Participatory Budgeting Performance Metrics

• Participatory budgeting is a democratic process where citizens vote on public projects, and these metrics quantify how close a losing project is to success.
• The proposed measures—cost-reduction, approval-augmentation, and approval-reallocation—provide actionable insights into strategic interventions and the sensitivity of different aggregation rules.
• Empirical evaluations on real-world PB elections validate these metrics by demonstrating their computational feasibility and importance for transparency and advocacy.

Participatory Budgeting (PB) allows citizens to allocate communal resources by voting on public projects, typically within a constrained financial budget. While most analysis focuses on which projects are funded, understanding the “closeness” of losing projects to success is crucial for transparency, strategic advocacy, and iterative policy design. Recent research introduces a precise formalism for evaluating the performance of losing projects by quantifying the minimal interventions needed for victory: cost reduction, approval augmentation, and approval reallocation. These metrics are tailored for three prominent PB aggregation rules—greedy Approval Voting (greedyAV), Phragmén’s sequential rule, and Equal-Shares PB (Eq/Ph)—and are empirically evaluated on real-world datasets to characterize their computation, sensitivity, and practical implications (Boehmer et al., 2023).

1. Formal Project-Performance Measures in PB

Three orthogonal measures quantify a losing project’s distance from success, each grounded in a natural “what-if” scenario:

1. Cost-Reduction ($\Delta_{\cost}(p)$): The minimal decrease in project cost required for victory. Formally,

$\Delta_{\cost}(p) = \cost(p) - c^*_p,$

where $c^*_p$ is the largest cost at which $p$ would enter the winning set under the rule in question.

2. Approval-Augmentation ($\Delta_{\app}(p)$): The minimal number of additional voter approvals needed for $p$ to win, optimistically selecting the best-positioned non-supporters:

$\Delta_{\app}(p) = \min\{\ell : \exists S\subseteq V^-(p), |S|=\ell,\, p \in f(P,V',B)\},$

where $V'(p)$ denotes non-supporters whose approvals are augmented.

3. Approval-Reallocation ($\Delta_{\realloc}(p)$): The minimal number of existing supporters who, if they withdrew approvals from rival projects (supporting only $p$), would cause $p$ to win. Formally,

$\Delta_{\realloc}(p) = \min\{\ell : \exists T\subseteq A(p), |T|=\ell,\, p \in f(P,V'',B)\},$

where supporters in $T$ submit singleton ballots.

These metrics are defined for a given PB rule $f$, election instance $E=(P,V,B)$, and project costs.

2. Algorithmic Computation and Complexity

The computability of the three measures depends both on the PB aggregation rule and the variant (optimistic, pessimistic, randomized).

• Cost-Reduction ($\Delta_{\cost}(p)$): For all considered rules—greedyAV, Phragmén, and Eq/Ph—$\Delta_{\cost}(p)$ can be computed in polynomial time. The key is to simulate the rule’s selection process, identify rounds where $p$ could be funded, and track the maximal feasible cost.
• Approval-Augmentation ($\Delta_{\app}(p)$):
  • Optimistic Variant: Polynomial time for all rules; for greedyAV, rank projects by approval and advance $p$ accordingly. For proportional rules, track marginal supporter contributions per round.
  • Pessimistic Variant: For Phragmén and (likely) Equal-Shares, deciding success is coNP-hard as it requires all possible combinations of $\ell$ non-supporters to be checked.
  • Randomized Variant: Uses Monte Carlo sampling to estimate, for a given $\ell$, the probability of victory by randomly augmenting non-supporter approvals.
• Approval-Reallocation ($\Delta_{\realloc}(p)$): Even for greedyAV, computing whether a given $\ell$ suffices is NP-complete. Practically, the randomized “50% victory probability” variant is approached via Monte Carlo simulation.
• Add-Singletons ($\Delta_{\text{single}}(p)$): Measures the effect of augmenting only with new singleton voters. This is polynomial-time for greedyAV and Phragmén, but can become non-monotonic and computationally harder for Eq.

The following table summarizes computational tractability:

Measure	greedyAV	Phragmén / Eq	Randomized
$\Delta_{\cost}(p)$	Polytime	Polytime	-
$\Delta_{\app}^{\text{opt}}(p)$	Polytime	Polytime	-
$\Delta_{\app}^{\text{pess}}(p)$	tractable	coNP-hard	-
$\Delta_{\realloc}(p)$	NP-complete	NP-complete	Monte Carlo

3. Empirical Characterization and Real-World Evaluation

Experiments conducted on 551 real PB elections (Pabulib dataset) involving up to 200 projects and 25,000 voters per election establish robust empirical relationships:

• Correlations Among Measures: The “add-approval” (optimist, randomized, singleton) metrics exhibit extremely high correlation ($\mathrm{PCC} > 0.95$), indicating close practical agreement. Pessimist and cost-reduction measures are less strongly correlated with approval-based metrics.
• Bootstrap Funding Probability Curves: The randomized add-approval curve for a project displays a sharp transition from low to high probability of funding, with the 50% threshold closely matching the computed minimal intervention, validating it as a salient summary statistic.
• Case Analyses: Detailed studies reveal projects that are “borderline” by different metrics, identifying whether cost reduction, additional outreach, or better mobilizing existing supporters is the most effective intervention.
• Combined Sensitivity: In some cases, marginal simultaneous improvements in cost and approval suffice for funding, while other projects display nonlinearity and require substantial change in one dimension due to structural rule effects.

4. Interpretative Insights and Decision-Maker Guidance

The multi-metric framework yields actionable guidance:

• Transparency: Explicitly reporting all three measures for each losing project makes PB outcomes auditable and constructive. Borderline projects identified by small-valued measures warrant special attention.
• Strategic Advocacy: The approval-augmentation measure—especially the randomized 50% variant—naturally translates to “number of voters to persuade,” while cost-reduction directly quantifies financial leeway for negotiation or project redesign.
• Understanding Rule Sensitivity: Under greedyAV, additional approvals efficiently shift outcomes, while proportionality in Phragmén and Eq/Ph rules makes both the distribution of approvals and rivalry among supporters critical factors. The rivalry-based measure captures when a project loses due to divided supporter funds—a typical phenomenon in proportional settings.
• Tailored Recommendations: Depending on which measure is smallest for a given project, the recommended intervention could be cost engineering (if $\Delta_{\cost}$ is low), targeted voter outreach ($\Delta_{\app}$), or coalition-splitting and rivalry management ($\Delta_{\realloc}$).

5. Correlations, Sensitivities, and Practical Selection of Metrics

Empirical results provide the following correlation highlights for Eq/Ph rules:

Measure Pair	Pearson Corr. Coefficient
Optimist–Pessimist	0.87
Optimist–Randomized	0.98
Optimist–Cost	0.76
Rivalry–Cost	0.63

This suggests that, while approval-based measures are largely redundant for practical purposes, rivalry and cost-reduction reveal complementary aspects. Pessimistic variants are more conservative and harder to compute but may be warranted in adversarial scenarios.

6. Limitations and Practical Recommendations

• Computational Feasibility: For large-scale elections, use of the randomized 50% variant and direct cost-reduction calculation are practical; pessimistic and rivalry metrics may require sampling or specialized FPT algorithms.
• Reporting Practice: Routine publication of $\Delta_{\app}^{50\%}(p)$, $\Delta_{\cost}(p)$, and (where feasible) $\Delta_{\realloc}(p)$ can help ensure equitable and comprehensible PB processes.
• Sensitivity to Rule Mechanics: The chosen PB rule determines which “distance” is operationally meaningful. Proportional rules necessitate attention to how supporter funds are allocated and may make approval-augmentation less straightforward.

7. Conclusion

Formal project-performance measures for losing PB proposals—cost-reduction, approval-augmentation, and approval-reallocation—provide a rigorous, practical, and interpretable framework for “thinking budget evaluation” in participatory budgeting. They underpin efficient, transparent, and data-driven strategies for project advocacy and policy feedback. Experimental results demonstrate their empirical validity, computational tractability (for core variants), and informative power in real-world PB implementations (Boehmer et al., 2023).