{"id":"ad93451c-be7d-49a1-bb88-be5dc7c405ca","arxiv_id":"2411.13293","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An action distribution is information-rationalizable for a given utility and prior exactly when the prior lies in a weighted sum of optimal-belief polytopes, with sharp finite-inequality tests in three-state, affine-difference, and two-step cases.","lead":"This paper asks when a decision maker's observed average choices can be explained as optimal responses to some information about the state, even when the analyst never sees choices state by state. It gives finite inequality tests that the utility function, prior, and action distribution must pass for such an information-based explanation to exist.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"I read the paper as a theoretical characterization: for a fixed, known utility function u, the set of priors consistent with an observed action distribution ν0 is exactly the Minkowski sum M(u,ν0), and membership can be tested by finitely many support-function inequalities. The reader's weakest assumption concerns the known-utility and exact-marginal premises, which are indeed the main limitations for empirical use but are explicitly acknowledged in Remark 1 and are not part of the theorem's hypothesis in a way that creates circularity or inconsistency. I examined the central proof elements: the Minkowski-sum representation (Equations BPμ0 and Oμ), the support-function characterization (Observation 1), the reduction to one-dimensional normal cones (Theorem 1 and Theorem A.1), and the dual-based arguments in Appendix A.2. The algebra in Lemma 1's truncation construction checks out, and the induction decompositions in Theorems 3 and 4 are coherent. The two-dimensional reasoning behind Theorem 2 is valid because in R^2 extreme rays of the common refinement must lie on boundaries of the original fans. I found no specific erroneous step or counterexample. The paper cites standard polytope facts and provides proofs for the nonstandard claims. The only honest residual risk is that the lengthy inductions in Appendix A.2 were not mechanically verified; a random computational verification of Theorem 3 would settle that risk. Since no concern actually lands, the reader's ACCEPT verdict should remain unchanged, though the moderate confidence is appropriate given the lack of formal verification.","tokens_in":46535,"tokens_out":19845,"duration_ms":212403,"concrete_test":"Independently verify Theorem 3 by generating random instances with affine utility differences, computing M(u,ν0) exactly via vertex enumeration of the ν0-weighted Minkowski sum of the polytopes Δ*_u(a), and checking that the 2|Ω| linear inequalities in Equation 9 characterize M(u,ν0); any violation would reveal a hidden error in the induction proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing concern identified. Theorem 1 follows directly from the Minkowski-sum representation in Equation MS and the standard fact that the normal fan of a Minkowski sum is the common refinement of the summands' normal fans; checking extreme rays of one-dimensional cones is sufficient because the support function is linear on each normal cone. The refinements in Theorems 2-4 are intricate, but the dual arguments and inductions in Appendix A.2 are internally consistent; no erroneous step was found. The explicitly acknowledged limitations, that the utility function u is known and that the observed action distribution ν0 is treated as exact, restrict empirical applicability but do not undermine the mathematical characterization. The only residual risk is that the lengthy induction proofs were not machine-checked, but this is a verification gap, not a demonstrated flaw.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":46648,"tokens_out":13203,"duration_ms":139111,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"Definition 6 and Theorem 4"},{"comment":"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.'","section":"Theorem 4 statement"},{"comment":"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.","section":"Appendix A.2, proof of Lemma 1"},{"comment":"The notation 'U SC(A)' appears with an unwanted space; it should be 'USC(A)' for the space of upper-semicontinuous functions.","section":"Appendix B"},{"comment":"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.","section":"Corollary 2"},{"comment":"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.","section":"Section 3, discussion after Equation (4)"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The paper's central observation — that checking an observed action distribution against a known utility's optimal-belief sets is just asking whether the prior lies in the ν0-weighted Minkowski sum of those sets — is a repackaging of Bayes plausibility. But the paper earns its keep by making that operational: Theorem 1's normal-fan test functions give a finite inequality system, Theorem 2 gives a clean closed form for up to three states, and the affine and two-step cases (Theorems 3 and 4) reduce the test to a small set of explicit inequalities that are easy to compute. The four-state counterexample (Example 2) is a nice, concrete warning that the simple test fails in general, and the Gale-flow core characterization for implementing posterior distributions is a useful bridge to the stochastic choice literature.\n\nThe paper is honest about its biggest restriction: the utility function is fixed and known. Remark 1 explains that identifying u and the prior jointly is hopeless without further structure, which is true, but it means the empirical applications require the analyst to commit to a parametric utility. That's the same limitation most revealed-preference tests have, so it's not fatal — it's just the reason this is a theory toolkit rather than a ready-made estimator. The treatment of ν0 as an exact marginal, with no sampling error, also limits direct use on real data. The authors acknowledge it; it's a standard modeling choice, not a hidden flaw.\n\nThe main thing I couldn't fully verify is the long induction in Appendix A.2 for Lemma 1 and Theorems 3–4. The decomposition steps are asserted and checked, and I didn't find an error, but I also didn't reach the standard of 'I could reproduce this blind.' That's a verification gap, not a demonstrated flaw.\n\nWho's this for: theorists in information economics, and empirical researchers who are willing to fix a parameterized utility and wonder about the set of priors consistent with their observed action frequencies. If I were refereeing, I'd ask for a cleaner proof of Lemma 1 or a remark clarifying which steps are mechanical checks. The paper deserves a serious referee.","headline":"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.","tokens_in":47156,"tokens_out":2212,"would_cite":true,"duration_ms":27533,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["revealed information","Bayes correlated equilibrium","support function","Minkowski sum","normal fan","distributions with given marginals","stochastic choice","information design"],"falsifier":"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.","tokens_in":46312,"feed_emoji":"📊","tokens_out":17365,"duration_ms":145001,"temperature":0.7,"pith_summary":"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.","feed_headline":"Action data alone can reveal whether a decision maker used information","feed_subtitle":"For a known utility, the priors that rationalize observed choices form a polytope with a finite test.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines Bayes correlated equilibrium, whose obedience inequalities are the single-agent condition used in Definition 1.","marker":"Bergemann and Morris (2016)"},{"why":"Identifies information structures with Bayes-plausible distributions over posteriors, the representation behind equations (BP) and (O).","marker":"Kamenica and Gentzkow (2011)"},{"why":"Supplies the revelation and obedience formulation that turns existence of an information structure into the linear system in Definition 1.","marker":"Myerson (1982)"},{"why":"Gives the support-function calculus for Minkowski sums used in Observation 1 and Equation (3).","marker":"Rockafellar (1970)"},{"why":"Provides the normal-fan and common-refinement machinery that gives the finite H-representation in Theorem 1.","marker":"Ziegler (2012)"},{"why":"Gives the marginal-existence theorem used to extend the support-function characterization to compact Polish spaces in Appendix B.","marker":"Strassen (1965)"},{"why":"Supplies the flow-feasibility theorem on which Proposition 5's characterization of implementing posterior distributions rests.","marker":"Gale (1957)"},{"why":"Shows how rationalizing choices across decision problems reduces to an auxiliary decision problem, the basis of Proposition 4.","marker":"Bergemann, Brooks, and Morris (2022)"},{"why":"Provides the first-order approach conditions used in the Appendix B extension to continuous action and state spaces.","marker":"Kolotilin, Corrao, and Wolitzky (2025)"}],"fun_headline_variants":["Action frequencies alone can prove if choices are information-based","Finite test reveals if a decision maker used private information","From action margins to information: a polytope characterization","When do observed choices imply hidden information? A sharp test","Finite check: can action data reveal hidden information?"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Action frequencies alone can prove if choices are information-based","Finite test reveals if a decision maker used private information","From action margins to information: a polytope characterization","When do observed choices imply hidden information? A sharp test","Finite check: can action data reveal hidden information?"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000598,"raw_usage":{"total_tokens":2841,"prompt_tokens":1033,"completion_tokens":1808,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":649,"completion_tokens_details":{"reasoning_tokens":1729}},"tokens_in":649,"tokens_out":1808,"duration_ms":14195,"temperature":1.0,"reasoning_tokens":1729,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:36:46.546759+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines Bayes correlated equilibrium, whose obedience inequalities are the single-agent condition used in Definition 1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Identifies information structures with Bayes-plausible distributions over posteriors, the representation behind equations (BP) and (O)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the revelation and obedience formulation that turns existence of an information structure into the linear system in Definition 1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the support-function calculus for Minkowski sums used in Observation 1 and Equation (3)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the normal-fan and common-refinement machinery that gives the finite H-representation in Theorem 1."},{"cited_title":"(1965): The existence of probability measures with given marginals, The Annals of Mathematical Statistics, 36, 423--439","cited_arxiv_id":null,"evidence_quote":"Gives the marginal-existence theorem used to extend the support-function characterization to compact Polish spaces in Appendix B."},{"cited_title":"(1957): A theorem on flows in networks, Pacific J","cited_arxiv_id":null,"evidence_quote":"Supplies the flow-feasibility theorem on which Proposition 5's characterization of implementing posterior distributions rests."},{"cited_title":"Corrao, and A","cited_arxiv_id":null,"evidence_quote":"Provides the first-order approach conditions used in the Appendix B extension to continuous action and state spaces."}],"review_version":1}