Pith. sign in

REVIEW 4 minor 140 references

Interpretive agreement is convex combination of signal models, completed by cosine similarity.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · grok-4.5

2026-07-11 05:48 UTC pith:2YXOP4X7

load-bearing objection Clean geometric order on interpretive disagreement with a sharp convex-hull characterization and unique cosine completion; solid theory paper.

arxiv 2607.05558 v1 pith:2YXOP4X7 submitted 2026-07-06 econ.TH

Agreement and Diversity in Interpretation

Francesco Bilotta , Luca Braghieri , Collin Raymond , Mark Whitmeyer This is my paper
classification econ.TH
keywords interpretive disagreementsubjective modelssignal structuresinclusion preordercosine similarityspeculative tradeex-ante Pareto frontierBregman divergences
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

When people share a prior, payoffs, and reservation value but disagree about how signals are generated, their willingness to commit to joint plans can be used to rank how close their interpretations are. The paper shows that one pair of signal models supports a larger set of mutually acceptable plans than another if and only if each model in the first pair is a mixture of the two models in the second. That ranking does not depend on the prior. The only rotation-invariant scalar that completes the ranking is cosine similarity between the joint state-signal distributions. The same geometry shrinks the room for pure bets against each other, enlarges a normalized Pareto frontier of joint payoffs, and expands the set of single-model stories that can rationalize the joint behavior. The order is independent of Blackwell informativeness and selects quadratic rather than Kullback-Leibler-type distances.

Core claim

A pair of subjective signal structures is more agreeable than another if and only if each structure in the more agreeable pair is a convex combination of the two structures in the less agreeable pair; this inclusion preorder is prior-independent, and its unique rotation-invariant strict completion ranks pairs by cosine similarity of the induced joint distributions over states and signals.

What carries the argument

The inclusion preorder: the cone of surplus vectors that both agents accept ex-ante under one pair of models contains the corresponding cone under another pair, which dualizes to the convex-hull condition on the models themselves.

Load-bearing premise

The reservation payoff is a fixed common number, so every possible surplus vector can be realized by some finite decision problem; if the reservation must instead come from an action that is already optimal under the prior, full cone inclusion is only sufficient, not necessary.

What would settle it

Construct two pairs of binary experiments that are not related by convex combination yet induce nested cones of jointly acceptable surplus vectors under some full-support prior, or find a rotation-invariant completion of the inclusion preorder that is not ordered by cosine similarity.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper studies joint decision-making by two agents who share prior, state space, actions, payoffs, reservation payoff, and Bayesian updating, but may disagree about signal likelihoods. It defines an inclusion preorder over pairs of subjective models: (m, m') is more agreeable than (m̂, m̂') if the cone of jointly acceptable surplus vectors under the former contains that under the latter, uniformly across decision problems. Proposition 1 shows this is equivalent to each model in the more agreeable pair being a convex combination of the two models in the less agreeable pair, and that the comparison is prior-independent. Proposition 5 shows that the unique rotation-invariant strict completion of the (incomplete) inclusion preorder is ranking by cosine similarity of the induced joint distributions. Applications establish that greater agreement shrinks speculative-trade transfer intervals (Proposition 2), expands a suitably normalized ex-ante Pareto frontier (Proposition 3), and enlarges the set of single-model rationalizations (Proposition 4). The order is independent of Blackwell dominance and selects quadratic over KL-type Bregman divergences among rotation-invariant comparisons.

Significance. If the characterizations hold, the paper supplies a clean, decision-theoretic, prior-free partial order on interpretive disagreement that is grounded in joint participation constraints rather than ad-hoc statistical distance. The convex-hull representation (Proposition 1) and the uniqueness of the cosine completion under rotation invariance (Proposition 5) are transparent and rest on standard dual-cone and separating-hyperplane arguments that are fully written out in Appendix A. The applications give the order immediate economic content for speculative trade, Pareto frontiers, and external rationalizability. The careful treatment of the reservation-payoff spanning assumption (footnote 6 and Online Appendix B.2) and the explicit comparison with Blackwell and Bregman divergences further strengthen the contribution. The results are machine-checkable from the written proofs and require no free parameters.

minor comments (4)
  1. In the abstract and introduction the phrase "selects quadratic over KL-type Bregman divergences" is slightly loose; Proposition 9 shows that the only rotation-invariant Bregman divergence on the sphere is a multiple of squared Euclidean distance (hence cosine). A one-sentence clarification would prevent misreading.
  2. Figure 1 and Figure 2 are helpful, but the captions could more explicitly state the coordinates (θ11, θ22) and the meaning of the dashed segments so that a reader skimming the figures alone can recover the geometry.
  3. A few minor typos appear (e.g., "suprlus" near the end of §4.2, "won" for "down" in the same paragraph). A careful proof-reading pass would catch them.
  4. The online appendix material on introspection-proofness (B.1) and the prior-optimal reservation variant (B.2) is useful; a brief pointer in the main-text conclusion would help readers locate it.

Circularity Check

0 steps flagged

No significant circularity: inclusion preorder and cosine uniqueness are derived characterizations, not restatements of inputs.

full rationale

The paper is pure decision theory with no data, fitted parameters, or empirical predictions. The inclusion preorder is defined directly from ex-ante participation cones C(p_m, p_m') (Definition 2). Proposition 1 then derives the equivalent convex-hull representation on models via dual-cone geometry and the fact that probability vectors force coefficients to sum to one; the prior-cancellation argument shows the comparison is intrinsic to the experiments. Proposition 5 defines a rotation-invariant strict completion by three axioms (extension of inclusion, rotation invariance of cones, strictness for strict rotated inclusions) and proves uniqueness by reducing to the aperture of the normal cone, which is exactly the angle (hence cosine) between the belief vectors. Both results are standard convex-geometry characterizations; neither is forced by construction from its own definition, nor does any load-bearing step rest on a self-citation uniqueness theorem or an ansatz. Applications (speculative-trade nesting, normalized Pareto expansion, rationalizing-model enlargement) are direct corollaries. The reservation-spanning assumption is explicitly flagged and relaxed in Online Appendix B.2 without circularity. No steps match the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 2 invented entities

Pure decision-theoretic paper. No free parameters. Background axioms are standard finite Bayesian decision theory plus the maintained modeling choices that isolate interpretive disagreement (common prior, common payoffs, dogmatic likelihoods). The only paper-specific modeling choice that materially affects the main theorem is the fixed scalar reservation payoff that restores full spanning of surplus space.

axioms (4)
  • domain assumption Agents share a common prior p over a finite state space Ω, a common finite action set and utility, a common reservation payoff ū, and Bayesian updating; they may differ only in signal likelihoods m(s|ω).
    Stated in Section 2.1; isolates interpretive disagreement from preference or prior heterogeneity.
  • domain assumption Finite decision problems with a fixed scalar reservation payoff span the entire surplus space R^{Ω×S}.
    Used to equate cone inclusion with uniform behavioral implications (Section 2.2 and Proposition 1); relaxed in Online Appendix B.2.
  • standard math Standard convex analysis: dual of a cone generated by two vectors is the intersection of the corresponding half-spaces; separating hyperplane theorem for compact convex sets.
    Invoked throughout Appendix A proofs of Propositions 1–5 and 9.
  • domain assumption Disagreement is dogmatic: agents do not entertain a common model of uncertainty over signal structures that could restore agreement.
    Footnote 7 and Introduction; rules out learning or higher-order beliefs about models.
invented entities (2)
  • Inclusion preorder ⪰_I on pairs of subjective models no independent evidence
    purpose: Decision-theoretic ranking of interpretive agreement via nested cones of jointly acceptable surplus vectors.
    Defined in Definition 2; characterized in Proposition 1. No independent empirical handle outside the paper’s geometry.
  • Rotation-invariant cosine completion ⪰_RIC no independent evidence
    purpose: Unique complete ranking that extends cone inclusion and is invariant to orthogonal reparameterizations of surplus space.
    Defined in Definition 4 and Proposition 5. Selected by the paper’s own axioms rather than external evidence.

pith-pipeline@v1.1.0-grok45 · 36654 in / 2539 out tokens · 22745 ms · 2026-07-11T05:48:42.970083+00:00 · methodology

0 comments
read the original abstract

We study joint decision-making when agents agree on all primitives other than signal likelihoods. We propose a decision-theoretic measure of interpretive disagreement: a pair of subjective models is more agreeable than another if, uniformly across decision problems, it supports a larger set of signal-contingent plans that both agents weakly prefer ex-ante to the common reservation payoff. We show that this measure is prior independent and can be represented as an inclusion preorder over pairs of subjective models: each model in the more agreeable pair is a convex combination of the two models in the less agreeable pair. We then show that the measure's unique rotation-invariant scalar completion is cosine similarity. Applications show that greater agreement reduces speculative-trade wedges, expands a normalized version of the ex-ante Pareto frontier, and enlarges the set of single-model rationalizations. Our order is independent of Blackwell dominance and selects quadratic over KL-type Bregman divergences.

Figures

Figures reproduced from arXiv: 2607.05558 by Collin Raymond, Francesco Bilotta, Luca Braghieri, Mark Whitmeyer.

Figure 1
Figure 1. Figure 1: Experiment space [0, 1]2 with axes θ11 (horizontal) and θ22 (verti￾cal). The inclusion chain is (A, A′ ) ≺I (B, B′ ) ≺I (C, C′ ). models under which an outside observer could rationalize the agents’ jointly acceptable behavior as optimal. These applications show that the inclusion order has economic content beyond the geometry used to define it. 4.1. Speculative Trade. Recall that a surplus vector x ∈ R Ω×… view at source ↗
Figure 2
Figure 2. Figure 2: Blackwell and agreement use different geometries. The arrows show the garbling chain A → E → E ′ → A′ , so (A, A′ ) is more spread out than (E, E′ ) in the Blackwell sense. The dashed segment is the fixed-interpretation slice conv{A, A′}. Here the Blackwell chain bends away from this segment: E and E ′ are Blackwell-between the endpoints of (A, A′ ), but are not convex￾between them. 6.2. Relation to Diverg… view at source ↗

discussion (0)

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

Reference graph

Works this paper leans on

140 extracted references · 1 linked inside Pith

  1. [1]

    Bohren, J Aislinn and Hauser, Daniel N , year=

  2. [2]

    2000 , publisher=

    Billot, Antoine and Chateauneuf, Alain and Gilboa, Itzhak and Tallon, Jean-Marc , journal=. 2000 , publisher=

    2000

  3. [3]

    2008 , publisher=

    Rigotti, Luca and Shannon, Chris and Strzalecki, Tomasz , journal=. 2008 , publisher=

    2008

  4. [4]

    2005 , publisher=

    Rigotti, Luca and Shannon, Chris , journal=. 2005 , publisher=

    2005

  5. [5]

    2014 , publisher=

    Brunnermeier, Markus K and Simsek, Alp and Xiong, Wei , journal=. 2014 , publisher=

    2014

  6. [6]

    2014 , publisher=

    Gilboa, Itzhak and Samuelson, Larry and Schmeidler, David , journal=. 2014 , publisher=

    2014

  7. [7]

    1978 , publisher=

    Harrison, J Michael and Kreps, David M , journal=. 1978 , publisher=

    1978

  8. [8]

    1951 , publisher=

    Kullback, Solomon and Leibler, Richard A , journal=. 1951 , publisher=

    1951

  9. [9]

    1967 , publisher=

    Bregman, Lev M , journal=. 1967 , publisher=

    1967

  10. [10]

    Aumann, Robert J , journal=

  11. [11]

    1983 , publisher=

    Sebenius, James K and Geanakoplos, John , journal=. 1983 , publisher=

    1983

  12. [12]

    Malmendier, Ulrike and Nagel, Stefan and Yan, Zhen , year=

  13. [13]

    2018 , publisher=

    Hansen, Stephen and McMahon, Michael and Prat, Andrea , journal=. 2018 , publisher=

    2018

  14. [14]

    Kaminski, Alexa and Macaulay, Alistair and Song, Wenting , year=

  15. [15]

    Braghieri, Luca , title =

  16. [16]

    Howes, Cooper and i Carreras, Marc Dordal and Coibion, Olivier and Gorodnichenko, Yuriy , year=

  17. [17]

    2021 , volume =

    Heidhues, Paul and K. 2021 , volume =

    2021

  18. [18]

    2023 , volume =

    Frick, Mira and Iijima, Ryota and Ishii, Yuhta , title =. 2023 , volume =

    2023

  19. [19]

    Aislinn and Hauser, Daniel N

    Bohren, J. Aislinn and Hauser, Daniel N. , title =. 2021 , volume =

    2021

  20. [20]

    2021 , note =

    Esponda, Ignacio and Pouzo, Demian and Yamamoto, Yuichi , title =. 2021 , note =

    2021

  21. [21]

    Aislinn , title =

    Bohren, J. Aislinn , title =. 2016 , volume =

    2016

  22. [22]

    2020 , volume =

    Frick, Mira and Iijima, Ryota and Ishii, Yuhta , title =. 2020 , volume =

    2020

  23. [23]

    2016 , volume =

    Esponda, Ignacio and Pouzo, Demian , title =. 2016 , volume =

    2016

  24. [24]

    2021 , volume =

    Fudenberg, Drew and Lanzani, Giacomo and Strack, Philipp , title =. 2021 , volume =

    2021

  25. [25]

    2025 , journal =

    Esponda, Ignacio and Pouzo, Demian , title =. 2025 , journal =

    2025

  26. [26]

    1991 , volume =

    Nyarko, Yaw , title =. 1991 , volume =

    1991

  27. [27]

    2025 , howpublished =

    Tamuz, Omer , title =. 2025 , howpublished =

    2025

  28. [28]

    2024 , volume =

    Frick, Mira and Iijima, Ryota and Ishii, Yuhta , title =. 2024 , volume =

    2024

  29. [29]

    2022 , volume =

    Frick, Mira and Iijima, Ryota and Ishii, Yuhta , title =. 2022 , volume =

    2022

  30. [30]

    Aislinn and Hauser, Daniel N

    Bohren, J. Aislinn and Hauser, Daniel N. , title =. 2025 , volume =

    2025

  31. [31]

    1995 , publisher=

    Kandel, Eugene and Pearson, Neil D , journal=. 1995 , publisher=

    1995

  32. [32]

    1999 , publisher=

    Bamber, Linda Smith and Barron, Orie E and Stober, Thomas L , journal=. 1999 , publisher=

    1999

  33. [33]

    1994 , publisher=

    Morris, Stephen , journal=. 1994 , publisher=

    1994

  34. [34]

    1982 , publisher=

    Milgrom, Paul and Stokey, Nancy , journal=. 1982 , publisher=

    1982

  35. [35]

    Au, Pak Hung , title =

  36. [36]

    Aislinn Bohren and Alex Imas , year =

    Cuimin Ba and J. Aislinn Bohren and Alex Imas , year =

  37. [37]

    Mira Frick and Ryota Iijima and Yuhta Ishii , year =

  38. [38]

    Aislinn Bohren and Daniel N

    J. Aislinn Bohren and Daniel N. Hauser , year =

  39. [39]

    Chambers and Yusufcan Masatlioglu and Collin Raymond , year =

    Christopher P. Chambers and Yusufcan Masatlioglu and Collin Raymond , year =

  40. [40]

    Davide Bordoli , year =

  41. [41]

    Braghieri, Luca , journal=

  42. [42]

    von Beringe, Konstantin and Whitmeyer, Mark , journal=

  43. [43]

    Morris, Stephen and Shin, Hyun Song , journal=

  44. [44]

    Milgrom, Paul and Shannon, Chris , title =

  45. [45]

    Yu, Feiyang and Moehring, Alex and Banerjee, Oishi and Salz, Tobias and Agarwal, Nikhil and Rajpurkar, Pranav , journal=

  46. [46]

    Green, Ben and Chen, Yiling , journal=

  47. [47]

    Eiermann, Martin and Fitzpatrick, Maria and Sadowski, Katharine and Wildeman, Christopher , journal=

  48. [48]

    de Clippel, Geoffroy and Zhang, Xu , title =

  49. [49]

    Quah and Bruno Strulovici , journal =

    John K.-H. Quah and Bruno Strulovici , journal =

  50. [50]

    2012 , number =

    Ray, Debraj and Robson, Arthur , title =. 2012 , number =

    2012

  51. [51]

    2010 , number =

    Xia, Bianjun , title =. 2010 , number =

    2010

  52. [52]

    2004 , Month =

    Hopkins, Ed and Kornienko, Tatiana , Title =. 2004 , Month =

    2004

  53. [53]

    Seel, Christian and Strack, Philipp , journal=

  54. [54]

    2016 , Month =

    Gentzkow, Matthew and Kamenica, Emir , Title =. 2016 , Month =

    2016

  55. [55]

    Kolotilin, Anton and Mylovanov, Tymofiy and Zapechelnyuk, Andriy and Li, Ming , title =

  56. [56]

    2015 , Month =

    Boleslavsky, Raphael and Cotton, Christopher , Title =. 2015 , Month =

    2015

  57. [57]

    Yucheng Ding and Tianle Zhang. 2018

    2018

  58. [58]

    Whitmeyer, Mark , journal=

  59. [59]

    2022 , publisher=

    Chan, David C and Gentzkow, Matthew and Yu, Chuan , journal=. 2022 , publisher=

    2022

  60. [60]

    Peter Klibanoff and Massimo Marinacci and Sujoy Mukerji , journal =

  61. [61]

    , title =

    Epstein, Larry G. , title =. 2006 , month =

    2006

  62. [62]

    Gul, Faruk and Pesendorfer, Wolfgang , title =

  63. [63]

    Siniscalchi, Marciano , title =

  64. [64]

    Machina and David Schmeidler , journal =

    Mark J. Machina and David Schmeidler , journal =

  65. [65]

    Machina , journal =

    Mark J. Machina , journal =

  66. [66]

    2012 , Month =

    Ortoleva, Pietro , Title =. 2012 , Month =

    2012

  67. [67]

    Grether , journal =

    David M. Grether , journal =

  68. [68]

    Kleiner, Andreas and Moldovanu, Benny and Strack, Philipp and Whitmeyer, Mark , journal=

  69. [69]

    Stahl , journal =

    Dale O. Stahl , journal =

  70. [70]

    Tyrrell , year=

    Rockafellar, R. Tyrrell , year=

  71. [71]

    Blackwell, David. 1951

    1951

  72. [72]

    Radner, Roy and Stiglitz, Joseph , Year =

  73. [73]

    Edwards, Ward , Year =

  74. [74]

    Schrag , journal =

    Matthew Rabin and Joel L. Schrag , journal =

  75. [75]

    Wakker, Peter , journal =

  76. [76]

    Safra, Zvi and Sulganik, Eyal , journal =

  77. [77]

    1950 , publisher=

    Wald, Abraham , booktitle=. 1950 , publisher=

    1950

  78. [78]

    1995 , publisher=

    Morris, Stephen , journal=. 1995 , publisher=

    1995

  79. [79]

    Deimen, Inga and Mora, Ernesto Rivera , journal=

  80. [80]

    1966 , publisher=

    Berk, Robert H , journal=. 1966 , publisher=

    1966

Showing first 80 references.