pith. machine review for the scientific record. sign in

arxiv: 2604.17505 · v1 · submitted 2026-04-19 · 💻 cs.GT · cs.AI· cs.LG· cs.MA

Recognition: unknown

Learning Unanimously Acceptable Lotteries via Queries

Davin Choo , Paul W. Goldberg , Nicholas Teh
Authors on Pith no claims yet

Pith reviewed 2026-05-10 05:02 UTC · model grok-4.3

classification 💻 cs.GT cs.AIcs.LGcs.MA
keywords lotteryunanimous acceptancequery modelbinary feedbackstakeholder elicitationrandomized algorithmslearning-augmented algorithmsfeasibility certification
0
0 comments X

The pith

Algorithms can locate a lottery that all stakeholders accept or prove none exists by asking only yes-or-no questions about proposed lotteries.

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

The paper establishes that in a setting with a finite menu of options and stakeholders each having private acceptability thresholds, it is possible to search for a unanimously acceptable lottery using queries that propose lotteries and receive binary accept or reject responses. A sympathetic reader would care because many real-world decisions, such as approving AI systems, require consensus without forcing full disclosure of individual standards. The deterministic algorithms ensure correctness while bounding the queries, and randomized variants lower the expected number by introducing randomness in proposals. Adaptivity means not every stakeholder needs to be queried on every possible constraint.

Core claim

The central discovery is that deterministic and randomized algorithms exist which, through adaptive or non-adaptive querying of lotteries to stakeholders for binary feedback, either output a lottery that is acceptable to every stakeholder or correctly certify that no such lottery exists. These algorithms achieve query complexity that is linear in the number of stakeholders with logarithmic dependence on precision parameters, and lower bounds establish that this dependence is necessary in the worst case. Learning-augmented versions further improve performance when given accurate predictions about which stakeholders are binding or which lottery is promising.

What carries the argument

The binary query model, in which the algorithm proposes a lottery over the finite options and receives a single accept or reject signal from each queried stakeholder based on their private threshold.

If this is right

  • If a unanimously acceptable lottery exists, the algorithm will find one.
  • Adaptivity reduces the total queries by not eliciting all constraints from all stakeholders.
  • Randomization decreases the expected number of queries compared to deterministic full elicitation.
  • Lower bounds prove that at least a linear number of queries in the number of stakeholders is required in the worst case.
  • Learning-augmented algorithms improve query complexity when predictions are accurate without worsening the worst case.

Where Pith is reading between the lines

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

  • This query approach could apply to other consensus problems where preferences are threshold-based rather than full rankings.
  • If stakeholders have correlated thresholds, the expected query savings from randomization might be even larger in practice.
  • The method suggests that for AI deployment decisions, partial elicitation via lottery proposals is sufficient for unanimous approval checks.

Load-bearing premise

Stakeholders answer queries truthfully according to fixed private thresholds for what lotteries they accept.

What would settle it

An instance where a unanimously acceptable lottery exists but the algorithm either fails to find it or incorrectly certifies infeasibility, or where the query count exceeds the claimed bounds.

Figures

Figures reproduced from arXiv: 2604.17505 by Davin Choo, Nicholas Teh, Paul W. Goldberg.

Figure 1
Figure 1. Figure 1: Example 2.3 (m = 3): halfspace acceptability regions A1, A2, A3 on the probability simplex. The vertices e1, e2, e3 are pure lotteries, arrows indicate the acceptable side of each constraint, and x ∗ is a feasible lottery in T3 i=1 Ai (gray). Deterministic algorithms must be correct on all instances. Our randomized algorithms are always correct; randomness only affects the number of queries. We analyze the… view at source ↗
Figure 2
Figure 2. Figure 2: Each point on this line segment is a lottery. The red (resp., green) region indicates the set of rejected [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
read the original abstract

Many high-stakes AI deployments proceed only if every stakeholder deems the system acceptable relative to their own minimum standard. With randomization over a finite menu of options, this becomes a feasibility question: does there exist a lottery over options that clears all stakeholders' acceptability bars? We study a query model where the algorithm proposes lotteries and receives only binary accept/reject feedback. We give deterministic and randomized algorithms that either find a unanimously acceptable lottery or certify infeasibility; adaptivity can avoid eliciting many stakeholders' constraints, and randomization further reduces the expected elicitation cost relative to full elicitation. We complement these upper bounds with worst-case lower bounds (in particular, linear dependence on the number of stakeholders and logarithmic dependence on precision are unavoidable). Finally, we develop learning-augmented algorithms that exploit natural forms of advice (e.g., likely binding stakeholders or a promising lottery), improving query complexity when predictions are accurate while preserving worst-case guarantees.

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

0 major / 3 minor

Summary. The paper studies the problem of finding a lottery over a finite menu of options that is unanimously acceptable to all stakeholders, where each stakeholder has a private acceptability threshold and provides only binary accept/reject feedback on queried lotteries. It develops deterministic and randomized adaptive algorithms that either output such a lottery or certify infeasibility, gives matching worst-case lower bounds (linear in the number of stakeholders and logarithmic in precision), and introduces learning-augmented variants that exploit predictions (e.g., likely binding stakeholders) while retaining worst-case guarantees.

Significance. If the results hold, the work advances query-complexity analysis in multi-agent mechanism design and AI alignment settings that require unanimous acceptance. The explicit benefits of adaptivity (avoiding full elicitation for many stakeholders) and randomization (lower expected cost), together with the learning-augmented algorithms that improve on accurate predictions without sacrificing robustness, are practically relevant. The paper ships matching upper/lower bounds and worst-case guarantees for the learning setting; these are clear strengths.

minor comments (3)
  1. §2 (Model): the definition of the feasibility LP and the query model should explicitly state how lotteries are represented (e.g., as vectors in the simplex) and how the binary response maps to half-space constraints for each stakeholder.
  2. §4 (Randomized algorithm): the expected-query analysis relies on a specific sampling distribution over the simplex; a short paragraph clarifying why this distribution yields the claimed logarithmic dependence on precision would improve readability.
  3. The lower-bound proofs (Theorem X) are stated for the worst-case deterministic and randomized query models; a brief remark on whether the same linear-in-n lower bound holds under the learning-augmented model would strengthen the comparison.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. We appreciate the recognition of the paper's contributions on adaptive and randomized query algorithms for unanimously acceptable lotteries, the matching lower bounds, and the learning-augmented variants with robustness guarantees.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper develops deterministic and randomized query algorithms to solve a feasibility problem over lotteries with binary stakeholder feedback, proves matching upper and lower bounds on query complexity using standard information-theoretic and LP arguments, and presents learning-augmented variants that use external advice while retaining worst-case guarantees. No self-definitional reductions, fitted parameters renamed as predictions, or load-bearing self-citations appear; all claims rest on explicit algorithmic constructions and adversarial lower-bound arguments that are independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard assumptions from algorithmic game theory and query complexity; no free parameters, invented entities, or non-standard axioms are visible in the abstract.

axioms (2)
  • domain assumption Stakeholders provide consistent binary accept/reject responses based on fixed private thresholds.
    Implicit in the query model and feasibility question stated in the abstract.
  • domain assumption The menu of options is finite.
    Required for the lottery to be well-defined and for feasibility to be decidable.

pith-pipeline@v0.9.0 · 5462 in / 1292 out tokens · 47569 ms · 2026-05-10T05:02:41.231680+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

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

  1. Efficient Ensemble Selection from Binary and Pairwise Feedback

    cs.GT 2026-05 unverdicted novelty 7.0

    The paper develops efficient algorithms for ensemble selection from binary and pairwise feedback, achieving (1-1/e) guarantees with query savings for coverage and PTAS-style results via submodular relaxation for theta...

Reference graph

Works this paper leans on

53 extracted references · 2 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    Portioning using ordinal preferences: Fairness and efficiency

    St\'ephane Airiau, Haris Aziz, Ioannis Caragiannis, Justin Kruger, J\'er\^ o me Lang, and Dominik Peters. Portioning using ordinal preferences: Fairness and efficiency. Artificial Intelligence, 314: 0 103809, 2023

    2023

  2. [2]

    Constrained Markov Decision Processes

    Eitan Altman. Constrained Markov Decision Processes. Chapman and Hall, 1999

    1999

  3. [3]

    The complexity and approximability of finding maximum feasible subsystems of linear relations

    Edoardo Amaldi and Viggo Kann. The complexity and approximability of finding maximum feasible subsystems of linear relations. Theoretical Computer Science, 147 0 (1): 0 181--210, 1995

    1995

  4. [4]

    Mlops deployment best practices for real-time inference model serving endpoints with amazon sagemaker, 2023

    Amazon Web Services . Mlops deployment best practices for real-time inference model serving endpoints with amazon sagemaker, 2023. URL https://aws.amazon.com/blogs/machine-learning/mlops-deployment-best-practices-for-real-time-inference-model-serving-endpoints-with-amazon-sagemaker/

    2023

  5. [5]

    Queries and concept learning

    Dana Angluin. Queries and concept learning. Machine Learning, 2 0 (4): 0 319--342, 1988

    1988

  6. [6]

    Anthropic R esponsible S caling P olicy (version 2.2), 2025

    Anthropic . Anthropic R esponsible S caling P olicy (version 2.2), 2025. URL https://www-cdn.anthropic.com/872c653b2d0501d6ab44cf87f43e1dc4853e4d37.pdf

    2025

  7. [7]

    Constitutional AI: Harmlessness from AI Feedback

    Yuntao Bai, Saurav Kadavath, Sandipan Kundu, Amanda Askell, Jackson Kernion, Andy Jones, Anna Chen, Anna Goldie, Azalia Mirhoseini, Cameron McKinnon, Carol Chen, Catherine Olsson, Christopher Olah, Danny Hernandez, Dawn Drain, Deep Ganguli, Dustin Li, Eli Tran-Johnson, Ethan Perez, Jamie Kerr, Jared Mueller, Jeffrey Ladish, Joshua Landau, Kamal Ndousse, K...

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  8. [8]

    Preference elicitation and query learning

    Avrim Blum, Jeffrey Jackson, Tuomas Sandholm, and Martin Zinkevich. Preference elicitation and query learning. Journal of Machine Learning Research, 5: 0 649--667, 2004

    2004

  9. [9]

    A pomdp formulation of preference elicitation problems

    Craig Boutilier. A pomdp formulation of preference elicitation problems. In Proceedings of the 18th AAAI Conference on Artificial Intelligence (AAAI), pages 239--246, 2002

    2002

  10. [10]

    Bshouty, Paul W

    Nader H. Bshouty, Paul W. Goldberg, Sally A. Goldman, and H. David Mathias. Exact learning of discretized geometric concepts. SIAM Journal on Computing , 28 0 (2): 0 674--699, 1998 a

    1998

  11. [11]

    Bshouty, Sally A

    Nader H. Bshouty, Sally A. Goldman, H. David Mathias, Subhash Suri, and Hisao Tamaki. Noise-tolerant distribution-free learning of general geometric concepts. Journal of the ACM , 45 0 (5): 0 863--890, 1998 b

    1998

  12. [12]

    Truthful aggregation of budget proposals with proportionality guarantees

    Ioannis Caragiannis, George Christodoulou, and Nicos Protopapas. Truthful aggregation of budget proposals with proportionality guarantees. Artificial Intelligence, 335: 0 104178, 2024

    2024

  13. [13]

    Servedio, and Tianqi Yang

    Xi Chen, Anindya De, Yizhi Huang, Shivam Nadimpalli, Rocco A. Servedio, and Tianqi Yang. Halfspaces are hard to test with relative error. arXiv preprint arXiv:2511.06171, 2025

    work page arXiv 2025

  14. [14]

    Clarkson

    Kenneth L. Clarkson. Las vegas algorithms for linear and integer programming when the dimension is small. Journal of the ACM, 42 0 (2): 0 488--499, 1995

    1995

  15. [15]

    Preference elicitation in combinatorial auctions

    Wolfram Conen and Tuomas Sandholm. Preference elicitation in combinatorial auctions. In Proceedings of the 3rd ACM Conference on Electronic Commerce (EC), pages 256--259, 2001

    2001

  16. [16]

    Holliday, Bob M

    Vincent Conitzer, Rachel Freedman, Jobst Heitzig, Wesley H. Holliday, Bob M. Jacobs, Nathan Lambert, Milan Moss\' e , Eric Pacuit, Stuart Russell, Hailey Schoelkopf, Emanuel Tewolde, and William S. Zwicker. Position: Social choice should guide ai alignment in dealing with diverse human feedback. In Proceedings of the 41st International Conference on Machi...

    2024

  17. [17]

    Helly's theorem and its relatives

    Ludwig Danzer, Branko Grünbaum, and Victor Klee. Helly's theorem and its relatives. In Proceedings of Symposia in Pure Mathematics, volume 7, pages 101--180, 1963

    1963

  18. [18]

    Settling the score: Portioning with cardinal preferences

    Edith Elkind, Matthias Greger, Patrick Lederer, Warut Suksompong, and Nicholas Teh. Settling the score: Portioning with cardinal preferences. Artificial Intelligence, 352: 0 104487, 2026

    2026

  19. [19]

    Regulation (eu) 2024/1689 of the E uropean P arliament and of the C ouncil ( A rtificial I ntelligence A ct), 2024

    European Parliament and Council of the European Union . Regulation (eu) 2024/1689 of the E uropean P arliament and of the C ouncil ( A rtificial I ntelligence A ct), 2024. URL https://eur-lex.europa.eu/eli/reg/2024/1689/oj/eng

    2024

  20. [20]

    Project-fair and truthful mechanisms for budget aggregation

    Rupert Freeman and Ulrike Schmidt-Kraepelin. Project-fair and truthful mechanisms for budget aggregation. In Proceedings of the 38th AAAI Conference on Artificial Intelligence (AAAI), pages 9704--9712, 2024

    2024

  21. [21]

    Truthful aggregation of budget proposals

    Rupert Freeman, David Pennock, Dominik Peters, and Jennifer Wortman Vaughan . Truthful aggregation of budget proposals. Journal of Economic Theory, 193: 0 105234, 2021

    2021

  22. [22]

    A comprehensive survey on safe reinforcement learning

    Javier Garc \'i a and Fernando Fern \'a ndez. A comprehensive survey on safe reinforcement learning. Journal of Machine Learning Research, 16: 0 1437--1480, 2015

    2015

  23. [23]

    Datasheets for datasets

    Timnit Gebru, Jamie Morgenstern, Briana Vecchione, Jennifer Wortman Vaughan, Hanna Wallach, Hal Daum\' e III, and Kate Crawford. Datasheets for datasets. Communications of the ACM, 64 0 (12): 0 86--92, 2021

    2021

  24. [24]

    Goldberg and Francisco J

    Paul W. Goldberg and Francisco J. Marmolejo Coss \' o. Learning convex partitions and computing game-theoretic equilibria from best-response queries. ACM Transactions on Economics and Computation , 9 0 (1): 0 3:1--3:36, 2021

    2021

  25. [25]

    Goldberg and Stephen Kwek

    Paul W. Goldberg and Stephen Kwek. The precision of query points as a resource for learning convex polytopes with membership queries. In Proceedings of the 13th Annual Conference on Computational Learning Theory (COLT), pages 225--235, 2000

    2000

  26. [26]

    Algorithmic I mpact A ssessment tool, 2020

    Government of Canada . Algorithmic I mpact A ssessment tool, 2020. URL https://www.canada.ca/en/government/system/digital-government/digital-government-innovations/responsible-use-ai/algorithmic-impact-assessment.html

    2020

  27. [27]

    Guide to peer review of automated decision systems, 2025

    Government of Canada . Guide to peer review of automated decision systems, 2025. URL https://www.canada.ca/en/government/system/digital-government/digital-government-innovations/responsible-use-ai/guide-peer-review-automated-decision-systems.html

    2025

  28. [28]

    Über mengen konvexer körper mit gemeinschaftlichen punkten

    Eduard Helly. Über mengen konvexer körper mit gemeinschaftlichen punkten. Jahresbericht der Deutschen Mathematiker-Vereinigung, 32: 0 175--176, 1923

    1923

  29. [29]

    Improving online algorithms via ML predictions

    Ravi Kumar, Manish Purohit, and Zoya Svitkina. Improving online algorithms via ML predictions. In Proceedings of the 32nd Annual Conference on Neural Information Processing Systems (NeurIPS), pages 9684--9693, 2018

    2018

  30. [30]

    Competitive caching with machine learned advice

    Thodoris Lykouris and Sergei Vassilvitskii. Competitive caching with machine learned advice. Journal of the ACM, 68 0 (4), 2021

    2021

  31. [31]

    A subexponential bound for linear programming

    Ji r \' Matou s ek, Micha Sharir, and Emo Welzl. A subexponential bound for linear programming. Algorithmica, 16 0 (4--5): 0 498--516, 1996

    1996

  32. [32]

    Model cards for model reporting

    Margaret Mitchell, Simone Wu, Andrew Zaldivar, Parker Barnes, Lucy Vasserman, Ben Hutchinson, Elena Spitzer, Inioluwa Deborah Raji, and Timnit Gebru. Model cards for model reporting. In Proceedings of the 4th ACM Conference on Fairness, Accountability, and Transparency (FAccT), pages 220--229, 2019

    2019

  33. [33]

    Roger B. Myerson. Optimal auction design. Mathematics of Operations Research, 6 0 (1): 0 58--73, 1981

    1981

  34. [34]

    John F. Nash. Equilibrium points in n -person games. Proceedings of the National Academy of Sciences, 36 0 (1): 0 48--49, 1950

    1950

  35. [35]

    Vazirani, editors

    Noam Nisan, Tim Roughgarden, Eva Tardos, and Vijay V. Vazirani, editors. Algorithmic Game Theory. Cambridge University Press, 2007

    2007

  36. [36]

    OpenAI P reparedness F ramework (version 2), 2025

    OpenAI . OpenAI P reparedness F ramework (version 2), 2025. URL https://cdn.openai.com/pdf/18a02b5d-6b67-4cec-ab64-68cdfbddebcd/preparedness-framework-v2.pdf

    2025

  37. [37]

    Osborne and Ariel Rubinstein

    Martin J. Osborne and Ariel Rubinstein. A Course in Game Theory. MIT Press, 1994

    1994

  38. [38]

    Training language models to follow instructions with human feedback

    Long Ouyang, Jeffrey Wu, Xu Jiang, Diogo Almeida, Carroll Wainwright, Pamela Mishkin, Chong Zhang, Sandhini Agarwal, Katarina Slama, Alex Ray, John Schulman, Jacob Hilton, Fraser Kelton, Luke Miller, Maddie Simens, Amanda Askell, Peter Welinder, Paul F Christiano, Jan Leike, and Ryan Lowe. Training language models to follow instructions with human feedbac...

    2022

  39. [39]

    Sculley, Sebastian Nowozin, Joshua V

    Yaniv Ovadia, Emily Fertig, Jie Ren, Zachary Nado, D. Sculley, Sebastian Nowozin, Joshua V. Dillon, Balaji Lakshminarayanan, and Jasper Snoek. Can you trust your model's uncertainty? evaluating predictive uncertainty under dataset shift. In Proceedings of the 33rd Annual Conference on Neural Information Processing Systems (NeurIPS), pages 14003--14014, 2019

    2019

  40. [40]

    Puterman

    Martin L. Puterman. Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley Series in Probability and Statistics. John Wiley & Sons, 1994

    1994

  41. [41]

    Lawrence, editors

    Joaquin Qui \ n onero-Candela, Masashi Sugiyama, Anton Schwaighofer, and Neil D. Lawrence, editors. Dataset Shift in Machine Learning. The MIT Press, 2008

    2008

  42. [42]

    White, Margaret Mitchell, Timnit Gebru, Ben Hutchinson, Jamila Smith-Loud, Daniel Theron, and Parker Barnes

    Inioluwa Deborah Raji, Andrew Smart, Rebecca N. White, Margaret Mitchell, Timnit Gebru, Ben Hutchinson, Jamila Smith-Loud, Daniel Theron, and Parker Barnes. Closing the AI accountability gap: defining an end-to-end framework for internal algorithmic auditing. In Proceedings of the 3rd ACM Conference on Fairness, Accountability, and Transparency (FAccT), p...

    2020

  43. [43]

    Near-optimal bounds for online caching with machine learned advice

    Dhruv Rohatgi. Near-optimal bounds for online caching with machine learned advice. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1834--1845, 2020

    2020

  44. [44]

    Small-dimensional linear programming and convex hulls made easy

    Raimund Seidel. Small-dimensional linear programming and convex hulls made easy. Discrete & Computational Geometry, 6: 0 423--434, 1991

    1991

  45. [45]

    A combinatorial bound for linear programming and related problems

    Micha Sharir and Emo Welzl. A combinatorial bound for linear programming and related problems. In Proceedings of the 9th Annual Symposium on Theoretical Aspects of Computer Science (STACS), pages 567--579, 1992

    1992

  46. [46]

    Herbert A. Simon. A behavioral model of rational choice. The Quarterly Journal of Economics, 69 0 (1): 0 99--118, 1955

    1955

  47. [47]

    Position: A roadmap to pluralistic alignment

    Taylor Sorensen, Jared Moore, Jillian Fisher, Mitchell Gordon, Niloofar Mireshghallah, Christopher Michael Rytting, Andre Ye, Liwei Jiang, Ximing Lu, Nouha Dziri, Tim Althoff, and Yejin Choi. Position: A roadmap to pluralistic alignment. In Proceedings of the 41st International Conference on Machine Learning (ICML), 2024

    2024

  48. [48]

    Voting in divisible settings: A survey

    Warut Suksompong and Nicholas Teh. Voting in divisible settings: A survey. In Proceedings of the 40th AAAI Conference on Artificial Intelligence (AAAI), pages 39789--39796, 2026

    2026

  49. [49]

    Sutton and Andrew G

    Richard S. Sutton and Andrew G. Barto. Reinforcement Learning: An Introduction. MIT Press, 2 edition, 2018

    2018

  50. [50]

    Introduction to AI assurance, 2024

    UK Department for Science, Innovation and Technology . Introduction to AI assurance, 2024. URL https://www.gov.uk/government/publications/introduction-to-ai-assurance

    2024

  51. [51]

    Artificial I ntelligence R isk M anagement F ramework (ai rmf 1.0), 2023

    US Department of Commerce . Artificial I ntelligence R isk M anagement F ramework (ai rmf 1.0), 2023. URL https://nvlpubs.nist.gov/nistpubs/ai/nist.ai.100-1.pdf

    2023

  52. [52]

    Theory of Games and Economic Behavior

    John von Neumann and Oskar Morgenstern. Theory of Games and Economic Behavior. Princeton University Press, 1944

    1944

  53. [53]

    Probabilistic computations: Toward a unified measure of complexity

    Andrew Chi-Chin Yao. Probabilistic computations: Toward a unified measure of complexity. In Proceedings of the 18th Annual Symposium on Foundations of Computer Science (FOCS), pages 222--227, 1977

    1977