Streamlining Equal Shares

TL;DR

This work addresses underspending in the Method of Equal Shares (MES) for participatory budgeting by introducing Exact Equal Shares (EES) and a principled, efficient completion method called add-opt. It proves EES preserves strong proportionality guarantees (EJR1) under uniform utilities and provides fast implementations: $O(mn)$ for cardinal utilities and $O(m^2n)$ for uniform utilities, with add-opt enabling a near-exhaustive exploration of budget-induced outcomes via a linear-time subroutine GreedyProjectChange. The authors then extend to uniform utilities with add-opt, and popularize the add-opt-skip variant, which focuses on not-yet-funded projects to achieve high spending efficiency with far fewer calls to the base method. Empirical evaluation on 250 real PB datasets from Pabulib shows EES+add-opt-skip achieves spending efficiency comparable to MES+add-one while dramatically reducing computation and avoiding worst-case non-monotone underspending cases, supporting practical deployment in large-scale PB settings.

Abstract

Participatory budgeting (PB) is a form of citizen participation that allows citizens to decide how public funds are spent. Through an election, citizens express their preferences on various projects (spending proposals). A voting mechanism then determines which projects will be approved. The Method of Equal Shares (MES) is the state of the art algorithm for a proportional, voting based approach to participatory budgeting and has been implemented in cities across Poland and Switzerland. A significant drawback of MES is that it is not \textit{exhaustive} meaning that it often leaves a portion of the budget unspent that could be used to fund additional projects. To address this, in practice the algorithm is combined with a completion heuristic - most often the ``add-one" heuristic which artificially increases the budget until a heuristically chosen threshold. This heuristic is computationally inefficient and will become computationally impractical if PB is employed on a larger scale. We propose the more efficient \textsc{add-opt} heuristic for Exact Equal Shares (EES), a variation of MES that is known to retain many of its desirable properties. We solve the problem of identifying the next budget for which the outcome for EES changes in $O(mn)$ time for cardinal utilities and $O(m^2n)$ time for uniform utilities, where $m$ is the number of projects and $n$ is the number of voters. Our solution to this problem inspires the efficient \textsc{add-opt} heuristic which bypasses the need to search through each intermediary budget. We perform comprehensive experiments on real-word PB instances from Pabulib and show that completed EES outcomes usually match the proportion of budget spent by completed MES outcomes. Furthermore, the \textsc{add-opt} heuristic matches the proportion of budget spend by add-one for EES.

Figures (10)

• Figure 1: A real-world example where considering all budgets significantly increases spending efficiency, i.e., the fraction of the actual budget spent. The graph on the left shows the spending efficiency of the winning project set for a given virtual budget. The graph on the right shows which projects are selected for a given virtual budget. Here, the add-one heuristic stops as soon as the budget is exceeded, which happens when the Asphalt project is selected. This leads to less than 15% spending efficiency. Increasing the budget further leads to the Asphalt project being dropped in favor of the Gravel project, as depicted on the right, increasing the spending efficiency to 60%
• Figure 2: Spending efficiency of MES and EES with add-one heuristic. Each point in the scatter point represents a Pabulib data set.
• Figure 3: max is the result of running MES + add-one and EES + add-opt-skip and taking the result with the higher spending efficiency.
• Figure 4: Graph showing executions and spending efficiency for identified cases where the optimal virtual budget occurs after the point at which the true budget is first overspent. In these cases, EES with add-opt-skip achieves much higher spending efficiency than MES with add-one, increasing from an average of 78% to 86% for the 3 identified instances with cost utilities and from 78% to 90% for the 14 identified instances with cardinal utilities.
• Figure 5: Spending efficiency of EES with add-opt-skip vs. MES with add-one: cardinal utilities.

Theorems & Definitions (45)

• Definition 3.1: Extended Justified Representation
• Theorem 3.2
• Definition 4.1
• Proposition 4.1: $\spadesuit$
• Example 4.2
• Lemma 4.3
• proof
• Proposition 4.3
• proof
• Theorem 4.4