Envy-free divisions, equivariant maps, and mapping degrees

We consider problems that are inspired by questions of economy and game theory. In mathematical terms, we want to partition the segment into $n$ segments and give the parts to the players so that every player is satisfied and does not envy any other player.
This problem has several mathematically stated variants, for some of them we have a positive answer, for some negative, some of them remain open. We also review two-dimensional and other versions of envy-free partition or equipartition problems and the related abstract results in equivariant topology.