Cooperative envy-free division

TL;DR

This paper uses advanced topological methods to prove the existence of envy-free division solutions in a cooperative cake-cutting problem where players' choices depend on entire allocation configurations.

Contribution

It introduces a novel model where players' preferences depend on the whole configuration, extending envy-free division theory with topological tools.

Findings

Existence of envy-free division when players' preferences depend on entire allocations

Applicable when the number of players is a prime power

Preferences are assumed to be closed

Abstract

Relying on configuration spaces and equivariant topology, we study a general "cooperative envy-free division problem". A group of players want to cut a "cake" $I=[0,1]$ and divide among themselves the pieces in an envy-free manner. Once the cake is cut and served in plates on a round table (at most one piece per plate), each player makes her choice by pointing at one (or several) plates she prefers. The novelty is that her choice may depend on the whole "allocation configuration". In particular, a player may choose an empty plate (possibly preferring one of the empty plates over the other), and take into account not only the content of her preferred plate, but also the content of the neighbouring plates. We show that if the number of players is a prime power, in this setting an envy-free division exists under standard assumptions that the preferences are closed.