  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.
