In the Approval Participatory Budgeting problem an agent prefers a set of projects $W'$ over $W$ if she approves strictly more projects in $W'$. A set of projects $W$ is in the core, if there is no other set of projects $W'$ and set of agents $K$ that both prefer $W'$ over $W$ and can fund $W'$. It is an open problem whether the core can be empty, even when project costs are uniform. the latter case is known as the multiwinner voting core. We show that in any instance with uniform costs or with at most four projects (and any number of agents), the core is nonempty.

翻译：在批准参与性预算编制问题中,代理商如果以W美元批准严格地说更多的项目,则更喜欢一套W$以上超过W美元的项目。 一套W美元的项目属于核心项目,如果没有其他一套W美元的项目和一套K美元代理商,两者都希望W美元超过W美元,而且能够为W美元提供资金。即使项目费用一致,核心是否是空的,这是一个未决问题。 后一种情况被称为多赢者投票核心。 我们表明,在任何情况下,无论费用统一还是最多四个项目(和任何数目的代理商),核心都不是空的。