Envy-free division using mapping degree
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- ENVY-FREE DIVISION USING MAPPING DEGREE
Accepted author manuscript, 785 KB, PDF document
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, Frédéric 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.
Original language | English |
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Journal | Mathematika |
Volume | 67 |
Issue number | 1 |
Pages (from-to) | 36-53 |
ISSN | 0025-5793 |
DOIs | |
Publication status | Published - 2021 |
- 51F99, 52C35, 55M20, 55M35
Research areas
ID: 256721704