The Core of Approval Participatory Budgeting with Uniform Costs (or with up to Four Projects) is Non-Empty

TL;DR

This paper proves that in approval participatory budgeting with uniform costs or up to four projects, the core is always non-empty, resolving an open problem in multiwinner voting scenarios.

Contribution

It establishes the non-emptiness of the core in cases with uniform costs or at most four projects, advancing understanding of core stability in participatory budgeting.

Findings

Core is non-empty with uniform costs.

Core is non-empty with up to four projects.

Addresses open problem in multiwinner voting core.

Abstract

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.