Envy-free cake division without assuming the players prefer nonempty pieces

TL;DR

This paper proves that envy-free connected cake divisions without the nonempty piece condition are possible for prime numbers and four, extending classical results and introducing a new topological combinatorial lemma.

Contribution

It extends envy-free cake division results to cases without the nonempty piece assumption for prime and four players, using a novel combinatorial topological lemma.

Findings

Envy-free division possible for prime and four players without nonempty condition

Introduces a new combinatorial lemma related to Sperner's lemma

Extends classical cake-cutting theorems to broader settings

Abstract

Consider $n$ players having preferences over the connected pieces of a cake, identified with the interval $[0,1]$. A classical theorem, found independently by Stromquist and by Woodall in 1980, ensures that, under mild conditions, it is possible to divide the cake into $n$ connected pieces and assign these pieces to the players in an envy-free manner, i.e, such that no player strictly prefers a piece that has not been assigned to her. One of these conditions, considered as crucial, is that no player is happy with an empty piece. We prove that, even if this condition is not satisfied, it is still possible to get such a division when $n$ is a prime number or is equal to $4$. When $n$ is at most $3$, this has been previously proved by Erel Segal-Halevi, who conjectured that the result holds for any $n$. The main step in our proof is a new combinatorial lemma in topology, close to a conjecture by Segal-Halevi and which is reminiscent of the celebrated Sperner lemma: instead of restricting the labels that can appear on each face of the simplex, the lemma considers labelings that enjoy a certain symmetry on the boundary.