arXiv:2409.15005v3 Announce Type: replace-cross Abstract: In participatory budgeting (PB), voters decide through voting which subset of projects to fund within a given budget. Proportionality in the context of PB is crucial to ensure equal treatment of all groups of voters. However, pure proportional rules can sometimes lead to suboptimal outcomes. We introduce the Method of Equal Shares with Bounded Overspending (BOS Equal Shares), a robust variant of Equal Shares that balances proportionality and efficiency. BOS Equal Shares addresses inefficiencies implied by strict proportionality axioms, yet the rule still provides fairness guarantees, similar to the original Method of Equal Shares. Our extensive empirical analysis on real-world PB instances shows excellent performance of BOS Equal Shares across several metrics. In the course of the analysis, we also present and examine a fractional variant of the Method of Equal Shares which allows for partial funding of projects.