{"type":"ScholarlyArticle","title":"On Lower Bounds for Maximin Share Guarantees","authors":[{"name":"Halvard Hummel","slug":"halvard-hummel","given":"Halvard","family":"Hummel","affiliation":"Norwegian University of Science and Technology"}],"abstract":"We study the problem of fairly allocating a set of indivisible items to a set of agents with additive valuations. Recently, Feige et al. (WINE'21) proved that a maximin share (MMS) allocation exists for all instances with n agents and no more than n + 5 items. Moreover, they proved that an MMS allocation is not guaranteed to exist for instances with 3 agents and at least 9 items, or n ≥ 4 agents and at least 3n + 3 items. In this work, we shrink the gap between these upper and lower bounds for guaranteed existence of MMS allocations. We prove that for any integer c > 0, there exists a number of agents n_c such that an MMS allocation exists for any instance with n ≥ n_c agents and at most n + c items, where n_c ≤ ⌊0.6597^c · c!⌋ for allocation of goods and n_c ≤ ⌊0.7838^c · c!⌋ for chores. Furthermore, we show that for n ≠ 3 agents, all instances with n + 6 goods have an MMS allocation.","doi":"10.24963/ijcai.2023/306","url":"https://ijcai.org/proceedings/2023/306","pdf":"https://ijcai.org/proceedings/2023/0306.pdf","markdown":"https://ijcai.org/proceedings/2023/306.md","bibtex":"https://ijcai.org/proceedings/2023/bibtex/306","volume":{"slug":"2023","title":"Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2023","url":"https://ijcai.org/proceedings/2023/","year":2023,"location":"Macao, SAR","dates":"19-25 August 2023","editors":"Edith Elkind","isbn":"978-1-956792-03-4","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2023"},"section":"Game Theory and Economic Paradigms","track":"Main Track","number":306,"pages":{"first":2747,"last":2755},"keywords":["Game Theory and Economic Paradigms: GTEP: Fair division"],"videos":[],"references":[{"position":1,"raw":"[Aigner-Horev and Segal-Halevi, 2022] Elad Aigner-Horev and Erel Segal-Halevi. 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