Representative Ranking for Deliberation in the Public Sphere
Authors: Manon Revel
Smitha Milli
Tyler Lu
Jamelle Watson-Daniels
Maximilian Nickel
Venue: ICML 2025 poster (2025)
Keywords: social choice
recommender systems
justified representation
deliberative democracy
ranking
Primary area: social_aspects
TL;DR: Algorithmic ranking with guarantees on representation from social choice theory

Abstract: Online comment sections, such as those on news sites or social media, have the potential to foster informal public deliberation, However, this potential is often undermined by the frequency of toxic or low-quality exchanges that occur in these settings. To combat this, platforms increasingly leverage algorithmic ranking to facilitate higher-quality discussions, e.g., by using civility classifiers or forms of prosocial ranking. Yet, these interventions may also inadvertently reduce the visibility of legitimate viewpoints, undermining another key aspect of deliberation: representation of diverse views. We seek to remedy this problem by introducing guarantees of representation into these methods. In particular, we adopt the notion of *justified representation* (JR) from the social choice literature and incorporate a JR constraint into the comment ranking setting. We find that enforcing JR leads to greater inclusion of diverse viewpoints while still being compatible with optimizing for user engagement or other measures of conversational quality.

Representative Ranking for Deliberation in the Public Sphere
Manon Revel 12
Smitha Milli
Tyler Lu
Jamelle Watson-Daniels' Max Nickel
Abstract
to bring online conversations closer towards the ideals of deliberation, leveraging algorithmic tools in doing so.
Online comment sections, such as those on news sites or social media, have the potential to foster informal public deliberation, However, this po- tential is often undermined by the frequency of toxic or low-quality exchanges that occur in these settings. To combat this, platforms increasingly leverage algorithmic ranking to facilitate higher- quality discussions, e.g., by using civility clas- sifiers or forms of prosocial ranking. Yet, these interventions may also inadvertently reduce the visibility of legitimate viewpoints, undermining another key aspect of deliberation: representation of diverse views. We seek to remedy this problem by introducing guarantees of representation into these methods. In particular, we adopt the notion of justified representation (JR) from the social choice literature and incorporate a JR constraint into the comment ranking setting. We find that enforcing JR leads to greater inclusion of diverse viewpoints while still being compatible with opti- mizing for user engagement or other measures of conversational quality.
What does it take for a discussion to count as delibera- live? Deliberative democracy ideals generally demand at minimum (i) certain aspects of conversational quality (c.g.. respect, reasoned arguments, etc) and (ii) the representation of diverse voices. For example, among other ideals, Fishkin & Luskin (2005) claim that deliberative discussion must be (i) conscientious, i.e., "the participants should be willing to talk and listen with civility and respect," and (ii) compre- hensive, i.c., "all points of view held by significant portions of the population should receive attention." Similarly, in their review of the 'second generation of deliberative ideals,' Bächtiger et al. (2018) highlight the ongoing importance of (i) respect, and (ii) the evolving view of equality as the equal- ity of opportunity political influence (Knight & Johnson, 1997), which requires that people with diverse viewpoints be able to receive attention to their perspectives.
Algorithmic moderation and ranking have become key tools for facilitating conversational quality in online dis- cussions (Kolhatkar al., 2020). For example, Google Jig- saw's Perspective API comment classifiers have been used by prominent news organizations, such the New York Times and Wall Street Journal, to filter or rerank comments to improve civility (Lees et al., 2022; Saltz et al., 2024). More recently, a line of work on bridging systems advocat- ing ranking aims at bridging different perspectives and re- ducing polarization (Ovadya & Thorburn, 2023). Bridging- based ranking has been used in several high-impact ap- plications such as ranking crowd-sourced fact checks on X, Instagram, Threads, Facebook, and TikTok (Wojeik et al., 2022; Meta, 2025; Tok, 2025) and selecting comments in collective response systems like Pol.is and Remesh (Small et al., 2021a; Konya et al., 2023a; Huang et al., 2024).
1. Introduction
In theories of deliberative democracy (Habermas, 1991; Co- hen, 2005; Fishkin, 2009; Bächtiger al., 2018), thought- ful and constructive public discourse is key for a well- functioning polity. Online platforms, such comment sec- tions on news sites or social media, present an opportunity to expand this kind of public sphere deliberation (Helberger, 2019; Landemore, 2024). These digital spaces enable in- teractions between individuals who might never meet in person, potentially exposing them to a broader array of viewpoints. However, in reality, these online discussions often deteriorate into low-quality or toxic exchanges (Nel- son et al., 2021; Kim et al., 2021). In this work, we aim
While these interventions may improve conversational qual- ity, the goal of ensuring representation in online discussions is less studied. In fact, solely focusing on conversational quality might inadvertently result in the censorship of cer- tain groups, in the sense that their comments may be filtered out or do not appear high enough in the comment rank- ing to gain visibility. For example, civility classifiers have been shown to disproportionately flag comments written in English (Sap et al., 2019) and comments
Equal contribution 'Meta FAIR, New York, NY, USA "Berkman Klein Center, Harvard University, Cambridge, MA, USA 'Meta, New York, NY, USA. Correspondence to: Manon Revel <manon @ metiimommmtt..mm.... Smitha Milli < @ mem..
Proceedings of the nnnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 20255 by the author(s).

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Representative Ranking for Deliberation in the Public Sphere
SCORE FUNCTION
Comments
+ set comments
Users
Maximize score function f over all sets of k comments
Uneoprowad
Maximize score function f over all sets of k comments
that satisfy justified representation
approved by
Our Approach
approved by
Standard Approach
Polorr should sopeci proteaars
Warr heney palia control
Fnur poline donen? smork
Frese pilee cosch stark protecien
Unare
No police at proseita
without representation guaranteas!
with representation guzranteosl
Chosen feed S*
Chosen feed Sij
OUTPUT
REPRESENTATIVENESS
OUTPUT
REPRESENTATIIVEESS
Fig. 1. Our approach compared the standard approach ranking comments. In this example, a platform with 6 users wants to select k comments to highlight while optimizing given score function / (e.g. engagement, civility, diverse approval, etc). This example is inspired by our experiments ranking comments related campus protests in See 6. In the standard approach, the platform simply selects the & comments with the highest score (in this case, diverse approval across the red and blue users). However, this leads only two users having comment that they approve of in the selected set S The two remaining red and blue do receive representation despite the fact that they each form a group large enough to warrant representation by proportionality (each of size [n/k] 2 [6/3] 2) and are minimally cohesive (have at least one item that they agree upon in common). On the other hand, our approach, the platform picks the & comments that maximize the score while satisfying justified representation, and guarantees that all cohesive groups of size at least n/k are represented in the resulting set S'R--
discussing encounters with racism (Lee et al., 2024). In the context of diverse approval, the groups being bridged have a large effect on what content gets shown. Commonly, these groups correspond to political groups like left- or right- leaning users (Wojeik al., 2022). Therefore, it is plausible that diverse approval could give more visibility to moderate comments, while failing to provide representation of others who hold more ideologically diverse views.'
the JR constraint affects the platform's ability to max- imize different classes of score functions. Despite negative results in worst-case settings, we show that in the natural setting where users' opinions cluster into a few groups, enforcing representation does not come price to other objectives such civility user engagement ($ 5).
3. Method and empirical analysis. Leveraging approxi- mation algorithms from social choice, we implement a JR constraint for ranking comments on Remesh² related to campus protests. We find that enforcing JR significantly enhances representation without com- promising other measures of conversational quality. Specifically, we evaluate the price of JR for metrics beyond simple engagement, including diverse approval and content- based bridging classifiers ($ 6), and find a low price in all settings, as predicted by our theoretical results ($ 5).
In this paper, our goal is to broaden the scope of delibera- tive ideals examined in the algorithmic facilitation of online discussions. particular, we extend the existing focus on conversational quality also include ideals of representa- tion. In brief, contributions the following:
1. Theoretical framework. We contribute a theoretical framework in which the platform selects the top-ranked comments by maximizing a score function (e.g. ci- vility, bridging objectives) subject to a representation constraint known as justified representation (JR) (Aziz et al., 2017) from the social choice literature ap- proval voting in multi-winner elections.
2. Related work
Unlike aggregative models of electoral democracy (Manin, 1997), which rely on citizens' periodic votes to express po- litical consent (Landemore, 2020), deliberative democracy
2. Theoretical analysis. We theoretically analyze how
!Or, imagine an online platform popular with fans of the Celtics and Knicks (two U.S. baskethall teams) where the only posts that get diverse approval are those about Kadeem Allen (a former player in both franchises). Bridging-based ranking la Ovadya & Thorburn (2023) may overly represent the Kadeem Allen posts, downgrading legitimate interest of the broader fan-bases.
*Remesh (Konya et al., 2023a) is popular collective response system (Ovadya, 2023) used frequently by governments, non- profits, and corporations clicit opinions from collective scale.
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Representative Ranking for Deliberation in the Public Sphere
anchors political legitimacy in discursive processes through which citizens continuously participate and consent to the polity's 's activity (Habermas, 2015). Deliberative democracy scholars have identified certain ideals that democratic de- liberation should strive to uphold (e.g. respect, absence of coercive power, etc), and Bächtiger et al. (2018) review how conceptualizations of these ideals have evolved over the years. More recently, scholars have begun exploring the potential for online spaces to democratize and scale deliber- ative discourse (Buchstein, 1997; Helberger, 2019; Gelauff et al., 2023).
of any demographic or social labels, making it compatible with the privacy and legal constraints of real-world plat- forms (Holstein et al., 2019; Veale & Binns, 2017). Second, JR automatically adapts to the groups that are relevant in different contexts. For instance, the groups relevant to rep- resent in the discussions about the NYC budget planning process differ from those in a post about a basketball game between the Celtics and the Knicks. On real-world plat- forms that have diverse range of content, the fact that JR naturally focuses on the relevant groups is significant advantage.
Algorithmic interventions for comment ranking focus pri- marily on promoting conversational quality (Kolhatkar & Taboada, 2017; Saltz et al., 2024; Lees et al., 2022; Ovadya & Thorburn, 2023; Jia et al., 2024; Piccardi et al., 2024), but without guarantees on representation of viewpoints another key ideal of deliberation. To incorporate such guar- antees into these methods, we focus on selecting the set of top k comments, which allows us to leverage notions of pro- portional representation. While other research on diversity in recommender systems also focuses on selecting optimal sets of items, diversity in this context is typically defined based on the content of items, e.g., choosing videos from variety of topics (Kunaver & Požrl, 2017; Zhao et al., 2025). In contrast, our goal here is to select a set of comments that ensures representation, thereby focusing on the perspective of users rather than the content diversity of the items.
The primary theoretical question we analyze is: what is the price of enforcing JR, with respect to the ability to optimize other score functions? In the recommender system context, it essential that a representation constraint be compatible with other objectives of interest for the platform (e.g. civility of comments). The need to optimize other objectives well is also why we start with the fundamental JR axiom rather than its stronger extensions (Aziz et al., 2017; et al., 2017: Skowron et al., 2017; Brill & Peters, 2023; Fish et al., 2024; Peters& & Skowron, 2020). Nevertheless, in our empirical experiments, we find that all but one of our JR committees also satisfies the stronger EJR+ axiom (Brill & Peters, 2023), consistent with other studies that find minimal differences between these axioms in real-world settings (Boehmer et al., 2024).² 3
The closest related theoretical work is by Elkind et al. (2022; 2024) who investigate the price of JR with respect to utili- tarian social welfare (the number of selected comments that each user approves of, summed over users). Other related work includes Skowron et al. (2017) who study the per- formance of various approval- based committee rules with respect to social welfare and coverage and Bredereck et al. (2019) who establish that maximizing social welfare or cov- erage with respect to JR ha-hard. In contrast to Elkind et al. (2022), we analyze the price of JR for more general classes of score functions, which are relevant in the con- text of recommender systems where objectives beyond user approval are also important, as well as constraints on the user approval matrix. We show that in the common setting when users' preferences cluster into a few groups, the price of JR, even with respect to arbitrary objective functions, is typically negligible ($ 5). These findings also relate to Faliszewski al. (2018) who use clustering as an empirical heuristic to achieve fully proportional representation.
To formalize representation, we borrow the social-choice- theoretic concept of justified representation (JR), developed by Aziz et al. (2017) in the context of approval voting in multi-winner elections. Justified representation is an axiom that formalizes a notion of proportionality: it guarantees that every large enough group of users that has shared pref- erences (approves of at least one comment in common) is allocated at least one of the top comments. While prior work has focused on selecting or generating comments subject to JR extensions of JR (Halpern et al., 2023; Fish al., 2024; Bernreiter et al., 2024), these works do not consider the optimization of other exogenous objectives (e.g. content- based classifiers for civility or other attributes) that are a key component of real-world comment ranking. Therefore, we study a constrained optimization problem where the plat- form selects the top comments based on optimizing a score function (e.g. civility, diverse approval, engagement, etc) subject to JR.
For the comment ranking setting, in particular, JR affords several advantages over notions of demographic or social representation more commonly used in the algorithmic fair- ness literature (Barocas et al., 2023; Chasalow & Levy, 2021). We discuss these advantages further in Appendix A, but here, we highlight two benefits. First, unlike most al- gorithmic fairness methods, JR does not require inference
3. Problem Setting
In this section, we define the comment ranking problem, the axiom of justified representation and the price function used to examine how ensuring representation might affect
"See also Bardal et al. (2025) for similar findings beyond the approval voting setting.
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the ability maximize other objectives.
is approval dependent if, for all items not in any approval set of An (that U Au), then we have Ai, 0.
3.1. Model
For m E N, we write [m]] = (I,...,,,) Let [ma] be the set of all comments and [r|] be set of users. The platform must select subset S S [mm] of k comments to highlight.* 4 Bach user u E [n] approves a set Au S [ma]] of comments and is the profile of approved comments
Engagement. A simple example of an approval dependent scoring function is the utilitarian scoring rule in which each item's score equals the total number of users that approve it:
An. = -A......, } across all users. In practice, user's approval set might be defined as the set of comments that they upvote, like or react to. In all, each problem is characterized by a tuple Im..A..k An. ke) of n approval sets over m comments, from which & comments are selected.
Seng (i, A..) = Iu A.
(2)
In prior work rooted in the multi-winner election setting, the utilitarian scoring rule has been referred to as 'social welfare' (Elkind et al., 2022). In the context of comment ranking, the approval profile A., would, in practice, likely be defined by user engagement. For example, a user might be said to 'approve' of a comment if they upvote or like it. Therefore, in this setting, it may be more accurate to think of the utilitarian scoring rule as an engagement- maximizing scoring rule that social media platforms are incentivized to optimize for. As such, we refer this scoring rule as the engagement scoring rule.
3.2. Scoring Rule
Comments are ranked based on scoring rule: each com- ment é E [m]] is assigned a score fi., 2 0. The score of set S C [m]] is additive; Am. = Eies f(i, An)) For occasionally write f(i) or f(S) and drop the potential dependence on the approval profile An We use S (m..A to denote the subset of [mm] with highest score for the instance Im,A..kk
notational simplicity,
Diverse approval is another example of an approval depen- dent scoring rule, where the score of a comment reflects the level of approval it receives across diverse groups of users. These groups could be pre-defined, e.g., based on demographic characteristics, or more commonly. learned from the data. Diverse approval is motivated by research finding that, compared to pure engagement, diverse approval tends to correlate with comments that are less toxic, more informative, and higher quality (Ovadya & Thorburn, 2023; Wojcik et al., 2022). Diverse approval has been used to select and rank user-generated content in high-impact ap- plications such as ranking crowd-sourced fact checks on social media platforms (Wojeik et al., 2022; TokT 2025; Meta, 2025), and selecting comments in collective response systems like Pol.is and Remesh (Small et al., 2021a; Konya et al., 2023a; Huang et al., 2024).
f(S" (Im.Anik))=
max f(S).
(L)
s[m)).s)
3.2.1. 1. GENERAL SCORING RULES
In general, we consider arbitrary scoring rules f that are potentially independent of the approval profile An. For example, scoring rule fc could be classifier that assigns score to comment based solely on its textual content. A notable example are the Perspective API classifiers, a suite of models that have been used by prominent news organizations like the New York Times and Wall Street Journal, for algorithmic filtering and ranking of comments. The Perspective API models assess comments for attributes such toxicity, compassion, reasoning, and more, relying solely on the comment text for classification.
Typically, in a diverse approval metric, users are partitioned into Y non-overlapping groups GI...., G, S [m]. The number of groups quite small in practice (Wojeik et al., 2022: Huang et al., 2024). One simple diverse approval met- ric, which we refer to maximin diverse approval (MDA), scores each comment by its minimum approval rate across groups:
3.2.2. APPROVAL DEPENDENT SCORING RULES
We also consider a natural class of scoring rules that are approval dependent in that the score of comment i cannot decrease if a new user approves of it, holding all else equal. Formally, we write A', Z(t) A, if A', = (i). Then, we define an approval dependent scoring rule the following.
fDA (i. An) min 1 Hu: u E G E A.H. sE(e] |G,,
(3)
Definition 1 (Approval Dependent). A scoring rule (i.,An)
"Work on diversity in recommender systems also focuses on sets of items, but with the goal of showing "diverse" items where diversity is typically defined based on the content of the items, e.g.. showing videos spanning different topics genres (Kunaver & Požrl, 2017: Zhao al., 2025). contrast, our focus is on selecting sets items (comments) to ensure that the top items provide a degree of proportional representation to users.
Other variants have also been used such as the product of the approval rate across groups (Small et al., 202 Huang et al., 2024) or softmax version of minimax diverse ap- proval (Konya et al., 2023a). Note that these examples satisfy property that is strictly stronger than approval dependency, that of approval monotonicity where an ap- proval monotonic function is an approval dependent func- tion such that for all approval profiles A²,, and An with
'If there multiple highest-score sets, we pick one randomly. Our results do not depend on the bie-breaking rule for S*
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Representative Ranking for Deliberation in the Public Sphere
Al (1) A, for some and A A, for all v e [n] {u), then A(,) 2 Al).
3.4. The Price of Justified Representation Justified representation is not guaranteed to be compatible with the objectives of general scoring rules." We formally analyze the question: what price must be paid in the to- tal score if we require that selected items satisfy JR? We define the optimal JR set Syn Am f) as the set that maximizes the score among all sets that satisfy justified representation":
3.3. Justified Representation
The problem of selecting k items from a set [m] based on individual's 's approval ballots is well studied in the context of multi-winnes social choice, where the k "items" are can- didates to be selected for committee." In order to ensure fairness, in the sense that all large groups with cohesive preferences receive some amount of representation, (Aziz. et al., 2017) axiomatically formalized the idea of justified representation. Following (Elkind et al., 2022)'s 's exposition, and (Bredereck et al., 2019)'s notion of justifying sets, as- sume we are given an instance Am..A.. of approval sets over comments, from which the platform must select k comments.
arg max f(S) Sc(m]. (|=
(4)
s.t. S satisfies JR.
Next, we define the price of JR as the ratio between the maximum (unconstrained) score and the maximum JR score, both for a specific instance, and as the maximum over all instances for a given k.
Definition (The Price of Justified Representation). We define the price of justified representation on an instance (Eq. 5) and over all instances with set of size k (Eq. 6) as
Definition 2 (Cohesiveness)) A group of users G S [r]] is said to be cohesive if all the users in G approve at least one common item: NueG A, * 0.
(Im. f))
Definition 3 (Representativeness), A set of items S S [m]] is further said to represent a group G of users if at least one user from G approves of at least one item in S : Bu E G such that A, S # 0.
P(ImiAn f) = J(S Am
(5)
f(k,f) =
mmxx P(Im.An Imm.A.kk
(6)
Definition 4 (n/k- justifyinng set). A set of items S S [m]] is a n/kk justiffinnggi set it represents every cohesive group of at least n/k users.
This formulation follows that of Elkind et al. (2022) who analyze the price of JR P(k, feng) for specifically the en- gagement scoring function feng- In later sections, we will also analyze a probabilistic version of the price of JR in Sec. 5.2 and prove high probability bounds over the distri- bution of instances in addition our worst-case analysis of the price of JR.
The concept of Justified Representation (JR) ensures that large enough cohesive groups are minimally represented.
Definition (Justified Representation), An n/ j/t-justifyyin set S [m]] is said to satisfy justified representation over Imn.A..k if [S] = k.
Finally, when either the instance and scoring function is clear from context, we short-hand S (Im.A,. f) and SjR (Im.An.k: f) by S* and SjR+ respectively. Similarly, when the scoring function clear from context, we short- hand P(Im,Ankk. f) and P(k, f) as P(Im.An.)) and P(k). respectively.
In the comment ranking setting, cohesive group is set of users that approves of at least one comment in common. If cohesive group contains at least n/k users, then by the principle of proportionality, since we are selecting k comments to highlight, this group is considered deserving representation. A cohesive group G C [n]] is said to be represented by set of comments S C [mm] if at least one user in the group Gi approves of a comment in S.
4. The General Price of JR We first study the price of JR without imposing any con- straints on the approval profiles of users. We analyze the price of JR for general scoring functions and for approval- dependent scoring functions. Unfortunately, we find that for either class of functions, the price of JR can be quite high. All proofs are deferred to Appendix C.
"An intuitive approach would be to select the items with the largest approval score Hu E [v] i E A. However, this method tends to favor items supported by the majority, potentially excluding minority groups from representation. Let a world with 60 people who approve the 10 items in set and another people who approve of a distinct set A' of 10 items. If a committee of size 10 is selected based on approval scores, the winning committee, A would fail to represent 10% of the world. Instead, committee composed of 6 items from A and 4 items from A' would respect an intuitive notion proportionality.
First, we establish that for general scoring functions, which includes functions that are independent of the approval pro- file such as content classifiers, the price of JR need not be bounded.
"We show how JR confficts with diverse approval in Fig. B.I. "Our results hold irrespective of random tie-breaking.
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Proposition 1. There exists a function J such that P(k,) f) is unbounded.
by clustering or matrix factorization of the user approval matrix, resulting in groups that are relatively distinct and homogeneous (Wojcik et al., 2022; Small et al., 2021b).
The result is not so surprising as, in general, if the scoring function can be independent of the approval profile (as with a classifier fc), we need not expect that it gives representa- tion to cohesive groups which are defined by their common approval. This exemplifies the potential negative externali- ties of using content classifiers that do not account for the users' own approval of different comments.
5.1. Homogeneity as Cohesiveness
We first illustrate the price of JR in the setting where within- group homogeneity is operationalized the JR sense. That is, users are partitioned into y non- overlapping groups G1...., G- C [m] and each of these groups is cohesive.
cohesiveness in
We next turn to analyzing the price of JR for approval de- pendent rules (Def. 1) which require that if an item has a positive score that someone must have approved of it. We show that the price of JR is k for such rules. In fact, the result is true even if we restrict ourselves to bridging scoring rules used in practice like maximin diverse approval.
Theorem Assume that k > 1 and users are partitioned into Y cohesive groups GI....,G, Gry where 1 $ k. For any scoring rule f. the price of justified representa- tion P(k, J') is at most بーद. h., and is exactly that for certain scoring rules such as maximin diverse approval fDA-
Assuming there are only a few divided groups, i.c. y is small, the price of JR of k/ -7) close to 1 (the lowest possible value).
Theorem 1. Assume y > and there exists an item with positive score. For any scoring function that is approval- dependent, P(k, J) is at most k and is equal to k for certain -eeeeddnn scoring rules such as maximin diverse approval fDA.
5.2. Homogeneity as Low Dispersion
Operationalizing homogeneity as cohesiveness was illustra- tive, but in real-world scenarios, it may be unlikely for all the members in a divided group to approve on one comment in common. For more realistic model, we now operational- ize within-group homogeneity in the context of common statistical model of preferences. In particular, we assume that users' preferences over items are drawn from a Mal- lows mixture model (Awasthi et al., 2014; Liu & Moitra, 2018), where each divided group has its own component in the mixture. We then operationalize homogeneity by the level of dispersion @ [0, of individual preferences from their group-specffic central ranking. In this setting, we find that with high probability, the price of JR is at most O (k/ (k- - 7 [(log k)/ log(1 (1))))) which approaches our previous bound of O(k/(k - 7)) as within-group homo- geneity increases, i.e., as Φ + O. Let us begin by defining the Mallows model.
For the comment ranking setting, a price of k may still be high. For example, suppose we use maximin diverse ap- proval to score comments and select 10 comments to high- light. Our results imply that, in the worst-case, enforcing representation can mean that the optimal JR set n attains 1 1/10-th the diverse approval that the optimal unconstrained set S* does-an undesirable outcome if diverse approval is a priority. In the next section, we show that adding a natu- ral assumption on the clustering of user preferences yields much lower bounds on the price of JR for all scoring rules.
5. The Price of JR in a Clustered Setting
Given the potential high price of JR when users' approval profiles can be arbitrary, we now turn to asking whether there are natural restrictions where the price of JR is low. We analyze one such setting- the setting in which users' preferences cluster into a few groups. This scenario is not only common in practice but also is the setting where bridg- ing interventions, which aim to bridge over few polarized groups, are most relevant. In particular, we assume that users can be partitioned into y groups GI...., G., S [m] on the basis of their approval profiles. We refer to these groups as divided groups to contrast them from the cohesive groups that are relevant for justified representation.
Definition 7 (Mallows model). The Mallows model M (Ø, σr) is described by central reference ranking a and a dispersion parameter & E [0, 1]. The probability of observ- ing a preference ranking π ~ M o) is given by
Pr(t; Φ σ) = o (( 0) (Z))),
(7)
where d is the Kendall-tau distance, which measures the number of pairs of items (z, y) that two rankings disagree on (i.e. x preferred y in one ranking and y to x in the other ranking). Finally 2 = 1- (1+ Φ) (1 + Φ + +0) 0== is the normalization constant.
We show that when ? is small and the groups display high bithin-group homogeneity, the price of JR is low. We oper- ationalize within-group homogeneity in two distinct ways, showing that the price of JR remains consistently low un- der both interpretations. These results are notable, because, in practice, when bridging interventions are implemented, (a) y small and (b) divided groups are often determined
+z).....( ..)....... (1+
Note that when the dispersion parameter ò is equal to zero, then the model generates the reference ranking with proba- bility one. At the other extreme, when the dispersion param- eter Φ is equal to one, then the model becomes a uniform
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distribution over rankings. Thus, low values of Φ indicate greater homogeneity.
the real-world setting we explore ranking comments on a collective response system (Konya al., 2023b), where users shared their thoughts on campus protests and the right to assemble. The simulations investigate the implications of our theoretical findings based on the Mallows mixture model and can be found in Appendix D.2.
We model preferences as being drawn from a Mallows mix- ture model where each divided group has its own compo- nent.
Definition 8 (Mallows mixture model), The Mallows mix- ture model (bb o, A) combines y Mallows models. Let Φ (δ). ................................ ②4) be a collection of dispersion parameters, (0) ( ..) be collection of reference rankings, and A (A1,...,,A,) X, be the mixing proportions where for all i S 7, we have X; 2 0 and λ +... Xy = 1. The mixture model is defined by the probability mass function
The Greedy CC algorithm and its price. Solving for the optimal JR set S NP-hard in general (Bredereck et al., 2019; Elkind al., 2022). For small instances, we may an ILP formulation, but for large instances (e.g. on social media platforms), we would expect practitioners to resort to approximation algorithms. To understand the effects we might expect in practice, we too employ an approximation algorithm in all our experiments, along with an associated approximate price of JR. Specifically, we use the GreedyCC algorithm (Elkind et al., 2022; 2023) to find set SJR satisfies JR and approximately maximizes the score function f. In brief, our implementation of GreedyCC greedily adds items to the selected set until a nz nn/k-justiffyin set is found; the remaining items are then selected to maximize the score function f. A full description of the algorithm is given in Alg. D.I.
7
Pr(T;0,0,N1 = Σ d Pr(T;;
5Gredd that
A sample from the Mallows model represents a user u's ranked preference T. over all items. For the approval voting setting, we assume that user approves of the top T items in their ranking n, i.e., Av = n. (r='((r]) ((r)). We can obtain the following analytical upper bound the price of JR, which is independent of any particular algorithmic strategy.
Given instance Am,,,, also define associated price of GreedyCC based on comparing the score of the set GGrody returned by GreedyCC and the optimal set S (which is not constrained to satisfy JR):
Theorem 3. Suppose that each user u ranking nu is drawn from the Mallows mixture model (1 o X) and their ap- proval set A, is equal to the top T items in their ranking, denoted by π. ([r). Let Omax = max; 65 be the maxi- mum dispersion parameter for any group (component of the mixture). Then, for any scoring function f, and for all @max E [0, 1]. with probability 1 ô over the i.idd draws of user rankings Tu. [ri], that maps to the approval profile An, we have
f(5 (Im,An,k Groody (Im.An f))
PGreddy (Im,An.ks f) =
(10)
Remesh dataset. We now investigate the impact of enfore- ing JR in real-world setting: ranking comments about campus protests and the right to assemble." "I These com- ments were generated as part of two sessions on Remesh, a popularly-used collective response system (CRS) (Ovadya, 2023), each with about 300 participants (see Appendix E. for summary statistics about the dataset). Collective re- sponse systems are used by governments, companies, and non-profits to elicit the opinions of a target population at scale, in a more participatory way than traditional polling, In particular, on a CRS, participants give their opinion to different questions via free-for formform responses, and then vote on whether they agree with other participants' comments. !2
k
P( max (0, k 9)
(9)
where
1 - Ø max
k: q == 7 log 88)
/
log 0max
i
m 7)
log m log(m - 7)
if =1
Observe that as Φmax + 0 and the groups become increas- ingly homogenous, we recover the previously established bound of k/(k- - -7) in the setting where divided groups are cohesive (Thm. 2).9
Experimental setup. We examine the impact of ranking
"The data are available at https: https:://itttb.coomm pateryapolaaizedd-a seseddissuss-datt
6. Experiments
"Individuals only give feedback only small of com- ments, however, we use inferences of the full approval matrix. In particular, we take the probabilistic agreement inferences (Konya et al., 2022) conducted by Remesh (in which each user and com- ment are given probability of that user agreeing with that com- ment), and threshold these inferences by 0.5 to get binary ap- proval matrix.
We empirically investigate the impact of ensuring JR through both real-world experiments and simulations. 10 In
Note the ceiling in the bound.
1urr code huilds upon the abcvot ing Python package for approval-base -baee multi-winner voting rules (Lackner et al., 2023).
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Representative Ranking for Deliberation in the Public Sphere
What are your thoughts on the way university campus administrators should appreach the issue of Israel Gaza demonstrations
What are some of the reasons you have not participated attended protest?
What measures any) could be taken to ensure appropriate protests protected?
What are you impressions of how campus adminittrators handled the protests?
What are features or characteristics that make protest appropriate? Pi PUsxim Pif2-114
PU4,)=2007- Pilo01108 PUci110
PiGegeaa PUax-111 PUgl-La
Pife--1822 PU6x)-182 P.lew1as
56 Unrepresmned Users
30-
20-
10-
Eng- Jeng
Uiverse Approval
Pers. API Jt
Eng Jos
Uivese Approval
Pers. API Je
Eng.
Diverse Approval /OA
Fers. API JC
Eng.
Liveree Approval Jux
Pers. API
Eng.
Diverse Approval
Pers.
API
Jc
Joa
Jc
Without JR (S7)
With JR (5 (1
Fig. 2. Representation when selecting & Remesh comments, with and without a JR constraint. individual who approves at least one comment in a set S is said to be represented. The JR constraint significantly increases the number of individuals who are represented. The prices of are reported with the shorthand P(f) plreddy (A: f). Due to space constraints, the figure only shows the five questions for which S" did not satisfy JR under diverse approval. Fig. E.2 shows results for the remaining five questions.
the comments using the following three scoring functions, with and without JR constraint.
S" (fDDA). or the Perspective API set S* (fc). respectively, After enforcing JR with GreedyCC, these percentages re- duced to 5%, 4%, and 2%, respectively (Table E.4).
1. Engagement feng (Eq. 2).
Ranking with Perspective API always satisfied JR. With- out explicitly enforcing JR, the engagement S (feng). di- verse approval S (fDx), and the Perspective API set S* (fc) satisfied JR 30%, 50%, and 100% of the time, respectively, This was surprising as the Perspective API does not explic- itly aim to satisfy JR. Moreover, both diverse approval and the Perspective API classifiers are meant to to"bridge" across groups (Ovadya & Thorburn, 2023). However, Perspective API satisfied JR at a much higher rate and also provided better overall representation (Fig. 2). This could be be- cause diverse approval focuses on bridging across a few pre-defined groups (in this case, the three political groups), whereas the bridging attributes targeted by the Perspective API might appeal to broader range of groups.
2. Maximin diverse approval fDA where participants' self- reported political ideologies were used to form the three groups (left-, center-, and right-leaning users) that were considered in the function SDA (see Eq. 3).
3. A Perspective API classifier fc that scores how "bridg- ing" comment is based upon its text (Saltz et al., 2024). The score of each comment is its average score for the seven "bridging" attributes in Perspective API: affinity, compassion, curiosity, nuance, personal story, reasoning, and respect.'
For each question in the Remesh dataset and each score function f, we compare the optimal set S (J) of k = 8 com- ments (without JR constraint) and the set JR (f) which optimizes the score function while ensuring JR through the GreedyCC approximation algorithm (Alg. D.1).
Enforcing JR improves representation across political groups. As shown in Appendix E.6, enforcing JR with GreedyCC improves representation for all three political groups 100%, 100%. and 70% of the time, respectively (although in the case of Perspective API, the set already satisfied JR without GreedyCC). This is notable as the JR criterion does not rely on any explicit ideology labels of users.
6.1. Results
Enforcing JR leads to more users being represented. We say that individual is represented in the set S* or Grreedy if there is at least one item in the set that they approve of. For all ten questions and all three scoring functions, enfore- ing a JR constraint increases overall representation (Fig. 2). Without enforcing JR, 18%, 15,, and 4% users were unrepresented in the engagement S (foug, diverse approval
Enforcing JR comes at low price. We know that the price of JR P(k, feng) for engagement is in O @(vk) (Elkind et al.,
15 All our results rely on Remesh's inferred approval votes and biases in these inference could impact our results. For example, if the inferences are biased towards predicting approval for longer texts (we do not know if this is the casc), which are also scored higher by the Perspective API, then the Perspective API might appear to be provide more representation than it actually does.
htt.tttppp/eeeeeeeeeeeeeeeeeeeee wwwpeerrsppprpppeeee perspect ieserspectisspp....... Leeapi......... com/ s aaabbottthh--pp-attriibteee aeattrrruuee--ad-aanniie Landdd-angagges for details each attribute. We also report results broken down by individual autributes in Appendix E.6.
"We filter the data to remove any empty or duplicate comments.
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Representative Ranking for Deliberation in the Public Sphere
2022), for maximin diverse approval SDA is equal k (Thm. 1), and for general score function like the Perspective API may be unbounded (Thm. 1). Thus, in the worst case, we would expect that P(k, feug) < P(k, fDA) < P(k, fe). We do indeed find this ordering when evaluating the price of GreedyCC, however, the prices are all also much lower than the worst-case bounds. Letting pGreedy (f) be the price of GreedyCC pGreddy (Im.An.8. f) averaged over the ten questions (instances) in the dataset, we found that:
proval matrix specifying which users approve which com- ments. It is not always clear which users should be included in the approval matrix. For example, should it be all users on the platform or only those who saw the post? The answer to this question may be different for different platforms.
Second, on real-world platforms, users will not "vote" on all comments, and thus, their approval of various comments will need to be inferred (Halpemn et al., 2023). To provide users with genuine representation, it is essential to ensure these inferences are accurate. (Our Remesh experiments in Section 6 also relied on inferences of the approval matrix and should be interpreted with this in mind.) Moreover, on many platforms, users' approval will need to be inferred from engagement data such as upvotes or likes. If the chosen engagement significantly diverges from actual user approval, the validity of the process could be compromised. Investi- gating the impact of biased approval votes, in a similar vein to the work of Halpern et al. (2023) or Faliszewski et al. (2022), could be an interesting direction for future work
poreedy (Seng) = 1.05
< pCrreedy (JDA)= = 1.06
Preedyy (fc) 1.18.
These low prices may be explained by the presence of small n/ j-jjutiifying sets. For all questions, GreedyCC was able to find an n/ jujjuisiygngn set with only at most 7 = 2 com- ments (Table E.3), which from Theorem 2, implies that the price of JR can be at most k/(k- (k- 8/(8- -) = 1.33. 16 A small n/ / j-justifying set would also be expected in set- tings where user opinions cluster into only a few groups (the same setting we analyze in Section 5).
Finally, it is crucial to test the effects of enforcing JR through A/B tests, as offline results may differ from those observed in real-world deployments. In our Remesh exper- iments, found that enforcing JR came at little cost to other conversational quality measures and also user engage- ment. However, these were "offline experiments" involving the re-ranking of historical data, without deploying our new algorithm to users, and further testing is still needed.
The JR feeds also satisfied EJR+. Finally, out of the 48 JR feeds in our experiment, 47 of them also satisfied EJR+ (Brill & Peters, 2023), strengthened version of JR. Although more experimentation is needed, these findings may indicate that for comment ranking, in other real- world applications of JR (Boehmer et al., 2024), the practical differences between these axioms may be minimal.
Future work. In this work, we focus on analyzing JR (although empirically, we find that all but one of our JR committees also satisfies EJR+). In future work, it would be interesting to analyze stronger axioms such as EJR (Aziz et al.. 2017), EJR+ (Brill & Peters, 2023) or BJR (Fish et al., 2024). Additionally, our work focused on the classical JR setting, where a set of k top comments is selected. However, it could be interesting to explore extensions of JR that apply directly to the ranking setting (Skowron et al., 2017; Israel & Brill, 2024), although it is not immediately clear how to integrate scoring function into these extensions.
7. Discussion
The online public sphere presents an opportunity to scale informal public deliberation. However, current tools for prosocial moderation and ranking of online comments may also inadvertently suppress legitimate viewpoints, under- mining another key aspect of deliberation: inclusion of diverse viewpoints. In this paper, we introduced a general framework for algorithm ranking that incorporates a repre- sentation constraint from social choice theory. We showed, in theory and practice, how enforeing this JR constraint can result in greater representation while being compatible with optimizing for other measures of conversational quality or even user engagement. Our work sets the groundwork for more principled algorithmic interventions that can uphold conversational norms while maintaining representation.
Acknowledgements
We thank Hoda Heidari, Seth Lazar, Aviv Ovadya, Ariel Proccacia, Luke Thorburn, and Glen Weyl for their feedback on this project. We thank Andrew Konya and Lina Qiu for sharing the Remesh inferred approval votes with us and making them publicly available for future research.
Limitations. Nevertheless, there are still additional ques- tions that should be considered before deploying our frame- work on real-world platform.
Impact Statement
Online comment sections have the potential to serve as important platforms for public discourse and deliberation, though their effectiveness is often undermined by the low quality of conversations. Algorithms can potentially sig-
First, the JR representation constraint depends on an ap-
16The proof of Theorem 2 follows from the fact that, if the population of users can he partitioned into 7 cohesive groups, then there is an n/ /k-justifying set of size 7.
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Representative Ranking for Deliberation in the Public Sphere
nificantly improve these spaces by facilitating more con- structive public sphere deliberation. This work takes step in this direction by directly incorporating ideals of repre- sentation, as formalized in social choice theory, into the comment ranking process. We demonstrate that this ap- proach can maintain conversational quality while ensuring diverse viewpoints are represented. Nonetheless, as dis- cussed in our limitations section, before deploying this on a real-world platform, more testing is necessary ensure that implementations of this approach genuinely provide users with representation.
Bredereck, R., Faliszewski, P., Kaczmarezyk, A., and Nie- dermeier, R. An Experimental View on Committees Pro- viding Justified Representation. In Proceedings of the -renttyEighth International Joint Conference on Artifi- cial Intelligence, IJCAI-19, pp- 109-115. International Joint Conferences on Artificial Intelligence Organization, 7 2019. doi: 10.24996/jjcaii2019//1 URL https: / //doi. 10 wttp://///.....//..a..aaa.a..aa. 2019/16.
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Piccardi. T., Saveski, M., Jia, C., Hancock, J. Tsai, J. L., and Bernstein, M. Social Media Algorithms Can Shape Affective Polarization via Exposure Antidemocratic Attitudes and Partisan Animosity, 2024. URL https: //arxiv. wtg/ abs/ 2411. 14652.
Wojeik, S., Hilgard, S., Judd, N.. Mocanu, D., Ragain, S., Hunzaker, M. B. F., Coleman, K., and Baxter, J. Bird- watch: Crowd Wisdom and Bridging Algorithms can Inform Understanding and Reduce the Spread of Misin- formation, 2022. URL https: http//axivvv.r... 2210 15723.
Saltz, E., Jalan, Z., and Acosta, T. Re-Ranking News Com- ments by Constructiveness and Curiosity Significantly Increases Perceived Respect, Trustworthiness, and Inter- est, 2024. URL https http//axiivv.////////.////e wrg. /tt/////ww...r////////44444 05429.
Zhao, Y., Wang, Y., Liu, Y., Cheng, X., Aggarwal, C. C., and Derr, T. Fairness and Diversity in Recommender Sys- tems: A Survey, ACM Trans. Intell. Syst. Techmol., 16(1), January 2025. ISSN 2157-6904. doi: 10. 1444//664/2... URL hww.httts//aee /doi. http://.......01.1.....0.1...... 36655/664222 928.
Sap, M., Card, D., Gabriel, S., Choi, Y., and Smith, N. The Risk of Racial Bias Hate Speech Detec- tion. In Korhonen, A., Traum, D., and Marquez, L. (eds.), Proceedings of the 57th Annual Meeting of the Asso- ciation for Computational Linguistics, pp. 1668-1678, Florence, Italy, July 2019. Association for Computa- tional Linguistics. doi: 10.18663/w1/111-1113. URL https: //aaaaaanhhloogyo/ wrtp:y/-///1111199111
Skowron, P., Lackner, M., Brill, M., Peters, D., and Elkind, E. Proportional Rankings. In Proceedings of the Twenty-Sixe Sixth International Joint Conference on Arti- ficial Intelligence, IJCAI-17, pp. 409-415, 2017. doi: 10.24963//jeca:22017/8. URL https:////// /taes://doo... org/ 10. 10.24963/ijjcn 2017
Small, C., Bjorkegren, M., Erkkilä, T., Shaw, L.,, and Megill, C. Polis:
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Small, C., Bjorkegren, M., Erkkilä, T.. Shaw, L., and Megill, Scaling Deliberation by Mapping High Dimen- sional Opinion Spaces. RECERCA. Revista de Pensament Anàlisi, 26(2), jul. 2021b. doi: 10.6035/reeerrca.5516
C. Polis:
URL
index. httpp///eeeeeeaatttaaee//ee////a
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Representative Ranking for Deliberation in the Public Sphere
A. How does JR compare to demographic-based representation?
is worth discussing how our JR-based approach differs from other, perhaps more familiar, notions of demographic or social representation from the algorithmic fairness literature (Barocas et al., 2023; Chasalow Levy, 2021). JR bottom-up measure where the groups that are represented depend entirely on the users' own approval of different comments. This has several advantages for the comment ranking setting.
First, is the feasibility for real-world applications. In algorithmic fairesss, groups are typically pre-defined based demographic, such as race or religion. In contrast to JR, the requirement of (inference of) user demographic labels make applying many algorithmic fairness methods infeasible or difficult to implement in industry settings, including on social media platforms, because conflicts with privacy and legal constraints (Holstein et al., 2019; Veale & Binns, 2017).
Second, JR has the flexibility to automatically represent different groups for different settings. For example, the groups that are relevant represent when ranking comments on an article about the 2025 New York City budget planning process different from the groups that are relevant to represent on a article about a basketball game between the Celtics and Knicks. On a news site or social media platform where the relevent groups to represent for each post can be vastly different, it is difficult to scale up algorithmic fairness approaches that require pre-specifying the set of dimensions or demographics to consider.
Third, JR may accommodate intersectionality (Crenshaw, 1991) more natural way than many algorithmic fairness approaches. Algorithmic fairness approaches to intersectionality attempt to provide guarantees to wide set of sub- groups (Gohar & Cheng, 2023), e.g., by defining intersectional groups the combination of different demographic attributes (e.g. age, gender, race) (Kearns et al., 2018). However, this approach can lead to issues where the subgroups longer correspond to meaningful entities. For instance, intersecting many dimensions can result in subgroups that are too specific and lack a meaningful reason for grouping (e.g.. "Jewish white males aged 18-35 who have a bachelor's s degree and live in a rural ). In contrast, JR focuses on cohesive groups-groups. of users who can agree on at least one comment in toemon inherently embedding some requirement for meaningfulness.
Finally, in cases where representation across pre-defined groups (e.g. based on demographic labels) that important, these groups still be considered through the score function f. We have already given example, the diverse approval score function fDA defined in Equation (3), that uses explicitly defined groups in its computation.
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B. An Example of Conflict between Diverse Approval and JR
Here, we provide an example where minimax diverse approval ÍDA leads to sets that do not satisfy JR. The example is similar to that of Figure There are two groups, Gi and G2, that are being bridged in the diverse approval objective function fDA (see Equation (3)). However, the only items that receive approval across both groups are because of two individuals F and G who approve of them (and no others). Thus, maximizing diverse approval leads to set that F and G are represented in, but no one else.
11
G2
A
B
C
D
E
F
G
H
1
K
L
1
x
3
X
4
5
and k 3. There 7 cohesive groups G1 = {A, B, C, D2 F, F} and G2 = {G, H, J, K, L.} such that |Gız2 = > n/k. Note that the sub-groups such as { K, L} are also cohesive and larger than n/k, so they ought contains {1, 2}. On the other hand, the comments' maximin diverse approval scores (across the groups G1 and G2) are f(1) f(2) = 0
Fig. B.I. When diverse approval fails at representativennss An approval profile with
(A, B, C, D}
be represented by JR. turn, any JR set
and f(3) = (4) (5) = 1/6. In turn, the set with the highest score 5" (3.4,5) 4,5.
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Representative Ranking for Deliberation in the Public Sphere
C. Proofs
C.1. Proof of Proposition
Proof. m > & items and y groups each of size n k such that every group member in group G, approves of one item st unique the group. Each group G, is cohesive and of size n/k JR set shall include all these unique items s; for all E [r]] Recall that general scoring rule (e.g. bridging) may be based on the output of classifier whose input the items (e.g. comments) and henceforth, be independent of the approval profile. In turn, we can construct an instance where the general scoring rule is such that f(s;) = ε for all these items and a larger constant e for at least one of the remaining items. Then f (S) = ks while f(S") 2 is positive constant. We can make e arbitrarily small for an unbounded P(k).
C.2. Proof of Theorem 1
We first prove the following lemma. This statement can be found in (Bredereck et al., 2019) (Theorem 3), but the authors do not provide a proof for the result in their paper so we write one below.
Lemma 1. If item i" is approved by one person, then there exists a JR set that contains i*
Proof. Let us distinguish two cases. Case I: There exists a set of size strictly less than & that satisfies JR. Then, trivially, take an arbitrary set S that satisfies JR. If item i is not already in S, add i" (and complete with any remaining item if necessary to create set of size k). This set of size l satisfies JR.
Case 2: The only sets that satisfy JR are of size k. This happens if and only if 3 a set T of size k such that Vi E T, 3 group of na/k users g = {uu.- ... Bn/kk} } such that i E Au and i & A' E [n] \ In other words, we are in Case 2 in there exists a set T of size k such that every item / E T is approved by exactly n/k people why do not approve of any other item in T. Note that such I satisfies JR.
Recall that is approved by at least one person who belongs to one of the groups g of size n/k. That is, there exists a user group g of size n/ l such that u E 9 and i E Ag- We will denote by ig the item in T that is approved by all users in g
and users in [r]] \ g.
If 1" & T, define S = T i-
Then, S is a set of size ls that satisfies JR given that i" represents the same group as
Let's next prove Theorem 1.
Proof. Let 1" be the item with the highest score s". By additivity of the scoring rule, for any set S of size k and any approval
profile An, fSS, An) S kst. In particular,
f(S",An) S ks"
(11)
Next, by approval dependency of the scoring rule S, there must be at least one user who approves item i (otherwise, its score is 0 well as that of any other item and, by approval dependency, there does not exist any item approved by anyone, a trivial case for which the price is not defined). By 1, there exists JR set Syn that contains item i*. This may not be the optimal JR set R but, by definition of the optimal set, f(SjR. An) 2 f(SJn An).
Further note that, by additivity again,
J(SJR, 2 S*
(12)
Combining Equations (11) and (12), we get that for any approval profile An, ((SIRIA)) (())((A)) S k.
C.3. Proof of Theorem 2
Proof. First, we prove that P(k) S J. Let S* {51, 84 } be the score maximizing k-set with respect to f, where the items are ordered by their scores such that f(s1) 2 (32) 2 2 (sx). Since each of the cohesive groups GI.... G. unanimously approves of at least one item, we can construct a set R with R I 7, containing one unanimously approved item from each group. Since the groups GI....,Gr G., partition the set of users, every user approves of at least one item in R. Therefore, any set of size i which contains R satisfies JR. The remaining k - 7 can be chosen to maximize the scoring rule
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Representative Ranking for Deliberation in the Public Sphere
f. Consequently, there exists set S which satisfies JR and contains the k 7 highest scoring items, 81, 82, 8k-y- This implies that f(S) is at least = f(S)), which shows that P(k) S k-7"
We can adapt the above to show that the bound can be tight. Let each group be of equal size zt > dt and each group unanimously approve of exactly one distinct item so that R = Y (in turn, users do not approve of any items other than their group's 's designated item). Note that every JR set must contain the set R. Therefore, if the items are such that for all r E R, fr) = 0 and for all s E S*, f()( = e for some positive constant e (the top k highest item scores are equal), then the maximum score for JR set is {(S)R) = (K- 7) while the optimal (non-JR) score is f(5") kc. This establishes that for certain scoring functions.
P(k) -
We show that diverse approval is one such scoring function. Let y = nike (WLOG assume n + k divides nk) groups with one consensus item per group (suppose it is item 2; for group i = 1, 1,.... 7) approved only by the members of that group. Further, there is one unique member in each group that approves of an additional k "shared" items (for concreteness suppose these items are y1.....: V/s therefore each group has member that approved of all these items). There are hence m = 7 items such that the first y items I1, 21,.... 2n approved by n/k + 1 persons within each group and have diverse approval score of O, and the last k: items V1..... Us are approved by 7 persons with a diverse approval score of I+n Each divided group contains a cohesive sub-group of size n/k (these individuals all approve of item 2;)) so JR committee needs to include 21,...,, :.g. A set S can become a JR set by completing it with k y of the remaining items yi,.... Wk--y In turn, f(S)R) = (Is (0)--))x k and f(S*) = k k knn
k4m
C.4. Proof of Theorem 3
There are several generative interpretations of the Mallows model (Lu & Boutilier, 2014) but one that is particularly useful for analyzing homogeneity is based on the Repeated Insertion Model (RIM) (Doignon et al., 2004). In our analysis, we make use of modified version of RIM in which items from the reference ranking are inserted starting with the last ranked item and ending with the first ranked item. It can be shown this bottom up RIM sampling induces the same distribution as the usual top down RIM sampling.
Bottom Up Sampling of Mallows क (( a)
1. Let or denote the item ranked i-th in the reference ranking
2. Initialize и as an empty ranking.
3. Loop i m, m 1,...;1: 1:
Insert 0, into π at rank position j where 1 J S m - +i 1 with probability 01-1/(1 +
This generative version of Mallows gives us a nice way to express and bound the probability the top items are ranked below a given threshold.
Lemma 2. Let π ~ M(ò,o). a). for all S m, the probability that none of the top s items in the reference ranking appear in the top T items of is
(1 (000--mm-))/CC -0-)--1111)) 6mm if <
Pr(m(1)>,,,(( > (2) r...., ((})) 5) (1 T/m)²
(13)
ifo=1
Proof. Let o; be the i-th ranked item in reference ranking a and (8) be the ranking obtained after inserting items om..... de. Therefore the final sampled ranking æ (11), Let 5 = {a. < < 8} and U = (o, s + S i S m).
Case 1: s 2 m r. Then there is not enough items in U to fit in the top items of π because [U| S r. Therefore an item from S must be ranked in one of the top positions of Hence Pr(vi
n 7) = 0.
Case 2: < m 7. We proceed by induction on Base case we have = and it is required that σN be inserted at position
7 ++1 1 or below which occurs with probability (Φ" + +... + 6th 10 )/(1 + Φ + ..... -1)
Suppose Ineq. (13) holds for all s up to some s 2 1. Let = m, m -....,, , 1 be the loop iteration index as in the Bottom Up Sampling procedure and define f = i ++1 1. The first item from S gets inserted when 1 (i.e. 8). Let
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Representative Ranking for Deliberation in the Public Sphere
Dt min (onns> srrssa 5) be the rank position of the highest ranked item from S at iteration with index i - 8+ 1. We show DE is non-increasing in l. We argue by induction on l. In base case when & = 1, 91 is the position item dT is in (T(*), Suppose 9r is non-increasing for all up to some t. Consider & + 1 when we insert 0----2 if it is inserted above position de then 96+1 < ge and if it is inserted at or below ge then -8-1 = ge. Therefore each item in S must be inserted at position r + or below with probability (Ø² +... 6 1/(1 + ... + "n- Applied for all items in we multiply these insertion probabilities.
+ m
Pr(vi Ss (s)) ())
Φ ++1+1
1 m-+++
F=1
m
m i i+1 1
x 6ー
if <
1
6m
(14)
T T/m,
o ==1
[=1
where the inequalities are obtained when the quotient is maximized at i = 1. We get the upper bound in Ineq. (13) by noticing the terms Ineq. (14) no longer depends on i.
Proof of Theorem 3. We begin by showing that there exists an n m//k-justifying set of size with probability at least 1 8. The bound on the price of justified representation is derived directly from this result. Consider the set S = or ((s)) U
02 ([s]) U... U o, ([s)) comprised of the top s items in each group's 's reference ranking. The probability that there exists an n/ justtustiygnng set of size y s is greater than or equal to the probability that the set S is an n/k justifying set, which we can lower bound as follows:
Pr(3S' s.t. [S"| = 7s and S' is an n/ justifying set)
2 Pr(S is n/k justifying set)
2 Pr(l(u: A, n = 0) < <nkk)
= 1 - Pr(l(u: (u: Au n S = 03 2 n/k)
1- - Pr
Σ Xu N n/k
numl
( (1 "max)) )01(1))1111)1)) ((11
6m i <
2 1 - k.
(15)
if
where X, 1fu Ag n = 0} is the random variable indicating whether user has no item they approve of in the set S.
By Lemma 2, we can bound Pr(X,, 1) as in Ineq, (13). Thus, Ineq. (15) holds by Markov's inequality. When @max < 1,
solving 1- k 6 (1 "m==)=== / )max)" ) 2 1 for the parameter s. yields 2 log(k/8) log((1 (111100 (0m1@mmmmm <mmx)/ / ( (1
"m>²" ))). turn, with probability at least
5, there exists an n/k justifying group of size <ooo(k//)) log((1
*max) (f (1 ( When @max the same calculation provides that, with probability at least 1 8, there exists an n/k justifying group of size (l/loo((/))))) log(m/ - +))]-
When S < k, any set W of size & that contains S satisfies justified representation. For the remaining items in W, we can pick the max (0, k- q) items with the highest bridging score. Thus, the highest bridging score f(S)R) attainable by set that satisfies justified representation is at least max(0.k-)) f(S*) where S* the bridging-optimal set (unconstrained I for justified representation). Therefore, have the desired result that the price of justified representation is such that P(Im,,A, ky f) S k/ max(0, k - g).
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D. Simulations
In the following sections, we investigate the implications of our theoretical findings in simulations in the Mallows mixture model.
D.1. The GreedyCC CC approximation algorithm
First, we review the GreedyCC algorithm used in both sets of experiments. Given that solving for the optimal JR set Sin is NP-hard general (Bredereck et al., 2019; Elkind et al., 2022), we would expect practitioners to resort to approximation algorithms. Thus, understand the effects we might expect in practice, in all our experiments, we too employ an approximation algorithm, along with an associated approximate price of JR. Specifically use the GreedyCC algorithm, which has also been used in prior work (Elkind al., 2022), find set S that satisfies JR and approximately maximizes the score function f. In brief, our implementation of GreedyCC greedily adds items to the selected set until a n/ /K-justifyinng set is found; the remaining items are then selected to maximize the score function f. A full description of the algorithm is outlined in Alg. D.1.
Given an instance Am we also define an associated price of GreedyCC based on comparing the conversational score of the set Grreody returned by GreedyCC and the optimal set S (which is not constrained to satisfy JR): JR
S(S" (Im..n.k.. Creeddy An
PGreatty (Im,,A. R. 1)
(16)
S(S pGreody AIm,
pGreedy fk, Am
max
(17)
Algorithm D.1 The GreedyCC algorithm proceeds in two stages. In the first stage, it identifies an n/ justiffsti set by greedily selecting comments maximize coverage. "Onee n/ k-jjstiifyiggg set is found, the algorithm proceeds the second stage, where fills any remaining slots with the comments with highest conversational norm scores.
1: Initialize V = [n] as the set of all voters
2: Initialize S C [m]] as the set of all items that are approved by at least n/k voters in V
3: Initialize W = ()
4: while [W| < do
5: if' [S] > 0 then
6: Add 1" arg max fu V EE Au) to W {Stage }}
7:
else
8:
Add i = arg ggw [(i, An) to W (Stage 2}
9:
end if
10: < the set of voters who do not yet approve of an item in W
11: 5 the set of all items that are approved by at least n/k voters V
12:
end while
13:
return W
D.2. Mallows Mixture Model Simulations
In simulations with the Mallows model, we empirically investigate the price of GreedyCC and compare our findings to the theoretical bounds derived in Section 5.2. Our theoretical bounds showed that when the population is clustered, in the sense that the population can be divided into a few groups that exhibit high within-group homogeneity, then the price of JR can be shown to be low. However, is this still true when using an approximation algorithm like GreedyCC, which is what a practitioner would use in practice?
Experimental setup. To investigate this, we created a Mallows mixture model
M2((o, o), [01 (1/2, 1/2) that simulates a polarized environment with two groups We draw inspiration from Esteban Ray (1994)'s axiomatic framework for measuring polarization. Esteban & Ray (1994) where they consider polarization to be present three intuitive properties are satisfied. We operationalized these three properties in our simulation follows:
The coverage of a set comments S the total number of users who approve any comment in S.
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Representative Ranking for Deliberation in the Public Sphere
Criteria I: There must be small number of significantly sized groups. The mixture model had two components and each component had probability of 1/2
Criteria 2: There must be high degree of heterogeneity across groups. We chose the central rankings or and of the two components to be exactly the opposite.
Criteria There must be a high degree of homogeneity within groups. Each component had the same dispersion paremeter é, and we varied this parameter to see the impact wihtin-group homogeneity, which our theoretical results depended upon.
As in Section 5.2, we generate approval votes for user's preference and taking the top items to be approved. Based upon the approval vote thresholding. we let / denote the random variable corresponding to an instance simulated from the Mallows mixture model, and denote by D the dataset of s i.i.d. instances from the Mallows mixture model. We evaluated (1) the expected price of GreedyCC, over instances, estimated using the sample mean E, [pGreedy (I, f)), and (2) the worst-case price of GreedyCC observed in the dataset, P. p.oreedy (k, f):
([o ø), (0r, ø2). 1//2, by thresholding
E [.Grody (1,
Σ
prreddy (I, f)/s,
(18)
IED
prrody (k, = max
Proddy (I.
(19)
ZED
Results. Figure D.I shows how the price of GreedyCC changes as the dispersion parameters e changes, for two scoring
functions: engagement Seng (i, An) (Equation 2) and maximin diverse approval ÍMDA An. where the two groups in the definition of fMDA (Equation 3) are the two components in the Mallows mixture model.
First, despite GreedyCC not being optimal, we find that the price of GreedyCC is low (close to one-th the lowest possible value) for both engagement and diverse approval when there low dispersion, as predicted by our theoretical results. For both engagement and diverse approval, we find that as the dispersion increases, the price of GreedyCC increases until the dispersion is very high (at which point cohesive groups unlikely exist), and the price decreases again.
Second, we find that the price of GreedyCC is higher for diverse approval than for engagement. For diverse approval, some instances result in price of GreedyCC that nearly our theoretical bound from Theorem 3. The gap in price between diverse approval and engagement is consistent with our result Theorem 1, which found that the price of JR for maximin diverse approval P(k, fMDA) equals k, whereas Elkind et al. (2022) demonstrated that the price for engagement P(k,, feug) is (v))). These results were shown, in the general setting, without any restrictions on the approval profiles. On the other hand, the theoretical bound we derived for the polarized setting in Theorem 3, and plotted in Figure D. holds for all score functions. But given the empirical disparity in price between diverse approval and engagement, it seems plausible that a similar fomereffuntion-ppiee
gap could be identified even in the polarized setting.
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Representative Ranking for Deliberation in the Public Sphere
Price of GreedyCC in Mallows Mixture Model Diverse Approval
Engagement
1.
1.2-
1.2-
1.0
1.0
E. pGreedy (I, f)]
0.0
0.2
10.4
0.6
1.88
10
0.0
11.2
0.4
0.6
0.5
10
+1 Standard Deviation
PGreedy (k, f'
2.0-
2.0
Bound from Theorem 3
t.
1.0
1.0
6.0
0.2
0.1
0.6
0.8
100
0.0
0.2
0.4
0.6
0.8
1.0
Mallows Dispersion Parameter (Φ)
Fig. D.I. The price of GreedyCC in Mallows mixture model. The price of GreedyCC shown for both the engagement scoring rule Seng (left) and diverse approval fnx (right) with a 25 (top) and = 50 (bottom). The simulations are for instance Z100,A100: 10 where the approval profile Aino is simulated through Mallow mixture model with Kendall-tau distance, y polarized groups with opposite reference rankings, and approval threshold ofr 25. The average price, B, [pGresty (T) (solid lines), and the maximum observed price P. Gready (I) (dotted lines), are both computed over 000 simulations for values of & in [0.1, 1] and 0.01-stepsize. The probabilistic bound from Theorem 3 (dash-dot black lines) is computed with 95% confidence, i.e., 8 = 0.05.
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E. Experiments Ranking Comments on Remesh
In this section, we now provide supplementary details and results for our Remesh experiments from Sec. 6.
E.1. Dataset Summary Statistics
For these experiments, we used data from Remesh sessions conducted in the summer of 2024 that engaged representative sample of Americans regarding their opinion of campus protests. The data are available at ht.tps: / / thub. com/ cooonyappolzze zed issues sasuues-ata... The following table lists the 10 questions that participants were asked about, across the two sessions, and the total number of participants and comments for each question.
ID
Questions
Participants (n)
Comments (m)
What are your thoughts on the way university campus administrators should approach the issue of Israel/Gaza demonstrations?
307
306
2 What are your impressions of how campus administrators handled the protests?
308
308
3 What should guide university campus administrators handling of protests?
301
298
What are your impressions of the campus protests?
310
310
5 How has your personal experience with protests influenced your view- point on the right to assemble?
105
103
6 What characteristics or actions, in your view, deem protest inappropri- ate?
306
297
7 What are some of the reasons you have not participated in or attended a protest?
201
200
8
What are features or characteristics that make a protest appropriate?
307
306
9
What measures (if any) could be taken to ensure appropriate protests are protected?
305
297
10 What measures (if any) could be taken to restrict or limit inappropriate protests?
303
284
Table E.I. Remesh Survey Questions and Number Participants (n) and Comments (m)
E.2. Perspective API Classifiers
We used three scoring functions to rank comments: engagement feng. diverse approval fDA, and a bridging score based on Google Jigsaw's Perspective API fc. For the Perspective API score function fc, we used the average of the scores for the seven "bridging" attributes available in Perspective API (see https: /wweellelllle/eee//eeee//eeee/ee wwweellpers..omm Perspecti ivAPII com/ s/ (abbutttaae
tat-aaariiuutee -arooutts--nn-aaaaaaaaaa Ianguagesgggss (taiggeeslaanagggg Language- _US): Nuance, Compassion, Personal Story, Reasoning, Curiosity, Affinity, Respect. Below, we show the score distribution of the Remesh comments, for each attribute.
Histograms of Perspective API Bridging Attribute Scores
Nuance
Compassion
Personal Story
Reasoning
Curiosity
Affinity
Respect
Frequency
400-
300-
D
Score of Comment
Fig. E.I. Scores from the Perspective API
For our main results reported in Figure 2, Table E.2, Table E.4 and Figure E.2, we use the average score across these seven attributes as the Perspective API score function fc. However, we also report results with each individual attribute in
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Appendix E.6.
E.3. Price of GreedyCC when Ranking Comments using Diverse Approval In Table E.2, we provide the price of GreedyCC pGrssy AA.., f) for each question (instance Z) and score function f. We also indicate how often the optimal set (f) (which was not explicitly constrained to satisfy JR) satisfied JR.
Engagement
Diverse Approval
Perspective API Index
Questions
pGreedy (I, Seng)
Seng feed is JR
pGreedy (I, SDA)
SDA feed is JR
PGreedy (I,
Jc feed is JR
1
1.03
False
1.05
False
1.11
True
2
1.02
False
1.02
True
1.11
True
3
1.00
True
1.00
True
1.06
True
4
1.03
False
1.03
False
1.11
True
5
1.09
False
1.08
True
1.07
True
6
1.11
True
1.11
True
1.15
True
7
1.06
False
1.08
False
1.08
True
8
1.03
False
1.03
False
1.14
True
9
1.07
False
1.10
False
1.19
True
10
1.07
True
1.08
True
1.16
True
Table E.2. Price of GreedyCC and frequency of satisfying JR for each score function and question. score
E.4. Size of the n/ n/K--usttffying Set Found by GreedyCC
In Section 5.2, we show that when the user population clusters into few groups, then the price of JR will be low because is possible to find a small n/ /--justifying set (and thus, the remaining items for the set can be chosen to maximize the score function). Indeed, this logic how the GreedyCC approximation algorithm that we use operates. As detailed in Algorithm D. the GreedyCC proceeds in two stages. In the first stage, it finds an n //l-justifyying set. In the second stage, selects the remaining comments that maximize the score function. Thus, the smaller the n /ejjustiifying set, the lower the price of GreedyCC will tend to be.
A small n/k- jujttffiii set is expected in empirical settings where user opinions cluster into only a few groups (as is common for politically-chareed questions). Table E.3 shows the size of the n/ k-justifying set found by GreedyCC for all ten questions in the Remesh dataset. For all questions, the n/ j-jjutiffyng set has at most only two comments. The small size of the n/ ik-justifying sets could then explain why the price of GreedyCC (Table E.2) across all questions and score functions is so low (close one all cases).
Questions n/k justifying set
2
2
2
3
1
4
2
5
7
8
2
9
2
10
2
Table E,3. The size of the /t/kk-jsstfyyinn set found by GreedyCC for each question.
23

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Representative Ranking for Deliberation in the Public Sphere
E.5. The Impact of JR Representation of Participants
In Figure 2, we showed the proportion of participants who were unrepresented when ranking Remesh comments both with and without JR constraint. Due to space constraints, only included the five Remesh questions for which S* (/DA) did not satisfy JR in Figure under diverse approval. Here, present the results for the remaining five questions (Table E.4 and Figure E.2).
Engagement
Diverse Approval
Perspective API Index
Questions
S
Grredy JR
S
S
S*
Greedy JJ
6
2
31
4
23
3
8
4
22
10
15
5
16
1
2
6
2
6
10
0
10
5
7
4
17
3
3
25
8
28
3
10
12
1
10
Average across all feeds
18
5
15
5
Average across the five feeds Figure Average across the five feeds in Figure E.2
4
5
Table E.4. The percentage of users who unrepresented when ranking with each score function, both with the JR constraint (S and without the JR constraint (S*)
What characteristics or actions, your deem protest inappropriate?
How has your personal experience protests influenced your viewpoint on the right to assemble? Pilm/-116- Pifoe-e PUC--1ur
What measures (if any) could be taken restrict limit inappropriate protests? Pil PUexi-10e P11
What should guide university campus administrators handling of protests?
What are your impressions of the campus protests? P04--110 PUa0-123 P10-11
P04,)+1.11 P160-141 PUci-sas
PU---112 Paa--U PUg--1s
% Unropeeenntd Users
30.
10-
Diveree Approval
Pers API
Diverse Approval
Pers API
Diverse Approval
Plers. API
Liverse Approval
Pers API
Liverrse Approval
Pers API Jc
Without JR (S*)
With JR (8 Credr)
Fig. E.2. Representation when selecting comments using diverse approval with and without a JR constraint (missing feeds).
In Figure E.3, we plot the average score per comment for the set that maximizes the scoring function f as function of the percentage of users represented by the selected set. This is shown for both the unconstrained optimal set (*) and the Grreddy (ff) that satisfies JR and approximately maximizes the score function We effectively calculate (()) and
set S SJR
scoeyy (f)) respectively.
E.6. Results Per Political Ideology
In this section, we report the representativeness results across the different political groups for the three scoring functions: engagement Seng- diverse approval SDA. and the Perspective API bridging classifier fc-
l' acknowledge anonymous reviewer for suggesting this plot.
24

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Representative Ranking for Deliberation in the Public Sphere
Engagement
Diverse Approval
Perspective API
260
Average Score (feng) of Set
Average Score
(Fax) of
0.80
Average Score
(fc) of Set
0.80
168
Without JR (S")
0.65
0.65
With JR (5)R (
75
0.50
0.50
100
80
100
95
100
% of Represented Users
% of Represented Users
% of Represented Users
Fig. F.3. The score-optimal sets 5" and the JR sets Greedy for each Remesh question (see Table E.1) and score function (engagement, diverse approval, and the Perspective API bridging score). Each set is plotted based on average score the the set and the percentage of users represented in the set. Compared to the score-optine score-optimal sets S", the JR sets Syx notably increase the percentage of represented users without significantly reducing score of the selected items.
Ranking Remesh Comments by Engagement
What characteristics actions, your view. deem protest inappropriate?
What reasons you have not participated in or attended protest? (Gomo((S fc)=1.0007
What measures (if any) could be taken to ensure appropriate protests protected?
How has your experience with protests influenced your viewpoint right assemble? pámey(8 fc)=1.06
What are features or characteristics that make protest appropriate? pámetr(8, (t) fc)-1.03
(S fc)--.11
plrntt (8. fc)-1.09 1.09
% Unipprsooedd In Group
60-
10-
Right
Left
Right
Left
Right
Right
Left
Right
What are your thoughts on the way university ad aimiisttatoor should approach the issue of
What measures (if any) could be taken restrict inappropriate protests?
What should guide university campus administratod handling of protests?
What are you impressions of how
What your impressions the campus protests?
administrators
Israel Gaza demonstrations?
the protests?
[S fc)==107
pliwlyj, fc]=100
parety(S, fc0-1.03
p6 (c))
pirett fc)03
Unrspreeened In Group
30
10-
Left
Center
Right
Left
Center
Right
Left
Center
Right
Left
Center
Right
Left
Center
Right
Without JR (5")
With JR (S
Fig, E.4. Representation of political groups when ranking with the diverse approval feng
25

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Representative Ranking for Deliberation in the Public Sphere
Ranking Remesh Comments by Diverse Approval
What characteristics your deem a procest inappropriate?
What are same of the reasons you have
What measures (if any) could be taken appropriate protests are protected?
How has your personal experience with influenced your viewpoint the right to assemble?
What characteristics that make protest appropriate?
participated or attended protest? (8 (8))c0000
(8, fc)-1.11
[8. fc]-1.1
(6 (c)))s fc]-1.08
páree fc]-1.03
Unroprsented in Group
50-
30
Left
Center
Right
Left
Center
Hight
Left
Center
Right
Left
Center
Hight
Left
Center
Right
What are your thoughts on the way university campus administrators should approach issue
What measures any) could be taken to restrict limit inappropriate protests? [Sassle,, (8, 1c0=1.08
What should guide university campus admin istrators handling protests?
you impressions of how campus
What your impressions campus protests?
administrators handled protests?
Israel/ Gaza demonstrations
prossrs, fe0-1.0
ploslr(s, fc)=1.03
phosars fc]=1.02
prosstij, fc1=1.05
56 Unispresented In Group
20-
10-
Right
Right
Left
Right
Right
Right
Without (S')
Fig. F.5. Representation of political groups when ranking with the diverse approval fDA
26

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Representative Ranking for Deliberation in the Public Sphere
Ranking Remesh Comments by Perspective API
What characteristics or actions, your deem a protest inappropriate?
What are some of the you participated attended protest? (S 1100
What measures (if any) could taken appropriate protests are protected?
How has your personal experience with protests influenced your viewpoint on the right to assemble? plantri8, fc)-1.07
What features characteristics that make a protest appropriate? páoatys (8, f.1-1.14
pSrredy fcc--.15
plrmay fc1-1.19
5 Unropresonnedd In Growp
12.0
2.--
Center
Right
Left
Center
Right
Left
Center
Right
Left
Center
Right
Left
Center
Right
What your thoughts the way university campus administrators should
What measures (if any) could restrict or limit
What should guide university campus administrators handling of protests?
What are you impressions how administrators handled the protests'
approach the issue of Israel Gaza demonstrations?
What are your impressions of the campus protests? pirely fc]-111
inappropriate protests?
p6relyyg fc]--.16
pireet fc.m0066
pirel (8. fc]=1.11
% Unimprosented in Group
12.0
10.0-
3.0-
Center
Right
Left
Center
Right
Left
Center
Right
Left
Center
Right
Left
Center
Right
Without JR (S*)
With JR (S
Fig, E.6. Representation of political groups when ranking with the Perspective fc
E.7. Individual Attributes from Perspective API
In Figure E.7, we show the representation of users and the price of GreedyCC when ranking with each of the seven individual Perspective API attributes: nuance, compassion, personal story, reasoning, curiosity, affinity, and respect.
27

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Representative Ranking for Deliberation in the Public Sphere
What features characterisiies
appropriate?
What characteristics
2003
Stoy
Reasoving
1.10
1.28
Curceity
4.15
1.16
Affeity
1.0
1.10
ressons you particisated proteat?
measures any)
to ensure appropriate protests are protested?
Comassioo
1.2
Perscesl Story
1.00
1.26
Reasoving
1.17
1.3
Curiesty
1.03
1.16
Affeitr
1.21
Regpos
0026
How has your personal experience with protests influenced your viewpoint on the right to assemble?
What mezsures (if any) could be taken to restrict or liit inappropriate protests?
Norcc
1.12
1.
Comssstion
10
129
Price GrodyyCC
Peraonal Skory
1.13
1J8
Ressoving
100
1.19
Regedd
1.01
1.09
What should guide university campus administrators handling of protests?
What are you impressions of how campus administrators handled the protests?
Naacce
1.1
123
Cancssion
Price GroedyCC
Perscoal
112
Curieity
Affiior
1.00
1.00
Regest
1.00
1.04
02
0.6
1.0
2/2
1.0
0.0
10.0
12.6
15.0
17.6
30.0
What your thoughts on the way university campus administrators should approach the of Israel Gaza demonstrations?
What the campus protests?
Nuscce
Story
Revoning
1.15
Curieity
1.15
Aflaie
1.1
1.30
Resped
Without (R5))
With JR (5mh))
Fig. E.7. Representation of users when ranking with individual Perspective API attributes, both with the JR constraint (S JR and without the JR constraint (S*).
28

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Representative Ranking for Deliberation in the Public Sphere
E.8. The Difference in Comments Selected With and Without JR
Lastly, we show an example of the feeds chosen by the three scoring functions; engagement feng- diverse approval fDA, and the Perspective API fe; for the question "What measures (if any) could be taken ensure appropriate protests are protected?"' Overall, the comments chosen by the Perspective API are much longer and more substantial than the comments chosen by engagement or diverse approval.
JJ (Jane)
St (fee)
Extra comments picked by GreedyCC
Highest scoring comments
Next-highest scoring comments
Crowd control; dialogue.
Body search and metal detectors for specific designated assembly location and strong police presence [...].
More of police presence.
There inappropriate protests, only inappropriate protesters.
think punishing those who make them inappropriate would help.
Put harsher punishments for people them, fines, jail time, other things.
This is difficult answer.[...] Height- ened police presence should help deter violence though.
Not issuing permits and leaders of the protest publicly denouncing.
Police force to shut these down.
Table E.5. The angagement-maximiinn set S* (feng) and the JR-constrained engagement set GGredd (feng) for the question "What measures (if any) could be taken to restrict or limit inappropriate protests?".
29

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Representative Ranking for Deliberation in the Public Sphere
JJ (Fon)
S" (fnA)
Extra comments picked by GreedyCC
Highest DA comments
Next-highest DA comments
Body search and metal detectors for specific designated assembly location and strong police presence I...l. Crowd control; dialogue.
No one really wants this. Again, it's about people being smart. Without that, we're just spitting the wind.
Make rules regulations stronger
T think punishing those who make them inappropriate would help.
Government approval of the protest.
This is difficult to answer. [...] Height- ened police presence should help deter violence though. Depends.
This a ridiculous question. reject the idea that there is such thing as an "inappropriate protest" at conceptual level. The premisc of the question is flawed and patently absurd.
Proper police presence and good super- vision ensure existing laws are not broken nor selectively enforced.
Table F.6. The diverss-approvall-axiine set S" (JDx) and the JR-constrained diverse approval set grredd (fDx) for the question "What measures (if any) could be taken to restrict or limit inappropriate protests?"
30

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Representative Ranking for Deliberation in the Public Sphere
Alimaly (fc)
St (fe)
Extra comments picked by GreedyCC
Highest scoring comments
Next-highest scoring comments
Body search and metal detectors for specific designated assembly location and strong police presence [...].
don't think we should limit any protests, the right to assemble must be protected all costs. []]t is impor- tant that we keep the playing feild level in case we are ever on a side deemed "wrong" we wouldn't 't want to be denied that right.
don't t know these days. People are will- ing to do anything. It would be nice if no violent, but if you lived where I do, you would give up (as have.)
Crowd control; dialogue.
Again, think this goes to broader cul- "right"
[...] Not having fake news spread over social media and people be less tional before protesting, level-headed, and respectful. Consequences for vio- lent and hateful actions by the protesters as if they doing things that going against the law, their job to then deal with their consequences and respect po- lice. Notify police before and see if they are available.
tural issues. We hold fast to protest. However, think a protest could be restricted or limited there are already plans for harms against others such the group having message of hate postings about planning violence, of any type, on social media.
In this democracy. it is difficult to to determine what kind of protest is "inappropriate" unless the definition is destruction property, violence, riot- ing, and that kind of thing. I person- ally do not believe in absolute freedom of speech, and in particularly feel that Nazi/white supermacist protests are ways out of line. [...]
think a greater oversight protest per- mits is needed, and should be okay to limit people from protesting if they have history of violent behavior in pre- vious protests. Free speech is okay but using it as an excuse for violence and destruction is not.
Unfortunately there aren't many steps to he taken. Bad apples always around and use the protests act inappropriately by
means
ate chaos ing a thief, vandal, assaulter, etc. Some- times these people are even planted by those opposing the protest diminish the validity of the original cause. Hav- ing protests during daylight hours might help some.
think people are breaking loss. They should be arrested. However, sometimes there is overstepping as you can see in other countries where people just arrested protesting.
Table E.7. The Perspective API set S* (fc) and the JR-constrained Perspective API set (fc) for the question "What measures (if any) could taken restrict or limit inappropriate protests?"
31


Reviews:
[{"review_id": "Cic9XAoowR", "reviewer": "Reviewer_WMX6", "summary": "This work applies the principle of \"justified representation\" as a means to algorithmically surface public comment for the end-goal of public deliberation. This is, in part, motivated by the ideals of deliberative democracy and normative reasons for selected public comments to satisfy some notion of \"representativeness.\" Mathematically, the work studies formalizations of justified representation (and the price of justified representation) from prior work and shows that, assuming that preferences can be modeled as mixtures of Mallows noise model, stronger bounds on the price of JR can be achieved. Finally, the authors run a previously-proposed (approximation) algorithm on real data and show that enforcing JR, even approximately, tangibly improves representation for a variety of score functions $f$.\n\n## Update after rebuttal\nMy initial review was overall positive; my main concern was about the roles that section 5/6 played in the contribution of the work - the rebuttal provided narrative clarification and I would be happy with seeing this paper in the conference with that discussion incorporated.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewer’s thoughtful comments and expertise. The reviewer’s main question is about their interpretation of Sections 5 and 6, which we address in the following. In Section 5, we first introduce novel theoretical results for the price of JR for arbitrary score functions, and then evaluate these results in section 6. Please note that the aspect of arbitrary score functions is key here for the theoretical results and connects both sections.\n\nOur main contribution is to introduce JR into the problem of facilitating deliberation online, particularly on social media platforms. In the social media context, JR is unlikely to be adopted if it also comes at a substantial decrease in user engagement or a substantial increase in the toxicity of comments. Therefore, we analyzed the price of JR, not only with respect to engagement (like Elkind et al did), but also with respect to arbitrary score functions. \n\nOur theoretical results in Section 5 hold for arbitrary score functions, which is why it logically follows to present them before our experiments where we evaluate the price of JR not just for engagement but also for diverse approval and the content-based bridging classifiers. In this way, our experiments also go beyond Elkind et al who only studied engagement. Although the worst-case price for arbitrary score functions can be arbitrarily high (as we showed in Sec 4), the observed price for all three score functions is remarkably low. As the reviewer points out, our theoretical results in Section 5 may explain these findings—a connection we intended to highlight and will clarify in our revision. \n\nOur empirical results also contribute to the literature on prosocial interventions in recommender systems, such as bridging-based ranking. The most common way to operationalize “bridging” is via diverse approval. However, surprisingly, we found that the content-based bridging classifiers provided much better representation than diverse approval. Indeed, all the Perspective API bridging feeds always satisfied JR by default. This is surprising as there is a line of literature showing that *toxicity* classifiers (including Perspective API’s) can be biased against various groups (e.g. Sap et al 2019, Lee et al 2024). Yet, it appears that the newer *bridging* classifiers target attributes with broad appeal, suggesting that these content-based classifiers may be a promising direction for future research in prosocial ranking.\n\nWe will also make clear that the bound in Theorem 5.4 is not about any particular algorithmic strategy.\n\nThank you again to the reviewer for your careful review of our paper.", "claims_and_evidence": "* Generally yes, see below for further comments", "methods_and_evaluation_criteria": "* Typically one would think that the experiments help to evaluate/validate the theory, but that doesn't appear to be the case here, and it's not actually a priori obvious that sections 5 and 6 are rather different and mostly unrelated contributions. That is, section 6 is not validating the theory given in section 5; instead, it is providing experimental evidence with real-world data for the theory work from the prior Elkind work (which does not have experiments). Section 5 is a distinct conceptual contribution about conditions under which we might expect JR to still do well in a utilitarian sense (and could, e.g., even be an explanatory mechanism for what is observed in section 6, though the paper itself doesn't make this connection as far as I can tell). This is not necessarily a problem with the paper itself but perhaps more so with presentation (e.g. if someone was not familiar with Elkind 22 and didn't know that they didn't run real-data experiments, then they might be confused what the message of section 6 is).  \n* The actual data being used in experiments is highly relevant to the work.", "theoretical_claims": "* I did not check proofs in detail. \n* Based on the statement of Theorem 5.4, I think it could be narratively useful to highlight that the bound is (a) in the worst case over all instances, and (b) therefore not about any particular algorithm or algorithmic strategy; in fact is is unclear how or whether to design algorithms that would have been specific to the Mallows mixture idea. \n* A lower bound would be nice (especially wrt identifiability, whether $\\gamma$ needs to be known, etc) but I am happy to defer that to future work.", "experimental_designs_or_analyses": "* See above. Main comment is that section 6 is _not_ a direct extension of section 5 and it would be nice to make that clearer; however the experiments themselves are reasonable for what they are.", "relation_to_broader_scientific_literature": "* This work contributes to concretely realizing the ideal of achieving better (democratic) deliberation, doing so by developing ideas from social choice. \n* More specifically, the sufficiency conditions implied by the Mallows mixture analysis are new for this problem setting/ and for JR (as far as I can tell), and build on prior social choice work that studies Mallows mixtures. \n* The experiments are followups to prior theory work on JR, and show that previously-proposed algorithms can be practically useful.", "essential_references_not_discussed": "N/A to my knowledge", "other_strengths_and_weaknesses": "The paper is overall well-written and easy to follow.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "* Can you clarify that my interpretation of the roles of sections 5/6 are correct? If so, could you comment on whether you were hoping to achieve something specific by structuring the paper this way?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "BLzTtQKZgb", "reviewer": "Reviewer_Axyg", "summary": "The authors take the problem of content ranking in online social deliberation, and adds justified representation (JR) constraints to the quality optimization problem to ensure diversity and representation. Theoretically, they show that under assumptions of user clusterization, the extra constraint leads to low cost in terms of the quality metric (e.g. civility, engagement). Empirically, they show that adding a greedy implementation of JR significantly improves the representation by existing ranking methods, while imposing only a modest cost in quality.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you to the reviewer for their comprehensive review. We address their questions below.\n\n**Connection Between Theoretical and Empirical Results**\n\nThe reviewer asks about how our theoretical results connect to the empirical findings. Even without considering Theorem 5.4, the bound from Theorem 5.1 already provides a close upper bound to the observed prices in practice. We will highlight this connection in our revised text. For Theorem 5.1, the bound holds if there exists an $n/k$-justifying set of size less than or equal to gamma (which the partition of gamma cohesive groups implies). In all ten questions, the GreedyCC algorithm finds an $n/k$-justifying set of size less than or equal to two (see Table E.3). Our bound from Theorem 5.1 would then imply that the price for all three score functions is at most 8/(8-2) = 1.33. The average empirical prices that we observe (see “Enforcing JR comes at a low price” in Section 6) are 1.05 for engagement, 1.06 for diverse approval, and 1.18 for Perspective API.\n\n**Handling Partially Observed Votes**\n\nThe reviewer also asks how the theoretical results hold when each user only reads a very small portion of items. In practice, the full approval matrix will need to be inferred from the partially observed votes—which is what we do in our Remesh experiments (see Footnote 10). Since this is a standard recommender system task, we do not focus on this approval inference in our paper. However, when implementing this on a real platform, it is very important to ensure that the inferred approvals are faithful to users’ actual approval. We touch on this issue in our discussion section: “on many platforms, users’ approval will need to be inferred from some form of engagement such as upvotes or likes. If the chosen engagement significantly diverges from actual user approval, the validity of the process could be compromised.”\n\n**Experimental Results and Stronger Proportionality Axioms**\n\nWith regards to experimental results, the reviewer asks how our experiments with GreedyCC may hold up against stronger proportionality axioms. In response, we have checked how many of the JR feeds in our experiments also satisfy EJR+ (Brill and Peters, 2023), a much stronger extension of JR. We found that 47 out of 48 feeds also satisfy EJR+ (and will add this result to the paper).\n\nI. Diverse Approval\n- 5/10 DA feeds are JR, all of them are also EJR+\n- 10/10 JR feeds are also EJR+\n\nII. Engagement\n- 3/10 engagement feeds are JR, all of them are also EJR+\n- 9/10 JR feeds are also EJR+\n\nIII. Perspective API\n- 10/10 Perspective API feeds are JR, all of them are also EJR+\n- 10/10 JR feeds are also EJR+\n\nThis suggests that, although GreedyCC is only optimizing for coverage, it nevertheless satisfies stronger axioms in practice. This result has also been corroborated in other empirical research which has found that JR and EJR+ tend to empirically coincide despite their large theoretical gap (e.g. “Approval-Based Committee Voting in Practice: A Case Study of (Over-)Representation in the Polkadot Blockchain” by Boehmer et al (2024)).\n\n> Potentially important as a future direction: The “one member approves one item” condition in JR is too weak (as the authors have acknowledged in the paper). How can we improve this, either theoretically or empirically?\n\nEmpirically, the results just described suggest that our approach may satisfy axioms stronger than JR in real-world datasets. We concur with the reviewer that theoretically studying and guaranteeing stronger extensions of JR would be an interesting direction for future research.\n\n**Other Comments**\n\n> “Have you tried other simple heuristics other than JR, and how well do they work?”\n\nWe have not tried other heuristics that are weaker than JR, since the GreedyCC algorithm on its own is already quite simple, and as noted above, it has the advantage of satisfying stronger axioms empirically as well.\n\n> “It would be helpful to visualize where these methods lie (with & without GreedyCC) on the quality-representation tradeoff plane.”\n\nThis is a great idea, and we will add these figures to the appendix. Right now, the trade-offs can be understood through the representation results + the prices of JR shown in Figure 2, but we agree that an explicit visualization would make this clearer.\n\nWe thank the reviewer again for their thoughtful comments.", "claims_and_evidence": "All claims are backed up by evidence. There are very few logical steps that I find unconvincing (which I detail in later sections).", "methods_and_evaluation_criteria": "The authors aim to address an import problem (online public deliberation) with a very interesting approach (content ranking), and give impressive theoretical & empirical evidence to support their approach. I find the approach reasonable and practical.", "theoretical_claims": "- My main uncertainty on the theoretical results lie in the assumptions and how they can be tested on empirical data. Specifically:\n  - For theorem 5.1, what is the value $\\gamma$ on each of the questions in your dataset?\n  - For theorem 5.2, what are the $\\gamma$ and $\\phi$ values that (statistically speaking) best explains your dataset? This can be done with, for example, a maximum likelihood estimation.\n- In reality, there are often cases where each user only reads a very small portion of entries (e.g. someone only reading tweets from people you follow; or someone only upvoting one or two comments before leaving) - do the theoretical bounds remain practical in that case? Would (n/k)-sized cohesive groups exist in that case?\n- I did not check the proofs.", "experimental_designs_or_analyses": "- **[important]** GreedyCC is, in some sense, specifically optimized for the coverage metric. It first searches for a tiny core of comments that satisfy minimum coverage (at least one comment per group), and then fill in the rest with no consideration for representation. Does it remain representative when we use other representation metrics that focuses on not just minimum coverage, but also proportionality? If not, is there another JR approximation algorithm that does well under proportionality?\n- It would be helpful to visualize where these methods lie (with & without GreedyCC) on the quality-representation tradeoff plane.\n- Have you tried other simple heuristics other than JR, and how well do they work?", "relation_to_broader_scientific_literature": "Content ranking, including content ranking with pro-social constraints, is a problem widely studied outside the algorithmic mechanism design literature - for example by RecSys researchers. This paper shows that JR constraint and its variants may be a valuable addition to their toolkit.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "- The work is not ground-breaking in the sense that it draws from previous literature on JR and its approximation algorithms. But it gives significant contribution by applying the idea in a important domain, and give a range of theoretical & empirical evidence to back up such an application.", "other_comments_or_suggestions": "- Typo: The y axis of Fig 2 & E.2 are not percentages despite the \"%\" sign in the axis labels.\n- Potentially important as a future direction: The “one member approves one item” condition in JR is too weak (as the authors have acknowledged in the paper). How can we improve this, either theoretically or empirically?", "questions_for_authors": "I have listed my questions in previous sections. The ones most likely to change my mind are marked with \"[important]\".", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "NweO2U2HT3", "reviewer": "Reviewer_SQBX", "summary": "The authors propose a comment ranking approach for public deliberation that incorporates justified representation, a concept from the social choice literature. The goal is to rank high quality comments, without losing the representation of groups that are present in the discussion. The approach relies on user approval mechanisms, making it more feasible than algorithms that require external user information. The authors can show that their implementation of enforcing justified representation, using the greedycc algorithm improves representation at a low cost.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewer’s careful reading of our paper and thank them for their constructive comments. We will correct all the typos that they identified.\n\nWith regards to the Perspective API bridging score $f_C$, it is the average of the scores for the seven available bridging attributes: \n\n* “Affinity”: References shared interests, motivations or outlooks between the comment author and another individual, group or entity\n* “Compassion”: Identifies with or shows concern, empathy, or support for the feelings/emotions of others.\n* “Curiosity”: Attempts to clarify or ask follow-up questions to better understand another person or idea.\n* “Nuance”: Incorporates multiple points of view in an attempt to provide a full picture or contribute   useful detail and/or context.\n* “Personal Story”: Includes a personal experience or story as a source of support for the statements made   in the comment.\n* “Reasoning”: Makes specific or well-reasoned points to provide a fuller understanding of the   topic without disrespect or provocation.\n* “Respect”: Shows deference or appreciation to others, or acknowledges the validity of another person.\n\nThe score for each individual attribute is a probability between 0 and 1. We provide results broken down by each individual attribute in Appendix E.7. \n\nAdditional references for the Perspective API bridging classifiers:\n* https://developers.perspectiveapi.com/s/about-the-api-attributes-and-languages?language=en_US\n* https://medium.com/jigsaw/announcing-experimental-bridging-attributes-in-perspective-api-578a9d59ac37\n* Saltz et al. Re-Ranking News Comments by Constructiveness and Curiosity Significantly Increases Perceived Respect, Trustworthiness, and Interest. arXiv 2024.\n* Schmer-Galunder et al. Annotator in the Loop: A Case Study of In-Depth Rater Engagement to Create a Prosocial Benchmark Dataset. AIES 2025.\n\nWe thank the reviewer once again for their consideration of our paper.", "claims_and_evidence": "The claims made are supported by clear evidence. The experiments resemble a real-world application. There could be some potential bias, because the process relies on user approval (likes or upvotes), which could easily be manipulated in political discussions. This potential drawback is mentioned by the authors themselves and should be investigated in the future.", "methods_and_evaluation_criteria": "Yes, the proposed methods and evaluation criteria make sense.", "theoretical_claims": "I didn't identify any issues in the proofs in the Appendix.", "experimental_designs_or_analyses": "The experimental designs and analyses are sound. More experiments are needed to investigate potential effects in the future, but since the paper only outlines the approach and theoretical assumptions, this is left for future work.", "relation_to_broader_scientific_literature": "The authors contribute to the literature on algorithmic fairness by proposing an approach to fair representation that is more feasible in practice than, for example, approaches based on demographics. It is shown that by approximation, sets of comments that satisfy JR can be found at low cost on real-world discussions.", "essential_references_not_discussed": "I did not identify essential references that are missing.", "other_strengths_and_weaknesses": "The paper is well written and easy to follow. The experiments and theoretical assumptions are clear. The results concerning the use of a score based on the Perspective API are surprising and could also be investigated in the future. It remains unclear if people would acutally feel represented by the algorithms choice and if the algorithm is able to guarantee content-wise diversity. This could be investigated in the future.", "other_comments_or_suggestions": "Small errors:\n- line 142 (right column): \"for all item i\" should be \"all items i\"\n- line 203 (right column): \"It seems that there could be perverse sets that satisfy JR.\" I am not sure if the wording is right here this seems off\n- line 225 (right column): \"the price of JR need to be bounded\" should be \"needs to be bounded\"\n- line 226: \"need not be compatible\" should rather be \"does not need to be\" or \"does not have to be..\"", "questions_for_authors": "- Can you give some details about fc (the Perspective API scoring function)? It's an average of all the bridging attribute scores. What is the range of the individual bridging scores?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "biridL9jhb", "reviewer": "Reviewer_q4GL", "summary": "The paper studies a setting of algorithmic comment ranking/selection incorporating fairness. There is a given set of comments, together with \"likes\" and based on these likes, a representative set of comments needs to be selected. The paper studies the impact of a group fairness concept called \"justified representation\" (JR) from the computational social choice literature. While in the worst-case, JR might inhibit the maximization of objective functions, in experiments on real-world comment data, as well as restricted settings, JR is compatible with approximately optimal function maximization.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their comprehensive and thoughtful comments.\n\nThe reviewer seems to be evaluating our paper primarily as a contribution to social choice. However, the main goal of our work was not to contribute to social choice, but rather to facilitate deliberation online (particularly on social media platforms). Since this area is lacking any representation axioms, we ground our algorithmic approach in social choice, specifically JR, and demonstrate the efficacy of this approach both theoretically and empirically under real-world conditions. We believe this is an important and novel contribution that can open up new directions in the research on prosocial recommender systems. This goal explains many of the decisions we made, and we will try to make this context clearer in our revision.\n\n**Our focus on JR**\n\nThe reviewer asked why we did not focus on stronger variants of JR. Practically speaking, a representation constraint is unlikely to be implemented on a social media platform if it also comes at a substantial decrease in user engagement or a substantial increase in the toxicity of comments. This is why in this initial work we started with the basic JR axiom, which is most likely to be compatible with the optimization of other score functions (engagement, civility classifiers, etc).\n\nHowever, in response to the reviewer’s concern, we also checked how many of the JR committees in our experiments satisfied EJR+ (and will add this to the paper). Across the 48 JR feeds, only one JR feed did not also satisfy EJR+. The paper that the reviewer mentioned by Boehmer et al. makes similar observations (e.g., \"All four proportional rules [including three rules not guaranteed to satisfy EJR+] return committees satisfying EJR+ for all tested instances.’’).\n\nOverall, we believe focusing on JR was a reasonable first step in our work due to (i) the need to demonstrate that the representation axiom is compatible with other arbitrary score functions, and (ii) the observed alignment of JR and EJR+ in real-world settings despite the theoretical gap. Nonetheless, we very much agree with the reviewer that studying stronger axioms would be a promising direction for future research.\n\n**Theoretical results**\n\nWe respectfully disagree with the reader’s assessment of our theoretical results. Coming back to our motivation of integrating JR into online platforms, the goal of our theoretical analysis was to show that, in natural settings, JR is compatible with the optimization of other score functions the platform may care about. While we agree that it is not surprising that the worst-care price of JR becomes higher for arbitrary score functions, these worst-case results primarily serve as motivation for our more novel contribution where we show that, in natural settings, the price is still low (Section 5). This result is crucial, as high prices would likely deter the incorporation of representation constraints on these platforms.\n\nTheorem 5.1 could be re-written in terms of n/k-justifying sets—that is essentially the proof of the theorem. However, our goal was to show a natural condition for why one might expect a small n/k-justifying set, and thus a low price of JR, on social media. In the social media context, when bridging interventions (diverse approval) are done, users are already partitioned into non-overlapping groups that are supposed to be distinctive in their preferences (these groups are typically learned by basically clustering the vote matrix). Theorem 5.1 says that if those groups are also cohesive in the JR sense, then we have a low price. We will clarify this reasoning and motivation, while acknowledging the prior related results, in our revised text.\n\nWe also note that, in practice, each group may not necessarily be fully cohesive in the JR sense. Thus, Theorem 5.1 was also primarily meant to serve as intuition for the extended Mallows model result (Theorem 5.4).\n\n**Set selection and ranking**\n\nThe reviewer is puzzled about the focus on set selection for comment ranking. We do not see a discrepancy here. In recommender systems, a very common method for showing a diverse set of items (e.g. videos that span your interests) is to re-rank items greedily so that the set of top items is diverse (see Algorithm 1 in “Fairness and Diversity in Recommender Systems: A Survey” by Zhao et al). Moreover, real-world recommender systems proceed in multiple steps that are actually set selection problems (e.g. retrieval -> early stage-ranking -> late-stage ranking). In our revision, we will clarify the connection between set selection and ranking.\n\n**Related Work & Impact Statement**\n\nWe appreciate the many references that the reviewer shared. We will incorporate them into our paper with a more comprehensive section on related work in social choice. We will also add an impact statement. Lastly, we will correct the typos the reviewer found.\n\nOnce again, we thank the reviewer for their thoughtful consideration of our paper.", "claims_and_evidence": "They seem both clear and convincing.", "methods_and_evaluation_criteria": "Yes, I would say so.", "theoretical_claims": "I did for the most part. Did not fully check the Mallows stuff.", "experimental_designs_or_analyses": "Yes, checked them", "relation_to_broader_scientific_literature": "The paper builds upon the concept of justified representation from computational social choice. This paper provides a novel application for justified representation and provides a few new theoretical results. In particular, the problem of maximizing an arbitrary function subject to JR was not really studied before. In general, the theoretical ideas  and results in the paper are quite similar to already present stuff in the computational social choice literature.", "essential_references_not_discussed": "I do not believe so", "other_strengths_and_weaknesses": "In general, I like the idea of the paper. Applying social choice fairness concepts to machine learning problems is a very interesting topic, and of growing importance in the last few years. The problem of comment ranking seems also quite important and well motivated. \n\nThere are a few things I am not too satisfied with, though: \n(i) firstly, the whole paper builds upon the notion of JR. However, JR is an incredibly weak axiom. As the authors note themselves, in a lot of their experiments 2 comments are already enough to satisfy JR (this is inline with other experimental works in social choice, see for instance the cited Bredereck et al. paper). Further, JR does not meet the intuitive notion of proportionality suggested by the paper itself (\"Let a world with 60 people who approve the 10 items in set A and another 40 people\nwho approve of a distinct set A↑ of 10 items. If a committee of size 10 is selected based on approval scores, the winning committee,\nA would fail to represent 40% of the world. Instead, a committee composed of 6 items from A and 4 items from A↑ would respect\nan intuitive notion of proportionality.\") In this example, JR would only require 1 item to be selected from each group (in the computational social literature the property the paper suggests here is also known as \"lower quota\"). There are significantly stronger yet still intuitive fairness axioms than JR, which could have been used instead, for instance priceability[1] or EJR+[2] (I would also encourage the authors to look at the recent work of Boehmer et al. [3] on proportional representation in a real-world approval-based committee voting setting). I believe using such stronger axioms would significantly improve the quality of the experiments and results.\n(ii) The theoretical results sadly seem quite weak. It is already known from the literature that the price of JR is \\sqrt(k), so quite bad. It is not really surprising that for arbitrary functions this gets worse. Further, Theorem 5.1 is also not really that new, see for instance Proposition 3 of Lackner and Skowron [4].\n(iii) One thing I found quite confusing is that the paper styles itself as being about ranking comments. The paper itself, however, is only about set selection, and I am not sure the results would entirely transfer to the ranking setting. In particular, Theorem 5.1 needs the set structure, if I see correctly. \n\n\nReferences\n[1] Proportionality and the Limits of Welfarism. Dominik Peters and Piotr Skowron 2020\n[2] Robust and verifiable proportionality axioms for multiwinner voting. Markus Brill and Jannik Peters 2023\n[3] Approval-based committee voting in practice: a case study of (over-) representation in the Polkadot blockchain. Niclas Boehmer et al. 2024\n[4] Utilitarian welfare and representation guarantees of approval-based multiwinner rules. Martin Lackner and Piotr Skowron 2020.", "other_comments_or_suggestions": "Line 94: e.g,\nLine 104: For cultural reasons, I think using such country specific examples should be avoided. No one outside of the US would understand this example.\nRelated Work: The related work section is currently missing computational social choice works entirely, even though the paper is on computational social choice. This seems wrong to me.\nLine 152: \\cdots -> \\dots (also elsewhere)\nLine 143 (right): \\cup -> \\bigcup (also elsewhere)\nLine 199 (left): the use of quantors in the text is quite ugly\nLine 195 (right): \\subset -> \\subseteq (also elsewhere)\n\"It appears as though there could be perverse sets that satisfy JR where\nmost people do not approve of the selected comments. However,\nresearch has found that common algorithms used to\nsatisfy JR do not lead to such perverse outcomes\" Yes, common algorithms selecting JR outcomes are usually good. However, this does not mean that all JR outcomes are usually good (which might also be quite relevant here)\nTheorem 4.2: \\gamma is never used in the theorem statement\nTheorem 5.1: I was wondering if you cant replace this whole construction by requiring that there is a small justifying set? I believe this is the only thing you need for the theorem.\nI find Figure 2 quite overwhelming at the moment. I believe it could be a bit simplified or spread out.\nYou are citing the Bredereck et al paper twice.\nThe Elkind et al paper has a journal version\n\"W¨”uthrich\" -> \"Wüthrich\"\n\"Landemore, H. 39Can Artificial Intelligence Bring Deliberation\nto the Masses?\" Cut the 39\n\nWhile searching for related work I came across the paper \"Combining Voting and Abstract Argumentation\nto Understand Online Discussions\" by Bernreiter et al., they seem to have a quite similar motivation to this work, but the results are different. Might still be worth to cite them though.  \n\nYou are currently missing an impact statement. I believe this paper might actually be one that needs one.", "questions_for_authors": "Can Theorem 5.1 be rephrased using justifying sets?\n\nWhy does the paper focus on JR and not stronger axioms?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper addresses the problem of fair representation in comment ranking by introducing a constraint, justified representation (JR), drawn from social choice theory. The authors show, both theoretically and empirically, that incorporating this constraint improves representation without compromising other metrics such as conversational quality or user engagement.

This work lies at the intersection of deliberation theory, social choice, and algorithmic ranking.

Following the rebuttal phase, which helped clarify remaining misunderstandings, all reviewers agreed that the paper's claims are supported by clear and convincing evidence. The methods and evaluation are solid, and the experiments are sound and valid. While the paper is well-positioned within the literature, one reviewer noted that the contribution might be somewhat limited from a purely social choice perspective. Nonetheless, the work offers a valuable bridge between deliberation and social choice.

The paper could be further improved by clarifying the interpretation of Sections 5 and 6, and by including a visualization of the methods with & without GreedyCC on the quality-representation tradeoff plane.

Overall, I believe this is a valuable contribution to the conference and recommend acceptance.

The reviewer discussion phase was particularly helpful to make explicit suggestions for improvement, which I strongly encourage the authors to incorporate into the revised version. These suggestions focus on making the limitations of the proposed methodology more explicit:

- The JR axiom does not fully imply the "idealized" notion of proportionality described in the footnote. However, as the authors note, this seems to have little impact in their experiments.

- The paper would benefit from a more thorough contextualization with respect to the broader social choice literature, particularly regarding alternative fairness notions and the rationale for not adopting them in this work.

- The authors should be more transparent about the limitations of their results. For example, acknowledging that the idea of mixing rules to achieve approximate guarantees was already present in earlier work on this topic.
Decision: Accept (poster)