We initiate the study of the proportionality degree for participatory budgeting, with a particular focus on two popular methods: the Method of Equal Shares (MES) and Phragmen's Sequential Rule. Among other results, we derive tight bounds (up to small constant factors) on the proportionality degree of these two rules, which showcase that, despite MES satisfying stronger axiomatic guarantees, the two rules have the same proportionality degree from a quantitative perspective. We complement our theoretical findings with an extensive experimental evaluation on real-world participatory budgeting datasets, the results of which closely mirror those of our developed theory. Our experiments also provide more insights into the comparisons between the rules.

翻译：我们首次对参与式预算中的比例度展开研究，重点分析了两种主流方法：等额份额法（MES）与法格曼序贯规则。研究结果中，我们推导出这两种规则比例度的严格界（至多相差小常数因子），表明尽管等额份额法满足更强的公理化保证，但从量化视角看两种规则具有相同的比例度。我们通过在实际参与式预算数据集上的大量实验评估来补充理论发现，实验结果与所发展理论高度吻合。实验还进一步揭示了两种规则间的对比特性。