{"id":"20c32f2c-e3fc-4fd3-a2ba-f25101093068","arxiv_id":"2506.06319","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In a search market with savvy and inexperienced consumers, firms conceal values just below a reservation point, and in large markets they conceal all low values so that inexperienced consumers stop after a single visit.","lead":"This paper models how firms choose what to reveal about their products when some consumers shop around for free and others pay a cost to search. It finds that with a mix of the two, even intense competition does not force full honesty, and more rivals can make less experienced shoppers stop searching after one product.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4 is proven only under weak convexity of F^(n−1); whether the paradox of choice survives for non-convex priors is untested, so the headline result may be special-case dependent.","rationale":"The reader's weakest-assumption diagnosis matches my own: the weakly convex F^(n−1) prior is the load-bearing restriction behind the structural form, the uniqueness proofs, and the large-market limit. The paper explicitly scopes the model to this class in Section 2 and acknowledges limitations in Section 6.2, so this is not an internal inconsistency; however, the abstract and introduction state the paradox of choice without this caveat, and the proof of Proposition 4 genuinely relies on convexity in several non-removable places. The reader's conditional verdict (CONDITIONAL) is appropriate: the main theorem appears correct within the stated class, but the breadth of the claim is unverified beyond it. My concrete test would settle whether the no-disclosure-at-bottom result is robust or an artifact of the convexity assumption. I found no stronger internal flaw: the equilibrium construction, the search-equation simplification to (10), the single-crossing arguments in Proposition 4, and the large-market limit all check out under the maintained assumption, and the uniform example provides a transparent special case. The deferred auxiliary proofs are a secondary concern about completeness, not about the central logic.","tokens_in":47481,"tokens_out":11007,"duration_ms":122208,"concrete_test":"Take F(v)=v^θ on [0,1] with θ<1 and n=2 (so F^(n−1)=F is strictly concave), or a mixture of uniforms making F^(n−1) non-convex. Numerically solve the firm's Bayesian-persuasion problem for increasing n using the Dworczak–Martini optimality conditions (or a fine grid search over mean-preserving contractions), coupled with the reservation-value equation (1), to find a symmetric fixed point. Check whether the equilibrium lower disclosure threshold v_L equals 0 for all sufficiently large n and whether inexperienced consumers stop after the first firm. If v_L>0 persists as n→∞, the paradox of choice fails outside the convex class; if v_L=0, the convexity assumption is not the bottleneck and the paper's scope concern is mitigated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The single most load-bearing concern is that the main paradox-of-choice result (Proposition 4) is established only under the weakly convex F^(n−1) assumption, and the proof mechanisms producing no disclosure at the bottom depend on that assumption. The structural characterization in Proposition A1 and the key Lemma A5 (full disclosure below vL) rely on Claims 6 and 8, which use convexity or affineness of F^(n−1) on regions of strictly increasing payoff. The large-market limit in Lemma A6, the uniqueness of the contact point in Lemma A4, and the fixed-point uniqueness in Proposition 3 all inherit this structure. Section 6.2 explicitly acknowledges that for general F 'we would no longer expect full disclosure' away from the reservation value, but it does not characterize the large-market equilibrium in that case. Thus the paper's central claim—that competition alone is insufficient to restore transparency—is unproven for natural priors such as any F with a decreasing density when n=2 (where F^(n−1)=F is concave). This is a scope limitation rather than a detected internal error: within the convex class, the chain of lemmas appears coherent and the uniform example provides independent support. But without a check outside the class, the headline contribution may overstate its domain. The deferred proofs mentioned in the text (e.g., footnote 39 and Lemma 5) are secondary; they do not threaten the argument as much as the convexity dependence.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies a search market with n firms that choose information structures for their products, a fraction α of consumers who face a positive search cost, and a fraction 1−α of savvy consumers who search all firms. The authors characterize the unique symmetric pure-strategy perfect Bayesian equilibrium under the standing assumption that F(·)^(n−1) is weakly convex. The equilibrium features partial disclosure around the inexperienced consumers' reservation value: low valuations are pooled with higher valuations, while sufficiently high values may be disclosed. The central result, Proposition 4, states that in sufficiently large markets there is no disclosure below the reservation value, inexperienced consumers always stop after one visit, and the equilibrium does not converge to full disclosure as n grows, a phenomenon the authors call the paradox of choice. The paper also derives comparative statics in search cost and welfare effects for the two consumer types.","tokens_in":47752,"tokens_out":12584,"duration_ms":132488,"significance":"If the characterization is correct, the paper makes a substantive contribution: it shows that in a market with heterogeneous search costs, competition alone does not restore full transparency, and it provides a rational-choice foundation for choice overload. The analysis is self-contained and uses the Dworczak-Martini multiplier characterization carefully; the fixed-point construction in Sections 3 and 4 is coherent, and the uniform example supports the structural results. The paper also ships extensive appendix proofs, and the main result is not fitted to a pre-specified conclusion. The principal weakness is scope: the headline results are proven only for weakly convex F^(n−1), and Section 6.2 explicitly leaves general distributions to future work. Because the abstract and introduction state the large-market conclusion without this qualifier, the paper risks overstating the domain of its main contribution.","major_comments":[{"comment":"Proposition 4, the paradox-of-choice result, is established only under the standing assumption that F(·)^(n−1) is weakly convex. The structural characterization in Proposition A1, the limiting arguments in Lemma A6, and the proof of Proposition 4 all rely on this convexity in a load-bearing way. In particular, Claims 6 and 8 in the proof of Proposition A1 use convexity of F^(n−1) to rule out alternative affine regions, and the no-disclosure-at-the-bottom construction in Lemma A6 uses the single-peakedness of D(v, β) that follows from convexity. Natural priors, such as a decreasing density when n=2 (so F^(n−1) is concave), are outside the class. Section 6.2 acknowledges that for general F the equilibrium away from the reservation value need not be full disclosure, but it does not characterize the large-market equilibrium in that case. Since the abstract states that 'in large markets, firms always conceal low valuations' without this qualification, the paper should either extend the large-market result beyond the weakly convex class, provide a concrete counterexample showing the paradox fails outside the class, or explicitly condition all headline statements on the convexity assumption.","section":"Section 2 / Proposition 4 / Section 6.2"},{"comment":"The proof that an equilibrium distribution G(·) must be atomless is deferred with the statement 'A formal proof is available upon request.' Atomlessness is used to establish continuity of G(·) and is then relied on throughout Claims 1–8, including the intermediate-value argument in Claim 3 and the mean-preserving-contraction arguments in Lemma A2. This is a load-bearing step, and the full proof should be included in the appendix rather than left to the reader.","section":"Appendix, proof of Proposition A1, footnote 39"},{"comment":"The convergence of the equilibrium to full disclosure as r→0 is proved by considering two cases: F^(n−1) strictly convex and F^(n−1) affine. The standing assumption, however, is weak convexity, which also permits functions that are neither globally strictly convex nor globally affine, such as piecewise affine distributions with kinks. The argument that D(v, β) is strictly concave and therefore has a unique maximizer and at most two roots in Lemma A4 and the proof of Proposition 2 does not cover these weakly convex but non-strictly-convex cases. Please either extend the proof to the full weakly convex class or state precisely which additional condition is needed for the stated results.","section":"Proof of Proposition 2 / Lemma A4"}],"minor_comments":[{"comment":"The sentence 'Meanwhile, with any s < 0 the unique equilibrium is full disclosure' should presumably read 'with s = 0', since search costs are otherwise assumed positive.","section":"Section 5.1, infinite-market discussion"},{"comment":"The threshold r(n, α) also depends on the prior F, so the notation r(n, α, F) would be clearer and would avoid an apparent omission relative to the statement 'for each (n, α, s, F(·))'.","section":"Proposition 3"},{"comment":"Assumption 2 is introduced without an Assumption 1; the numbering should be adjusted, or the earlier model assumptions should be numbered.","section":"Section 6.1"},{"comment":"The thresholds denoted s and s are used in the proof via identities such as s = μ − r, but the statement of Proposition 5 does not define these thresholds. Please state the definitions of the two thresholds explicitly in the proposition.","section":"Proposition 5"}],"recommendation":"major_revision","confidential_remarks":"The convexity concern raised in the stress-test is real and is acknowledged by the authors in Section 6.2. I would recommend major revision rather than rejection because the center of the paper is coherent within the stated class, and the requested changes are scope qualifications or additional robustness analysis rather than a detected internal contradiction. The deferred proof of atomlessness should also be completed before the paper is accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a real advance, and the headline result is real within a stated class. The binary-type model—savvy consumers with zero search cost plus a positive-cost type—is new relative to Hwang et al. and Au-Whitmeyer, and the lower-censorship characterization is sharp. The paradox of choice (inexperienced consumers search in small markets but stop after one visit in large ones) is a genuine equilibrium finding, not a modeling artifact. The fixed-point argument in Sections 3 and 4 is coherent; I found no contradiction in the central equations, and the uniform example provides an independent check. The welfare corollaries—rich get richer, poor get poorer—follow cleanly from the informativeness ordering. The citation pattern is fair; the concurrent Hwang-Hwang work is discussed and distinguished, not buried.\n\nThe soft spot is exactly the one flagged. Proposition 4 is proved only under weak convexity of F^(n-1), and that assumption is load-bearing: it guarantees the affine payoff region, the uniqueness of the contact point, and the no-disclosure-at-bottom limit. For n=2, any prior with a decreasing density makes F concave, and the paper is silent on that case. Section 6.2 acknowledges the gap but does not characterize it. So the claim 'competition alone is insufficient to ensure transparency' is established only for the convex class, and the abstract should say so. This is a scope limitation, not an internal error—within the class, the proof chain holds up.\n\nTwo smaller issues. Two auxiliary proofs are deferred (the atomlessness argument in footnote 39 and the MPC-of-reservation-value claim in Lemma 5), one 'available upon request.' Neither threatens the main theorem, but for a posted version they should be supplied. Also, the uniqueness statement is scoped to symmetric pure-strategy equilibria with passive beliefs; that is fine, but it should be in the main text, not just the appendix.\n\nVerdict: this paper deserves a serious referee. I would send it to review, with a request to (i) either extend the result beyond convex F^(n-1) or clearly scope the abstract, and (ii) include the deferred proofs. A referee can do that in one round. I'd cite it if I worked on competitive information design.","headline":"A real result with a real scope condition: the paradox of choice holds under weak convexity of F^{n-1}, and the paper never shows it survives outside that class.","tokens_in":48267,"tokens_out":2237,"would_cite":true,"duration_ms":24105,"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":"In a search market with both savvy and inexperienced consumers, competition alone does not force firms to reveal low valuations; once the market is large enough, firms conceal everything below the reservation value and inexperienced…","keywords":["information design","search market","Bayesian persuasion","consumer search","heterogeneous search costs","partial disclosure","informational Diamond paradox","competition and disclosure"],"falsifier":"One concrete check is to compute the equilibrium for a prior $F$ on which $F^{n-1}$ is not weakly convex, such as a distribution with a flat density in the middle, and see whether all valuations below $r=\\mu-s$ still disappear from the support as $n$ grows; the paper only proves this under convexity. A second check within the paper's assumptions is to simulate the large-market equilibrium and inspect firms' signal distributions for any positive mass below $r$, which Proposition 4 says should be zero.","tokens_in":47274,"feed_emoji":"🛒","tokens_out":11651,"duration_ms":115083,"temperature":0.7,"pith_summary":"Firms selling products can choose how much to reveal about what a consumer would think of the product. In a market where some consumers visit every seller for free and others pay a search cost, the paper shows that equilibrium disclosure is partial, and that more competition does not cure it. The central result is that in a sufficiently large market every firm conceals all valuations below the inexperienced consumers' reservation value, so those consumers stop at the first firm they visit. This matters because it reverses the usual conclusion that competition forces transparency: even as the number of firms grows without bound, full disclosure is not approached, and the less experienced shoppers can be made worse off by more choice. Changes in search costs are also not a reliable cure, since they improve informativeness when costs are low but reduce it when costs are high.","feed_headline":"Expect concealment, not transparency, in large search markets","feed_subtitle":"As firms multiply, they pool low values into vague signals, so inexperienced shoppers stop early and lose out.","key_machinery":"The load-bearing object is the firm's payoff as a function of its realized posterior mean $v$: it is $G(v)^{n-1}$ below the reservation value $r$ and jumps upward at $r$, because an inexperienced consumer stops and buys immediately when $v\\ge r$ but keeps searching otherwise. This upward jump creates a kink in the firm's objective and is the reason full disclosure is not an equilibrium. The equilibrium is pinned down by two thresholds, a lower disclosure threshold $v_L$ and an upper contact point $v_H$, with $G^{n-1}$ affine on the interval between $r$ and $v_H$; a multiplier condition from convex duality (Dworczak and Martini, 2019) verifies that this structure is a best response. The reservation value itself is endogenous and solves the Weitzman search equation, which, given the equilibrium structure, reduces to an equation in $v_L$ and $r$ only.","core_discovery":"The paper's central discovery is a precise characterization of the unique symmetric equilibrium of a disclosure game with $n$ firms, a mass of savvy consumers who visit all firms, and a mass of inexperienced consumers with search cost $s$. The equilibrium distribution of posterior means takes a two-threshold form: truthful disclosure below a lower threshold $v_L$ and above an upper threshold $v_H$, with all intermediate valuations pooled into signals just above the reservation value. Around the reservation value the firm's payoff jumps up, and that jump makes it optimal to pool values just below $r$ with values just above it. In large markets the gain from capturing an inexperienced consumer who stops immediately dominates the loss from concealing low values from savvy consumers, so the lower threshold is $v_L=0$, all values below $r=\\mu-s$ are concealed, and the equilibrium does not converge to full disclosure as $n\\to\\infty$. The same equilibrium logic implies that inexperienced consumers search actively only when the market is small.","pith_inferences":["If the mechanism is robust, the concealment interval should survive in richer environments, and the next natural step is to let firms choose prices: the price dimension may amplify or offset the disclosure distortion, and the welfare ranking of small versus large markets could change.","The model supplies a rational-choice microfoundation for the 'choice overload' phenomenon studied in psychology: the debilitation of large choice sets arises here from sellers' strategic concealment, not from cognitive limits, and could be tested by measuring how the dispersion of disclosed product information falls with the number of competing sellers.","The gap/no-gap contrast between positive and zero minimum search costs raises the possibility of a discontinuity at arbitrarily small frictions: a negligible search cost or a tiny mass of inexperienced consumers may flip the large-market equilibrium between full disclosure and pervasive concealment, which is a sharper prediction than the paper's qualitative statement.","If $F^{n-1}$ is not convex, the frictionless full-disclosure benchmark itself fails, so even without search frictions some concealment may survive; the paper's machinery identifies which regions of the value distribution the reservation-value jump will distort."],"forward_implications":["If Proposition 4 is right, market designers cannot rely on entry or competition to discipline sellers' information: adding more firms leads to more concealment of low-quality draws, not less.","Inexperienced consumers' search behavior is discontinuous in market size: for $n$ below a threshold they visit multiple firms with positive probability, while for $n$ at or above the threshold they always stop at the first firm, even though their search cost is unchanged.","Welfare splits by type: savvy consumers' surplus rises with $n$, while inexperienced consumers' surplus is strictly higher in any small market than in any large one, falling to the single-draw value $\\mu-s$.","Search-cost policy has non-monotone effects: reducing $s$ increases informativeness and both consumer surpluses when $s$ is already low, but when $s$ is high the same reduction lowers informativeness and hurts savvy consumers.","As the number of firms grows large, the equilibrium distribution converges to a simple limit: full disclosure of an upper tail from some $v_H^\\infty$ to $1$, a flat pool at $r=\\mu-s$, and zero mass below $r$."],"supporting_citations":[{"why":"Establishes the frictionless-search benchmark in which full disclosure is the unique equilibrium when $F^{n-1}$ is convex; the paper's Lemma 1 adopts this result as the savvy-only case.","marker":"Hwang et al. (2023)"},{"why":"Establishes the informational Diamond paradox with only inexperienced consumers; the paper's Lemma 2 uses it as the inexperienced-only benchmark.","marker":"Au and Whitmeyer (2023)"},{"why":"Supplies the multiplier optimality conditions used to verify that the candidate disclosure distributions are best responses.","marker":"Dworczak and Martini (2019)"},{"why":"Provides the underlying search-market model with savvy and inexperienced consumers whose search costs are heterogeneous, which the paper extends by adding information design.","marker":"Stahl (1989)"},{"why":"Gives the Pandora's-rule reservation value and the search equation that governs inexperienced consumers' stopping decisions.","marker":"Weitzman (1979)"},{"why":"Shows that full disclosure emerges in large competitive markets under rotation-ordered disclosure strategies, serving as the main contrast for Proposition 4.","marker":"Ivanov (2013)"}],"fun_headline_variants":["In large markets, firms conceal low valuations","Disclosure fails to improve with market competition","Big search markets hide low values from buyers","Partial disclosure remains even with savvy consumers","Concealment dominates when search markets scale"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire characterization assumes the prior distribution $F$ is such that $F^{n-1}$ is weakly convex, so that the frictionless benchmark is full disclosure; the paper itself notes in Section 6.2 that this assumption is restrictive, and the large-market concealment result is not shown to hold outside this class.","fun_headline_variants_meta":{"raw":{"variants":["In large markets, firms conceal low valuations","Disclosure fails to improve with market competition","Big search markets hide low values from buyers","Partial disclosure remains even with savvy consumers","Concealment dominates when search markets scale"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000588,"raw_usage":{"total_tokens":2704,"prompt_tokens":834,"completion_tokens":1870,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":450,"completion_tokens_details":{"reasoning_tokens":1804}},"tokens_in":450,"tokens_out":1870,"duration_ms":14439,"temperature":1.0,"reasoning_tokens":1804,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:08:10.217033+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One concrete check is to compute the equilibrium for a prior $F$ on which $F^{n-1}$ is not weakly convex, such as a distribution with a flat density in the middle, and see whether all valuations below $r=\\mu-s$ still disappear from the support as $n$ grows; the paper only proves this under convexity. A second check within the paper's assumptions is to simulate the large-market equilibrium and inspect firms' signal distributions for any positive mass below $r$, which Proposition 4 says should be zero.","supporting_citations":[{"cited_title":"Kim, and R","cited_arxiv_id":null,"evidence_quote":"Establishes the frictionless-search benchmark in which full disclosure is the unique equilibrium when $F^{n-1}$ is convex; the paper's Lemma 1 adopts this result as the savvy-only case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the multiplier optimality conditions used to verify that the candidate disclosure distributions are best responses."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the underlying search-market model with savvy and inexperienced consumers whose search costs are heterogeneous, which the paper extends by adding information design."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Pandora's-rule reservation value and the search equation that governs inexperienced consumers' stopping decisions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows that full disclosure emerges in large competitive markets under rotation-ordered disclosure strategies, serving as the main contrast for Proposition 4."}],"review_version":1}