{"date_created":"2020-07-30T10:45:51Z","abstract":[{"text":"In this paper we study envy-free division problems. The classical approach to some of such problems, used by David Gale, reduces to considering continuous maps of a simplex to itself and finding sufficient conditions when this map hits the center of the simplex. The mere continuity is not sufficient for such a conclusion, the usual assumption (for example, in the Knaster--Kuratowski--Mazurkiewicz and the Gale theorem) is a certain boundary condition.\r\n  We follow Erel Segal-Halevi, Fr\\'ed\\'eric Meunier, and Shira Zerbib, and replace the boundary condition by another assumption, which has the economic meaning of possibility for a player to prefer an empty part in the segment\r\npartition problem. We solve the problem positively when $n$, the number of players that divide the segment, is a prime power, and we provide counterexamples for every $n$ which is not a prime power. We also provide counterexamples relevant to a wider class of fair or envy-free partition problems when $n$ is odd and not a prime power.","lang":"eng"}],"status":"public","publication":"arXiv","article_number":"1907.11183","date_updated":"2023-09-07T13:12:17Z","article_processing_charge":"No","date_published":"2019-07-25T00:00:00Z","year":"2019","day":"25","author":[{"last_name":"Avvakumov","id":"3827DAC8-F248-11E8-B48F-1D18A9856A87","full_name":"Avvakumov, Sergey","first_name":"Sergey"},{"first_name":"Roman","full_name":"Karasev, Roman","last_name":"Karasev"}],"language":[{"iso":"eng"}],"department":[{"_id":"UlWa"}],"main_file_link":[{"url":"https://arxiv.org/abs/1907.11183","open_access":"1"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","related_material":{"link":[{"url":"https://doi.org/10.1112/mtk.12059","relation":"later_version"}],"record":[{"id":"8156","status":"public","relation":"dissertation_contains"}]},"project":[{"_id":"26611F5C-B435-11E9-9278-68D0E5697425","name":"Algorithms for Embeddings and Homotopy Theory","call_identifier":"FWF","grant_number":"P31312"}],"oa":1,"publication_status":"submitted","external_id":{"arxiv":["1907.11183"]},"month":"07","title":"Envy-free division using mapping degree","_id":"8185","doi":"10.48550/arXiv.1907.11183","type":"preprint","citation":{"ama":"Avvakumov S, Karasev R. Envy-free division using mapping degree. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.1907.11183\">10.48550/arXiv.1907.11183</a>","ista":"Avvakumov S, Karasev R. Envy-free division using mapping degree. arXiv, 1907.11183.","ieee":"S. Avvakumov and R. Karasev, “Envy-free division using mapping degree,” <i>arXiv</i>. .","chicago":"Avvakumov, Sergey, and Roman Karasev. “Envy-Free Division Using Mapping Degree.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.1907.11183\">https://doi.org/10.48550/arXiv.1907.11183</a>.","mla":"Avvakumov, Sergey, and Roman Karasev. “Envy-Free Division Using Mapping Degree.” <i>ArXiv</i>, 1907.11183, doi:<a href=\"https://doi.org/10.48550/arXiv.1907.11183\">10.48550/arXiv.1907.11183</a>.","short":"S. Avvakumov, R. Karasev, ArXiv (n.d.).","apa":"Avvakumov, S., &#38; Karasev, R. (n.d.). Envy-free division using mapping degree. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.1907.11183\">https://doi.org/10.48550/arXiv.1907.11183</a>"},"oa_version":"Preprint"}