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Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study

Salim Hafid, Manon Berriche, Jean-Philippe Cointet

Year: 2026Venue: arXiv preprintArea: cs.CYType: PreprintEmbeddings: 73

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Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 92%

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Summary

This paper addresses the challenge of algorithmically selecting representative opinions for online deliberation platforms. It benchmarks existing strategies (e.g., consensus, diversity) against democratic criteria like proportional representation and diversity. The authors propose a novel algorithm, DiverseBJR, based on social choice theory (specifically Balanced Justified Representation), which aims to balance proportional representation with content diversity. Empirical results on real-world data from Remesh indicate that DiverseBJR achieves the best trade-off between these competing desiderata compared to other methods.

Entities (10)

Salim Hafid · person · 99%DiverseBJR · algorithm · 98%Balanced Justified Representation · theoretical-concept · 95%Justified Representation · theoretical-concept · 95%Diversity · metric · 93%Proportional Representation · metric · 93%Remesh · platform · 92%Pol.is · platform · 90%Social Choice Theory · field · 90%Community Notes · system · 88%

Relation Signals (7)

DiverseBJR → basedon → Balanced Justified Representation

confidence 95% · We also build on social choice theory to propose a novel algorithm that incorporates both diversity and a balanced notion of representation... DiverseBJR aims to preserve the balanced and justified representation properties

DiverseBJR → optimizesfor → Proportional Representation

confidence 92% · our social-choice-inspired selection rule achieves the strongest trade-off between proportional representation and diversity.

DiverseBJR → optimizesfor → Diversity

confidence 92% · DiverseBJR aims to preserve the balanced and justified representation properties while additionally promoting diversity in the selected opinions

Consensus-seeking strategies → leadsto → Erasing minority voices

confidence 90% · enforcing consensus-seeking algorithmic strategies can imply ignoring or flattening conflicting preferences, which may lead to erasing minority voices

Pol.is → usesalgorithmtype → Bridging Algorithm

confidence 88% · Another example is the popular Pol.is platform which has used bridging for large-scale online deliberations

Community Notes → usesalgorithmtype → Bridging Algorithm

confidence 88% · For instance, Community Notes, the community-based fact-checking system on X (ex-Twitter) relies on a bridging algorithm.

Remesh → usedfordata → DiverseBJR

confidence 85% · Our analysis using real-world deliberation data from Remesh reveals that existing algorithms exhibit clear trade-offs

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Abstract

Abstract:During deliberation processes, mediators and facilitators typically need to select a small and representative set of opinions later used to produce digestible reports for stakeholders. In online deliberation platforms, algorithmic selection is increasingly used to automate this process. However, such automation is not without consequences. For instance, enforcing consensus-seeking algorithmic strategies can imply ignoring or flattening conflicting preferences, which may lead to erasing minority voices and reducing content diversity. More generally, across the variety of existing selection strategies (e.g., consensus, diversity), it remains unclear how each approach influences desired democratic criteria such as proportional representation. To address this gap, we benchmark several algorithmic approaches in this context. We also build on social choice theory to propose a novel algorithm that incorporates both diversity and a balanced notion of representation in the selection strategy. We find empirically that while no single strategy dominates across all democratic desiderata, our social-choice-inspired selection rule achieves the strongest trade-off between proportional representation and diversity.

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Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study Salim Hafid 1 Manon Berriche 1 Jean-Philippe Cointet 1 Abstract During deliberation processes, mediators and fa- cilitators typically need to select a small and rep- resentative set of opinions later used to produce digestible reports for stakeholders. In online delib- eration platforms, algorithmic selection is increas- ingly used to automate this process. However, such automation is not without consequences. For instance, enforcing consensus-seeking algorith- mic strategies can imply ignoring or flattening conflicting preferences, which may lead to eras- ing minority voices and reducing content diversity. More generally, across the variety of existing se- lection strategies (e.g., consensus, diversity), it remains unclear how each approach influences de- sired democratic criteria such as proportional rep- resentation. To address this gap, we benchmark several algorithmic approaches in this context. We also build on social choice theory to propose a novel algorithm that incorporates both diversity and a balanced notion of representation in the se- lection strategy. We find empirically that while no single strategy dominates across all democratic desiderata, our social-choice-inspired selection rule achieves the strongest trade-off between pro- portional representation and diversity. 1. Introduction Deliberative democracy argues that proper deliberation yields rational collective outcomes (Habermas, 2015; Rawls, 1997). The essence of democratic legitimacy is seen as the ”capacity of those affected by a collective decision to de- liberate in the production of that decision” (Dryzek & List, 2003). To bring such collective decision-making to the masses, platforms for participatory democracy run public online consultations and deliberations for various use-cases such as peacebuilding, participatory budgeting, and pub- lic policy design (e.g., Polis (Small et al., 2021), Remesh 1 m ́ edialab, Sciences Po Paris, France. Correspondence to: Salim Hafid <salim.hafid@sciencespo.fr>. Preprint. February 18, 2026. (Konya et al., 2022), make 1 ). However, such processes in- creasingly involve large numbers of free-form contributions (e.g., opinions, critiques, proposals) from diverse partici- pants (Zhu et al., 2025). It thus becomes impossible for participants to process all inputs during deliberation. In this context, a crucial task is to select a small set of opinions that faithfully represents participants, and that can serve as a digestible input. This representative subset is typically selected before, during, and after the process by human fa- cilitators, and is shared with participants or experts in order to prepare the next discussion cycle, or used to produce reports for stakeholders, decision-makers, or for the gen- eral public. However, while essential for all stakeholders, producing such representative subsets is resource-intensive and therefore costly to scale-up (Landemore, 2020; Fishkin et al., 2019). To address this problem, a vast literature from recommender systems (Deldjoo et al., 2024) and collective response sys- tems (Ovadya, 2023) has developed automated selection strategies that optimize for various desiderata such as pop- ularity (Klimashevskaia et al., 2024) and fairness (Zhao et al., 2025). In the case of online deliberation, designing an algorithmic mechanism for selecting a subset of repre- sentative opinions requires making an explicit choice with regard to the type of opinions that the algorithm ought to favour. In other words, it requires defining what makes an opinion representative or desirable for selection: is it the most popular? the most consensual? the most disruptive? In deliberation contexts where multiple, often incompati- ble perspectives coexist (Crawford, 2016), answering these questions is not trivial. Nonetheless, their corresponding answers are embedded in algorithmic choices which directly influence deliberation results and yet are typically not made transparent to participants (Revel & P ́ enigaud, 2025). In parallel, a large body of literature from social choice the- ory has developed axioms whose goal is to formalize when collective outcomes adequately reflect the preferences of significant minorities. Such axioms offer precise and well- studied fairness guarantees against the systematic erasure of minority viewpoints, and are thus directly relevant for deliberation contexts. Recent efforts towards incorporating 1 https://make.org/en 1 arXiv:2602.15439v1 [cs.CY] 17 Feb 2026 Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study such guarantees into algorithmic systems for participatory democracy are promising but remain scattered (Fish et al., 2024; Revel et al., 2025). To our best knowledge, research has yet to comparatively evaluate the variety of existing algorithmic strategies for opinion selection using a compre- hensive set of metrics specifically tailored to democratic desiderata for online deliberation. To address this gap, this work contributes the following: (1) A systematic study of how algorithmic mechanisms for opinion selection impact specific democratic desider- ata in online deliberation. Our analysis using real-world deliberation data from Remesh 2 reveals that existing algo- rithms exhibit clear trade-offs, and that enforcing constraints based on social choice theory leads to strong gains in repre- sentation while reasonably covering the opinion space and limiting redundancy among selected opinions. We make our code publicly available for both researchers and democracy practitioners to facilitate future usage and benchmarking of algorithms in this context. 3 (2) A novel selection algorithm based on social choice the- ory to promote diversity without breaking proportional representation constraints. We find empirically that, when the selected subset is small (k ≈ 1− 5), our algorithm offers the best compromise between proportional representation and diversity, making it well suited to deliberation settings where (i) the subset must represent the diversity of expressed opinions, (i) the subset must remain small and digestible. With this work, we aim to strengthen the systematic study of algorithmic impacts in contexts of digital democracy, and to contribute to existing efforts towards incorporating fairness guarantees into algorithmic systems in such contexts. 2. Related Work Deliberative democracy platforms and bridging systems: Platforms for online participatory democracy rely on au- tomated methods for condensing large-scale participations (comments, opinions, proposals) into small-size, digestible inputs. One such example is Remesh (Konya et al., 2023b), a platform for collective dialogues previously used by the UN in peacebuilding contexts (Bilich et al., 2023). On Remesh, each participant expresses their personal opinion in free-form text, and votes on the opinions of a few other participants. This results in a sparse approval matrix, which is computationally made dense by inference of missing votes through a model combining language-model embeddings 2 Remesh is a ”collective response system” (Ovadya, 2023) that enables participants to discuss divisive issues and surface shared insights. 3 To reproduce the paper’s results or benchmark algorithms on any deliberation dataset, seehttps://github.com/Salim HFX/Algorithms-for-Opinion-Selection-in-Del iberation and latent-factors (Konya et al., 2022). Once the approval matrix is full, the platform selects a subset of representa- tive statements through a bridging-based algorithm (Ovadya & Thorburn, 2023). In short, bridging algorithms aim at highlighting items which receive approval across groups that typically disagree. The underlying assumption is that focusing on items which bridge across existing divides can reduce unproductive conflict and promote cooperation. Vari- ous online platforms have shown benefits of bridging-based selection. For instance, Community Notes, the community- based fact-checking system on X (ex-Twitter) relies on a bridging algorithm. 4 . Another example is the popular Pol.is platform which has used bridging for large-scale online deliberations (Small et al., 2021). While bridging-based al- gorithms have proven useful for opinion selection in online deliberation, they have, to our best knowledge, yet to be systematically compared to other algorithmic strategies to quantify how they rank on democratic desiderata such as proportional representation, coverage, and diversity. Diversity in algorithmic selection: Diversity has long been the interest of a large body of work in recommender systems (Zhao et al., 2025; Zhou et al., 2025) and algorithmic fair- ness (Pessach & Shmueli, 2022; Shrestha & Yang, 2019). This is in part due to popularity bias (Abdollahpouri et al., 2019), a phenomenon whereby systems (e.g., online press, streaming services) tend to select popular items by default and ignore or rarely select less popular, niche items in the ”long tail”. In our work, niche items might represent minor- ity voices or novel disruptive ideas whose inclusion in the selection is arguably required for the deliberation process to be seen as democratic (Shortall et al., 2021). To solve this, recommender systems have mainly focused on devel- oping (i) measures to assess diversity, for instance through distance-based metrics such as Intra-List Diversity (Ziegler et al., 2005), Jaccard similarity, Hamming distance, or co- sine similarity, (i) algorithms and learning objectives that promote diversity, where many involve optimizing maximin- based objectives for learning, clustering, or re-ranking, and are typically operationalized via greedy approximation pro- cedures. In this work, we build on this literature to (1) de- velop diversity-related metrics that best serve the use-case of online deliberation and incorporate them in our compara- tive analysis, (2) develop a novel algorithm that promotes diversity while also guaranteeing criteria of proportional representation based on social choice theory. Theoretical guarantees from social choice theory: De- spite the proven empirical efficiency of bridging-based and diversity-based algorithmic selection, the outcomes in such approaches typically lack theoretical guarantees. For ex- ample, can a selection algorithm guarantee that minority 4 https://communitynotes.x.com/guide/en/un der-the-hood/ranking-notes 2 Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study opinions will not be ignored? Social choice theory aims precisely at answering such questions, by studying the for- mal aspects of aggregating human preferences (Brandt et al., 2016). As observed by (Halpern et al., 2023), selecting a subset of opinions based on agreements and disagreements is ”equivalent to electing a committee based on approval votes”, a process known as approval-based committee elec- tions. In this context, a vast and robust literature has defined various desired properties for election processes. One pop- ular property previously used in contexts of online deliber- ation (Revel et al., 2025; Halpern et al., 2023) is Justified Representation (JR) (Aziz et al., 2017). In simple terms, the satisfaction of JR requires that every large enough group of cohesive voters has an approved candidate in the elected committee. 5 In our context, JR would require that every large enough group of cohesive voters has an approved opinion in the selected subset. However, while an algo- rithm satisfying JR would ensure that opinions supported by enough participants end up in the selected set, it does not necessarily guarantee that the matching between partici- pants and opinions is balanced in the final set. For example, in a scenario where the task is to select 7 representative opinions, selecting very few highly popular opinions (e.g., 2) might be enough to satisfy JR, leaving the remaining 5 opinions to be selected without any constraints of propor- tional representation. To solve this, many stronger variants of JR have been proposed (S ́ anchez-Fern ́ andez et al., 2017; Kalayci et al., 2025). Perhaps the most intuitive is Balanced Justified Representation (BJR) (Fish et al., 2024). In the same example, under BJR constraints, it becomes impos- sible for a small amount of popular opinions to satisfy the property. Instead, representation must be equally balanced across all selected opinions, meaning that every selected opinion should represent approximately the same amount of participants. 6 Assuming a BJR-satisfying subset of opin- ions, one limitation remains: the subset is not necessarily maximally diverse, i.e., it might include redundant opinions. In the context of online deliberation where the size of the subset must be small, selecting redundant opinions is not desired. While existing works from approval-based com- mittee election have investigated combining the quality of the elected committee with diversity constraints (Bredereck et al., 2018; B ́ eal et al., 2025; Relia, 2022), such works have focused on specific diversity attributes (e.g., gender, age) and have assumed their availability in data. In this work, we build on social choice theory to (1) operationalize ex- isting properties (JR and BJR) into algorithms for opinion subset selection; (2) define and operationalize a novel selec- tion property (DiverseBJR) that accounts for both balanced 5 Explanations of notions such as ”large enough group” and ”cohesive voters”, as well as definitions of all relevant proper- ties—including JR—will be given in Section 3.2. 6 We discuss more in detail the limitations of JR and how JR and BJR differ with a concrete example in Appendix B. representation and diversity, where diversity is based ex- clusively on voting data and does not require access to any additional attributes; (3) evaluate the resulting algorithms against strategies which lack theoretical guarantees (e.g. bridging) in the context of online deliberation. Scope: A related line of research has used LLMs to produce abstractive summaries during or after deliberation sessions (Tessler et al., 2024; Zhu et al., 2025; Fish et al., 2024). This use-case is outside the scope of our study. While arguably easier to streamline, the LLM-based summary of a delib- eration session comes with its own set of biases: beyond general reformulation and trust biases in LLMs (Akpinar et al., 2026; Bouyzourn & Birch, 2025), recent work has revealed persistent weaknesses in LLM-based deliberation summarization, including the under-representation of minor- ity voices (Zhu et al., 2025). In this work, we focus exclu- sively on algorithmically selecting representative opinions, not on reformulating them into a single cohesive summary. 3. Methodology 3.1. Theoretical framework We adopt the (Elkind et al., 2022) framework for multi- winner approval voting , adapted to online deliberation by (Revel et al., 2025). In this setting, the goal is to selectkrep- resentative microposts by maximizing a given scoring rulef (e.g., engagement, consensus). In our case, the microposts are opinions expressed during a deliberation process. Problem formalization: Let[m] = 1,...,mbe the set of all opinions and[n]the corresponding set of users. The task is to select a subsetS ⊆ [m]ofkopinions.A n = A 1 ,...,A n is the preference profile across all users, where each useru ∈ [n]approves a set of opinionsA u ⊆ [m]. The setting is then described by the tupleI m,A n ,k =< m,A n ,k >, wherekopinions must be selected based onn approval profiles over m opinions. Scoring rule: To select a subsetSofkopinions from [m], a scoring rule is used, whereby each opinioni ∈ [m] is a assigned a scoref (i,A). Scoring rules are additive: f (S,A n ) = Σ i∈S f (i,A n ), and may or may not depend on the approval profileA n . For instance, random selection as a scoring rule is not approval-dependent, whereas examples of approval-dependent rules include engagement (also de- scribed as the utilitarian or social welfare rule (Elkind et al., 2022)):f engagement (i,A n ) = u : i ∈ A u , and cover- age:f coverage (i,A n ,V ) = u ∈ V : i ∈ A u , whereV is the set of uncovered users. 3 Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study 3.2. DiverseBJR: Diversity-aware Balanced Justified Representation To investigate the effects of directly enforcing guarantees of proportional representation into our task, we use two existing axioms from social choice theory: Justified Rep- resentation (Revel et al., 2025; Aziz et al., 2017), and Bal- anced Justified Representation (Fish et al., 2024), which we define below. We adapt their formulation from approval- based committee election to our problem setting. We also define a novel criterion: Diversity-aware Balanced Justified Representation (DiverseBJR). DiverseBJR aims to preserve the balanced and justified representation properties while additionally promoting diversity in the selected opinions, an aspect not explicitly accounted for by either JR or BJR. Definition 3.1 (Justified Representation (JR) (Revel et al., 2025; Aziz et al., 2017)). Let[m]be the set of all opinions expressed by a set of users[n]. A subset of opinionsS ⊆ [m] with|S| = ksatisfies JR if, for every groupGof at leastn/kusers, where all users inGapprove at least one common opinion in[m], there is at least one user fromG who approves at least one item in S . In words, a selected subset of opinions satisfies JR if every large enough group of users that has shared preferences (approves of at least one opinion in common) is allocated at least one opinion in this subset. Definition 3.2 (Balanced Justified Representation (BJR) (Fish et al., 2024), adapted to binary approvals). Let[m] be the set of all opinions expressed by a set of users[n]. A subset of opinionsS ⊆ [m]with|S| = ksatisfies Balanced Justified Representation (BJR) if there exists a mapping ω : [n]→ Ssuch that (i) each opinionq ∈ Sis assigned either ⌊n/k⌋or⌈n/k⌉users, and (i) there is no groupT ⊆ [n] with|T| ≥ n/kand opinionq ∈ [m]such that every user u∈ Tapprovesqwhileq /∈ ω(T ), whereω(T ) :=ω(u)| u∈ T. In words, a selected set of opinions satisfies BJR if there exists a way to assign every user to one of the selected opinions so that the users are split as evenly as possible across the selected opinions, and there can be no group of at leastn/kusers which unanimously approves some opinion without at least one member of that group being assigned to that opinion. Definition 3.3 (ε-neighbor). LetA ∈ 0, 1 n×m be the approval matrix and leta i ∈ 0, 1 n denote the approval vector of opinioni. Letd : 0, 1 n ×0, 1 n → [0, 1]be any distance function. Forε∈ [0, 1], an opinionj ̸= iis an ε-neighbor of opinion i (w.r.t. d) iff d(a i ,a j )≤ ε. Definition 3.4 (DiverseBJR). Let[m]be the set of all opin- ions expressed by a set of users[n]. Letdist : [m]× [m]→ R ≥0 be a distance function andε > 0a diversity thresh- old. A subset of opinionsS ⊆ [m]with|S| = ksatisfies DiverseBJR if there exists a mappingω : [n]→ Ssuch that (i) each opinionq ∈ Sis assigned either⌊n/k⌋or⌈n/k⌉ users, and (i) there is no groupT ⊆ [n]with|T| ≥ n/k and opinionq ∈ [m]such that every useru ∈ Tapproves qwhileq /∈ ω(T ), whereω(T ) := ω(u) | u ∈ T(BJR constraints). Additionally, (i) S is locallyε-diverse among sets satisfying the BJR constraints, i.e., there do not exist p ∈ Sandq ∈ [m]\ Ssuch thatqis not anε-neighbor of any opinion inS \pand(S \p)∪qalso satisfies the BJR constraints. In words, a selected set of opinions satisfies DiverseBJR if it satisfies BJR and the selected opinions are ε-diverse, i.e., there is no way to replace one selected opinion by an unse- lected opinion that is not anε-neighbor of any of the other selected opinions while still satisfying the BJR constraints. The notion ofε-diversity relates to the notion of approx- imate clones from social choice theory (Procaccia et al., 2025), whereby the selected outcome should be approxi- mately invariant to duplicating an item within ε distance. Note that in this work, we do not attempt to prove exis- tence of DiverseBJR for every instance where BJR holds. We instead (1) precisely define DiverseBJR, (2) extend the existing GreedyCC procedure (Elkind et al., 2022) to con- struct a diversity-aware approximation of BJR, and (3) use a separate greedy simulator as a sufficient feasibility test, en- suring that the imposed diversity constraints do not violate BJR under the greedy policy. 7 4. Experiments 4.1. Evaluation metrics LetA ∈ 0, 1 n×m be the binary approval matrix across all users, whereA u,s = 1iff useruapproves opinions. LetG =G 1 ,...,G γ be a partition of users into political groups, withG g ⊆ [n]and|G g |the size of a given politi- cal groupg. LetS ⊆ [m]be the selected set of opinions with|S| = k. We operationalize the following democratic criteria as quantitative metrics: (i) Individual- and Group-level Representation. Let ̄r u (S)be the ”unrepresentation” indicator for a useru, whereuis unrepresented if they approve none of the selected opinions inS: ̄r u (S) = 1 P s∈S A u,s = 0 . Following (Revel et al., 2025), we measure overall ”unrepresentation” across users as U all (S) = 100· 1 n n X u=1 ̄r u (S).(1) 7 Alternatively, BJR constraints could be checked exactly via ILP, but ILP solving is NP-hard and doesn’t scale. In large online democracy initiatives (e.g., Pol.is consultations with tens of thou- sands of votes (Hsiao et al., 2018)), exact verification is impractical, making approximations necessary in real-world settings. 4 Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study U all (S)relates to classic social welfare, it accounts for total approvals among selected opinions. However, as shown by (Aziz et al., 2017), it does not assess proportionality of representation (a single highly-approved opinion can dominate the sum). Therefore, we also measure the median- voter welfare across political groups. This metric relates to the Rawlsian max-min (maximizing the utility of the worst- off user) used in fairness-aware recommendation systems (Deldjoo et al., 2024). In comparison to Rawlsian max-min, our median measure is robust to outliers (a single highly disadvantaged group can dominate the Rawlsian objective, forcing the algorithm to sacrifice overall quality to ”protect” the worst-case), and is faster to compute. Let the political group-level unrepresentation be defined asU g (S) = 100· 1 |G g | P u∈G g ̄r u (S), we measure ”unrepresentation” across political groups as medianU (S) = median U g (S) : g = 1,...,γ .(2) (i) Consensus. Following recent work in consensus- building via AI-assisted collective dialogues (Konya et al., 2025), we define consensus as the maximin cross-political- group agreement. It is the maximum, over selected opinions, of the minimum approval share across groups. More intu- itively, the measured consensus can be seen as an answer to the question ”Is there at least one selected opinion which satisfies all political groups?” Consensus(S) = max s∈S min g∈[γ] 1 |G g | X u∈G g A u,s .(3) (i) Diversity: Coverage gap (CG). CG quantifies how well the selectedkopinions cover the full opinion space. It is defined as the largest distance between any non-selected opinion and its most similar selected opinion, and is closely related to maximin coverage objectives in recommender systems (Zhao et al., 2025). More intuitively, the least- covered opinion differs from every selected opinion by at least the coverage gap (as a fraction of votes). CG(S) = max o∈[m] min s∈S d(o,s).(4) Note that JR/BJR do not guarantee a low CG. 8 For instance, a very novel but highly impopular opinion could be ignored under JR/BJR. Measuring CG enables surfacing precisely such scenarios (identified by a large CG). In our task, the goal is both to guarantee proportional representation in the selected subset (JR/BJR), but also to surface distinct per- spectives by covering a large spectrum of expressed views (small CG) (Small et al., 2021; Hsiao et al., 2018). (iv) Diversity: Opinion Redundancy. While CG measures how well the selected subsetScovers the full opinion space, 8 Conversely, a small CG does not imply JR/BJR. the opinion redundancy metric measures how many distinct viewpoints exist withinSitself. As discussed in Section 2, redundancy is not desired inS. We define opinion redun- dancy as the fraction of redundant opinions inS. This metric is inspired by outcome-level diversity measures from rec- ommender systems (Deldjoo et al., 2024; Zhao et al., 2025) which we tailor to discrete opinion similarity to match our data. For|S| = k, and given a distance metricdand a di- versity thresholdε, letC 1 ,...,C m denote the clone groups inSformed by grouping opinions that are pairwise linked by the relationd(i,j)≤ ε(where chains are allowed). We define opinion redundancy as Redundancy score(S) = P m g=1 (|C g |− 1) k .(5) In all metrics and baselines, we measure distances using normalized Hamming. In our data, 0 encodes explicit dis- approval (not a missing vote), so shared zeros are as in- formative as shared ones. Hamming therefore captures similarity in both support and opposition. By contrast, Jaccard ignores shared zeros, which is suitable for miss- ingness but not for meaningful disapproval. We thus use d H (a i ,a j ) = 1 n P n u=1 1[a ui ̸= a uj ]. Together, these five metrics operationalize democratic desiderata that are relevant for online deliberation: social welfare, proportional representation, consensus, coverage, and low redundancy. Moreover, no single strategy can op- timize them all: work in information retrieval shows that jointly maximizing relevance and diversity is NP-hard (Gol- lapudi & Sharma, 2009), while work from social choice shows that optimizing welfare or coverage under Justified Representation is also NP-hard (Bredereck et al., 2019). We therefore use them as a comprehensive evaluation tool for existing selection algorithms. 4.2. Baselines To assess adherence to democratic desiderata, we define for each metric a baseline algorithm that optimizes it by con- struction (e.g., bridging targets consensus). We then evalu- ate every algorithm on all metrics to characterize which prin- ciples it promotes and what it sacrifices (e.g., engagement does not target low redundancy). For notational simplicity, we denote implementations approximating each axiom as JR, BJR, and DiverseBJR. We consider three families of baseline selection algorithms: Consensus-based baselines: We include (i) Engagement, a utilitarian rule scoring each item by its number of approvals: f eng (i,A n ) =|u : i∈ A u |, and (i) Bridging, for which we use maximin cross-group agreement (CGA, also called Diverse Approval (Revel et al., 2025)). CGA ranks items by their minimum approval rate across political groups, and has been used in online deliberation (Konya et al., 2025; 5 Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study 2024; 2023a; Revel et al., 2025): f CGA (i,A n ) = min g∈[γ] 1 |G g | |u∈ G g : i∈ A u |. Diversity-based baselines: We operationalize this base- line as greedy minimization of the coverage gap defined in Equation 4. Representation-constraint baselines: We implement ap- proximations of JR and BJR as two distinct baselines using a GreedyCC-style (Elkind et al., 2022) two-stage selection procedure. For JR, we use GreedyCC (Algorithm D.1 in (Revel & P ́ enigaud, 2025)) with thresholdr = ⌈n/k⌉: in Stage 1, while there exists an opinion approved by at leastr currently unrepresented users, we repeatedly add the opin- ion maximizing coverage among those users; once no such opinion exists, Stage 2 fills the remaining slots up tokus- ing a scoring rulef(we use random scoring to minimize dependence on additional scoring rules). For BJR, we add an explicit greedy matching for balance: we fix per-slot capacitiesr j ∈⌊n/k⌋,⌈n/k⌉whose sum isn. For each slotj, if some opinion is approved by at leastr j currently unmatched users, we select the opinion with maximum approval among them and assign (i.e., remove)r j of its approvers; otherwise, we select an opinion byfand assign up tor j of its approvers. When choosing which users to assign/remove, we prioritize users with fewer remaining approved opinions (residual degree). As with JR, we use random scoring for f . DiverseBJR (ours): Building on the BJR-style balanced selection described above, we additionally enforce local ε-diversity using an opinion-distance. We first precom- pute, for each opinion, its set of directε-neighbors (un- der Hamming distance on approval vectors). Selection proceeds inkrounds with the same per-slot capacities r j ∈⌊n/k⌋,⌈n/k⌉and Stage 1/Stage 2 structure as our BJR baseline. In Stage 1, when there exists an opinion ap- proved by at leastr j currently unmatched users, we pick a maximum-coverage opinion, breaking ties to favor opin- ions with fewerε-neighbors and more “unique” approvers (users who do not approve any neighbor), and then assign r j approvers using the residual-degree rule. In Stage 2, we select by the scoring rulef(random), and then assign up tor j of its approvers using the residual-degree rule, while avoiding opinions marked ineligible for Stage 2 because they areε-neighbors of already selected opinions. After each selection, we attempt to mark the newly selected opin- ion’sε-neighbors as ineligible for Stage 2; we do so only if a greedy feasibility check indicates that the remaining rounds can still satisfy the BJR constraints given the remaining voters and candidate opinions. Full algorithm descriptions are given in Algorithm 1 (DiverseBJR) and Algorithm 2 (BJR feasibility checker). In experiments, we setε = 0.8. A sensitivity analysis over ε is reported in Appendix H. We compare all baselines to each other and to Diverse- BJR, our proposed method. We also include (i) a random- selection sanity check (N=100 seeds) to ensure results are non-trivial, and (i) a zero-shot ChatGPT baseline to probe LLM adherence (details in Appendix G). 4.3. Data We use the ”Polarized Issues Dataset” by Remesh. 9 The data contains real-world collective dialogues conducted with N300 representative samples of the US public, focusing on polarized topics such as the right to protest, and in- cludes participants’ political-leaning attributes, captured via self-reported choices drawn fromModerate, Slightly conservative, Very conservative, Slightly liberal, Very lib- eral. Across various questions (e.g., ”What are features or characteristics that make a protest appropriate?”), each participant expresses their personal opinion in free-form text, and votes on the opinions of a few other participants. Following Remesh’s process (see Section 2), missing votes are inferred probabilistically. Following (Revel et al., 2025), we threshold probabilities by 0.5 to get a binary approval matrix, thus matching our problem setting. To control task difficulty, we split questions by how well a random set of five opinions represents users: consensual if it leaves≤5% unrepresented (near-trivial), and controversial if it leaves ≥20% unrepresented (Fig. 1). We choose these empirical thresholds to increase difficulty while retaining sufficient data for analysis (Table 1). We also replicate all experiments on a different dataset with fully observed approvals. 10 Figure 1. Random selection for a consensual (blue) and controver- sial (orange) question. Atk = 5, the consensual question leaves ≤5% unrepresented, whereas the controversial question leaves ≥20% unrepresented, motivating more sophisticated selection. 9 The data is publicly available athttps://github.com /akonya/polarized-issues-data. We preprocess it only to remove duplicate or empty opinions. 10 This validates the robustness of our findings to Remesh’s vote inference and to the 0.5 threshold, see Appendix E. 6 Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study 4.4. Results For each metric, we plot performance with respect to the number of selected opinions across baselines. We focus on controversial questions, where selection is more challenging (Fig. 1). Results on consensual questions are presented in Appendix D. We omit the zero-shot ChatGPT baseline from the main results because it occasionally returns results with fewer than k indices (i.e., selects less than required), which makes the metrics not directly comparable. We report it separately in Appendix G. (1) DiverseBJR typically gives the largest gains in rep- resentation (overall and across political groups) across all baselines when selecting a small subset(k ≈ 2− 3), making it most efficient for deliberation where subset size matters. More generally, across both overall represen- tation (Fig. 2 - upper-left) and group representation (Fig. 2 - upper-right), representation-constraint algorithms (JR, BJR, DiverseBJR) substantially reduce unrepresented users and achieve the strongest early gains, particularly at very small k, consistent with prior work (Revel et al., 2025; Fish et al., 2024; Boehmer et al., 2025). However, they do not system- atically dominate for allk. At moderate k, Diversity can outperform JR and BJR . Consensus-oriented algorithms perform less favorably: Engagement is comparable to or worse than random, while Bridging consistently improves on Engagement with strong early gains but typically remains behind representation-constraint methods. Results on con- sensual questions are reported in Appendix D: trends for representation-constraint algorithms are consistent, while all baselines improve, especially Diversity and Bridging, an expected result since most opinions are widely approved in this set (Fig. 1), making representation easier to achieve without constraints. Note also that some baselines are non- monotonic because they re-select from scratch for eachk. In particular, BJR/DiverseBJR need not have thek + 1solution contain the k solution. (2) Coverage-wise, representation-constraint algorithms (JR, BJR, DiverseBJR) and Bridging generally outper- form Engagement, particularly at smallk. (Fig. 2 - center-left) Diversity typically achieves the lowest coverage gap, as expected since it minimizes CG by design. En- gagement has the largest gaps for mostkvalues, yield- ing the worst opinion-space coverage. On consensual questions (Appendix D), BJR and DiverseBJR generally perform worse for moderate-to-highk, typically trailing JR and Bridging. Overall, this indicates that although representation-constraint methods target proportional rep- resentation, they still deliver broad opinion-space coverage (particularly on controversial questions), matching the goal in online deliberation of guaranteeing proportional represen- tation (JR/BJR) while surfacing diverse perspectives (low CG) (Small et al., 2021; Hsiao et al., 2018). (3) JR/BJR (and DiverseBJR fork ≥ 2) typically select opinions with consensus levels similar to Engagement, but less consensual than Bridging, which maximizes con- sensus by design (Fig. 2 - center-right). This aligns with prior work showing that while JR-style selection can in theory reduce engagement, in practice it yields little engage- ment loss (Revel et al., 2025). Crucially, it suggests that enforcing proportional representation and diversity still al- lows selecting broadly acceptable opinions when they exist. Results are consistent on consensual questions (Appendix D). (4) For small k, representation-constraint algorithms typically select less redundant opinions than consensus- oriented algorithms. On controversial questions (Fig. 2 - bottom), DiverseBJR achieves low redundancy at small k, second only to Diversity (which minimizes redundancy by design). To illustrate the differing selection dynamics, Fig. A.3 visualizes Engagement versus DiverseBJR. Engage- ment selects closely clustered opinions, whereas Diverse- BJR spreads selection across the opinion space, reducing redundancy. Results are consistent on consensual questions (Appendix D), where redundancy rises quickly for all base- lines, as many opinions are similar. Takeaways: DiverseBJR provides the strongest trade-off at smallk: it improves proportional representation rel- ative to diversity baselines while preserving high cover- age and low redundancy. More generally, representation- constraint methods are beneficial for deliberation settings, where only a few opinions can be surfaced. Diversity achieves high coverage and low redundancy but offers no proportional-representation guarantees, whereas consensus- oriented methods can miss minority views and typically underperform on representation. Bridging is a strong com- promise but requires sensitive attributes (e.g., political lean- ings) and depends on their quality. Findings replicate on a separate dataset from a virtual UK citizen assembly (Ap- pendix E). Limitations and Future work: Algorithms can signifi- cantly improve the efficiency of online deliberation by help- ing participants, moderators and facilitators process large volumes of inputs. This work takes a step in this direction by incorporating notions of diversity and balanced repre- sentation into the algorithmic selection process. While our results clearly show how different strategies for algorith- mic selection perform on democratic desiderata, they are still offline results, and may differ from those observed in real-world deployments. In future work, it would be inter- esting to go beyond observational data by deploying these algorithms during an online deliberation, similar to what existing works have done for the related use-case of algorith- mic intervention on social media (Guess et al., 2023; Bail et al., 2018). 7 Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study IDSplitQuestionsParticipants(n)Opinions(m) Q1ControversialHow has your personal experience with protests influenced your viewpoint on the right to assemble? 105105 Q2ControversialWhat are features or characteristics that make a protest appropriate? 307306 Q3ConsensualWhat characteristics or actions, in your view, deem a protest inappropriate? 306299 Q4ConsensualWhat are some of the reasons you have not participated in or attended a protest? 201201 Q5Consensual What measures (if any) could be taken to restrict or limit inappropriate protests? 303299 Q6ConsensualWhat measures (if any) could be taken to ensure appropri- ate protests are protected? 305302 Table 1. Data summary Figure 2. Evolution of various democratic desiderata with the number of selected opinions (dotted lines are used for better distinction of overlapping zones). 8 Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study Reproducibility Statement We provide, in full details, our algorithms’ pseudocode and corresponding empirical parameters, as well as our evaluation protocols and metrics. We also release the code and evaluation scripts athttps://github.com/Sal imHFX/Algorithms-for-Opinion-Selection -in-Deliberationto enable independent replication of experiments and extension of our evaluation framework. Special attention was given to making code easy-to-use, with the goal of disseminating it to both researchers and civil-society practitioners. Impact Statement This work has direct societal consequences on the efficiency and safety of online democracy initiatives. It presents work whose goal is to advance fairness in algorithmic selection in contexts of online deliberation. It assesses how various existing algorithmic strategies perform on specific metrics reflecting democratic desiderata, and proposes a novel al- gorithm that incorporates notions of diversity and balanced representation into the algorithmic selection process. Given the increasingly large amount of online participatory democ- racy platforms and initiatives across various use-cases (e.g., peacebuilding, participatory budgeting, public policy de- sign), it becomes crucial to develop publicly available (i) principled evaluation benchmarks that can be used by prac- titioners and researchers, (i) fairness-aware algorithms that guarantee desired democratic properties and can be safely deployed in such high stakes online contexts. References Abdollahpouri, H., Burke, R., and Mobasher, B. 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Illustrative differences in selection between DiverseBJR and Engagement Differences in the selected opinions’ content: Differences in the selected opinions’ distribution: Figure 3. Visualization of the opinion space - Comparison of selecting 3 representative opinions based on Engagement (left) versus DiverseBJR (right). Embeddings are based on MDS (distance-preserving) - Color = distance to nearest selected opinion (normalized Hamming). DiverseBJR selects more distant (i.e., diverse) thoughts, whereas Engagement selects based on close clusters, therefore potentially selecting redundant opinions. B. Relationship between JR, BJR and DiverseBJR Previous work has shown that BJR is different from JR and does not imply it (Fish et al., 2024). To illustrate the limitations of JR as well as the need for a balanced-matching-based notion of justified representation (BJR), we cite the following standard example previously used by multiple works (Fish et al., 2024; Aziz et al., 2017): 12 Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study α α ′ β β ′ u 0 1100 u 1 1100 u 2 0011 In this example, as was previously observed (e.g., (Fish et al., 2024) Appendix A), JR is satisfied by the slateα,β,β ′ . However, the slate is unproportional as it ”represents two-thirds of the population by one-third of the slate, and vice versa” (Fish et al., 2024). In simpler terms, the selection of very few highly popular items is enough to satisfy JR and can in some scenarios give a false sense of representation in the selected set. For instance, in a task similar to ours, (Revel et al., 2025) find, when selectingk = 8opinions, that the size of JR-sets is typically≈1-2, thus giving no representation guarantees for≈ 75%of the final slate. To solve this issue, BJR introduces the notion of budgets in selection, whereby the users represented by the selected items must be balanced across items, thus avoiding a scenario where the inclusion of a single or a few very popular items suffices to satisfy desired representation guarantees. Next, we illustrate the benefits of enforcing diversity on top of BJR. DiverseBJR enables the selection of more diverse opinions without breaking BJR guarantees under the greedy policy. It does so through two distinct mechanisms: tie-breaking by diversity in Stage 1, and flaggingε-neighbors as ineligible for future picks in Stage 2 iff flagging does not jeopardize future BJR budgets (see Algorithm 1). To illustrate how the diversity-based tie-breaking mechanism works, let us consider the following approval matrix with n=3 voters and m=3 opinions: m 0 m 1 m 2 u 0 100 u 1 110 u 2 011 The task is to select a subset ofk = 2opinions. Letε = 0.7be theε-neighbors threshold (in our example, this means that any two opinions which share at least one common voter are seen asε-neighbors). 11 Let us work through the greedy selection algorithm for all of JR, BJR and DiverseBJR. • JR: – Stage 1:m 0 andm 1 tie on coverage, as both cover 2 unsatisfied voters (u 0 ,u 1 andu 1 ,u 2 , respectively). JR may arbitrarily selectm 0 orm 1 , after which JR will be satisfied: there is at least one user from every cohesive group of at least n/k users who approves at least one selected item. –Stage 2: selects eitherm 1 orm 2 depending on the scoring rulefgiven as input. This stage has no impact on the JR-guarantee which was already satisfied in Stage 1. –Final slate:m 0 ,m 2 orm 0 ,m 1 orm 1 ,m 2 . We assumem 0 ,m 1 as the final slate for the remainder of this example. For both BJR and DiverseBJR, BJR-budgets are set as [2, 1] (see Algorithm 1), meaning that the procedure will contain two rounds, where rounds 1 and 2 will aim at selecting opinions that satisfy 2 users and 1 user (respectively). • Standard BJR: – Round 1:m 0 andm 1 tie on coverage, as both cover 2 unsatisfied voters (u 0 ,u 1 andu 1 ,u 2 , respectively). BJR may arbitrarily select m 0 or m 1 , leaving only one user left unsatisfied. –Round 2: if Round 1 selectedm 0 , Round 2 can select eitherm 1 orm 2 depending on the chosen tie-break rule (both cover the remaining unsatisfied voteru 2 ); if Round 1 selectedm 1 , Round 2 must selectm 0 (to coveru 0 ). No user remains unsatisfied after Round 2. – Final slate: m 0 ,m 2 orm 0 ,m 1 . We assumem 0 ,m 1 as the final slate for the remainder of this example. • DiverseBJR: 11 Specifically, computing distances gives (normalized Hamming)d(m 0 ,m 1 ) = 2/3≈ 0.67,d(m 1 ,m 2 ) = 1/3≈ 0.33, d(m 0 ,m 2 ) = 1.0(Jaccard produces the same neighbor graph here with distances0.67,0.5,1.0), so withε = 0.7theε-neighbors are m 0 − m 1 and m 1 − m 2 (m 0 and m 2 are not neighbors). 13 Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study –Round 1: selectsm 0 after (m 0 ,m 1 ) tie-break (m 0 only has 1ε-neighbor, whereasm 1 has 2), leaving onlyu 2 unsatisfied. – Round 2: selectsm 2 after (m 1 ,m 2 ) tie-break which is solved based on diversity (m 2 only has one neighbor, whereas m 1 has 2), leaving no user left unsatisfied. – Final slate: m 0 ,m 2 Out of all three, DiverseBJR is the only one that guarantees the selection of the slatem 0 ,m 2 , which covers the same voters but is more diverse than the JR/BJR slate candidatesm 0 ,m 1 andm 1 ,m 2 , as it contains less redundancy in voters among the selected opinions. While seemingly trivial, this distinction can have very concrete impacts when considering our task of online deliberation: by selecting opinions which differ the most from already selected opinions (subject to BJR constraints), the final subset ofkopinions typically covers a larger fraction of the opinion-space, and avoids including redundant opinions. These intuitions were validated in our experiments where DiverseBJR displayed less redundancy and better coverage than BJR (see for instance Figure 2: center-left and bottom)). C. Detailed algorithms C.1. DiverseBJR approximation algorithm Algorithm 1 DIVERSEBJR. The algorithm follows a GreedyCC-style two stage procedure, similar to those used to operationalize the JR (Revel et al., 2025) and BJR (Fish et al., 2024) axioms. First, BJR-budgets are computed to ensure that each selected opinion is approved by⌊n/k⌋or⌈n/k⌉users. In the main loop, Stage 1 greedily selects opinions to satisfy the current BJR budget by maximizing coverage, using the diversity criterion only for tie-breaking. When no remaining opinion can satisfy the current budget, the algorithm proceeds to Stage 2, where it fills any remaining slots with a given scoring rule f. At each round, the algorithm attempts to promote diversity by marking near-duplicate opinions as ineligible, but only when a separate greedy feasibility test certifies that the remaining BJR budgets can still be satisfied under the greedy policy. In this way, the algorithm preserves balanced justified representation under the greedy policy while encouraging diversity in the final selected set of opinions. Require: Approval matrix A∈0, 1 n×m , set size k, scoring rule f , diversity threshold ε Ensure: A set W of k opinions 1: Define [n] =1,...,n (the set of voters) and [m] =1,...,m (the set of opinions) 2: Compute ε-neighbors for each opinion (distances are computed based on A) 3: Initialize W ←∅, V ← [n], ineligible-ε-neighbors←∅ 4: Compute BJR-budgetsr 1 ,...,r k by distributing the remainder:r j = ⌊n/k⌋ + 1forj ≤ n mod k, otherwise r j =⌊n/k⌋ 5: for t = 1 to k do 6: S t ←i∈ [m]\ W :|u∈ V : A u,i = 1|≥ r t (unselected opinions approved by at least r t voters in V ) 7: E t ← ([m]\ W )\ ineligible-ε-neighbors (eligible unselected opinions - not currently excluded byε-neighbor rule) 8:if S t ̸=∅ then 9:Pick p∈ S t with maximum approval, break ties by (i) fewer ε-neighbors, then (i) more ”unique” approvers 10:(voters who approve p and no ε-neighbor) (Stage 1) 11:else 12:if E t ̸=∅ then 13:Pick p∈ arg max i∈E t f (i) (filling remaining slots with the scoring rule: Stage 2) 14:else 15:Pick p∈ arg max i∈[m] f (i) (fallback: relax ineligibility) 16:end if 17:end if 18:Add p to W and remove up to r t approving voters from V 19:Add ε-neighbors of p to ineligible-ε-neighbors iff future BJR-budgets remain satisfiable (assessed via a separate 20:greedy simulator for BJR feasibility, see Algorithm 2) 21: end for 22: return W 14 Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study C.2. BJR Feasibility Checker (Greedy simulator) Algorithm 2 BJR FEASIBILITY CHECKER. The feasibility checker is a greedy simulator: given the remaining voters V, candidate opinionsI, and future BJR demandsR = (r t+1 ,...,r k ), it attempts to construct an explicit assignment by simulating the remaining rounds (processing larger demands first). In each round it selects an opinion with at leastr remaining approvers using the same tie-breaking and voter-removal rules as the main algorithm, removesrsuch voters, and continues. If all demands are met, the simulator returns True with a constructive certificate; otherwise it returns False, which is inconclusive. Require:Approval matrixA ∈ 0, 1 n×m , unrepresented voter setV ⊆ [n], candidate opinionsI ⊆ [m](not inWor hypothetically excluded), remaining BJR demandsR = (r t+1 ,...,r k ), number of trialsT ≥ 1, removalpolicy and tiebreak (same as used by the main algorithm) Ensure: True and an assignment mapA if a feasible greedy schedule is found; otherwise False Sort R in nonincreasing order (process largest demands first) for trial← 1 to T do V ′ ← V , I ′ ← I ,A← ∅clear assignment map if trial > 1 then optionally randomize tie-breaking (to produce alternative greedy trajectories) end if flagfail← False for each r in R do C ←i∈ I ′ :|v ∈ V ′ : A v,i = 1|≥ rcandidates with≥ r approvers in V ′ if C = ∅ then flag fail← True; breaktrial failed end if Choose i ⋆ ∈ C according to tiebreak (use same heuristic as main algorithm) Select S ⊆v ∈ V ′ : A v,i ⋆ = 1 with|S| = r according to removal policy (e.g. lowest residual degree) A[i ⋆ ]← S V ′ ← V ′ \ S, I ′ ← I ′ \i ⋆ end for if not flagfail thenall demands filled in this trial return True,Acertificate: greedy schedule exists end if end for return Falseinconclusive: greedy simulator failed to construct a schedule D. Results on additional questions The data is split in two question-types: controversial and consensual (defined in Section 4.3). Due to space constraints, results on only one controversial question (Q1) were shown in the main body of the paper. Results on all remaining questions are shown below. Note that when analyzing and discussing results in the main body of the paper, results on all questions (including ones presented here) were considered. D.1. Controversial questions Results on the second controversial question in the data (Q2) are presented in Figure 4 below. 15 Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study Figure 4. Q2: ”What are features or characteristics that make a protest appropriate?” (307 participants, 306 opinions) D.2. Consensual questions Results on consensual questions in the data (Q3 to Q6) are presented in the figures below. 16 Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study Figure 5. Q3: ”What characteristics or actions, in your view, deem a protest inappropriate?” (306 participants, 299 opinions) 17 Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study Figure 6. Q4: ”What are some of the reasons you have not participated in or attended a protest?” (201 participants, 201 opinions) 18 Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study Figure 7. Q5: ”What measures (if any) could be taken to restrict or limit inappropriate protests?” (303 participants, 299 opinions) 19 Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study Figure 8. Q6: ”What measures (if any) could be taken to ensure appropriate protests are protected?” (305 participants, 302 opinions) E. Replicating results using a Virtual Citizen Assembly dataset In this section, we control for our findings not being dependent on the probabilistic inference run by Remesh to fill missing votes (see Section 2 for a description of how Remesh’s process works). To do so, we reproduce our experimental pipeline on a distinct dataset, namely the Virtual Citizen Assembly data from the Habermas Machine study (Tessler et al., 2024). The study used a sample of≈200 participants that are demographically representative of the UK population, and a set of 9 questions related to UK public policy, e.g., ”Should we reduce voting age to 16?”. Participants were divided in groups of 4-5 participants. Each group received a question. Participants individually expressed their opinion on the question in free-text format, and then voted on all of the remaining group’s opinions. Votes were collected using the following 7-point Likert scale:”STRONGLYDISAGREE”, ”DISAGREE”, ”SOMEWHATDISAGREE”, ”NEUTRAL”, ”SOMEWHATAGREE”, ”AGREE”, ”STRONGLYAGREE”. We convert data to binary votes to match our problem setting, where we map”STRONGLYDISAGREE”, ”DISAGREE”, ”SOMEWHATDISAGREE”, ”NEUTRAL”to a value of zero, and”SOMEWHATAGREE”, ”AGREE”, ”STRONGLYAGREE”to a value of one. We preprocess data to only keep groups with exactly 5 participants for homogeneous evaluation. This results in a total of 174 approval matrices of 20 Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study shape 5x5. In practice, a set of 5 opinions does not need a representative subset, let alone an algorithm to do so. However, this data offers a valuable validation set on which algorithms can be evaluated. We reproduce all metrics and all baselines except the bridging baseline and the consensus metric (both of which require political leaning attributes not present in the publicly available version of the data). We also discard the Zero-shot ChatGPT baseline for the same reasons as with the Remesh data: it failed to reliably respect the requested cardinalityk(e.g., selecting 2 opinions when prompted to only select 1), which precludes fair comparison with other baselines (details in Appendix G). We compute median values to aggregate results on all 9 questions. Results for this Virtual Assembly validation set are shown in Fig. 9. Our results replicate the findings from the main experiments in Section 4.4: JR/BJR/DiverseBJR offer large and quick early gains in representation (upper-left panel - Fig 9), and BJR/DiverseBJR offer a better coverage gap (upper-right panel - Fig 9) and lower redundancy scores (lower-panel - Fig 9) than Engagement. A single unexpected result is observed with the Engagement on representation fork = 1(upper-left panel - Fig 9), where Engagement exhibits better performance than remaining baselines. However, Engagement quickly lags behind all others starting from k = 2. Figure 9. We find that our findings are replicated on a separate dataset originating from the Habermas Machine study (Tessler et al., 2024), involving a demographically representative sample of the UK public. In this dataset, the approval votes are complete, the resulting approval matrix is thus originally dense and does not require any inference (dotted lines in the plot are used for better distinction of overlapping zones). F. Computational Complexity Analysis Results for time and memory complexity are presented in Table 2. For the Diversity and DiverseBJR baselines which require pairwise distance comparisons, a vectorized matrix of distances is pre-computed once before the greedy loop (leading to theO(m 2 n)time complexity). A na ̈ ıve implementation where distances are computed on the fly would have led to a time complexity ofO(km 2 n). For DiverseBJR, the time complexity ofO(Tk 2 mn) +O(m 2 n) +O(kmn)includes the greedy BJR feasibility checker which, when invoked, costsO(Tkmn)per invocation andO(Tk 2 mn)in aggregate (each ofTtrials simulates up tokfuture rounds). Note that in practice, in our experiments,Tis a small constant (e.g., 5-10) and that the simulator which checks BJR is only triggered if at the end of a round neighbors exist and future demands remain. 21 Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study BaselineTime ComplexityMemory Complexity RandomO(k)O(1) EngagementO(nm + m logm)O(m) BridgingO(nm + γm)O(m) Diversity (Greedy Maximin) O(m 2 n) +O(kmn)O(m 2 ) JR / BJR (Greedy C)O(kmn)O(m + n) DiverseBJRO(Tk 2 mn) +O(m 2 n) +O(kmn)O(m 2 ) Zero-shot ChatGPT O(k)× LLM inferenceModel-dependent Table 2. Theoretical complexity of baseline methods.n= number of users,m= total number of opinions,k= number of selected opinions, γ = number of groups, T = number of BJR feasibility trials. G. Zero-shot LLM baseline Implementation: We used ChatGPT (”gpt-4.1-mini”) in a zero-shot setting as our LLM-baseline. We used a temperature of 0 and a nucleus sampling (top-p) of 1 for more deterministic results and enhanced reproducibility. We experimented with various prompting strategies but none mitigated the behavior where the model returned less than k indices. Results: The model occasionally returned less indices (i.e., selected less opinions) than the requestedk, especially for high kvalues (k ≈ 10− 20). Because representation, coverage, and redundancy are computed on the returned indices, variable subset sizes break comparability with other methods: fewer-than-k outputs can systematically understate redundancy and distort representation/coverage, yielding curves that reflect noise rather than selection quality. Since this behavior is not controllable post hoc without introducing additional assumptions (e.g., padding with a fallback rule) and would bias the comparison, we exclude the ChatGPT baseline entirely from the main results. We include one illustrative plot below for completeness (Fig. 10). The ChatGPT baseline performed worse than most other baselines for mostkvalues, but note that this plot should be interpreted with caution because it is biased by inconsistent slate cardinality. Prompt: ”Context: You are an assistant that converts a list ofsize of the opinionshuman free-form opinions on a single question into a set ofkstatements that are collectively representative of the whole set. Treat each input opinion both as a voter and as a candidate statement. Use semantic similarity to infer approvals, cluster opinions to detect groups. The question is: question, and the opinions are: opinions. Output: Your output should be a JSON object with the following shape:”selectedstatements”: [”idx”:<idx statement 1>, ”statement”:<statement 1>, ...,”idx”:<idx statement K>, ”statement”:<statement K>], where each statement in selectedstatements is a verbatim copy of the selected statement, and each idx is the corresponding integer index of the selected statements in the opinions list (not in the selected statements list). Please don’t write any single additional word, and don’t modify any character (don’t uppercase/lowercase, don’t remove punctuation).” 22 Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study Figure 10. Results of the zero-shot ChatGPT baseline, which we omit from the main results because it occasionally returns subsets with less than k opinions, see Appendix G H. Sensitivity analysis of the diversity threshold parameterε We plot the performance of DiverseBJR for various values ofεand compare it against BJR (the baseline it was designed to enhance) and against random selection (sanity check). Across all data, performance enhances with higher epsilon values (see Figures 11 to 16). The value of 0.8 which we set in our main experiments might still be optimizable. Crucially, DiverseBJR enhances performance over BJR for all evaluatedεvalues, making the choice ofεa case-by-case and probably data-specific problem, rather than a choice with impact on the condition the algorithm is designed to fulfill (enhancement of BJR). 23 Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study Figure 11. Q1: ”How has your personal experience with protests influenced your viewpoint on the right to assemble?” (105 participants, 105 opinions) 24 Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study Figure 12. Q2: ”What are features or characteristics that make a protest appropriate?” (307 participants, 306 opinions) 25 Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study H.1. Controversial questions Figure 13. Q3: ”What characteristics or actions, in your view, deem a protest inappropriate?” (306 participants, 299 opinions) 26 Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study H.2. Consensual questions Figure 14. Q4: ”What are some of the reasons you have not participated in or attended a protest?” (201 participants, 201 opinions) 27 Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study Figure 15. Q5: ”What measures (if any) could be taken to restrict or limit inappropriate protests?” (303 participants, 299 opinions) 28 Algorithmic Approaches to Opinion Selection for Online Deliberation: A Comparative Study Figure 16. Q6: ”What measures (if any) could be taken to ensure appropriate protests are protected?” (305 participants, 302 opinions) 29

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