{"type":"ScholarlyArticle","title":"Comparing Approximate Relaxations of Envy-Freeness","authors":[{"name":"Georgios Amanatidis","slug":"georgios-amanatidis","given":"Georgios","family":"Amanatidis","affiliation":"Centrum Wiskunde & Informatica (CWI) , Amsterdam , the Netherlands"},{"name":"Georgios Birmpas","slug":"georgios-birmpas","given":"Georgios","family":"Birmpas","affiliation":"Department of Informatics , Athens University of Economics and Business , Athens , Greece"},{"name":"Vangelis Markakis","slug":"vangelis-markakis","given":"Vangelis","family":"Markakis","affiliation":"Department of Informatics , Athens University of Economics and Business , Athens , Greece"}],"abstract":"In fair division problems with indivisible goods it is well known that one cannot have any guarantees for the classic fairness notions of envy-freeness and proportionality. As a result, several relaxations have been introduced, most of which in quite recent works. We focus on four such notions, namely envy-freeness up to one good (EF1), envy-freeness up to any good (EFX), maximin share fairness (MMS), and pairwise maximin share fairness (PMMS). Since obtaining these relaxations also turns out to be problematic in several scenarios, approximate versions of them have also been considered. In this work, we investigate further the connections  between the four notions mentioned above and their approximate versions. We establish several tight or almost tight results concerning the approximation quality that any of these notions guarantees for the others, providing an almost complete picture of this landscape. Some of our findings reveal interesting and surprising consequences regarding the power of these notions, e.g., PMMS and EFX provide the same worst-case guarantee for MMS, despite PMMS being a strictly stronger notion than EFX. We believe such implications provide further insight on the quality of approximately fair solutions.","doi":"10.24963/ijcai.2018/6","url":"https://ijcai.org/proceedings/2018/6","pdf":"https://ijcai.org/proceedings/2018/0006.pdf","markdown":"https://ijcai.org/proceedings/2018/6.md","bibtex":"https://ijcai.org/proceedings/2018/bibtex/6","volume":{"slug":"2018","title":"Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2018","url":"https://ijcai.org/proceedings/2018/","year":2018,"location":"Stockholm","dates":"13-19 July 2018","editors":"Jérôme Lang","isbn":"978-0-9992411-2-7","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2018"},"section":"Agent-based and Multi-agent Systems","track":"Main track","number":6,"pages":{"first":42,"last":48},"keywords":["Agent-based and Multi-agent Systems: Algorithmic Game Theory","Agent-based and Multi-agent Systems: Computational Social Choice"],"videos":[],"references":[{"position":1,"raw":"[Amanatidis et al., 2017a] G. 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