Pith. sign in

REVIEW 6 minor 51 references

Revealed Information

T0 review · 0 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read Fixing the decision maker's utility, an observed action distribution is information-rationalizable exactly when the prior lies in a weighted Minkowski sum of optimal-belief sets—a condition checkable by finitely many linear inequalities.

desk verdict Solid, useful theory: a finite test for information-consistency of action marginals, provided the utility is known; the special-case characterizations are the real payoff. read the letter →

arxiv 2411.13293 v2 pith:HODXP7KV submitted 2024-11-20 econ.TH econ.EMmath.OC

Laura Doval , Ran Eilat , Tianhao Liu , Yangfan Zhou This is my paper · ORCID
classification econ.THecon.EMmath.OC
keywords revealedinformationBayescorrelatedequilibriumsupportfunctionMinkowskisumnormalfandistributionswithgivenmarginalsstochasticchoicedesign
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 asks when an analyst who observes only how often a decision maker takes each action—not how often each action is taken conditional on the state—can conclude that the choices are consistent with the decision maker having learned something about the state before acting. Fixing the decision maker's utility function $u$, the set of priors $\mu_0$ for which an observed action distribution $\nu_0$ can be rationalized by some information structure is exactly the $\nu_0$-weighted Minkowski sum of the sets $\Delta^*_u(a)$ of beliefs at which each action is optimal, and membership in this set is equivalent to a finite system of linear inequalities. This matters because researchers rarely observe state-contingent choice data: the characterization lets an analyst test information-rationalizability, or identify the set of priors consistent with the data, without observing the information structure itself. Under mild restrictions on the number of states or on the utility function, the inequalities have closed forms and are directly computable.

What carries the argument

The load-bearing object is the $\nu_0$-weighted Minkowski sum $M(u,\nu_0)=\sum_a \nu_0(a)\Delta^*_u(a)$, where $\Delta^*_u(a)$ is the polytope of beliefs at which action $a$ maximizes expected utility. Its support function decomposes as the $\nu_0$-weighted sum of the support functions of the $\Delta^*_u(a)$, turning set membership into one inequality per direction. The common refinement of the normal fans of the $\Delta^*_u(a)$ supplies the finite set of directions that define the polytope's H-representation; for monotone-concave problems a dual linear program with adjacent obedience constraints identifies which test functions are needed, giving closed forms for affine and two-step utility differences.

What would settle it

Take a finite decision problem with at least four states and two actions, such as $\Omega=\{\omega_1,\ldots,\omega_4\}$, $u(a_1,\cdot)=0$, $u(a_2,\cdot)=(-9,-5,-1,5)$, and $\nu_0$ uniform. Compute the set of priors satisfying the claimed inequalities of Theorems 2 and 3, and independently compute the set of priors for which the linear program (O, $M_{\mu_0}$, $M_{\nu_0}$) in Definition 1 is feasible, say on a fine grid. If any prior passes the claimed inequalities but has no feasible joint distribution, or vice versa, the H-representation of $M(u,\nu_0)$ is falsified.

Watch

Extended reading notes

Core claim

The paper's central claim is that for a fixed utility $u$, the pair $(\mu_0,\nu_0)$ is BCE-consistent—there exists a joint distribution of states and actions with these marginals under which every recommended action is optimal—if and only if $\mu_0\in M(u,\nu_0)=\sum_{a\in A}\nu_0(a)\Delta^*_u(a)$. Equivalently, $\sum_a \nu_0(a)\max_{\mu\in\Delta^*_u(a)} p\cdot\mu \ge p\cdot\mu_0$ for all test directions $p$, and Theorem 1 shows only finitely many directions need be checked: the extreme rays of the one-dimensional cones in the common refinement of the normal fans of the polytopes $\Delta^*_u(a)$. With at most three states, or with affine or two-step utility differences, the test directions have closed forms, yielding the belief-martingale and payoff-martingale inequalities (Theorem 2) and systems of $2|\Omega|$ or $2(|A|-1)$ inequalities (Theorems 3 and 4). The paper also characterizes which Bayes-plausible distributions over posteriors implement a rationalizable $\nu_0$ as those satisfying coalitional inequalities $\sum_{a\in B}\nu_0(a)\ge\sum_{C\subseteq B}\tau_A(C)$ for all $B\subseteq A$, and it uses the characterization for comparative statics and for testing whether one information structure rationalizes choices across several decision problems.

Load-bearing premise

The characterization takes the decision maker's utility function as known or drawn from a parameterized family, and takes the observed action distribution $\nu_0$ to be the exact marginal of one obedient joint distribution; if $u$ is misspecified or the frequencies are noisy estimates, the finite inequality tests do not directly apply.

Editorial extensions

If this is right

  • An analyst who knows $u$ but not the prior can read the set of priors consistent with an observed $\nu_0$ directly off the polytope $M(u,\nu_0)$; if $\nu_0$ puts weight on a strictly dominated action, the set is empty and no information structure rationalizes the data.
  • For at most three states, the belief-martingale and payoff-martingale inequalities are explicit and finite: they bound the prior from below and require the prior's expected payoff differences to lie within what the observed action frequencies can support.
  • Under affine utility differences—which cover every binary-action problem—rationalizability reduces to $2|\Omega|$ linear inequalities, and moving the prior to a $d$-mean-preserving spread preserves rationalizability.
  • If the same decision maker is observed in several decision problems, one information structure rationalizes the joint action distribution exactly when that joint distribution is BCE-consistent in the auxiliary decision problem with summed payoffs, extending the test to multi-decision and public-persuasion settings.
  • For any rationalizable pair, the Bayes-plausible posterior distributions that implement it are exactly those satisfying the coalitional inequalities of Proposition 5, so the full set of implementing information structures is characterized, not merely its existence.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the observed action frequencies are estimates from finite samples, the sharp inequalities become a testing problem; a natural extension is to check the inequalities with a slack proportional to sampling error, or to compute a confidence set for the implied priors.
  • The affine-difference case is a generalized convex-order condition, so existing empirical tests of stochastic dominance or mean-preserving spreads could be repurposed as tests of information-rationalizability.
  • Because $M(u,\nu_0)$ is a Minkowski sum, mixing observed action distributions from different populations corresponds to Minkowski combinations of the corresponding prior sets, which suggests a way to aggregate or compare information-rationalizability across heterogeneous groups.
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

0 major / 6 minor

Summary. The paper studies when an observed marginal distribution over actions can be rationalized as the outcome of a decision maker who observes some information before acting. For a fixed utility function u, prior μ0, and action distribution ν0, the authors define BCE-consistency via the existence of an obedient joint distribution with those marginals. Their central result, Theorem 1, characterizes the set M(u,ν0) of priors consistent with ν0 as the ν0-weighted Minkowski sum of the optimal-belief sets Δ*_u(a), and shows that membership can be checked by finitely many support-function inequalities indexed by the extreme rays of the common refinement of the normal fans of the Δ*_u(a). Theorems 2–4 provide more explicit test-function sets under small state spaces or under monotone/concave and affine- or two-step-difference utility assumptions. Section 5 derives comparative statics and cross-decision-problem consistency; Section 6 characterizes the set of posterior distributions implementing a given marginal via Gale's flow theorem; Appendix B extends the main ideas to compact Polish spaces under a first-order approach.

Significance. If the results hold, the paper makes a genuine contribution: it converts the existence question for a single-agent Bayes correlated equilibrium with observed action marginals into a finite system of inequalities, which is exactly the kind of characterization needed for empirical work that observes only average choices. The main derivations are clean and appropriately use standard tools: support functions and normal fans from convex geometry, Strassen's theorem for the continuum extension, and Gale's flow theorem for the posterior-implementation result. The paper is honest about its scope: the utility function is fixed or parameterized, and the observed action distribution is treated as exact. These limitations are stated explicitly in Remark 1 and do not undermine the mathematical claims. The paper does not rely on fitted parameters or self-citations, and the finite inequality tests are, in principle, falsifiable.

minor comments (6)
  1. [Definition 6 and Theorem 4] The notation for the two values of d(aj+1,aj,·) is hard to parse: the manuscript writes 'dj+1,j < 0 < dj+1,j', which uses the same symbol for both values, and the formula for q↑_j then becomes ambiguous. Please introduce distinct notation such as \underline{d}_{j+1,j} and \overline{d}_{j+1,j} throughout the statement and proof.
  2. [Theorem 4 statement] The sentence 'The (µ0,ν0) ∈ ∆(Ω) × ∆(A) is BCE-consistent given u' contains a grammatical error; it should read 'The pair (µ0,ν0) ∈ ∆(Ω) × ∆(A) is BCE-consistent given u.'
  3. [Appendix A.2, proof of Lemma 1] The assertion that the dual minimizer p is single-peaked is stated without proof. It follows from the fact that each term is monotone in the state and p is the minimum of an increasing and a decreasing family, but spelling this out in one sentence would make the proof more transparent.
  4. [Appendix B] The notation 'U SC(A)' appears with an unwanted space; it should be 'USC(A)' for the space of upper-semicontinuous functions.
  5. [Corollary 2] The sentence 'Whenever a1 is optimal at the prior, the right-hand side of Equation 10 is 0' is imprecise: Equation 10 defines the lower bound LB, so the sentence should read 'the lower bound LB(µ0,d) equals 0'; the analogous comment applies to Equation 11 and the upper bound.
  6. [Section 3, discussion after Equation (4)] The proof of Theorem 2 in Appendix A.1 is concise, especially the d=2 case. A short expansion of why every facet normal of M(u,ν0) must already be a facet normal of some Δ*_u(a) in R^2, rather than a new direction created by the intersection of vertex cones, would help the reader see why |Ω|≤3 is the critical threshold.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central characterization is a direct derivation from stated primitives against external mathematical results.

full rationale

The paper's core claim, Theorem 1, is derived from the Minkowski-sum representation M(u,ν0)=Σ_a ν0(a)Δ*_u(a), which is an immediate rewriting of Definition 1 in terms of conditional beliefs: Bayes plausibility plus obedience exactly say that the prior is a ν0-weighted average of beliefs in the Δ*_u(a) sets, and conversely any such weighted average yields a feasible joint distribution. This is an equivalence by construction of the definitions, not a circular reduction of a prediction to a fitted input. The subsequent support-function characterization uses the standard external fact that the normal fan of a Minkowski sum is the common refinement of the summands' normal fans (Ziegler 2012), and the test-function refinements in Theorems 2-4 are proved in Appendix A.2 through dual linear programs and inductions; no fitted parameter is renamed as a prediction and no benchmark is reverse-engineered. Proposition 5 is derived from Gale's (1957) flow theorem, and the first-order-approach extension in Appendix B relies on Strassen (1965) and Kolotilin et al. (2025), all external. The paper's stated limitations—that the utility function u is known and that the observed action distribution ν0 is treated as exact—are explicit modeling assumptions, not hidden equivalences. There are no self-citations used to justify a load-bearing uniqueness claim or to smuggle in an ansatz. Accordingly, the paper is self-contained against external benchmarks and merits a circularity score of 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

Pure theory paper: the results are derivations from stated assumptions. No free parameters are fitted to data. The input objects are the primitives (u, μ0, ν0) and standard convex-geometric facts. The model assumptions are the BCE consistency definition, finite spaces, non-dominated support, and the monotone and concavity restrictions used in Section 4.

assumptions (7)
  • domain assumption The DM's behavior is captured by a Bayes correlated equilibrium: there exists a joint distribution π over actions and states with marginals μ0 and ν0 satisfying obedience constraints (Definition 1).
    This is the rationalization concept imported from Kamenica and Gentzkow (2011) and Bergemann and Morris (2016); the paper's goal is to characterize when such π exists given only marginals.
  • domain assumption The analyst knows the DM's utility function u; only the prior and the information structure are unknown (Remark 1).
    Without fixing or parameterizing u, any ν0 can be rationalized, as the paper notes in Remark 1; all theorems are conditional on u.
  • domain assumption Finite state and action spaces in the main text; compact Polish spaces in Appendix B under the first-order approach.
    Section 2 sets Ω and A finite; Appendix B relaxes to compact Polish spaces with additional regularity conditions.
  • domain assumption No action in the support of ν0 is strictly dominated, and every nonempty Δ*_u(a) is full-dimensional, with the general case handled in Appendix A.1.
    Section 2 and the discussion after Equation (MS); the sup-convention handles dominated actions, and Appendix A.1 handles lower-dimensional polytopes.
  • ad hoc to paper Assumption 1: utility has increasing differences and concavity*, meaning at most two optimal adjacent actions at any belief, used for Lemma 1 and Theorems 3 and 4.
    This is a substantive restriction on utility; it narrows the binding obedience constraints to adjacent pairs and makes p single-peaked in the dual. The paper presents it as a class of decision problems, not as a consequence of more primitive assumptions.
  • domain assumption Affine utility differences (Definition 5) or two-step utility differences (Definition 6) for the closed-form systems in Theorems 3 and 4.
    These functional-form assumptions define the tractable classes; binary actions make affine differences without loss, and absolute loss is a limit of two-step differences.
  • standard math Standard convex geometry: support functions of Minkowski sums, normal fans, common refinements, and facet-defining halfspaces, plus Strassen's theorem and Gale's flow theorem.
    Used in Observation 1, Theorem 1, Theorem A.1, Appendix B, and Proposition 5; these are external mathematical results, not derived in the paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Revealed Information." pith.science (2026). https://pith.science/paper/HODXP7KV

@misc{pith2026241113293,
  author       = {Pith},
  title        = {Pith review of: Revealed Information},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HODXP7KV}},
  note         = {Machine review of arXiv:2411.13293}
}
read the original abstract

An analyst observes the frequency with which a decision maker (DM) takes actions, but not the frequency conditional on payoff-relevant states. We ask when the analyst can rationalize the DM's choices as if the DM first learns something about the state before acting. We provide a support-function characterization of the triples of utility functions, prior beliefs, and (marginal) distributions over actions such that the DM's action distribution is consistent with information given the DM's prior and utility function. Assumptions on the cardinality of the state space and the utility function allow us to refine this characterization, obtaining a sharp system of finitely many inequalities the utility function, prior, and action distribution must satisfy. We apply our characterization to study comparative statics and to identify conditions under which a single information structure rationalizes choices across multiple decision problems. We characterize the set of distributions over posterior beliefs that are consistent with the DM's choices. We extend our results to settings with a continuum of actions and states assuming the first-order approach applies, and to simple multi-agent settings.

Figures

Figures reproduced from arXiv: 2411.13293 by the authors.

Figure 1
Figure 1. The set M(u, ν0) in Example 1 for ν0 = (1/3, 1/3, 1/3) finitely many halfspaces (the so-called H-representation). In fact, letting ext(X) denote the set of extreme points of a subset X of R Ω, we have that ext(M(u, ν0)) ⊂ X a∈A ν0(a)ext(∆∗ u (a)). (V -rep) In words, any extreme point of M(u, ν0) is a ν0-weighted combination of extreme points in the sets ∆∗ u (a), but the opposite may not hold. Although Equation MS c… view at source ↗
Figure 2
Figure 2. Illustration of Definition 2 in Example 1. Theorem 1 characterizes the set of priors µ0 such that (µ0, ν0) is BCE-consistent given u via a finite system of inequalities µ0 must satisfy. In practice, the analyst knows neither the prior µ0 nor the DM’s utility u. From this perspective, Theorem 1 describes the joint restrictions on the pairs (µ0, u) for which an information structure exists that rationalizes the given … view at source ↗
Figure 3
Figure 3. Illustrating Theorem 2 in Example 1 for ν0 = (1/3, 1/3, 1/3). implies Equation PM. Consider now the case in which the DM has three actions, A = {a1, a2, a3}, and once again, Equation PM for the pair a1, a2. Whereas the obedience constraints feature the comparison between these two actions when a1 or a2 is recommended, no such inequality arises when a3 is recommended. Still, Equation PM adds up over all actions, incl… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Theorem 2 does not hold when |Ω| ≥ 4. facets as illustrated in Figure 4b. Indeed, whereas the grey facets correspond to the test func￾tions in the statement of Theorem 2, the hatched red facet has normal vector (0, −2, −5), which corresponds to the test function p = (0…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

51 extracted references · 51 canonical work pages

  1. [1]

    Aliprantis, C. D. and K. C. Border (2013): Infinite Dimensional Analysis: A Hitchhiker's Guide, Springer-Verlag Berlin and Heidelberg GmbH & Company KG

    work page 2013

  2. [2]

    Arieli, I. and Y. Babichenko (2019): Private bayesian persuasion, Journal of Economic Theory, 182, 185--217

    work page 2019

  3. [3]

    Babichenko, F

    Arieli, I., Y. Babichenko, F. Sandomirskiy, and O. Tamuz (2021): Feasible joint posterior beliefs, Journal of Political Economy, 129, 2546--2594

    work page 2021

  4. [4]

    Azrieli, Y. and J. Rehbeck (2022): Marginal stochastic choice, arXiv preprint arXiv:2208.08492

    work page Pith review arXiv 2022

  5. [5]

    Molinari, T

    Barseghyan, L., F. Molinari, T. O'Donoghue, and J. C. Teitelbaum (2013): The nature of risk preferences: Evidence from insurance choices, American economic review, 103, 2499--2529

    work page 2013

  6. [6]

    Prince, and J

    Barseghyan, L., J. Prince, and J. C. Teitelbaum (2011): Are risk preferences stable across contexts? Evidence from insurance data, American Economic Review, 101, 591--631

    work page 2011

  7. [7]

    Molchanov, and F

    Beresteanu, A., I. Molchanov, and F. Molinari (2011): Sharp identification regions in models with convex moment predictions, Econometrica, 79, 1785--1821

    work page 2011

  8. [8]

    Brooks, and S

    Bergemann, D., B. Brooks, and S. Morris (2015): The limits of price discrimination, American Economic Review, 105, 921--957

    work page 2015

Show all 51 references
  1. [9]

    --- -.1pt --- -.1pt --- (2022): Counterfactuals with Latent Information, American Economic Review, 112, 343--368

    2022

  2. [10]

    Bergemann, D. and S. Morris (2016): Bayes correlated equilibrium and the comparison of information structures in games, Theoretical Economics, 11, 487--522

    2016

  3. [11]

    Border, K. C. (1991): Functional analytic tools for expected utility theory, in Positive Operators, Riesz Spaces, and Economics, ed. by C. D. Aliprantis, K. C. Border, and W. A. Luxemburg, Springer, 69--88

    1991

  4. [12]

    Caplin, A. and M. Dean (2015): Revealed preference, rational inattention, and costly information acquisition, American Economic Review, 105, 2183--2203

    2015

  5. [13]

    Dean, and J

    Caplin, A., M. Dean, and J. Leahy (2022): Rationally inattentive behavior: Characterizing and generalizing Shannon entropy, Journal of Political Economy, 130, 1676--1715

    2022

  6. [14]

    Caplin, A., D. J. Martin, and P. Marx (2023): Rationalizable Learning, Tech. rep., National Bureau of Economic Research

    2023

  7. [15]

    Chambers, C. P., C. Liu, and J. Rehbeck (2020): Costly information acquisition, Journal of Economic Theory, 186, 104979

    2020

  8. [16]

    Cohen, A. and L. Einav (2007): Estimating risk preferences from deductible choice, American economic review, 97, 745--788

    2007

  9. [17]

    Das, S., S. R. Dev, and S. Sarvottamananda (2024): A worst-case optimal algorithm to compute the Minkowski sum of convex polytopes, Discrete Applied Mathematics, 350, 44--61

    2024

  10. [18]

    De Oliveira, H. and R. Lamba (2022): Rationalizing dynamic choices, Available at SSRN 3332092

    2022

  11. [19]

    (2022): Posterior separable cost of information, American Economic Review, 112, 3215--3259

    Denti, T. (2022): Posterior separable cost of information, American Economic Review, 112, 3215--3259

    2022

  12. [20]

    Dewan, A. and N. Neligh (2020): Estimating information cost functions in models of rational inattention, Journal of Economic Theory, 187, 105011

    2020

  13. [21]

    Dickstein, M. J., J. Jeon, and E. Morales (2024): Patient costs and physicians' information, Tech. rep., National Bureau of Economic Research

    2024

  14. [22]

    Dickstein, M. J. and E. Morales (2018): What do exporters know? The Quarterly Journal of Economics, 133, 1753--1801

    2018

  15. [23]

    Dillenberger, D., R. V. Krishna, and P. Sadowski (2023): Subjective information choice processes, Theoretical Economics, 18, 529--559

    2023

  16. [24]

    Dillenberger, D., J. S. Lleras, P. Sadowski, and N. Takeoka (2014): A theory of subjective learning, Journal of Economic Theory, 153, 287--312

    2014

  17. [25]

    Enke, B. and T. Graeber (2023): Cognitive uncertainty, The Quarterly Journal of Economics, 138, 2021--2067

    2023

  18. [26]

    Ergin, H. and T. Sarver (2010): A unique costly contemplation representation, Econometrica, 78, 1285--1339

    2010

  19. [27]

    (1957): A theorem on flows in networks, Pacific J

    Gale, D. (1957): A theorem on flows in networks, Pacific J. Math, 7, 1073--1082

    1957

  20. [28]

    Galichon, A. and M. Henry (2011): Set identification in models with multiple equilibria, The Review of Economic Studies, 78, 1264--1298

    2011

  21. [29]

    Gualdani, C. and S. Sinha (2024): Identification in discrete choice models with imperfect information, Journal of Econometrics, 244, 105854

    2024

  22. [30]

    Kamenica, E. and M. Gentzkow (2011): Bayesian persuasion, American Economic Review, 101, 2590--2615

    2011

  23. [31]

    Khaw, M. W., Z. Li, and M. Woodford (2021): Cognitive imprecision and small-stakes risk aversion, The Review of Economic Studies, 88, 1979--2013

    2021

  24. [32]

    (2015): Identifying higher-order rationality, Econometrica, 83, 2065--2079

    Kneeland, T. (2015): Identifying higher-order rationality, Econometrica, 83, 2065--2079

    2015

  25. [33]

    (2018): Optimal information disclosure: A linear programming approach, Theoretical Economics, 13, 607--635

    Kolotilin, A. (2018): Optimal information disclosure: A linear programming approach, Theoretical Economics, 13, 607--635

    2018

  26. [34]

    Corrao, and A

    Kolotilin, A., R. Corrao, and A. Wolitzky (2025): Persuasion and matching: Optimal productive transport, 133

    2025

  27. [35]

    Kolotilin, A. and A. Wolitzky (2024): Distributions of Posterior Quantiles via Matching, Theoretical Economics, 19, 1399--1413

    2024

  28. [36]

    (2016): Random choice and private information, Econometrica, 84, 1983--2027

    Lu, J. (2016): Random choice and private information, Econometrica, 84, 1983--2027

    2016

  29. [37]

    Magnolfi, L. and C. Roncoroni (2023): Estimation of discrete games with weak assumptions on information, The Review of Economic Studies, 90, 2006--2041

    2023

  30. [38]

    Molchanov, I. and F. Molinari (2018): Random sets in econometrics, vol. 60, Cambridge University Press

    2018

  31. [39]

    Morris, S. E. (2020): No trade and feasible joint posterior beliefs, Tech. rep., Massachusetts Institute of Technology

    2020

  32. [40]

    Myerson, R. B. (1982): Optimal coordination mechanisms in generalized principal--agent problems, Journal of Mathematical Economics, 10, 67--81

    1982

  33. [41]

    (2024): Identifying prediction mistakes in observational data, The Quarterly Journal of Economics, 139, 1665--1711

    Rambachan, A. (2024): Identifying prediction mistakes in observational data, The Quarterly Journal of Economics, 139, 1665--1711

    2024

  34. [42]

    (2023): Revealed Bayesian expected utility with limited data, Journal of Economic Behavior & Organization, 207, 81--95

    Rehbeck, J. (2023): Revealed Bayesian expected utility with limited data, Journal of Economic Behavior & Organization, 207, 81--95

    2023

  35. [43]

    Rockafellar, R. T. (1970): Convex Analysis, Convex Analysis, 28

    1970

  36. [44]

    (1982): Convergence of Lebesgue integrals with varying measures, Sankhy \=a : The Indian Journal of Statistics, Series A , 380--402

    Serfozo, R. (1982): Convergence of Lebesgue integrals with varying measures, Sankhy \=a : The Indian Journal of Statistics, Series A , 380--402

    1982

  37. [45]

    Strack, P. and K. H. Yang (2024): Privacy Preserving Signals, Available at SSRN 4467608

    2024

  38. [46]

    (1965): The existence of probability measures with given marginals, The Annals of Mathematical Statistics, 36, 423--439

    Strassen, V. (1965): The existence of probability measures with given marginals, The Annals of Mathematical Statistics, 36, 423--439

    1965

  39. [47]

    Tamer, and J

    Syrgkanis, V., E. Tamer, and J. Ziani (2017): Inference on auctions with weak assumptions on information, arXiv preprint arXiv:1710.03830

    2017 arXiv

  40. [48]

    Toikka, and R

    Vohra, A., J. Toikka, and R. Vohra (2023): Bayesian persuasion: Reduced form approach, Journal of Mathematical Economics, 102863

    2023

  41. [49]

    (2007): Minkowski sums of polytopes: combinatorics and computation, Tech

    Weibel, C. (2007): Minkowski sums of polytopes: combinatorics and computation, Tech. rep., EPFL

    2007

  42. [50]

    Yang, K. H. and A. K. Zentefis (2024): Monotone function intervals: Theory and applications, American Economic Review, 114, 2239--2270

    2024

  43. [51]

    Ziegler, G. M. (2012): Lectures on polytopes, vol. 152, Springer Science & Business Media

    2012

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.