There is a strategy parents use when dividing food among 2 kids: First kid cuts the cake, 2nd kid chooses which piece. This enforces [anyway strongly incentivizes] dividing the food equally.
Generalizing this to 3 or N players:
Is there a winning strategy for any of the N players [winning meaning they get the biggest piece]?
Is there a way to set up the game to enforce/encourage equality?
You can change the rules so that any subset of the players can have any number of cuts of the cake, and any subset of the players can choose pieces for themselves or others any number of times, and I guess I would even allow switching of the pieces.
But lets make some simplifications like: each game is independent (so no long term teaming up of players). Also, the mass of each piece is time independent, e.g. there is no way for player 1 to cut a piece that is small and then it changes in time such that when it's his turn the piece is big.