ENVY-FREE DIVISION USING MAPPING DEGREE

Sergey Avvakumov, Roman Karasev*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

11 Scopus citations

Abstract

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 languageEnglish
Pages (from-to)36-53
Number of pages18
JournalMathematika
Volume67
Issue number1
DOIs
StatePublished - Jan 2021
Externally publishedYes

Funding

FundersFunder number
Horizon 2020 Framework Programme716424

    Keywords

    • 51F99
    • 52C35
    • 55M20
    • 55M35

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