{"type":"ScholarlyArticle","title":"Convexity of b-matching Games","authors":[{"name":"Soh Kumabe","slug":"soh-kumabe","given":"Soh","family":"Kumabe","affiliation":"The University of Tokyo; RIKEN AIP"},{"name":"Takanori Maehara","slug":"takanori-maehara","given":"Takanori","family":"Maehara","affiliation":"RIKEN AIP"}],"abstract":"The b-matching game is a cooperative game defined on a graph.\r\nThe game generalizes the matching game to allow each individual to have more than one partner.\r\nThe game has several applications, such as the roommate assignment, the multi-item version of the seller-buyer assignment, and the international kidney exchange.\r\n\r\nCompared with the standard matching game, the b-matching game is computationally hard.\r\nIn particular, the core non-emptiness problem and the core membership problem are co-NP-hard.\r\nTherefore, we focus on the convexity of the game, which is a sufficient condition of the core non-emptiness and often more tractable concept than the core non-emptiness. \r\nIt also has several additional benefits. \r\n\r\nIn this study, we give a necessary and sufficient condition of the convexity of the b-matching game.\r\nThis condition also gives an O(n log n + m α(n)) time algorithm to determine whether a given game is convex or not, where n and m are the number of vertices and edges of a given graph, respectively, and α(・) is the inverse-Ackermann function.\r\nUsing our characterization, we also give a polynomial-time algorithm to compute the Shapley value of a convex b-matching game.","doi":"10.24963/ijcai.2020/37","url":"https://ijcai.org/proceedings/2020/37","pdf":"https://ijcai.org/proceedings/2020/0037.pdf","markdown":"https://ijcai.org/proceedings/2020/37.md","bibtex":"https://ijcai.org/proceedings/2020/bibtex/37","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":37,"pages":{"first":261,"last":267},"keywords":["Agent-based and Multi-agent Systems: Cooperative Games","Agent-based and Multi-agent Systems: Algorithmic Game Theory"],"videos":[{"label":"Video","url":"https://videos.ijcai20.org/24284/file_24284.m3u8","length_sec":735,"provider":"hls"}],"references":[{"position":1,"raw":"[Aziz and de Keijzer, 2014] Haris Aziz and Bart de Keijzer. 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