Proportionality Degree in Participatory Budgeting

Abstract

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.

Figures (6)

• Figure 1: An illustration of the construction of the ordering $R$ in the proof of \ref{['lemma: maximum-payment']}. The figure shows round $\pi_{\ell_1}$, i.e., the first round in which a project in $\mathcal{T}\xspace \setminus T$ is selected, and then a subsequent round $\pi_{\ell_k}$, in which some other project in $\mathcal{T}\xspace \setminus P$ is selected. $p(i)$ denotes the payment of voter $i$. By "$any$", it is meant that the ordering of the voters within the corresponding set is any arbitrary order.
• Figure 2: The figure shows groups of voters (dots) organized into sets $Q_j$, where $j \in [0,|V|-2]$. Each group approves a distinct set $Z_j$ of projects, indicated by a blue rectangle. The set $V$ contains one voter from each $Q_j$, and each of these voters additionally approves a common set $T$ of projects (indicated as a bold blue rectangle). For $j \in \{0,\ldots,|V|-2\}$, the size of set $Q_j$ is $|V|-j$.
• Figure 3: The graphs show the results for 5 different representative datasets out of the 100. Note that the suffix 'Exh' represents the rule with budget exhaustion. For each dataset, the $X$-axis represents the subset size, while the $Y$-axis represents the average proportionality degree of all the sampled subsets of that size.
• Figure 4: The X-axis corresponds to the dataset number. On the Y-axis, the left graph represents the average proportionality degree of mes-Exh and Phragmén-Exh with greedy approval as the basis (i.e., zero). The right graph represents the average degree of mes-Exh, Phragmén, and Phragmén-Exh with mes as the basis.
• Figure 5: Pie chart showing the percentage of datasets for which each rule is better than the others.

Theorems & Definitions (7)

• proof
• proof
• proof
• proof
• proof
• proof
• proof : Proof of \ref{['thm:conclusion']}