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arxiv: 2511.00986 · v2 · submitted 2025-11-02 · 💻 cs.GT

Deliberation via Matching

Kamesh Munagala , Qilin Ye , Ian Zhang This is my paper

Pith reviewed 2026-05-18 01:53 UTC · model grok-4.3

classification 💻 cs.GT
keywords deliberative social choicemetric distortiontournament rulesmatching protocoluncovered setdistortion boundsocial choice theory
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The pith

Pairwise deliberation via matching achieves a tight distortion bound of 3 for tournament rules in metric social choice.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a deliberation protocol that, for each pair of candidates, matches voters who disagree on that pair and has each matched pair deliberate to update their preferences. These individual and deliberative preferences are then aggregated by the weighted uncovered set tournament rule. The authors establish that this protocol attains a distortion of at most 3 in the metric distortion framework, and that the bound is tight. This result matters to a sympathetic reader because, without any deliberation, general deterministic rules achieve distortion 3 while pure tournament rules have a lower bound of 3.11; the protocol shows that adding minimal pairwise deliberation lets tournament rules close the gap and match the best deterministic performance.

Core claim

The deliberation-via-matching protocol forms a maximum matching among voters who disagree on each candidate pair and has those matched pairs deliberate. The resulting preferences are aggregated by the weighted uncovered set. This protocol has a tight distortion bound of 3 within the metric distortion framework. Without deliberation, general deterministic social choice rules achieve distortion 3 while deterministic tournament rules have a lower bound of 3.11. The protocol therefore closes the gap, showing that pairwise deliberation lets tournament rules attain the same distortion as general deterministic rules. The proof proceeds via a bilinear relaxation of the distortion program whose value

What carries the argument

The deliberation-via-matching protocol, which forms maximum matchings among disagreeing voters for each candidate pair before feeding the updated preferences into the weighted uncovered set rule.

If this is right

  • Tournament rules attain the same distortion guarantee as general deterministic rules once they receive the minimal power of pairwise deliberation.
  • The distortion bound of 3 is tight, so no smaller constant works in the worst case.
  • The protocol admits a sampling-based implementation that approximates the deterministic guarantee with high probability and low per-voter complexity.
  • The supermodularity and convexity of the distortion objective for any three alternatives supplies a new analytic tool for bounding distortion in deliberative settings.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same matching idea might improve distortion bounds for other tournament-based rules or aggregation methods beyond the uncovered set.
  • In applied settings the protocol suggests a way to reduce full preference elicitation by limiting deliberation to small matched groups.
  • The bilinear relaxation technique could be adapted to analyze distortion for protocols that use different deliberation structures or partial updates.

Load-bearing premise

That the preferences produced by each matched pair's deliberation can be fed directly into the weighted uncovered set without further restrictions on how deliberation alters the votes.

What would settle it

A concrete metric instance with voters and candidates in which the worst-case ratio of the protocol's social cost to the optimal social cost exceeds 3.

Figures

Figures reproduced from arXiv: 2511.00986 by Ian Zhang, Kamesh Munagala, Qilin Ye.

Figure 1
Figure 1. Figure 1: Counter-monotonic coupling of 𝑋 (in blue) and 𝑌 (in red). Note the graphs partition [0, 1] into 𝐴𝐶/𝐶 𝐴 (by 𝑋) and 𝐵𝐶/𝐶𝐵 (by 𝑌). 5.4 Tight 𝑓 Constraints and the Optimal Matchings A fixed instance 𝐼 may admit various matchings for the candidate pair (𝐴, 𝐶) and thus potentially different values for 𝑊𝐴𝐶, the number of matchings where the outcome is 𝐴 ≻ 𝐶. Consequently, the values of 𝑓 (𝐴𝐶) need not be unique; … view at source ↗
Figure 2
Figure 2. Figure 2: The prefix-suffix structure of 𝐴 ≻ 𝐶 pairs. Lemma 5.7 (Prefix Property of Optimal Matchings). Fix an instance 𝐼 and an 𝐴-optimal matching for candidates (𝐴, 𝐶). Suppose this matching admits a mass 𝑊𝐴𝐶 of 𝐴 ≻ 𝐶 pairs. Then, there exists another 𝐴-optimal (𝐴, 𝐶) matching (hence also with mass 𝑊𝐴𝐶 of 𝐴 ≻ 𝐶 pairs) such that: (i) It takes place between the 𝑊𝐴𝐶 masses of highest 𝑋(𝑢) ⩾ 0 and highest 𝑋(𝑣) < 0; an… view at source ↗
Figure 3
Figure 3. Figure 3: A visualization of continuous tie-handling on [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The lines go from 0 to 1, capturing cumulative voter mass. The top line represents 𝑋 values in decreasing order and the bottom line represents 𝑌 values in increasing order. A voter appears at the same position in both lines. The pink masses 𝑝1 and 𝑝5 represent a set of 𝐴 ≻ 𝐶 matching pairs. This means 𝑝1 = 𝑝5. The same holds for the purple masses 𝑝2 and 𝑝4. The masses 𝑝1, 𝑝2, 𝑝3 correspond to non-negative … view at source ↗
Figure 5
Figure 5. Figure 5: Interval split for Case 2. The interpretation of this figure is similar to [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Distortion heatmap and the maximizer (arg max) partition of the (𝜆, 𝑤)-plane induced by 𝑑1, 𝑑2, 𝑑3. Each color shows the decision region arg max𝑖 𝑑𝑖(𝜆, 𝑤); dashed curves represent the decision boundaries 𝑑𝑖 = 𝑑 𝑗 . As 𝑑𝑖 quickly blows up, we only plot (𝜆, 𝑤) over [0.5, 0.7] × [0, 1.25]. The unique global minimum of D (𝜆, 𝑤) is 3, attained by (𝜆 ∗ , 𝑤∗ ) (this is also the unique intersection of all three de… view at source ↗
read the original abstract

We study deliberative social choice, where voters engage in small-group discussions to output collective preferences that are then aggregated by a social choice rule. We introduce a simple deliberation-via-matching protocol. In this protocol, for each pair of candidates, we form a maximum matching among voters who disagree on that pair, and have each matched pair deliberate. We then aggregate the resulting individual and deliberative preferences using the weighted uncovered set tournament rule. We show that this protocol has a tight distortion bound of $3$ within the metric distortion framework. In the absence of deliberation, general deterministic social choice rules can achieve this distortion, whereas deterministic tournament rules face a strictly larger lower bound of $3.11$. Our result closes this gap: Pairwise deliberation allows a tournament-based rule to attain distortion $3$. Conceptually, this shows that tournament rules can match the power of general deterministic social choice rules once they are given the minimal added power of pairwise deliberations. We prove this bound via a novel bilinear relaxation of the non-linear program capturing optimal distortion, whose vertices we can explicitly enumerate, leading to an analytic proof. Loosely speaking, our key technical insight is that the distortion objective, as a function of metric distances to any three alternatives, is both supermodular and convex. This characterization therefore provides a new analytical tool for studying the distortion of deliberative protocols, and may be of independent interest. Finally, although our analysis is for the full protocol, we show that this mechanism also admits a lightweight sampling-based implementation, yielding a high-probability approximation to the deterministic guarantee with arbitrary accuracy and low per-voter complexity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper introduces a deliberation-via-matching protocol for social choice: for each pair of candidates, a maximum matching is formed among voters who disagree on that pair, matched pairs deliberate, and the resulting individual and deliberative preferences are aggregated via the weighted uncovered set tournament rule. The central claim is a tight metric distortion bound of 3, which matches the known upper bound for general deterministic rules and improves upon the 3.11 lower bound for deterministic tournament rules without deliberation. The proof proceeds via an analytic argument that enumerates the vertices of a bilinear relaxation of the underlying distortion NLP, relying on the claim that the distortion objective (as a function of metric distances to any three alternatives) is supermodular and convex even after the deliberation step modifies the weighted tournament.

Significance. If the central claim holds, the result is significant: it demonstrates that a minimal pairwise deliberation mechanism suffices for tournament rules to attain the distortion performance of general deterministic social choice rules. The explicit vertex enumeration of the bilinear relaxation, grounded in a supermodular-convex characterization rather than fitted parameters or computational search, is a technical strength that supplies a parameter-free derivation and may serve as an analytical tool for other deliberative protocols. The sampling-based lightweight implementation, which yields high-probability approximation to the deterministic guarantee, adds practical value.

major comments (1)
  1. [Main proof section (bilinear relaxation and vertex enumeration)] Main proof section (bilinear relaxation and vertex enumeration): The central claim that the distortion bound of 3 is tight rests on the distortion objective remaining supermodular and convex after deliberation alters the input tournament. Deliberation can change relative orderings or weights on disagreed pairs, which modifies edges in the weighted tournament. Please supply the explicit verification (e.g., case analysis over three-alternative configurations) that this modification preserves the supermodularity/convexity property used to justify the relaxation; without it, the enumerated vertices may miss higher-distortion instances.
minor comments (2)
  1. [Abstract] Abstract: the claim that deterministic tournament rules face a lower bound of 3.11 would benefit from an explicit citation to the source establishing this bound.
  2. [Section on sampling implementation] The description of the sampling implementation would be clearer if it included a brief statement of the per-voter complexity and the precise sense in which arbitrary accuracy is achieved with high probability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive feedback and for recognizing the significance of our result on deliberation via matching. We respond to the major comment as follows.

read point-by-point responses

  1. Referee: Main proof section (bilinear relaxation and vertex enumeration): The central claim that the distortion bound of 3 is tight rests on the distortion objective remaining supermodular and convex after deliberation alters the input tournament. Deliberation can change relative orderings or weights on disagreed pairs, which modifies edges in the weighted tournament. Please supply the explicit verification (e.g., case analysis over three-alternative configurations) that this modification preserves the supermodularity/convexity property used to justify the relaxation; without it, the enumerated vertices may miss higher-distortion instances.

    Authors: We appreciate this observation and agree that making the preservation of supermodularity and convexity explicit strengthens the presentation. In the manuscript, the distortion objective is analyzed as a function of the metric distances to three alternatives, and the bilinear relaxation is constructed for the weighted tournament that results after the deliberation protocol is applied. The deliberation step adjusts the tournament weights on disagreed pairs according to the outcomes of the matched deliberations, but these adjustments are constrained by the underlying metric (e.g., voters' updated preferences remain consistent with the distances). Consequently, the objective function retains its supermodular and convex character with respect to the distance variables. To provide the requested verification, we will include in the revised manuscript an appendix containing a complete case analysis over all possible three-alternative configurations. This analysis will enumerate the possible pre- and post-deliberation tournament modifications and confirm that the key properties used for vertex enumeration continue to hold, thereby ensuring no higher-distortion instances are overlooked. revision: yes

Circularity Check

0 steps flagged

No circularity: analytic bound derived via explicit vertex enumeration of bilinear relaxation

full rationale

The paper establishes the tight distortion bound of 3 for the deliberation-via-matching protocol by formulating the distortion objective as a non-linear program, relaxing it to a bilinear program, and enumerating its vertices under the claimed supermodularity and convexity properties for any three alternatives. This constitutes an independent mathematical argument that does not reduce to self-definition, fitted parameters renamed as predictions, or load-bearing self-citations. Prior bounds of 3 and 3.11 for non-deliberative rules are referenced from external literature rather than the authors' own prior results, and the deliberation step is incorporated into the input tournament before the relaxation is applied. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard metric distortion model and the assumption that deliberation in matched pairs produces usable updated preferences; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption Voter-candidate preferences are embedded in a metric space
    Standard assumption in the metric distortion framework invoked throughout the abstract.
  • domain assumption Deliberation between matched disagreeing voters produces well-defined updated preferences
    Invoked when the protocol states that matched pairs deliberate and the resulting preferences are aggregated.

pith-pipeline@v0.9.0 · 5820 in / 1355 out tokens · 45835 ms · 2026-05-18T01:53:08.273296+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    the distortion objective, as a function of metric distances to any three alternatives, is both supermodular and convex. This characterization therefore provides a new analytical tool for studying the distortion of deliberative protocols

  • IndisputableMonolith/Foundation/BranchSelection.lean branch_selection echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    novel bilinear relaxation of the non-linear program capturing optimal distortion, whose vertices we can explicitly enumerate

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [1]

    Awareness of voter passion greatly improves the distortion of metric social choice

    Ben Abramowitz, Elliot Anshelevich, and Wennan Zhu. Awareness of voter passion greatly improves the distortion of metric social choice. In Ioannis Caragiannis, Vahab Mirrokni, and Evdokia Nikolova, editors,Web and Internet Economics, pages 3–16, Cham, 2019. Springer International Publishing

    work page 2019

  2. [2]

    Approximating optimal social choice under metric preferences.Artif

    Elliot Anshelevich, Onkar Bhardwaj, Edith Elkind, John Postl, and Piotr Skowron. Approximating optimal social choice under metric preferences.Artif. Intell., 264:27–51, 2018

    work page 2018

  3. [3]

    Voudouris

    Elliot Anshelevich, Aris Filos-Ratsikas, Nisarg Shah, and Alexandros A. Voudouris. Distortion in social choice problems: An annotated reading list.SIGecom Exch., 19(1):12–14, 2021

    work page 2021

  4. [4]

    Voudouris

    Elliot Anshelevich, Aris Filos-Ratsikas, and Alexandros A. Voudouris. The distortion of distributed metric social choice.Artificial Intelligence, 308:103713, 2022

    work page 2022

  5. [5]

    Randomized social choice functions under metric preferences.J

    Elliot Anshelevich and John Postl. Randomized social choice functions under metric preferences.J. Artif. Intell. Res., 58:797–827, 2017

    work page 2017

  6. [6]

    Argyle, Christopher A Bail, E

    Lisa P. Argyle, Christopher A Bail, E. Busby, Joshua R Gubler, Thomas Howe, Christopher Rytting, Taylor Sorensen, and David Wingate. Leveraging AI for democratic discourse: Chat interventions can improve online political conversations at scale.Proceedings of the National Academy of Sciences of the United States of America, 120, 2023

    work page 2023

  7. [7]

    Solomon E. Asch. Opinions and social pressure.Scientific American, 193(5):31–35, 1955

    work page 1955

  8. [8]

    Plurals: A system for guiding LLMs via simulated social ensembles

    Joshua Ashkinaze, Emily Fry, Narendra Edara, Eric Gilbert, and Ceren Budak. Plurals: A system for guiding LLMs via simulated social ensembles. In Naomi Yamashita, Vanessa Evers, Koji Yatani, 26 SharonXianghuaDing,BongshinLee,MarshiniChetty,andPhoebeO.ToupsDugas,editors,Proceed- ingsofthe2025CHIConferenceonHumanFactorsinComputingSystems,CHI2025,YokohamaJap...

    work page 2025

  9. [9]

    Bakker, Martin J

    Michiel A. Bakker, Martin J. Chadwick, Hannah R. Sheahan, Michael Henry Tessler, Lucy Campbell- Gillingham,JanBalaguer,NatMcAleese,AmeliaGlaese,JohnAslanides,MatthewM.Botvinick,and ChristopherSummerfield. Fine-tuninglanguagemodelstofindagreementamonghumanswithdiverse preferences. InProceedings of the 36th International Conference on Neural Information Pro...

    work page 2022

  10. [10]

    Can a few decide for many? The metric distortion ofsortition

    Ioannis Caragiannis, Evi Micha, and Jannik Peters. Can a few decide for many? The metric distortion ofsortition. InForty-firstInternationalConferenceonMachineLearning,ICML2024,Vienna,Austria, July 21-27, 2024. OpenReview.net, 2024

    work page 2024

  11. [11]

    Metric distortion bounds for randomized social choice

    Moses Charikar and Prasanna Ramakrishnan. Metric distortion bounds for randomized social choice. In Joseph (Seffi) Naor and Niv Buchbinder, editors,Proceedings of the 2022 ACM-SIAM Symposium onDiscreteAlgorithms,SODA2022,VirtualConference/Alexandria,VA,USA,January9-12,2022, pages 2986–3004. SIAM, 2022

    work page 2022

  12. [12]

    Metric distortion for tournament voting and beyond

    Moses Charikar, Prasanna Ramakrishnan, Zihan Tan, and Kangning Wang. Metric distortion for tournament voting and beyond. In Itai Ashlagi and Aaron Roth, editors,Proceedings of the 26th ACM Conference on Economics and Computation, EC 2025, Stanford University, Stanford, CA, USA, July 7-10, 2025, pages 790–818. ACM, 2025

    work page 2025

  13. [13]

    Breaking the metric votingdistortionbarrier

    Moses Charikar, Kangning Wang, Prasanna Ramakrishnan, and Hongxun Wu. Breaking the metric votingdistortionbarrier. InDavidP.Woodruff,editor,Proceedingsofthe2024ACM-SIAMSymposium on Discrete Algorithms, SODA 2024, Alexandria, VA, USA, January 7-10, 2024, pages 1621–1640. SIAM, 2024

    work page 2024

  14. [14]

    MortonDeutschandHaroldBenjaminGerard.Astudyofnormativeandinformationalsocialinfluences upon individual judgement.Journal of abnormal psychology, 51 3:629–36, 1955

    work page 1955

  15. [15]

    Sequential deliberation for social choice

    Brandon Fain, Ashish Goel, Kamesh Munagala, and Sukolsak Sakshuwong. Sequential deliberation for social choice. In Nikhil R. Devanur and Pinyan Lu, editors,Web and Internet Economics - 13th InternationalConference,WINE2017,Bangalore,India,December17-20,2017,Proceedings,volume 10660 ofLecture Notes in Computer Science, pages 177–190. Springer, 2017

    work page 2017

  16. [16]

    Fishkin.Democracy and Deliberation: New Directions for Democratic Reform

    J.S. Fishkin.Democracy and Deliberation: New Directions for Democratic Reform. Yale University Press, 1991

    work page 1991

  17. [17]

    Procaccia, and Sven Wang

    Bailey Flanigan, Ariel D. Procaccia, and Sven Wang. Distortion under public-spirited voting. In Kevin Leyton-Brown, Jason D. Hartline, and Larry Samuelson, editors,Proceedings of the 24th ACM Conference on Economics and Computation, EC 2023, London, United Kingdom, July 9-12, 2023, page 700. ACM, 2023

    work page 2023

  18. [18]

    Resolvingtheoptimalmetricdistortionconjecture

    VasilisGkatzelis,DanielHalpern,andNisargShah. Resolvingtheoptimalmetricdistortionconjecture. InSandyIrani,editor,61stIEEEAnnualSymposiumonFoundationsofComputerScience,FOCS2020, Durham, NC, USA, November 16-19, 2020, pages 1427–1438. IEEE, 2020

    work page 2020

  19. [19]

    Metric distortion of small-group deliberation

    Ashish Goel, Mohak Goyal, and Kamesh Munagala. Metric distortion of small-group deliberation. In MichalKouckýandNikhilBansal,editors,Proceedingsofthe57thAnnualACMSymposiumonTheory of Computing, STOC 2025, Prague, Czechia, June 23-27, 2025, pages 1568–1579. ACM, 2025. 27

    work page 2025

  20. [20]

    Metric distortion of social choice rules: Lower bounds and fairness properties

    Ashish Goel, Anilesh Kollagunta Krishnaswamy, and Kamesh Munagala. Metric distortion of social choice rules: Lower bounds and fairness properties. In Constantinos Daskalakis, Moshe Babaioff, and HervéMoulin,editors,Proceedingsofthe2017ACMConferenceonEconomicsandComputation,EC ’17, Cambridge, MA, USA, June 26-30, 2017, pages 287–304. ACM, 2017

    work page 2017

  21. [21]

    Metric distortion under probabilistic voting

    Mohak Goyal and Sahasrajit Sarmasarkar. Metric distortion under probabilistic voting. In Itai Ashlagi and Aaron Roth, editors,Proceedings of the 26th ACM Conference on Economics and Computation, EC 2025, Stanford University, Stanford, CA, USA, July 7-10, 2025, page 840. ACM, 2025

    work page 2025

  22. [22]

    Can deliberative minipublics influence public opinion? Theory and experimental evidence.Political Research Quarterly, 71(3):654–667, 2018

    Sean Ingham and Ines Levin. Can deliberative minipublics influence public opinion? Theory and experimental evidence.Political Research Quarterly, 71(3):654–667, 2018

    work page 2018

  23. [23]

    AnanalysisframeworkformetricvotingbasedonLPduality.ProceedingsoftheAAAI Conference on Artificial Intelligence, 34(02):2079–2086, Apr

    DavidKempe. AnanalysisframeworkformetricvotingbasedonLPduality.ProceedingsoftheAAAI Conference on Artificial Intelligence, 34(02):2079–2086, Apr. 2020

    work page 2079

  24. [24]

    Bot in the bunch: Facili- tating group chat discussion by improving efficiency and participation with a chatbot

    Soomin Kim, Jinsu Eun, Changhoon Oh, Bongwon Suh, and Joonhwan Lee. Bot in the bunch: Facili- tating group chat discussion by improving efficiency and participation with a chatbot. InProceedings ofthe2020CHIConferenceonHumanFactorsinComputingSystems,CHI’20,page1–13,NewYork, NY, USA, 2020. Association for Computing Machinery

    work page 2020

  25. [25]

    Pluralityveto: Asimplevotingruleachievingoptimalmetric distortion

    FatihErdemKizilkayaandDavidKempe. Pluralityveto: Asimplevotingruleachievingoptimalmetric distortion. In Luc De Raedt, editor,Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence, IJCAI 2022, Vienna, Austria, 23-29 July 2022, pages 349–355. ijcai.org, 2022

    work page 2022

  26. [26]

    Graph-theoreticalapproachestothetheoryofvoting.AmericanJournalofPolitical Science, 21(4):769–803, 1977

    NicholasR.Miller. Graph-theoreticalapproachestothetheoryofvoting.AmericanJournalofPolitical Science, 21(4):769–803, 1977

    work page 1977

  27. [27]

    Improved metric distortion for deterministic social choice rules

    Kamesh Munagala and Kangning Wang. Improved metric distortion for deterministic social choice rules. InAnnaR.Karlin,NicoleImmorlica,andRameshJohari,editors,Proceedingsofthe2019ACM Conference on Economics and Computation, EC 2019, Phoenix, AZ, USA, June 24-28, 2019, pages 245–262. ACM, 2019

    work page 2019

  28. [28]

    Online deliberation platform.https://stanforddeliberate.org/

    work page

  29. [29]

    Cappella

    Vincent Price, Lilach Nir, and Joseph N. Cappella. Normative and informational influences in online political discussions.Communication Theory, 16(1):47–74, 03 2006

    work page 2006

  30. [30]

    Procaccia and Jeffrey S

    Ariel D. Procaccia and Jeffrey S. Rosenschein. The distortion of cardinal preferences in voting. In Proceedings of the 10th International Conference on Cooperative Information Agents, CIA’06, page 317–331, Berlin, Heidelberg, 2006. Springer-Verlag

    work page 2006

  31. [31]

    Sally J. Scholz. Dyadic deliberation versus discursive democracy.Political Theory, 30(5):746–750, 2002

    work page 2002

  32. [32]

    A dynamic dyadic systems approach to interpersonal communication.Journal of Communication, 71(6):1001–1026, 09 2021

    Denise Haunani Solomon, Miriam Brinberg, Graham D Bodie, Susanne Jones, and Nilam Ram. A dynamic dyadic systems approach to interpersonal communication.Journal of Communication, 71(6):1001–1026, 09 2021

    work page 2021

  33. [33]

    Szpiro.Numbers Rule: The Vexing Mathematics of Democracy, from Plato to the Present

    George G. Szpiro.Numbers Rule: The Vexing Mathematics of Democracy, from Plato to the Present. Princeton University Press, 2010

    work page 2010

  34. [34]

    Characterizing online discussion using coarse discourse sequences.Proceedings of the International AAAI Conference on Web and Social Media, 11(1):357–366, May 2017

    Amy Zhang, Bryan Culbertson, and Praveen Paritosh. Characterizing online discussion using coarse discourse sequences.Proceedings of the International AAAI Conference on Web and Social Media, 11(1):357–366, May 2017. 28 A Explicit Dual Construction We now present an analytic proof based on the LP solutions to the bilinear program, via the dual certificate ...

    work page 2017