Participatory Budgeting Voting Rules

• Participatory budgeting voting rules are formal mechanisms that aggregate voter preferences into public project selections under strict budget constraints.
• They utilize diverse ballot structures and aggregation methods to balance utilitarian welfare with proportional, individual-minimum fairness guarantees.
• Recent advances focus on algorithmic efficiency, handling substitute projects, and ensuring extended justified and group representation.

Participatory budgeting (PB) voting rules are formal mechanisms for aggregating voters’ preferences over public-project selections subject to a hard budget constraint. The literature provides a spectrum of aggregation rules, axiomatic frameworks, input formats, and fairness guarantees. Distinct voting rules operationalize different trade-offs between utilitarian welfare, proportional or justified group representation, individual-minimum fairness, and computational tractability, with particular attention to both additive and more complex voter utility structures. Key desiderata and developments involve extended and full justified representation (EJR/FJR), substitute project handling, hybrid algorithmic approaches, epistemic interpretability, and practical stability under real-world constraints.

1. Formal Models, Ballot Structures, and Aggregation Frameworks

A generic PB instance consists of $N = \{1, \dots, n\}$ voters; $P = \{p_1, \dots, p_m\}$ projects with cost function $c: P \to \mathbb{R}_+$; and public budget $B \in \mathbb{R}_+$. Each voter $i$ provides a ballot encoding (approval $A_i \subseteq P$; or utility $u_i: P \to [0,1]$ or $[0, \infty)$), possibly with richer structures such as marginal utilities for dependencies or substitute groups (Fairstein et al., 2021). An outcome is any $X \subseteq P$ with $c(X) \le B$.

Voters' ballots may be:

• Approval (K-approval): Approving up to $k$ projects ($A_i \subseteq P$, $|A_i| \leq k$).
• Knapsack: Budget-feasible picks ($S_i \subseteq P$, $\sum_{p \in S_i} c(p) \leq B$).
• Rankings or Scored Points: Borda orderings or distributive point allocation.
• Marginal/substitutable utilities: Partitioning approved projects into substitute groups, with utility determined by diminishing marginal returns (Fairstein et al., 2021).
• Cumulative/weighted votes: Voters distribute fixed voting credits across projects (Pournaras et al., 20 May 2025).

Aggregation rules $R$ map $(N, P, c, B, \{A_i\})$ (plus any additional model details) to feasible $X \subseteq P$.

2. Key Participatory Budgeting Voting Rules

Rule/Family	Mechanism/Objective	Main Guarantees/Limitations
Utilitarian Greedy / AV	Maximize $\sum_i |A_i \cap X|$ (often greedy by support)	Max welfare, poor minority/proportionality (Faliszewski et al., 2023, Nelissen, 2023, Fairstein et al., 2024)
Chamberlin–Courant (CC)	Maximize number of covered voters	High representation, nonproportional if costs vary (Fairstein et al., 2022, Fairstein et al., 2024)
Proportional Approval Voting (PAV)	Maximize $\sum_{i} \sum_{k=1}^{|A_i \cap X|} 1/k$	EJR for unit costs; fails with variable costs (Peters et al., 2020, Fairstein et al., 2022)
Method of Equal Shares (MES, Rule X)	Iterated cost-sharing with per-voter budget $B/n$	Satisfies EJR (also for substitutes), polynomial-time (Peters et al., 2020, Fairstein et al., 2021, Faliszewski et al., 2023, Papasotiropoulos et al., 2024)
Substitute Rule X (SRX)	MES extended to respect substitute groups	EJR for substitutes, higher welfare in substitute-rich cases (Fairstein et al., 2021)
Maxmin PB (MPB)	Maximize $\min_i u_i(X)$	Individual fairness ($\max \min$), NP-hard (Sreedurga et al., 2022, Sreedurga, 2024)
BOS Equal Shares (BOS)	MES + controlled overspending for full exhaust	Approximate EJR, nearly exhausts budget, low exclusion (Papasotiropoulos et al., 2024)
Sequential CC/Monroe for PB	Unit-cost generalizations of MWV CC/Monroe to PB	Satisfy strong representation for unit costs; degrades with heterogeneous costs (Page et al., 2024)
Phragmén's Sequential	Budget-accumulation time-sharing	PB-PJR (proportional justified representation) up to one, priceability (Los et al., 2022)
Data-driven / Neural	Learned set transformer aggregation (compromise AV/CC/PAV)	Flexibly optimize arbitrary objectives, generalize to large real data (Fairstein et al., 2024)

Rules vary in computational complexity: AV/CC/PAV are NP-hard in general (though greedy AV is poly-time); MES and Rule X admit strongly polynomial implementations. Rules with fixed-parameter tractability exist for certain maxmin and dependency cases.

3. Proportionality Axioms, Justified Representation, and Core Notions

Modern axiomatic frameworks distinguish several levels of proportional (group) fairness:

• Extended Justified Representation (EJR): For all $T \subseteq P$ and all $(\alpha, T)$-cohesive groups $S \subseteq N$ (large enough and with sufficient support per project in $T$), at least one $i \in S$ gets utility at least $\sum_{c \in T} \alpha(c)$ (Peters et al., 2020, Los et al., 2022).
• EJR up to one project: Same, but utility threshold can be met by adding one additional project.
• Full Justified Representation (FJR): Every weakly $(\beta, T)$-cohesive $S$ contains $i_*$ with $u_{i_*}(X) \ge \beta$ (Peters et al., 2020).
• PJR and PB-PJR: Cohesive groups see proportional utility (see section 2 of (Los et al., 2022)).
• Laminar Proportionality (LP): For instance classes with hierarchical (laminar) structure.
• Maximal Coverage: Newly introduced, requires no redundant project at the cost of leaving any voter uncovered (Sreedurga et al., 2022, Sreedurga, 2024).
• Strong/B-JR, U-JR: Strengthened group-proportionality for approval ballots/unit or general costs (Page et al., 2024).
• District Fairness: Outcomes guarantee each district at least their stand-alone welfare (Hershkowitz et al., 2021).

MES/Rule X (and their substitute/project-dependent generalizations) achieve EJR and other strong group guarantees, including for cohesive minoritarian subgroups and in the presence of dependencies (Fairstein et al., 2021). Greedy/Utilitarian AV maximizes welfare but fails all above forms except possibly PJR for unit costs. PAV satisfies EJR only in unit-cost (multiwinner) settings; fails EJR generically for PB (Peters et al., 2020, Fairstein et al., 2022, Los et al., 2022).

4. Algorithmic and Strategic Considerations

• Computational Tractability: MES, Rule X, and SRX can be implemented in $O(m n \log n)$ or $O(m n^2)$ per iteration; BOS-Equal-Shares in $O(m n \min(m, b/\min c_j))$ (Papasotiropoulos et al., 2024).
• Non-Exhaustiveness and Completions: MES/Rule X may finish before exhausting the budget; completions (Add1U, utilitarian fill-in, $\varepsilon$-completion) increase budget usage with minor proportionality trade-offs (Faliszewski et al., 2023, Nelissen, 2023, Papasotiropoulos et al., 2024).
• Strategy and Manipulability: End-to-end models show that no shortlisting+allocation pair is immune to manipulation under mild conditions (Rey et al., 2020).
• Epistemic Perspective: MES, SeqPhragmén, and greedy approval rules do not coincide with maximum-likelihood estimators under any noise model—relative-Nash and certain utilitarian rules do, but fail exhaustiveness (Rey et al., 2023).
• Approximation to the Core: MES approximates the core within logarithmic factors in approval settings (Peters et al., 2020). The core itself (and FJR) is computationally intractable in general.

5. Trade-off Analyses and Empirical Evaluations

Extensive empirical analyses demonstrate that proportional rules (MES, Rule X, BOS) consistently yield more inclusive, representative, and equitable outcomes—at marginal welfare cost—relative to utilitarian or greedy rules. Representative findings include:

• MES/Add1U achieves up to 43% higher score-utility and half the exclusion/power-inequality versus utilitarian greedy; top-up completions preserve exhaustiveness and inclusion (Faliszewski et al., 2023, Pournaras et al., 20 May 2025, Nelissen, 2023).
• SRX (MES with substitutability) increases welfare 10–20% in rich-dependency inputs versus vanilla MES/Rule X, without losing proportionality (Fairstein et al., 2021).
• Sequential CC/Monroe for PB achieve PR with near-100% probability for unit costs, but drop sharply as cost heterogeneity rises (Page et al., 2024).
• Maxmin PB (MPB) achieves individual-worst-off fairness, covering nearly all voters, but at sum-welfare loss; computationally tractable for small diversity of ballots or cost granularity (Sreedurga et al., 2022, Sreedurga, 2024).
• District-fair PB can be achieved by randomization or small budget overspend, efficiently; deterministic strict fairness is NP-hard (Hershkowitz et al., 2021).

Empirical studies on real-world PB elections (e.g., Amsterdam, Warsaw, Aarau) confirm that MES and variants outperform greedy or cumulative/knapsack rules on proportional inclusion metrics (budget share, geographic spread, preference representation, Gini index), with minimal loss (often <10%) in aggregate voter satisfaction or mean utility (Faliszewski et al., 2023, Nelissen, 2023, Pournaras et al., 20 May 2025).

6. Input Format, Usability, and Practical Design

PB system performance and representation can depend acutely on the input format:

• k-Approval is fast, expressive, and induces robust proportional outcomes when coupled with MES/Rule X (Fairstein et al., 2023).
• Knapsack ballots enforce true budget trade-offs, lowering the average cost of selected bundles versus approval; K-ranking can serve as a faithful paper proxy for knapsack inputs (Gelauff et al., 2024).
• Expressive ballots (point distributions, rankings) enhance user satisfaction and perceived legitimacy, without overcomplicating MES outputs (Yang et al., 2023).
• Greedy rules are highly sensitive to input format and can disenfranchise minority coalitions with small tweaks in participation or ballot type (Fairstein et al., 2023, Yang et al., 2023).
• Stability of outcomes is highest for MES/Rule X, even under low turnout or noisy (inconsistent) voter input (Fairstein et al., 2023).

7. Open Directions and System Design Recommendations

• Rule selection: For general approval PB with heterogenous costs and an emphasis on proportionality, Method of Equal Shares (MES or Rule X) with Add1U or BOS completion offers strong group guarantees, high efficiency, and empirical robustness (Papasotiropoulos et al., 2024, Faliszewski et al., 2023, Pournaras et al., 20 May 2025).
• Dependencies and interactions: Substitutability-aware MES (SRX) is preferred where project dependencies are relevant (Fairstein et al., 2021); further research is needed for fully general interaction graphs and FPT variants.
• Egalitarian aims: Use maxmin (MPB) where individual-worst-off utility is the priority; expect increased computational cost and trade-off in total welfare (Sreedurga et al., 2022, Sreedurga, 2024).
• Combining objectives: Multi-objective and neural-learned rules enable tailored intermediate points between welfare and representation, supporting evolving community or institutional goals (Fairstein et al., 2024).
• Transparency and legitimacy: MES and proportional rules, despite greater conceptual complexity than greedy or cumulative voting, are perceived as fair and legitimate, especially when accompanied by appropriate explanations and visualization (Pournaras et al., 20 May 2025, Yang et al., 2023).
• District and category alignment: Splitting elections by districts or categories is unnecessary with proportional rules (MES, BOS), and running unified citywide elections increases welfare and inclusiveness (Faliszewski et al., 2023).

Participatory budgeting voting rules thus comprise a multifaceted landscape, rigorously characterized by formal representation axioms, computational considerations, and practical experience in real settings. The state-of-the-art combines rule design (MES, BOS, SRX), axiomatic guarantees (EJR, PB-PJR, FJR, core, maximal coverage), and practical implementation strategies to balance efficiency, proportionality, and participation in large-scale democratic resource allocation.