Avvakumov, SerhiiIST Austria; Karasev, Roman
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. 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 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 partition problems when $n$ is odd and not a prime power.
Avvakumov S, Karasev R. Envy-free division using mapping degree. arXiv.
Avvakumov, S., & Karasev, R. (n.d.). Envy-free division using mapping degree. arXiv.
Avvakumov, Sergey, and Roman Karasev. “Envy-Free Division Using Mapping Degree.” ArXiv, n.d.
S. Avvakumov and R. Karasev, “Envy-free division using mapping degree,” arXiv. .
Avvakumov S, Karasev R. Envy-free division using mapping degree. arXiv, 1907.11183.
Avvakumov, Sergey, and Roman Karasev. “Envy-Free Division Using Mapping Degree.” ArXiv, 1907.11183.
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