@article{de8c3b63041d4b78904ccd0f27b97816,
title = "ENVY-FREE DIVISION USING MAPPING DEGREE",
abstract = "In this paper we study envy-free division problems. The classical approach to such problems, used by David Gale, reduces to considering continuous maps of a simplex to itself and finding sufficient conditions for this map to hit the center of the simplex. The mere continuity of the map is not sufficient for reaching such a conclusion. Classically, one makes additional assumptions on the behavior of the map on the boundary of the simplex (e.g., in the Knaster–Kuratowski–Mazurkiewicz and the Gale theorem). We follow Erel Segal-Halevi, Fr{\'e}d{\'e}ric Meunier, and Shira Zerbib, and replace the boundary condition by another assumption, which has the meaning in economy as the possibility for a player to prefer an empty part in the segment partition 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 division problems when n is odd and not a prime power.",
keywords = "51F99, 52C35, 55M20, 55M35",
author = "Sergey Avvakumov and Roman Karasev",
note = "Publisher Copyright: {\textcopyright} 2020 The Authors. The publishing rights for this article are licensed to University College London under an exclusive licence.",
year = "2021",
month = jan,
doi = "10.1112/mtk.12059",
language = "אנגלית",
volume = "67",
pages = "36--53",
journal = "Mathematika",
issn = "0025-5793",
publisher = "John Wiley & Sons Inc.",
number = "1",
}