# Envy-free division using mapping degree

## TL;DR

This paper explores envy-free division problems using a topological approach based on mapping degree, providing positive results for prime power numbers of players and counterexamples otherwise, advancing understanding of fair division.

## Contribution

It introduces a novel boundary condition assumption in envy-free division, proving positive results for prime power cases and counterexamples for non-prime power cases.

## Key findings

- Positive solutions for prime power number of players
- Counterexamples for non-prime power cases
- Extension of topological methods to envy-free division

## 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 (for example, in the Knaster--Kuratowski--Mazurkiewicz and the Gale theorem). We follow Erel Segal-Halevi, Fr\'ed\'eric 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. In this arxiv version that appears after the official publication we have corrected the statement and the proof of Lemma 3.4.