Envy-free division using mapping degree
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Envy-free division using mapping degree. / Avvakumov, Sergey; Karasev, Roman.
In: Mathematika, Vol. 67, No. 1, 2021, p. 36-53.Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
T1 - Envy-free division using mapping degree
AU - Avvakumov, Sergey
AU - Karasev, Roman
PY - 2021
Y1 - 2021
N2 - 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.
AB - 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.
KW - 51F99
KW - 52C35
KW - 55M20
KW - 55M35
UR - http://www.scopus.com/inward/record.url?scp=85099846085&partnerID=8YFLogxK
U2 - 10.1112/mtk.12059
DO - 10.1112/mtk.12059
M3 - Journal article
AN - SCOPUS:85099846085
VL - 67
SP - 36
EP - 53
JO - Mathematika
JF - Mathematika
SN - 0025-5793
IS - 1
ER -
ID: 256721704