{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:34:26Z","timestamp":1787322866771,"version":"build-2736575974"},"reference-count":26,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"name":"Stanford Graduate Fellowship"},{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"publisher","award":["639945"],"award-info":[{"award-number":["639945"]}],"id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF-1655215"],"award-info":[{"award-number":["CCF-1655215"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF-1813188"],"award-info":[{"award-number":["CCF-1813188"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF-1815434"],"award-info":[{"award-number":["CCF-1815434"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2020,1]]},"abstract":"<jats:p>We consider a fair division setting in which $m$ indivisible items are to be allocated among $n$ agents, where the agents have additive utilities and the agents' utilities for individual items are independently sampled from a distribution. Previous work has shown that an envy-free allocation is likely to exist when $m=\\Omega(n\\log n)$ but not when $m=n+o(n)$, and left open the question of determining where the phase transition from non-existence to existence occurs. We show that, surprisingly, there is in fact no universal point of transition---instead, the transition is governed by the divisibility relation between $m$ and $n$. On the one hand, if $m$ is divisible by $n$, an envy-free allocation exists with high probability as long as $m\\geq 2n$. On the other hand, if $m$ is not \u201calmost\u201d divisible by $n$, an envy-free allocation is unlikely to exist even when $m=\\Theta(n\\log n\/\\log\\log n)$.<\/jats:p>","DOI":"10.1137\/19m1279125","type":"journal-article","created":{"date-parts":[[2020,7,2]],"date-time":"2020-07-02T15:41:31Z","timestamp":1593704491000},"page":"1505-1521","source":"Crossref","is-referenced-by-count":14,"title":["When Do Envy-Free Allocations Exist?"],"prefix":"10.1137","volume":"34","author":[{"given":"Pasin","family":"Manurangsi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Warut","family":"Suksompong","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2020,7,2]]},"reference":[{"key":"atypb1","doi-asserted-by":"crossref","unstructured":"G. Amanatidis, G. Birmpas, and E. 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Taylor,\n                      Fair Division: From Cake-Cutting to Dispute Resolution\n                      , Cambridge University Press, Cambridge, 1996.","DOI":"10.1017\/CBO9780511598975"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1145\/3355902"},{"key":"atypb7","first-page":"629","author":"Conitzer V.","year":"2017","journal-title":"New York"},{"key":"atypb8","doi-asserted-by":"crossref","unstructured":"J. P. Dickerson, J. Goldman, J. Karp, A. D. Procaccia, and T. 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Moulin,\n                      Fair Division and Collective Welfare\n                      , MIT Press, Cambridge, MA, 2003.","DOI":"10.7551\/mitpress\/2954.001.0001"},{"key":"atypb18","first-page":"2584","author":"Plaut B.","year":"2018","journal-title":"Philadelphia"},{"key":"atypb19","unstructured":"E. Segal-Halevi and S. Nitzan,\n                      Envy-Free Cake-Cutting among Families\n                      , preprint, arXiv:abs\/1607.01517, 2016."},{"key":"atypb20","doi-asserted-by":"crossref","unstructured":"E. Segal-Halevi and W. Suksompong,\n                      Democratic fair allocation of indivisible goods\n                      , in Proceedings of the 27th International Joint Conference on Artificial Intelligence, ijcai.org, 2018, pp. 482-488.","DOI":"10.24963\/ijcai.2018\/67"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1080\/00029890.1980.11995109"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20235"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1016\/j.mathsocsci.2016.03.007"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1016\/j.mathsocsci.2017.09.004"},{"key":"atypb25","first-page":"261","author":"Thomson W.","year":"2016","journal-title":"Cambridge"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0531(74)90075-1"}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/19M1279125","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:30:16Z","timestamp":1787319016000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/19M1279125"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1]]},"references-count":26,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,1]]}},"alternative-id":["10.1137\/19M1279125"],"URL":"https:\/\/doi.org\/10.1137\/19m1279125","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1]]}}}