Approval-Based Multiwinner Voting Rules

• Approval-based multiwinner voting rules are methods that select fixed-sized committees using voter approval ballots to achieve fair representation, diversity, and individual excellence.
• They encompass several families, such as Approval Voting, Proportional Approval Voting, and Chamberlin–Courant, each emphasizing different trade-offs between proportionality and utilitarian efficiency.
• These rules highlight key challenges in balancing robustness, computational tractability, and strategic behavior, with applications in political elections, participatory budgeting, and collaborative filtering.

Approval-based multiwinner voting rules select committees of fixed size based on approval ballots, where each voter indicates the subset of candidates they approve. This area forms a core subfield within computational social choice, with key objectives including proportional representation, diversity, welfare, tractability, and robustness. Notably, distinct families of rules emphasize "individual excellence" (most-approved candidates), "^^^^1^^^^" (voter blocs receive fair share), or "diversity" (broadest voter coverage). Contemporary research establishes sharp tradeoffs between these goals and assesses rules both axiomatical and algorithmically.

1. Formal Definitions and Core Rule Classes

Given a set of candidates $C = \{c_1, \ldots, c_m\}$, voters $N = \{1, \ldots, n\}$, and approval ballots $A_i \subseteq C$ for each $i \in N$, a multiwinner rule selects a committee $W \subseteq C$ with $|W| = k$. Prominent rules include:

• Approval Voting (AV): Selects the $k$ candidates with the highest approval counts ($\sum_{i} 1_{c \in A_i}$) (Lackner et al., 2020, Aziz et al., 2014).
• Satisfaction Approval Voting (SAV): Maximizes $\sum_{i} \frac{|W \cap A_i|}{|A_i|}$, equalizing across voter ballot sizes (Aziz et al., 2015, Aziz et al., 2014).
• Proportional Approval Voting (PAV): For $H(t) = 1 + 1/2 + \cdots + 1/t$, maximizes $\sum_{i} H(|W \cap A_i|)$, which embodies diminishing returns and is a canonical "proportional" Thiele rule (Brill et al., 2016, Aziz et al., 2014).
• Chamberlin–Courant (CC): Maximizes $|\{i: W \cap A_i \neq \emptyset\}|$, i.e., the number of covered voters (Lackner et al., 2017).
• Sequential PAV: Greedy variant, sequentially adds candidates yielding maximal marginal PAV gain (Skowron, 2018).
• Phragmén’s Sequential Rule: Minimizes maximum "load" voters take in supporting the winners, viewed as a credit or budget accumulation process (Skowron, 2018).

Other variants, such as Constrained SAV (CSAV) and Modified SAV (MSAV), adjust the SAV normalization (by committee size or the minimum of ballot and committee size) and retain polynomial-time computability (Aziz et al., 2015).

2. Axiomatic Analysis: Proportionality, Diversity, and Consistency

Axiomatic approaches categorize rules along three orthogonal principles:

• Individual Excellence (AV): Maximizes summed approvals; uniquely characterized via disjoint equality (Lackner et al., 2017).
• Proportionality (PAV): Achieves D'Hondt proportionality (lower quota on party lists); uniquely characterized among counting rules via D’Hondt’s axiom (Brill et al., 2016, Lackner et al., 2017).
• Diversity (CC): Ensures each large group gets at least one representative (disjoint diversity); CC is the unique rule embodying this for counting rules (Lackner et al., 2017).

Consistency (committee choices are preserved upon merging disjoint electorates) characterizes ABC rules as counting rules (Lackner et al., 2017). Other important axioms include committee monotonicity (chosen committees expand with increased $k$), candidate monotonicity (increasing support for a winner never removes them), and Pareto efficiency.

3. Quantitative Proportionality: Degree Hierarchy and Tradeoffs

Quantitative frameworks define a proportionality degree: for each ℓ-large, g-cohesive group (size at least ℓ·n/k, with at least g common approved candidates), rules guarantee average group satisfaction of at least $g$:

• PAV attains the optimal degree: $d_{PAV}(\ell) = \ell - 1$, tight by impossibility for $g(\ell,k) = \ell$ (Skowron, 2018).
• Sequential PAV: Roughly 70% of PAV's proportionality (e.g., $d_{SeqPAV}(\ell,10) \gtrsim 0.78\ell$; for larger $k$ plateauing at ~0.7).
• Phragmén’s Sequential Rule: $d_{PhragSEQ}(\ell) = \ell/2$, so only half of the possible proportionality (Skowron, 2018).
• Other convex Thiele rules (e.g., λ(i)=1/√i): Intermediate trade-offs between proportionality and utilitarian efficiency, parameterizable by λ.
• Maximal-load Phragmén: Proportionality degree bounded by 1, failing for ℓ > 1.

There is a clear efficiency–proportionality trade-off: rules optimizing proportionality (PAV) sacrifice utilitarian welfare; those optimizing utilitarian welfare (AV) can represent minorities poorly (Skowron, 2018, Lackner et al., 2018).

4. Robustness, Monotonicity, and Strategic Behavior

Recent results address how committee outputs respond to ballot perturbations and strategic manipulation:

• Robustness: AV is maximally robust—at most one committee member changes per single approval flip. All proportional/diversity rules can have entire committees replaced by a single approval change ($k$-level robustness) (Faliszewski et al., 27 Jan 2026).
• Monotonicity: All counting rules (AV, SAV, PAV, CC) satisfy strong support-monotonicity with population increase; but strong monotonicity variants are generally incompatible with strong forms of proportionality (e.g., perfect representation) (Sánchez-Fernández et al., 2017). Committee monotonicity is satisfied by AV, SAV, SeqPAV, and sequential Phragmén, but not by PAV, CC, Monroe, nor many others.
• Strategyproofness: Only AV (top-k approvals) is (inclusion- and cardinality-) strategyproof. All proportional rules (PAV, Phragmén, CC) are vulnerable to both individual and group manipulation (Peters, 2021, Caragiannis et al., 2024). A sharp impossibility holds: even weakened forms of proportionality and strategyproofness cannot be satisfied concurrently if $k \geq 3$ (Peters, 2021, Caragiannis et al., 2024). Moreover, strongly group-strategyproof approval-based rules with unanimity do not exist for $k \in \{1, ..., m-2\}$ (Caragiannis et al., 2024).

Empirical work further reveals that experimental and axiomatic distinctions frequently collapse on real or synthetic data, with rules such as PAV and sequential PAV being behaviorally indistinguishable for most practical distributions (Faliszewski et al., 2024). Deliberation mechanisms (e.g., structured discussion before voting) can substantially increase welfare and proportionality under simple rules like AV (Mehra et al., 2023).

5. Algorithmic and Computational Properties

Computational tractability varies sharply across rule families:

• Polynomial-time Winner Determination: AV, SAV, sequential variants (sequential PAV, sequential CC), greedy and load-balancing Phragmén, CSAV, MSAV (Aziz et al., 2015, Aziz et al., 2014).
• NP-hardness: Exact winner determination is NP-hard for PAV, CC, Monroe, leximax (optimal Phragmén), and minimax AV. Heuristic and approximation algorithms—greedy, sequential, ILP formulations—provide practical solutions (Aziz et al., 2014, Skowron, 2018, Lackner et al., 2020).
• Approximation Guarantees: Sequential Thiele methods achieve constant-factor approximation to their respective objectives, e.g., sequential CC is a $1 - 1/e \approx 0.63$ approximation for representation (Lackner et al., 2018).
• Manipulation and Control: Most rules are NP-hard to manipulate or control, though tractability may arise fixed-parameter-wise in small numbers of candidates or voters (Yang, 2023, Aziz et al., 2014, Faliszewski et al., 2021). AV, uniquely, is immune to many control types and tractable for winner determination.

Sample complexity for learning ABC rules from example elections is polynomial, yet even deciding compatibility of a committee with some scoring rule is coW[1]-hard; sequential Thiele rules are more learnable but still face NP-hardness for the winner-verification problem (Caragiannis et al., 2021).

6. Connections to Apportionment, Applications, and Extensions

• Apportionment Correspondence: PAV corresponds exactly to D'Hondt (Jefferson) seat allocation on party lists; Monroe to the largest remainder (Hamilton method); Sainte-Laguë rules emerge from alternate OWA weights (Brill et al., 2016). This unifies multiwinner elections with parliamentary seat allocation.
• Practical Applications: Used in political open-list elections, participatory budgeting, collaborative filtering, and group recommendations (Lackner et al., 2020). Contemporary deployments include blockchain validator selection and participatory processes in civic tech.
• Generalizations: Mixed-goods models (indivisible and divisible items) extend EJR and proportionality to settings blending discrete and continuous resources, with GreedyEJR-M and MES providing guaranteed proportionality degrees (Lu et al., 2022). Robustness to arbitrary noise and adversarial operations is quantified; only "modal committee" (MC) rule is universally robust, with AV robust for majority-concentric metrics (Caragiannis et al., 2020).

7. Open Problems and Research Directions

Active topics include:

• Efficient, strongly proportional, and monotone committee rules with tractable winner determination remain elusive.
• Existence and computation of core committees (groupwise envy-freeness) is open; no Thiele-type rule satisfies it generally (Lackner et al., 2020).
• Empirical characterization on real-world and rich synthetic data; robustness analysis under realistic noise; learning rule classes from observed winning committees (Faliszewski et al., 2024, Caragiannis et al., 2021).
• Incorporation of richer ballots (trichotomous, constraints), multi-attribute quotas, and dynamic or perpetual voting schemes.
• Algorithmic approaches for rapidly solving ILP/MIP formulations and handling very large-scale participatory datasets.
• The design and analysis of deliberative processes that interact with the selection mechanism (Mehra et al., 2023).

References: (Aziz et al., 2014, Aziz et al., 2015, Brill et al., 2016, Lackner et al., 2017, Sánchez-Fernández et al., 2017, Lackner et al., 2018, Skowron, 2018, Caragiannis et al., 2020, Lackner et al., 2020, Peters, 2021, Faliszewski et al., 2021, Caragiannis et al., 2021, Lu et al., 2022, Yang, 2023, Mehra et al., 2023, Faliszewski et al., 2024, Caragiannis et al., 2024, Faliszewski et al., 27 Jan 2026).