Cooperative envy-free division

Relying on configuration spaces and equivariant topology, we study a general “cooperative envy-free division problem” where the players have more freedom of expressing their preferences (compared to the classical setting of the Stromquist-Woodall-Gale theorem).
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 still exists under standard assumptions that the preferences are closed.