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Envy-free division using mapping degree Karasev, Roman


We discuss some classical problems of mathematical economics, in particular, so-called envy-free division problems. The classical approach to some of such problem 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 theorem and the Gale theorem) is a boundary condition. We try to replace the boundary condition by a certain equivariance condition under all permutations, or a weaker condition of ``pseudo-equivariance'', which has a certain real-life meaning for the problem of partitioning a segment and distributing the parts among the players. It turns out that we can guarantee the existence of a solution for the segment partition problem when the number of players is a prime power; and we may produce instances of the problem without a solution otherwise. The case of three players was solved previously be Segal-Halevi, the prime case and the case of four players were solved by Meunier and Zerbib. Going back to the true equivariance setting, we provide, in the case when the number of players is odd and not a prime power, the counterexamples showing that the topological configuration space / test map scheme for a wide class of equipartition problems fails. This is applicable, for example, to building stronger counterexamples for the topological Tverberg conjecture (in another joint work with Sergey Avvakumov and Arkadiy Skopenkov). Joint work with Sergey Avvakumov

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