Pith. sign in

REVIEW 1 major objections 3 minor 1 cited by

Strong Equilibria in Bayesian Games with Bounded Group Size

T0 review · 1 major / 3 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Truthful reporting in the Miller-Resnick-Zeckhauser peer-prediction mechanism remains stable against colluding groups up to an exact, computable size, and fails above that size.

desk verdict The paper supplies a genuinely new exact threshold for bounded-size collusion in the MRZ peer prediction mechanism, but Theorem 1's proof has a missing (n-1) factor in the always-lie corner that must be fixed before the result is fully supported. read the letter →

arxiv 2502.00260 v1 pith:RTJA5Y5P submitted 2025-02-01 cs.GT

Qishen Han , Grant Schoenebeck , Biaoshuai Tao , Lirong Xia This is my paper · ORCID
classification cs.GT MSC 91A1091A06
keywords ex-anteBayesiank-strongequilibriumpeerpredictioncollusionstrongNashproperscoringrulemechanismdesignboundedgroupdeviation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces two bounded-size coalition solution concepts for Bayesian games and uses them to measure how many agents must collude before truth-telling stops being stable in a standard peer-prediction mechanism. Under the first concept, ex-ante Bayesian $k$-strong equilibrium, no group of at most $k$ agents can deviate to make every member weakly better off and at least one strictly better off using ex-ante expected utilities; under the second, Bayesian $k$-strong equilibrium, the same must hold for interim utilities conditioned on every private signal. The main theorems give explicit formulas for the threshold $k$ in terms of the number of agents $n$, the common symmetric prior, and the strictly proper scoring rule, with exact behavior below and above the threshold: below it no deviation succeeds, above it the all-high or all-low constant-report deviation succeeds. These thresholds give a quantitative, mechanism-dependent measure of collusion robustness, and the paper sketches how the same solution concepts apply to voting and private Blotto games.

What carries the argument

The load-bearing device is the reduction of arbitrary group deviations to two canonical deviations in which every deviator always reports $h$ or every deviator always reports $\ell$. The proof compares the ex-ante or interim utility of a deviator under truth-telling with utility under these constant reports, producing thresholds where the fraction $(k-1)/(n-1)$ weights the gain from fellow deviators against the loss from truthful peers. For the upper bound, the average of deviators' mixed strategies is used; in the ex-ante case, convexity of the expected reward function $f(\beta_\ell,\beta_h)$ and concavity of the utility gap let the proof check only the corner strategies, while in the interim case Lemmas 1 and 2 show that the strategy square $[0,1]^2$ is covered by two triangles where agents with signal $h$ or signal $\ell$ respectively do not want to deviate.

What would settle it

Take the paper's own $n=100$ Brier example, where the theorems give $k_E=27$ and $k_B=45$. Enumerate every deviation by exactly 27 and 28 agents restricted to always reporting high, and by exactly 45 and 46 agents for the interim notion; if a group one below the threshold gains or a group one above fails to gain, the dichotomies are wrong. Because the formulas are closed-form, this finite check can be repeated for any fixed prior and strictly proper scoring rule.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is a pair of exact dichotomies. For the peer-prediction mechanism of Miller, Resnick, and Zeckhauser with any strictly proper scoring rule $PS$ and a symmetric binary-signal prior satisfying $q(h|h)>q(h|\ell)$, $q(h|\ell)>0$, and $q(\ell|h)>0$, truthful reporting $\Sigma^*$ is an ex-ante Bayesian $k$-strong equilibrium exactly for $k\le k_E$, where $k_E=\min(k_E^h,k_E^\ell,n)$ and $$k_E^h=\left\lfloor (n-1)\frac{\mathbb{E}_{s\sim q_\ell}[PS(s,q_\ell)-PS(s,q_h)]}{PS(h,q_h)-PS(\ell,q_h)}\right\rfloor+1$$ if $PS(h,q_h)>PS(\ell,q_h)$, with $k_E^h=n$ otherwise, and $k_E^\ell$ defined symmetrically. Theorem 2 gives the analogous exact threshold $k_B$ for Bayesian $k$-strong equilibrium, using ceilings and with $q(\ell|\ell)$ and $q(h|h)$ multiplying the denominators, for all $n\ge n_0$. Above the threshold, the all-report-$h$ or all-report-$\ell$ constant deviation succeeds; below it, no mixed-strategy deviation benefits all members. The two thresholds differ because interim agents evaluate gains conditional on each private signal, making them more conservative, and Proposition 1 records that ex-ante stability implies interim stability.

Load-bearing premise

The thresholds rely on colluders coordinating before they learn anything private and then being unable to share what they learn; if that timing assumption fails, collusion works differently and the formulas do not apply.

Editorial extensions

If this is right

  • A platform using the peer-prediction mechanism can compute the largest colluding group that truth-telling withstands from $n$, the prior, and the scoring rule; any group larger than the threshold has a concrete profitable deviation in which every deviator reports the same constant signal.
  • For every $k\le k_E$, no group of size $k$ can improve over truth-telling, and for $k>k_E$ the always-$h$ or always-$\ell$ deviation succeeds, so the two constant-report deviations are the canonical threats to check.
  • Because the thresholds depend on the scoring rule, the results give a quantitative design handle: among strictly proper scoring rules, one can compare $k_E$ or $k_B$ for a fixed prior and choose the rule with the largest threshold.
  • The interim notion $k_B$ is at least as large as $k_E$ in the characterized setting, so requiring every signal-conditioned deviator to gain makes collusion harder; conservative agents need larger groups.
  • The same bounded-size coalition solution concepts extend to other Bayesian environments, with $k$ interpolating between individual Bayesian Nash equilibrium at $k=1$ and fully general strong equilibrium at $k=n$.
  • The paper leaves implicit a mechanism-selection optimization: within a parametric family of strictly proper scoring rules, one could choose the rule maximizing $k_E$ or $k_B$ for a fixed prior, turning the threshold into an explicit design objective rather than only an evaluation criterion.
  • The threshold formulas also suggest an empirical prediction: in crowdsourcing platforms, observed profitable collusion groups should be near $k_E$ if workers collude ex ante, and measuring actual successful collusion sizes could validate or reject the no-communication timing model.
  • A natural extension is to derive analogous bounded-size collusion thresholds for multi-task peer prediction, where stronger robustness results are known, and to check whether the constant-report worst-case deviations still dominate there.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 3 minor

Summary. The paper introduces two bounded-coalition solution concepts for Bayesian games, the ex-ante Bayesian k-strong equilibrium and the Bayesian k-strong equilibrium, and applies them to the single-task peer prediction mechanism of Miller, Resnick, and Zeckhauser. The main results, Theorem 1 and Theorem 2, give explicit closed-form thresholds k_E and k_B such that truthful reporting is an equilibrium for all groups of size at most the threshold and fails for all larger groups. The paper also sketches applications to voting and the Private Blotto game. The proposed concepts are natural, and the claimed thresholds are parameter-explicit rather than asymptotic in the ex-ante case, with an explicit (if complicated) n0 for the interim case.

Significance. If the proofs are completed, the paper would provide a quantitative, mechanism-parameter-dependent measure of collusion robustness for a canonical single-task peer prediction mechanism, and it would introduce solution concepts that interpolate between Bayesian Nash equilibrium and full strong equilibrium. The distinction between ex-ante and interim coalitional incentives is conceptually useful and is supported by the paper's Proposition 1. The main theorems are falsifiable and do not rely on fitted constants. However, because the proof of Theorem 1 contains a concrete algebraic error in a corner case needed to establish the lower-bound direction, the central claim is not yet established as written.

major comments (1)
  1. [Appendix C, Step 2 (always-lie corner)] The displayed threshold for the always-lie deviation omits the (n-1) factor. The inequality immediately before the display gives Δu(1,0)<0 if and only if k > (n-1)·A/B + 1, where A is the numerator and B the denominator defined in that paragraph, not k > A/B + 1. The claimed threshold k' = A/B + 1 is therefore not the correct threshold, and the subsequent statement k' ≥ k_E is false as written: in the Brier example of Section 3.1/Example 4 with n=100, one has A=0.32 and B=0.24, so the literal threshold is about 2.33, while k_E=27. Inserting the missing (n-1) gives a threshold of about 133, so Theorem 1 may still be true, but the proof as printed does not rule out the always-lie deviation for k ≤ k_E. In addition, the proof of k' ≥ k_E is not supplied: Case (1) merely asserts that k' lies between k_h^E and k_l^E. A short argument using B ≤ q(h)(PS(ℓ,q_ℓ)-PS(h,q_ℓ)) + q(ℓ)(PS(h,q_h)-PS(ℓ,q_h)) would establish the corrected claim, but the manuscript should provide it explicitly.
minor comments (3)
  1. [Example 6] The example says 'a deviation group needs at least 27 deviators to succeed', but the derived condition is k > 27.4, so the minimal successful group size is 28. The value k_E=27 is the largest equilibrating group size, not the minimal colluding group size; the wording conflates the two.
  2. [Section 3.2 and Appendix D, Step 2] The proof sketch and the appendix state that Step 2 handles 'any k ≤ k_E', but the relevant bound for Theorem 2 is k ≤ k_B. This appears to be a typo, since the proof of Lemma 1 uses k ≤ k_B, but it should be corrected for readability.
  3. [Appendix D] There are several notational slips in the displayed expressions: Δℓ is defined as PS(ℓ,q_ℓ)-q(ℓ,q_h) where the second term should be PS(ℓ,q_h); the ℓ-side conditions (5) and (6) refer to ¯b_h where they presumably mean ¯b_ℓ; and Example 7 writes k_ℓ^E where it should write k_ℓ^B.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: thresholds are computed from stated assumptions and external proper-scoring-rule facts.

full rationale

The paper's central claims, Theorems 1 and 2, derive explicit closed-form thresholds k_E and k_B from the primitives of the peer prediction model: the symmetric binary-signal prior, the derandomized MRZ mechanism, and the definition of a strictly proper scoring rule. No parameter is fitted to the quantity being predicted, and no 'prediction' is renamed from an input. The proof of Theorem 1 reduces arbitrary deviations to corner strategies using a convexity argument (Claim 1 and the concavity of Delta u), and the always-report-h and always-report-l corners are the same inequalities used to define k_E; this is a direct derivation, not a circular definition. The always-lie corner introduces a separate threshold k' and claims k' >= k_E; whether that inequality is proven correctly is a question of proof correctness (the skeptic's note about a possibly missing (n-1) factor is an algebraic concern, not a circularity), because k' is not defined in terms of k_E and the theorem does not assume its own conclusion. Theorem 2's lower bound n0 is explicitly supplied in Appendix D as a function of the prior and scoring rule, and it is independent of n; it is a constructed sufficiency condition rather than a fitted constant. The paper cites external mathematical facts for proper scoring rules (Hendrickson and Buehler, Theorem 3) and for positive semidefiniteness (Meyer, Lemma 4), and these citations are independent support. Self-citations to prior work on strong equilibria and voting appear in the introduction, related work, and applications sections and are motivational or contextual; they are not used to justify the main theorems. No self-definitional, fitted-input, imported-uniqueness, or ansatz-by-citation pattern is present. The derivation chain is self-contained given its stated assumptions, so the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central results are derived without fitting any parameters; the prior q and scoring rule PS are inputs from the model. The only hidden premise is q(l|l)>0, needed by Theorem 2, which is not stated in Section 2.3. The solution concepts themselves are new definitions, not physical entities.

assumptions (5)
  • standard math Standard Bayesian game model with common prior, private types, and expected utility calculations.
    The paper builds on the standard Harsanyi model in Section 2.1; this is background theory, not contested.
  • domain assumption Symmetric binary-signal prior with informativeness: q(h|h)>q(h|l), q(l|l)>q(l|h), and no full correlation, stated as q(h|l)>0 and q(l|h)>0.
    Section 2.3 assumes these conditions so that truthful reporting is a Bayesian Nash equilibrium. If they fail, the thresholds are not meaningful.
  • domain assumption Coalition members coordinate before types are realized and cannot communicate after seeing their signals.
    Section 2.2 and Example 3 motivate this ex-ante coordination assumption. It excludes deviations based on type sharing, which are studied in Abraham et al.
  • domain assumption The mechanism uses a strictly proper scoring rule, and the original random-peer mechanism is derandomized by averaging over all peers.
    Section 2.3 and Remark 1 define the mechanism. Strict properness is needed for truth-telling to be individually optimal and for Lemma 3.
  • standard math The Hendrickson-Buehler characterization of strictly proper scoring rules via convex functions is used in Lemma 3.
    Appendix A cites Theorem 3 from [31]; this is standard background mathematics for scoring rules.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Strong Equilibria in Bayesian Games with Bounded Group Size." pith.science (2026). https://pith.science/paper/RTJA5Y5P

@misc{pith2026250200260,
  author       = {Pith},
  title        = {Pith review of: Strong Equilibria in Bayesian Games with Bounded Group Size},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RTJA5Y5P}},
  note         = {Machine review of arXiv:2502.00260}
}
abstract

We study the group strategic behaviors in Bayesian games. Equilibria in previous work do not consider group strategic behaviors with bounded sizes and are too ``strong'' to exist in many scenarios. We propose the ex-ante Bayesian $k$-strong equilibrium and the Bayesian $k$-strong equilibrium, where no group of at most $k$ agents can benefit from deviation. The two solution concepts differ in how agents calculate their utilities when contemplating whether a deviation is beneficial. Intuitively, agents are more conservative in the Bayesian $k$-strong equilibrium than in the ex-ante Bayesian $k$-strong equilibrium. With our solution concepts, we study collusion in the peer prediction mechanisms, as a representative of the Bayesian games with group strategic behaviors. We characterize the thresholds of the group size $k$ so that truthful reporting in the peer prediction mechanism is an equilibrium for each solution concept, respectively. Our solution concepts can serve as criteria to evaluate the robustness of a peer prediction mechanism against collusion. Besides the peer prediction problem, we also discuss two other potential applications of our new solution concepts, voting and Blotto games, where introducing bounded group sizes provides more fine-grained insights into the behavior of strategic agents.

Figures

Figures reproduced from arXiv: 2502.00260 by the authors.

Figure 1
Figure 1. The illustration of Lemma 1 and 2. The X-axis and [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Beyond the PPAD hardness of Auto-bidding Auctions

    cs.GT 2026-08 conditional novelty 6.0 of 10

    Under non-atomic value distributions, auto-bidding equilibria become separately monotone generalized Nash equilibria and PRIME solves them with last-iterate linear convergence.

Reference graph

Works this paper leans on

63 extracted references · 60 canonical work pages · cited by 1 Pith paper

  1. [1]

    Ittai Abraham, Danny Dolev, Rica Gonen, and Joe Halpern. 2006. Distributed computing meets game theory: robust mechanisms for rational secret sharing and multiparty computation. In Proceedings of the Twenty-Fifth Annual ACM Symposium on Principles of Distributed Computing (Denver, Colorado, USA). Association for Computing Machinery, New York, NY, USA, 53–62

    work page 2006

  2. [2]

    Ittai Abraham, Danny Dolev, and Joseph Y. Halpern. 2008. Lower Bounds on Implementing Robust and Resilient Mediators. In Theory of Cryptography , Ran Canetti (Ed.). Springer Berlin Heidelberg, Berlin, Heidelberg, 302–319

    work page 2008

  3. [3]

    Avidit Acharya. 2016. Information aggregation failure in a model of social mobility. Games and Economic Behavior 100 (2016), 257–272

    work page 2016

  4. [4]

    S Nageeb Ali, Maximilian Mihm, and Lucas Siga. 2018. Adverse selection in distributive politics. A vailable at SSRN 3579095 (2018)

    work page 2018

  5. [5]

    Robert J. Aumann. 1959. Acceptable points in general cooperative n-person games. In Annals of Mathematics Study 40 . Contributions to the Theory of Games, Vol. IV. Princeton University Press, 287–324

    work page 1959

  6. [6]

    David Austen-Smith and Jeffrey S Banks. 1996. Information aggregation, ratio- nality, and the Condorcet jury theorem. American political science review 90, 1 (1996), 34–45

    work page 1996

  7. [7]

    Salvador Barbera, Michael Maschler, and Jonathan Shalev. 2001. Voting for voters: a model of electoral evolution. Games and Economic Behavior 37, 1 (2001), 40–78

    work page 2001

  8. [8]

    Sourav Bhattacharya. 2013. Preference monotonicity and information aggrega- tion in elections. Econometrica 81, 3 (2013), 1229–1247. Strong Equilibria in Bayesian Games with Bounded Group Size

    work page 2013

Show all 63 references
  1. [9]

    Sourav Bhattacharya. 2023. Condorcet Jury Theorem in a Spatial Model of Elections. Ph. D. Dissertation. Indian Institute of Management Calcutta

    2023

  2. [10]

    Emile Borel. 1953. The Theory of Play and Integral Equations with Skew Sym- metric Kernels. Econometrica 21, 1 (1953), 97–100. http://www.jstor.org/stable/ 1906946

    1953

  3. [11]

    Noah Burrell and Grant Schoenebeck. 2023. Measurement Integrity in Peer Pre- diction: A Peer Assessment Case Study. InProceedings of the 24th ACM Conference on Economics and Computation (EC 2023)

    2023

  4. [12]

    Julio B Clempner and Alexander S Poznyak. 2015. Computing the strong Nash equilibrium for Markov chains games. Appl. Math. Comput. 265 (2015), 911–927

    2015

  5. [13]

    Julio B Clempner and Alexander S Poznyak. 2020. Finding the strong nash equilibrium: Computation, existence and characterization for markov games. Journal of Optimization Theory and Applications 186 (2020), 1029–1052

    2020

  6. [14]

    Marquis de Condorcet. 1785. Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix . Paris: L’Imprimerie Royale

  7. [15]

    Peter J Coughlan. 2000. In defense of unanimous jury verdicts: Mistrials, com- munication, and strategic voting. American Political science review 94, 2 (2000), 375–393

    2000

  8. [16]

    Anirban Dasgupta and Arpita Ghosh. 2013. Crowdsourced judgement elicitation with endogenous proficiency. In Proceedings of the 22nd International Conference on World Wide Web. Association for Computing Machinery, New York, NY, USA, 319–330

    2013

  9. [17]

    Xiaotie Deng, Biaoshuai Tao, and Ying Wang. 2024. Aggregation of An- tagonistic Contingent Preferences: When Is It Possible?. In International Conference on Web and Internet Economics . Springer, Forthcoming. See also https://arxiv.org/abs/2410.08127

    2024 arXiv

  10. [18]

    Yvo Desmedt and Edith Elkind. 2010. Equilibria of plurality voting with ab- stentions. In Proceedings of the 11th ACM conference on Electronic commerce . 347–356

    2010

  11. [19]

    Kate Donahue and Jon Kleinberg. 2023. Private Blotto: Viewpoint Competition with Polarized Agents. arXiv preprint arXiv:2302.14123 (2023)

    2023 arXiv

  12. [20]

    John Duggan and César Martinelli. 2001. A Bayesian model of voting in juries. Games and Economic Behavior 37, 2 (2001), 259–294

    2001

  13. [21]

    Timothy Feddersen and Wolfgang Pesendorfer. 1997. Voting behavior and infor- mation aggregation in elections with private information. Econometrica: Journal of the Econometric Society (1997), 1029–1058

    1997

  14. [22]

    Timothy Feddersen and Wolfgang Pesendorfer. 1998. Convicting the innocent: The inferiority of unanimous jury verdicts under strategic voting. American Political science review 92, 1 (1998), 23–35

    1998

  15. [23]

    Xi Alice Gao, Andrew Mao, Yiling Chen, and Ryan Prescott Adams. 2014. Trick or treat: putting peer prediction to the test. In Proceedings of the Fifteenth ACM Con- ference on Economics and Computation . Association for Computing Machinery, New York, NY, USA, 507–524

    2014

  16. [24]

    Wright, and Kevin Leyton-Brown

    Xi Alice Gao, James R. Wright, and Kevin Leyton-Brown. 2019. Incentivizing evaluation with peer prediction and limited access to ground truth. Artificial Intelligence 275 (2019), 618–638

    2019

  17. [25]

    Dino Gerardi. 2000. Jury verdicts and preference diversity. American Political Science Review 94, 2 (2000), 395–406

    2000

  18. [26]

    Huiyi Guo and Nicholas C Yannelis. 2022. Robust coalitional implementation. Games and economic behavior 132 (2022), 553–575

    2022

  19. [27]

    Guangsug Hahn and Nicholas C Yannelis. 2001. Coalitional Bayesian Nash implementation in differential information economies. Economic Theory 18 (2001), 485–509

    2001

  20. [28]

    Qishen Han, Grant Schoenebeck, Biaoshuai Tao, and Lirong Xia. 2023. The Wisdom of Strategic Voting. In Proceedings of the 24th ACM Conference on Eco- nomics and Computation (London, United Kingdom). Association for Computing Machinery, New York, NY, USA, 885–905

    2023

  21. [29]

    Tobias Harks and Max Klimm. 2012. On the existence of pure Nash equilibria in weighted congestion games. Mathematics of Operations Research 37, 3 (2012), 419–436

    2012

  22. [30]

    1967–1968

    John Harsanyi. 1967–1968. Game with incomplete information played by Bayesian players. Management Science 14 (1967–1968), 159–182; 320–334; 486– 502

    1967

  23. [31]

    Hendrickson and Robert J

    Arlo D. Hendrickson and Robert J. Buehler. 1971. Proper Scores for Probability Forecasters. The Annals of Mathematical Statistics 42, 6 (1971), 1916 – 1921. doi:10.1214/aoms/1177693057

    1971

  24. [32]

    Ron Holzman and Nissan Law-Yone. 1997. Strong equilibrium in congestion games. Games and economic behavior 21, 1-2 (1997), 85–101

    1997

  25. [33]

    Tatsuro Ichiishi and Adam Idzik. 1996. Bayesian cooperative choice of strategies. International Journal of Game Theory 25 (1996), 455–473

    1996

  26. [34]

    2006.Cooperative extensions of the Bayesian game

    Tatsuro Ichiishi and Akira Yamazaki. 2006.Cooperative extensions of the Bayesian game. Springer Japan, Tokyo, 273–296

    2006

  27. [35]

    Radu Jurca and Boi Faltings. 2007. Collusion Resistant, Incentive Compatible Feedback Payments. InProceedings of the ACM Conference on Electronic Commerce (EC). San Diego, CA, USA

    2007

  28. [36]

    Radu Jurca and Boi Faltings. 2009. Mechanisms for making crowds truthful. Journal of Artificial Intelligence Research 34, 1 (mar 2009), 209–253

    2009

  29. [37]

    Jaehoon Kim and Mark Fey. 2007. The swing voter’s curse with adversarial preferences. Journal of Economic Theory 135, 1 (2007), 236–252

    2007

  30. [38]

    Yuqing Kong, Katrina Ligett, and Grant Schoenebeck. 2016. Putting Peer Pre- diction Under the Micro(economic)scope and Making Truth-Telling Focal. In Web and Internet Economics, Yang Cai and Adrian Vetta (Eds.). Springer Berlin Heidelberg, Berlin, Heidelberg, 251–264

    2016

  31. [39]

    Yuqing Kong and Grant Schoenebeck. 2018. Equilibrium Selection in Information Elicitation without Verification via Information Monotonicity. In 9th Innovations in Theoretical Computer Science Conference (Leibniz International Proceedings in Informatics (LIPIcs), Vol. 94), Anna...

    2018

  32. [40]

    Yuqing Kong and Grant Schoenebeck. 2019. An Information Theoretic Frame- work For Designing Information Elicitation Mechanisms That Reward Truth- telling. ACM Transactions on Economics and Computation (TEAC) 7, 1 (February 2019), 2:1–2:33. doi:10.1145/3296670

    2019 doi

  33. [41]

    Cesar Martinelli. 2002. Convergence results for unanimous voting. Journal of Economic Theory 105, 2 (2002), 278–297

    2002

  34. [42]

    Andrew McLennan. 1998. Consequences of the Condorcet jury theorem for beneficial information aggregation by rational agents. American political science review 92, 2 (1998), 413–418

    1998

  35. [43]

    Adam Meirowitz. 2002. Informative voting and Condorcet jury theorems with a continuum of types. Social Choice and Welfare 19, 1 (2002), 219–236

    2002

  36. [44]

    Carl D. Meyer. 2000. Matrix analysis and applied linear algebra . SIAM

    2000

  37. [45]

    Nolan Miller, Paul Resnick, and Richard Zeckhauser. 2005. Eliciting Informative Feedback: The Peer-Prediction Method. Management Science 51, 9 (2005), 1359– 1373

    2005

  38. [46]

    Roger B Myerson. 1998. Extended Poisson games and the Condorcet jury theorem. Games and Economic Behavior 25, 1 (1998), 111–131

    1998

  39. [47]

    Wolfgang Pesendorfer and Timothy Feddersen. 1996. The swing voter’s curse. American Economic Review 86, 3 (1996), 408–424

    1996

  40. [48]

    Drazen Prelec. 2004. A Bayesian Truth Serum for Subjective Data. Science 306, 5695 (2004), 462–466. doi:10.1126/science.1102081 arXiv:https://www.science.org/doi/pdf/10.1126/science.1102081

    2004 doi

  41. [49]

    Dražen Prelec, H Sebastian Seung, and John McCoy. 2017. A solution to the single-question crowd wisdom problem. Nature 541, 7638 (2017), 532–535

    2017

  42. [50]

    Zinovi Rabinovich, Svetlana Obraztsova, Omer Lev, Evangelos Markakis, and Jeffrey Rosenschein. 2015. Analysis of equilibria in iterative voting schemes. In Proceedings of the AAAI conference on artificial intelligence , Vol. 29

    2015

  43. [51]

    Goran Radanovic and Boi Faltings. 2015. Incentive schemes for participatory sensing. In Proceedings of AAMAS

    2015

  44. [52]

    Mikhail Safronov. 2018. Coalition-proof full efficient implementation. Journal of Economic Theory 177 (2018), 659–677

    2018

  45. [53]

    Grant Schoenebeck and Biaoshuai Tao. 2021. Wisdom of the Crowd Voting: Truthful Aggregation of Voter Information and Preferences. In Advances in Neural Information Processing Systems , Vol. 34. Curran Associates, Inc., 1872– 1883

    2021

  46. [54]

    Grant Schoenebeck, Fang-Yi Yu, and Yichi Zhang. 2021. Information Elicitation from Rowdy Crowds. In Proceedings of the 30th The Web Conference (WWW 2021)

    2021

  47. [55]

    Victor Shnayder, Arpit Agarwal, Rafael Frongillo, and David C. Parkes. 2016. Informed Truthfulness in Multi-Task Peer Prediction. In Proceedings of the 2016 ACM Conference on Economics and Computation (Maastricht, The Netherlands). Association for Computing Machinery, New York...

    2016

  48. [56]

    Frongillo, and David C

    Victor Shnayder, Rafael M. Frongillo, and David C. Parkes. 2016. Measuring Performance of Peer Prediction Mechanisms Using Replicator Dynamics. In 25th International Joint Conference on Artificial Intelligence (IJCAI 2016) . New York, NY. http://nrs.harvard.edu/urn-3:HUL.InstR...

    2016

  49. [57]

    Victor Shnayder and David Parkes. 2016. Practical Peer Prediction for Peer Assessment. Proceedings of the AAAI Conference on Human Computation and Crowdsourcing 4, 1 (Sep. 2016), 199–208

    2016

  50. [58]

    Bo Waggoner and Yiling Chen. 2014. Output Agreement Mechanisms and Common Knowledge. Proceedings of the AAAI Conference on Human Computation and Crowdsourcing 2, 1 (Sep. 2014), 220–226

    2014

  51. [59]

    Jörgen Wit. 1998. Rational choice and the Condorcet jury theorem. Games and Economic Behavior 22, 2 (1998), 364–376

    1998

  52. [60]

    Jens Witkowski and David C Parkes. 2013. Learning the prior in minimal peer prediction. In Proceedings of the 3rd Workshop on Social Computing and User Generated Content at the ACM Conference on Electronic Commerce , Vol. 14. 5

    2013

  53. [61]

    Stefan Wojcik, Sophie Hilgard, Nick Judd, Delia Mocanu, Stephen Ragain, MB Hunzaker, Keith Coleman, and Jay Baxter. 2022. Birdwatch: Crowd wisdom and bridging algorithms can inform understanding and reduce the spread of misinformation. arXiv preprint arXiv:2210.15723 (2022)

    2022 arXiv

  54. [62]

    Huibing Yin, Uday V Shanbhag, and Prashant G Mehta. 2011. Nash equilibrium problems with scaled congestion costs and shared constraints. IEEE transactions on automatic control 56, 7 (2011), 1702–1708

    2011

  55. [63]

    Yichi Zhang and Grant Schoenebeck. 2023. Multitask Peer Prediction With Task- dependent Strategies. In Proceedings of the 32nd The Web Conference (WWW 2023). Han et al. A Properties of proper scoring rule. For a scoring rule𝑃𝑆 , we define ˜𝑃𝑆 : ΔS× ΔS→ R such that for two dist...

    2023

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.