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.

翻译：我们研究公平分配现象的存在,当我们有更多的角色可以分配的时候,我们研究的是公平分配问题,重点是那些接受一票的人之间无忌妒的分布。 玩家的需求条件可能从斯特罗姆奎斯特、伍德尔和苏经典的蛋糕切割和分租结果中被削弱。 我们把蛋糕分割问题的现有变式扩大到隐秘的玩家和那些抵制清除足够小的玩家的玩家。