proceedings/2020/bibtex/19","volume":{"slug":"2020","title":"Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2020","url":"https://ijcai.org/proceedings/2020/","year":2020,"location":"Yokohama","dates":"January 2021 (postponed from 11-17 July 2020)","editors":"Christian Bessiere","isbn":"978-0-9992411-6-5","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2020"},"section":"Agent-based and Multi-agent Systems","track":"Main track","number":19,"pages":{"first":132,"last":138},"keywords":["Agent-based and Multi-agent Systems: Algorithmic Game Theory","Agent-based and Multi-agent Systems: Computational Social Choice","Agent-based and Multi-agent Systems: Voting"],"videos":[{"label":"Video","url":"https://videos.ijcai20.org/26564/file_26564.m3u8","length_sec":880,"provider":"hls"}],"references":[{"position":1,"raw":"[Aziz et al., 2017] Haris Aziz, Markus Brill, Vincent Conitzer, Edith Elkind, Rupert Freeman, and Toby Walsh. Justified representation in approval-based committee voting. Social Choice and Welfare, 48(2):461-485, 2017.","title":"Justified representation in approval-based committee voting","authors":"Haris Aziz, Markus Brill, Vincent Conitzer, Edith Elkind, Rupert Freeman, Toby Walsh","year":"2017","venue":"Social Choice and Welfare","doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Aziz et al., 2018] Haris Aziz, Edith Elkind, Shenwei Huang, Martin Lackner, Luis Sánchez-Fernández, and Pi- otr Skowron. On the complexity of extended and propor- tional justified representation. In Proceedings of the 32nd AAAI Conference on Artificial Intelligence, pages 902- 909, 2018.","title":"On the complexity of extended and proportional justified representation","authors":"Haris Aziz, Edith Elkind, Shenwei Huang, Martin Lackner, Luis Sánchez-Fernández, Piotr Skowron","year":"2018","venue":"Proceedings of the 32nd AAAI Conference on Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Brams et al., 2007] Steven J Brams, D Marc Kilgour, and M Remzi Sanver. A minimax procedure for electing com- mittees. Public Choice, 132(3-4):401-420, 2007.","title":"A minimax procedure for electing committees","authors":"D Steven J Brams, M Remzi Marc Kilgour, Sanver","year":"2007","venue":"Public Choice","doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Brandl and Peters, 2019] Florian Brandl and Dominik Pe- ters. An axiomatic characterization of the Borda mean rule. Social Choice and Welfare, 52(4):685-707, 2019.","title":"An axiomatic characterization of the Borda mean rule","authors":"Peters Brandl","year":"2019","venue":"Social Choice and Welfare","doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Brill et al., 2017] Markus Brill, Rupert Freeman, Svante Janson, and Martin Lackner. Phragmén's voting methods and justified representation. In Proceedings of the 31st AAAI Conference on Artificial Intelligence, pages 406- 413, 2017.","title":"Phragmén's voting methods and justified representation","authors":"Markus Brill, Rupert Freeman, Svante Janson, Martin Lackner","year":"2017","venue":"Proceedings of the 31st AAAI Conference on Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Cheng et al., 2019] Yu Cheng, Zhihao Jiang, Kamesh Mu- nagala, and Kangning Wang. Group fairness in committee selection. In Proceedings of the 20th ACM Conference on Economics and Computation (EC), pages 263-279, 2019.","title":"Group fairness in committee selection","authors":"Yu Cheng, Zhihao Jiang, Kamesh Munagala, Kangning Wang","year":"2019","venue":"Proceedings of the 20th ACM Conference on Economics and Computation (EC)","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[Duddy et al., 2016] Conal Duddy, Ashley Piggins, and William S Zwicker. Aggregation of binary evaluations: a Borda-like approach. Social Choice and Welfare, 46(2):301-333, 2016.","title":"Aggregation of binary evaluations: a Borda-like approach","authors":"Conal Duddy, Ashley Piggins, William S Zwicker","year":"2016","venue":"Social Choice and Welfare","doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[Faliszewski et al., 2017] Piotr Faliszewski, Arkadii Slinko, and Nimrod Talmon. The complexity of multiwinner vot- ing rules with variable number of winners. arXiv preprint arXiv:1711.06641, 2017.","title":"The complexity of multiwinner voting rules with variable number of winners","authors":"Piotr Faliszewski, Arkadii Slinko, Nimrod Talmon","year":"2017","venue":null,"doi":null,"url":null,"internal_url":null},{"position":9,"raw":"[Fishburn and Pekeč, 2004] Peter C Fishburn and Aleksan- dar Pekeč. Approval voting for committees: Threshold approaches. Technical Report, 2004.","title":"Approval voting for committees: Threshold approaches","authors":"C Peter, Aleksandar Fishburn, Pekeč","year":"2004","venue":null,"doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Janson, 2016] Svante Janson. Phragmén's and Thiele's elec- tion methods. arXiv preprint arXiv:1611.08826, 2016.","title":"Phragmén's and Thiele's election methods","authors":"Svante Janson","year":"2016","venue":null,"doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Kilgour and Marshall, 2012] D Marc Kilgour and Erica Marshall. Approval balloting for fixed-size committees. In Electoral Systems, pages 305-326. 2012.","title":"Approval balloting for fixed-size committees","authors":"Marc Kilgour, Erica Marshall","year":"2012","venue":"Electoral Systems","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Kilgour et al., 2006] D Marc Kilgour, Steven J Brams, and M Remzi Sanver. How to elect a representative committee using approval balloting. In Mathematics and Democracy, pages 83-95. Springer, 2006.","title":"How to elect a representative committee using approval balloting","authors":"Steven J Marc Kilgour, M Remzi Brams, Sanver","year":"2006","venue":"Mathematics and Democracy","doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Kilgour, 2016] D Marc Kilgour. Approval elections with a variable number of winners. Theory and Decision, 81(2):199-211, 2016.","title":"Approval elections with a variable number of winners","authors":"Kilgour Marc","year":"2016","venue":"Theory and Decision","doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Mueller, 1969] John E Mueller. Voting on the propositions: Ballot patterns and historical trends in california. Ameri- can Political Science Review, 63(4):1197-1212, 1969.","title":"Voting on the propositions: Ballot patterns and historical trends in california","authors":"John E Mueller","year":"1969","venue":"American Political Science Review","doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Peters and Skowron, 2019] Dominik Peters and Piotr Skowron. Proportionality and the limits of welfarism. arXiv preprint arXiv:1911.11747, 2019.","title":"Proportionality and the limits of welfarism","authors":"Skowron Peters, Dominik Peters, Piotr Skowron","year":"2019","venue":null,"doi":null,"url":null,"internal_url":null},{"position":16,"raw":"[Phragmén, 1899] Edvard Phragmén. Till frågan om en pro- portionell valmetod. Statsvetenskaplig Tidskrift, 2(2):297- 305, 1899.","title":"Edvard Phragmén. Till frågan om en proportionell valmetod","authors":"Phragmén","year":"1899","venue":"Statsvetenskaplig Tidskrift","doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Sánchez-Fernández et al., 2017] Luis Sánchez-Fernández, Edith Elkind, Martin Lackner, Norberto Fernández, Jesús A Fisteus, Pablo Basanta Val, and Piotr Skowron. Proportional justified representation. In Proceedings of the 31st AAAI Conference on Artificial Intelligence, pages 670-676, 2017.","title":"Proportional justified representation","authors":"Luis Sánchez-Fernández, Edith Elkind, Martin Lackner, Norberto Fernández, A Jesús, Pablo Basanta Fisteus, Piotr Val, Skowron","year":"2017","venue":"Proceedings of the 31st AAAI Conference on Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Skowron et al., 2017a] Piotr Skowron, Martin Lackner, Markus Brill, Dominik Peters, and Edith Elkind. Propor- tional rankings. In Proceedings of the 26th International Joint Conference on Artificial Intelligence (IJCAI), pages 409-415, 2017.","title":"Proportional rankings","authors":"Skowron","year":"2017","venue":"Proceedings of the 26th International Joint Conference on Artificial Intelligence (IJCAI)","doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[Skowron et al., 2017b] Piotr Skowron, Martin Lackner, Edith Elkind, and Luis Sánchez-Fernández. Optimal av- erage satisfaction and extended justified representation in polynomial time. arXiv preprint arXiv:1704.00293, 2017.","title":"Optimal average satisfaction and extended justified representation in polynomial time","authors":"Skowron","year":"2017","venue":null,"doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Skowron, 2018] Piotr Skowron. Proportionality degree of multiwinner rules. arXiv preprint arXiv:1810.08799, 2018.","title":"Proportionality degree of multiwinner rules","authors":"Piotr Skowron","year":"2018","venue":null,"doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[Thiele, 1895] Thorvald N Thiele. Om flerfoldsvalg. Over- sigt over det Kongelige Danske Videnskabernes Selskabs Forhandlinger, 1895:415-441, 1895.","title":"Om flerfoldsvalg. Oversigt over det","authors":"Thorvald N Thiele","year":"1895","venue":"Kongelige Danske Videnskabernes Selskabs Forhandlinger","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}