The core in Participatory Budgeting can be empty

Abstract

The core is a strong fairness notion in multi-winner 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.