Fair distributions for more participants than allocations.

We study the existence of fair distributions when we have more players than pieces to allocate, focusing on envy-free distributions among those who receive a piece. The conditions on the demand from the players can be weakened from those of classic cake-cutting and rent-splitting results of Stromquist, Woodall, and Su. We extend existing variations of the cake-splitting problem with secretive players and those that resist the removal of any sufficiently small set of players.