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Where Should Society Draw the Line? A Social Choice Approach to Collective Consent

Chris Dong, Sonja Kraiczy, Rohit Vasishta, Markus Brill, Wesley H. Holliday, Niclas Boehmer

arXiv:2609.10759v1cs.GTecon.TH

TL;DR

Collective consent asks which options should receive societal approval given individuals’ attitudes, a question distinct from choosing the final outcome. The paper formalizes three blocking principles and develops solution concepts, including the Approval-Weighted Veto Core. Across five datasets, concepts that violate a principle in theory also violate it empirically.

  • Problem

    The paper addresses how society should derive consented options from individual attitudes, a question distinct from selecting a final outcome.

    Source: §1 · pp. 1–2

  • Method

    The paper axiomatizes sufficient support, minority protection, and relational blocking, then develops corresponding solution concepts and a veto-budgeting game that yields the Approval-Weighted Veto Core.

    Source: §1 · pp. 2–3

  • Results

    Across five datasets, solution concepts that violate a principle theoretically also violate it empirically.

    Source: §1 · p. 3

  • Takeaways & Limitations

    Composing the Approval-Weighted Veto Core with the disapproval Smith set provides a principled way to satisfy all three blocking principles simultaneously.

    Source: §3.2 · p. 9

    The analysis assumes voters report truthfully; the paper leaves the incentive-compatibility of its concepts for future work.

    Source: §5 · p. 12

Abstract

from arXiv · show

Society constantly has to determine the boundaries of what it deems acceptable, from legislative decisions to the guardrails governing autonomous systems. We initiate the axiomatic study of collective consent: given individuals' attitudes toward options, which options should receive societal consent? We organize our analysis around three principles: sufficient support, minority protection, and dominance by decisively better options. Each captures a distinct reason for withholding societal consent from an option. For each principle, we develop a corresponding solution concept that transparently implements the principle and is canonical in a mathematically precise sense. For example, for minority protection, the resulting concept is a consent-adapted version of Moulin's Proportional Veto Core. Balancing multiple principles simultaneously is more challenging. To address this, we develop a game-theoretic characterization of our veto core that naturally gives rise to a family of related concepts. From this family, we identify the Approval-Weighted Veto Core as particularly desirable. By making minorities' blocking power depend on the approval support of the options being challenged, it smoothly interpolates between proportional minority protection and majority support. Experiments on five datasets spanning high-stakes decision-making (such as political elections, ethical AI evaluations, and moral decision-making) show that solution concepts violating a principle in theory also violate it empirically.

1. Introduction

The paper frames collective consent as deciding which options merit societal consent from individual attitudes, while treating consent as a pre-selection rather than the final outcome. It organizes exclusion around sufficient support, minority protection, and decisively better alternatives.

1. Introduction

  • The central question is which options society should consent to given the attitudes of the individuals affected by those options.Collective consent concerns a pre-selection of defensible candidates; which candidates are ultimately enacted is decided downstream.Source: §1 · pp. 1–2

1.1. Preliminaries

  • The model takes each voter’s utility for every candidate and the voter’s approval set as input, and returns any subset of candidates as societally consented.The returned set has no fixed size; it may contain every candidate, none, or any subset in between.Source: §1 · p. 2, §1.1 · p. 4

1. Introduction

  • The paper formalizes three distinct grounds for withholding consent: insufficient support, disproportionate harm to a cohesive minority, and the availability of a clearly superior candidate.A candidate must be approved by at least a fixed fraction of voters; minority veto power should scale with coalition size; and unanimously approved candidates should outrank candidates some voters disapprove.Source: §1 · pp. 2–3
  • For each principle, the paper proposes a transparent solution concept, including the α-approval threshold, Disapproval Veto Core, and disapproval Smith set.The concepts identify voters or candidates responsible for exclusion, and the Disapproval Veto Core adapts Moulin’s proportional veto core [Moulin, 1981, 1982] using approval information.Source: §1 · p. 3
  • A veto-budgeting game yields a family of variants, including the Approval-Weighted Veto Core, which interpolates between proportional minority protection and majority support.The paper proposes composing the Approval-Weighted Veto Core with the disapproval Smith set to address all three blocking principles simultaneously.Source: §1 · p. 3
  • Across five datasets, solution concepts that violate a principle theoretically also violate it empirically.The datasets cover political elections, ethical AI evaluation, and moral decision-making scenarios.Source: §1 · p. 3

2. Blocking Framework and Solution Concepts

The paper formalizes three blocking principles and develops solution concepts that implement them. These range from an approval threshold for local blocking to coalition-based vetoes and pairwise comparisons for relational blocking.

2.1. Local Blocking: Sufficient Support

  • The α-threshold concept accepts every candidate whose approval score reaches α times the electorate size.Threshold concepts judge candidates independently of other candidates; they are characterized by independence of other candidates, equality, anonymity, and approval monotonicity.Source: §2.1 · pp. 4–5

2.2. Coalitional Blocking: Minority Protection

  • Coalitional blocking addresses a weakness of approval thresholds: the same approval score can conceal whether disapproval is scattered or reflects a cohesive minority’s shared objections.The cohesive group’s strong objections should be able to rule out a candidate even when other voters value it highly.Source: §2.2 · p. 5
  • A witness function identifies coalitions responsible for exclusions, while veto power measures the largest candidate-space share a coalition can force the solution concept to exclude.Proportional veto power scales with coalition size as s/n; Moulin [1981] showed this is the maximum achievable without empty consent sets.Source: §2.2 · p. 5
  • The Disapproval Veto Core (DVC) excludes a candidate when a nonempty coalition unanimously disapproves of it and unanimously ranks more than a coalition-size-proportional share of candidates above it.A coalition member who approves the candidate cannot help block it.Source: §2.2 · p. 6
  • Any solution concept with a witness function satisfying polarized minority protection, locality, upper-contour monotonicity, and approval-competition monotonicity must exclude at least the candidates blocked by DVC.This makes DVC the minimal blocking requirement under those axioms.Source: §2.2 · p. 6

2.3. Relational Blocking: Decisively Better Candidates

  • The disapproval Smith set uses a pairwise majority relation incorporating approval information, then retains candidates that reach every other candidate through that relation.It satisfies unanimous approval consistency: when unanimously approved candidates exist, the solution returns exactly those candidates.Source: §2.3 · p. 7

3. Veto Budgets for Combining Blocking Principles

The section moves from checking principle-specific concepts to combining minority protection and sufficient support through veto-budget variants. It identifies AWVC as a compromise and composes it with DSS to address all three blocking principles.

3. Veto Budgets for Combining Blocking Principles

  • The principle-specific checks find that DVC and fα for α < 1 violate exclusive unanimity.DVC and DSS have approval threshold 0, while fα satisfies its α-approval threshold.Source: §3 · p. 7

3.2. The Goldilocks Zone Between Proportional Veto Power and Majority Disapproval

  • AWVC guarantees majority threshold while its veto power stays near DVC for small coalitions and approaches 2-DVC near a majority.Its veto-power formula is s/(n − s), evaluated in the hardest case where all voters outside the coalition approve the candidate.Source: Figure 1 · p. 8, §3.2 · p. 9

3.1. Veto Budgeting Games and a Family of Disapproval Veto Cores

  • A veto-budgeting game characterizes DVC rejection through disapproval-restricted coalitions that can challenge every allocation sparing a candidate.The game interpretation also motivates variants formed by scaling voters’ veto budgets.Source: §3 · p. 7, §3.1 · p. 8
  • Scaling voter budgets by ω gives ω-DVC approval threshold α = max(0, 1 − 1/ω) and veto power v(s) = ωs/n.At ω = 1 this coincides with DVC; increasing ω raises the threshold but uniformly increases coalitional veto power, so 2-DVC doubles minorities’ veto power.Source: §3.1 · p. 8, §3.2 · p. 9

3.2. The Goldilocks Zone Between Proportional Veto Power and Majority Disapproval

  • AWVC preserves exclusion when another disapproving voter joins, whereas ω-DVC may restore consent after such additions.The paper reports that this ω-DVC behavior can regularly occur in experiments.Source: §3.2 · p. 9
  • Composing DSS with AWVC achieves all three blocking principles while retaining AWVC’s veto-power function.The resulting concept has veto power v(s) = s/(n − s) for s ≤ n.Source: §3.2 · p. 9

4. Empirical Evaluation

The evaluation tests the solution concepts on five datasets and finds that concepts lacking theoretical guarantees exhibit corresponding empirical failures. It also compares their selectiveness and minority-protection behavior.

Setup

  • The evaluation covers five datasets and compares the proposed concepts with Approval Relative Utilitarianism and Minimize Margin of Disapproval baselines.The datasets include CoVal, Synthetic, Political, Kidney Exchange, and Moral Machine instances.Source: §4 · pp. 9–10

Selectiveness

  • DVC is consistently the most permissive, while DSS and DSS◦AWVC sharply narrow selections, reaching near-singletons on Kidney.AWVC selects roughly as many candidates as the majority threshold concept.Source: §4 · p. 10, Figure 2 · p. 10

Sufficient Support

  • DVC selects candidates approved by less than half the voters in 52 of 319 CoVal elections, all 9 Political elections, and all 60 Kidney and Moral Machine instances.On Political and Kidney, the median minimum approval fraction falls to 0.04 and 0.08, respectively.Source: §4 · p. 10

Minority Protection

  • AWVC selects no candidates violating minority protection; its median worst-endowment is 1.91–2.55 across datasets versus DVC’s 1.00–1.75, while DSS◦AWVC reaches ∞ on CoVal and Moral Machine.An infinite median occurs where every selected candidate is unanimously approved and therefore unblockable under any ω-DVC.Source: §4 · p. 11

Decisively Better Candidates

  • On CoVal and Moral Machine, concepts without an exclusive-unanimity guarantee select non-unanimously approved candidates regularly; DVC has the most such violations.DVC produces 304 such candidates on CoVal and 2028 on Moral Machine.Source: §4 · p. 12

5. Discussion

The discussion frames collective consent as a social-choice problem and identifies open directions. The related-work section distinguishes this variable-sized, per-candidate consent setting from neighboring voting and deliberation approaches.

5. Discussion

  • The paper formalizes local, coalitional, and relational blocking and proposes canonical concepts; empirical results show that concepts without theoretical guarantees exhibit corresponding flaws in practice.Future work includes incentive compatibility under truthful-voter assumptions, computation for infinite candidate spaces, automated explanations, and concepts for other parameter choices.Source: §5 · p. 12

A. Related Work

  • Unlike single-winner voting, collective consent may legitimately approve multiple incomparable candidates or none, rather than identify one best candidate.The framework imposes no fixed consent-set size and judges each accepted candidate against a quality threshold.Source: A. Related Work · p. 16
  • Unlike multiwinner voting, the framework has no fixed committee size, requires each accepted candidate to clear a quality threshold, and gives disapproval asymmetric weight through minority protection.Kraiczy et al. [2025] study group veto over candidate combinations, whereas this paper decides consent per candidate.Source: A. Related Work · p. 16
  • The proportional veto core provides a starting point for adapting minority protection to collective consent, while prior approval-based work by Halpern et al. [2025] motivates addressing unanimously disapproved selections.In the illustrated case, PVC selects only the unanimously disapproved candidate, whereas ω-DVC and AWVC also block it and return the empty set.Source: A. Related Work · p. 16, Figure 3 · p. 17
  • Prior work combining preference and approval input often outputs rankings or approval sets constrained by rankings, whereas this paper focuses on consent and minority protection.Brams and Sanver [2009] propose rules selecting best candidates rather than societal consent sets.Source: A. Related Work · p. 17
  • Aggregation and deliberation are presented as complementary: when discussion in a consent assembly does not resolve disagreement, aggregation may help gauge collective consent.The deliberative approach is associated with Kyi et al. [2026].Source: A. Related Work · p. 17

B. A Collection Of Proofs

This appendix defines voter-restricted partial instances and gives a characterization of threshold concepts through four axioms.

B. A Collection Of Proofs

  • For a voter subset S, I|S denotes the partial instance containing only the utilities and thresholds of voters in S.The notation restricts both the utility profile and the threshold profile to S.Source: B. A Collection Of Proofs · p. 18

B.1. An Axiomatic Characterization of Threshold Concepts

  • A solution concept is a threshold concept if and only if it satisfies independence of other candidates, equality, anonymity, and approval monotonicity.The threshold rule selects candidates according to whether their approval score reaches a fixed threshold.Source: §B.1 · p. 18
  • The proof identifies the threshold from the smallest number of approving voters at which the rule returns a nonempty set, then establishes selection exactly when a candidate meets that threshold.It uses anonymity, independence of other candidates, equality, and approval monotonicity to extend the result across profiles and candidates.Source: §B.1 · p. 18

B.2. Witness-Based Characterizations

The witness-based results characterize when a solution concept must exclude candidates under veto-core rules. The appendix also establishes that the disapproval Smith set preserves unanimously approved candidates.

B.2. Witness-Based Characterizations

  • With a witness function, ω-polarized minority protection and the stated monotonicity and independence axioms imply that the solution concept is contained in ω-DVC.For finite candidate sets, the conditions include locality, candidate monotonicity, weak independence of other candidates, and approval competition monotonicity; the general result uses upper contour monotonicity.Source: §B.2 · pp. 19–20
  • The proof reconstructs the target profile from a polarized one through preference changes within contour sets, moving the candidate down, and then adding approval thresholds.Locality allows preferences outside the blocking coalition to be replaced while preserving the witness for blocking.Source: §B.2 · p. 20
  • The witness-based results also yield characterizations of Moulin’s PVC in finite and infinite settings when approvals and the weak-independence-of-approvals axiom are removed.The results support a parameterized family by allowing the minority’s blocked share to scale with ω.Source: §B.2 · pp. 19–20

B.3. Disapproval Smith Set

  • The disapproval Smith set satisfies unanimous approval consistency: when unanimously approved candidates exist, it returns exactly those candidates.The proof identifies the set as candidates with approval score n.Source: §B.3 · p. 20

B.4. Proofs for Solution Concepts and Blocking Principles

The proofs establish veto-power and approval-threshold guarantees for the proposed rules, including the Approval-Weighted Veto Core and its composition with the disapproval Smith set. They also identify when rule composition preserves exclusive unanimity.

B.4. Proofs for Solution Concepts and Blocking Principles

  • DVC and approval-threshold rules fα with α < 1 violate exclusive unanimity, whereas DSS and DSS composed with AWVC satisfy unanimous-approval consistency.The composition is unanimous-approval consistent when unanimously approved candidates are present.Source: §B.4 · pp. 20, 23
  • AWVC has veto power v(s) = s/(n−s), and DSS composed with AWVC retains that veto power, a 1/2 approval threshold, and unanimous-approval consistency.The proof section states the guarantees as Theorems 4 and 5.Source: §B.4 · p. 21
  • Composing rules f ◦ g preserves g’s blocking of a candidate set when f is defined on g’s outputs, satisfies exclusive unanimity, and g retains a unanimously approved candidate when one exists.These conditions let the second rule refine g’s output without reintroducing blocked candidates.Source: §B.4 · pp. 21–23

C. Proofs of Section 3.1

This section develops a veto-budget characterization of ω-DVC and related properties, computational methods, and extensions for minority protection, alongside experiments examining support-monotonicity violations by 2-DVC for disapproving voters.

C. Proofs of Section 3.1

  • Theorem 8 characterizes ω-DVC through veto-budget behavior: every allocation containing c admits a disapproval-restricted ε-deviation for some ε > 0.For continuous candidate sets, the proof constructs allocations whose outcome is a small neighborhood of c.Source: C. Proofs of Section 3.1 · pp. 23–24

C.1. Additional Details on the ω-DVC

  • AWVC preserves support monotonicity for disapprovers: adding a voter who disapproves c cannot make a previously blocked c unblocked.A profile illustrates that 2-DVC can reverse consent after a disapprover joins; candidate-wise scaled voter budgets support AWVC’s monotonicity.Source: §C.1 · pp. 25–26, Figure 4 · p. 26

D.1. Critical Endowment

  • For finite C, critical endowment ω*(c) is computable in O((mn)1+o(1)) time, where m is the number of candidates.The algorithm reduces blocking detection at tested endowments to a minimum-cut computation.Source: §D.1 · pp. 26–27

D.2. Critical ϵ and Generalization

  • Local excess budget ε*(c) extends the minority-protection measure to any candidate; for c in DVC, its negative value indicates the closest blocking coalition’s budget shortfall.The motivation is to distinguish the levels of minority protection offered by choice sets that are both contained in DVC.Source: §D.2 · p. 27

E. Computation

  • For finite candidate sets, all solution concepts are computable in polynomial time; DVC and AWVC use candidate-wise flow-based blocking tests.The tests adapt the veto-core network, with AWVC adjusting the blocking threshold using the candidate’s approval score.Source: E. Computation · p. 28

F. Modifying Threshold Based Rules

  • For arbitrary approval sets, the appendix derives thresholds and adjusts approved utilities before applying Rel Util and Minimize Margin of Disapproval, marking infeasible thresholds as cheats.Minimize Margin of Disapproval first minimizes total voter distance, then selects candidates with the fewest cheat marks.Source: F. Modifying Threshold Based Rules · pp. 28–29

G. Support Monotonicity for Disapproving Voters

  • A modified max-flow heuristic finds disapprovers closest to blocking a chosen candidate, then tests whether abstention by the other disapprovers lets them block it.The approach avoids recalculating the rule for every possible set of disapprovers.Source: G. Support Monotonicity for… · p. 29
  • 2-DVC may frequently violate support monotonicity for disapproving voters, and some violations appear under brute-force single-voter abstentions.The reported test examines whether removing disapprovers can cause a previously chosen candidate to become blocked.Source: G. Support Monotonicity for… · p. 29

H. Experiments

  • Running all experiments took around six hours on a 12-core machine, without GPUs or other specialized computing resources.Source: H. Experiments · p. 29

H.1. Dataset 1: The Moral Machine

The section describes how four empirical datasets are converted into voter utilities, approval sets, and election instances. Their sources include moral-dilemma judgments, political elections, ethical-response assessments, and kidney-allocation preferences.

H.1. Dataset 1: The Moral Machine

  • 60 Moral Machine elections use 137 country-level voters and 100 sampled policy candidates, with utilities and approvals generated under six variants based on utility thresholds or agreement on strong-opinion dilemmas.Country utility vectors are learned from pairwise responses; countries with fewer than 1,000 complete scenarios or below 50% training accuracy are excluded.Source: §H.1 · pp. 29–30

H.2. Dataset 2: Politics

  • Nine political elections combine approval and score ballots, cardinal evaluations, or rankings; ordinal-only ballots are converted to Borda-score utilities, with unranked candidates treated as tied.The passage cautions that results may be heavily influenced by how preferences were retrieved, including the treatment of unranked candidates.Source: §H.2 · p. 30

H.3. Dataset 3: The CoVal Dataset

  • CoVal provides 319 election instances with four AI-generated responses each; unacceptable flags determine disapproval, while personal rankings are converted to Borda-score utilities.Only annotators providing personal rankings are retained, and prompts with no disapproving voters are discarded.Source: §H.3 · p. 30

H.4. Dataset 4: Kidney Exchange

  • Kidney Exchange constructs 60 approval-based elections from 301 reported importance-weight vectors and 100 simulated recipients per election, using dot-product utilities and approval for positive utility.Recipient features are sampled independently from the uniform distribution over an eight-dimensional space.Source: §H.4 · p. 31

H.5. Dataset 5: Synthetic Data

The synthetic elections use impartial-culture, Mallows, and spatial preference models. The section also describes table measures for exclusive-unanimity violations and sufficient support.

H.5. Dataset 5: Synthetic Data

  • The synthetic datasets include impartial-culture, Mallows, and spatial elections, with 100 independent elections per configuration.The setup uses 50 voters and five candidates for impartial culture; Mallows dispersion varies across three values, and spatial profiles are generated on a line and in a unit square.Source: §H.5 · p. 31
  • Table 3 measures exclusive-unanimity violations: # counts elections selecting at least one non-unanimously approved candidate, while tot. counts such candidates across those elections.Its columns also report the dataset’s total elections N and the number Nu containing at least one unanimously approved candidate.Source: Table 3 · p. 31
  • Table 4 reports sufficient-support results, with its first column (#) counting elections in which the stated condition occurs.The supplied table description does not specify the remaining columns.Source: Table 4 · p. 32

I.1. Effect of Candidate Pool Size on the Behavior of Solution Concepts

As the candidate pool grows, solution concepts differ in selectivity, sufficient support, minority protection, and unanimity consistency. DVC becomes increasingly prone to sub-majority selections, while the DVC, AWVC, Majority Threshold, and ARU select non-unanimous candidates whenever a unanimously approved candidate exists.

I.1. Effect of Candidate Pool Size on the Behavior of Solution Concepts

  • ARU is the only concept that becomes less selective as m grows, rising from 0.19 at m = 5 to ≈0.37 to 0.40 by m = 40.The experiment uses 100 elections per candidate-pool size, split across six synthetic models.Source: §I.1 · p. 32
  • DVC’s rate of selecting sub-majority candidates rises with m, while the DVC’s share of selected candidates stays around 0.55–0.60.AWVC remains around 0.27–0.30; DSS◦AWVC falls from ≈0.15 to 0.20 at m = 5 to ≈0.035 by m = 50.Source: §I.1 · pp. 32–33
  • DVC and AWVC never select candidates with ω∗ < 1 as m grows, but DVC’s median ω∗ falls from 1.25 to 1.00 while AWVC stays around 1.9–2.0.DSS◦AWVC’s median rises from 2.6 at m = 5 to 7.1 at m = 50.Source: §I.1 · p. 33, Figure 7 · p. 33
  • DVC, AWVC, Majority Threshold, and ARU violate exclusive unanimity in 100% of elections containing a unanimously approved candidate.For DVC and Majority Threshold, non-unanimous selections total ≈170 to 190 at m = 40.Source: §I.1 · p. 33, Figure 8 · p. 34

I.2. Rank Consistency of ω∗and ϵ∗

The two candidate-protection measures, critical endowment ω∗ and critical excess budget ϵ∗, rank candidate safety similarly across the datasets. Their per-election Spearman correlation is close to −1, with variation in the small Political groups.

I.2. Rank Consistency of ω∗and ϵ∗

  • Across datasets, ω∗ and ϵ∗ have a clear monotone negative relationship, and per-election Spearman ρ is close to −1.Moral Machine mean ρ is −0.997 to −0.998 across six variants; Synthetic values range from −0.953 (Mallows ϕ = 0.5) to −0.996 (Mallows ϕ = 0.2).Source: §I.2 · p. 34, Figure 10 · p. 35
  • Political-election mean ρ ranges from −0.653 (Virginia 2012) to −1.0, but each group contains at most three elections.CoVal’s mean ρ is −0.971 (median −1.0) with m = 4 candidates per election; Kidney achieves ρ = −0.924.Source: §I.2 · pp. 34–35
  • The negative monotone relationship confirms that ω∗ and ϵ∗ rank candidates by safety in the same order.Figure 10 pools election-candidate pairs for which both measures are well-defined; its dashed ϵ∗ = 0 line separates blocked from safe candidates.Source: §I.2 · pp. 34–35

I.3. Solution Concepts on the CoVal Dataset

On CoVal, solution concepts differ in the approval support and critical endowments of their selected candidates. DSS-based concepts and MMOD tend toward highly approved or unanimously approved selections, while DVC is less protective by these measures.

I.3. Solution Concepts on the CoVal Dataset

  • DSS achieves a median minimum approval fraction near 1.0, while DVC has the widest spread and sometimes selects candidates below the 0.5 majority threshold.AWVC concentrates higher than DVC; ARU and MMOD have medians around 0.8 with occasional outliers below 0.5.Source: §I.3 · p. 35, Figure 11 · p. 36
  • DSS◦AWVC, DSS, and MMOD have median minimum critical endowments above 4, with 175 of 319 elections in which every selected candidate is unanimously approved.DVC is least protective; Majority Threshold and ARU sometimes select candidates with ω∗ < 1.Source: §I.3 · pp. 35–36, Figure 12 · p. 37
  • On the CoVal dataset, AWVC shifts the minimum critical-endowment distribution upward compared with DVC and Majority Threshold, with 43 elections in which every selected candidate is unanimously approved.Figure 12 separates elections with finite minimum critical endowments from those in which every selected candidate is unanimously approved.Source: §I.3 · p. 35, Figure 12 · p. 37

I.4. Experimental Variants: Synthetic and Moral Machine

Experimental variants show that selection behavior depends on the preference model and variant, while the concepts differ in support, minority protection, and unanimity consistency. The relational-blocking appendix also describes a disapproval Pareto alternative to DSS.

I.4. Experimental Variants: Synthetic and Moral Machine

  • Across synthetic models, selectivity varies: spatial models yield the most selective behavior overall, while DVC and Majority Threshold are more permissive as Mallows ϕ increases.Under Impartial Culture, DVC and Majority Threshold select the largest fractions, while AWVC and DSS◦AWVC are the most selective.Source: §I.4 · p. 37, Figure 13 · p. 38
  • Across synthetic models, AWVC and DSS◦AWVC never select sub-majority candidates, while DVC is most prone to them and has median minimum approval below 0.50.AWVC, DSS◦AWVC, and MMOD never produce ω∗ < 1 and have median endowments of 2.00–4.55; Majority Threshold selects candidates with ω∗ < 1 in 68 of 100 Spatial 1D elections, with median 0.67.Source: §I.4 · pp. 37–38, Figure 14 · p. 39, Figure 15 · p. 39
  • Across Moral Machine variants, DSS◦AWVC and AWVC are most selective, selecting roughly 10–15% of candidates, while DVC alone selects sub-majority candidates in every variant.DSS, DSS◦AWVC, and MMOD select only unanimously approved candidates and never violate unanimity; DVC, Majority Threshold, and AWVC violate it in all 10 elections of every variant.Source: §I.4 · p. 38, Figure 16 · p. 40, Figure 17 · p. 40

J. Relational Blocking Revisited

  • The disapproval Pareto set excludes candidates if and only if they are disapproval Pareto-dominated, and may suit real-world elections better than DSS when DSS outputs only one candidate.Disapproval Pareto dominance requires a candidate to be approved or weakly preferred by all voters, with at least one voter satisfying the specified strict condition against the dominated candidate.Source: J. Relational Blocking Revisited · p. 41
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