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 -