The core is a strong fairness notion in multiwinner voting and participatory budgeting (PB). It is known that the core can be empty if we consider cardinal utilities, but it is not known whether it is always satisfiable with approval-ballots. In this short note, I show that in approval-based PB the core can be empty for nearly all satisfaction functions that are based on the cost of a project. In particular, I show that the core can be empty for the cost satisfaction function, satisfaction functions based on diminishing marginal returns and the share. However, it remains open whether the core can be empty for the cardinality satisfaction function.

翻译：核心是多赢选举和参与式预算（PB）中的一个强公平性概念。已知在使用基数效用时，核心可能为空，但尚不确定在批准投票中是否总能满足核心条件。在这篇短文中，我证明在基于批准的参与式预算中，对于几乎所有基于项目成本的满意度函数，核心可能为空。特别地，我展示了对于成本满意度函数、基于边际效用递减的满意度函数以及份额函数，核心可能为空。然而，对于基数满意度函数，核心是否可能为空仍然是一个未解决问题。