{"id":"fb2570c3-4b67-40db-bc46-eedae9047b62","arxiv_id":"2411.17597","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"In a rational model with costly information processing, polarization and apparent confirmation bias are optimal responses to incentives, not cognitive errors.","lead":"This paper builds a rational Bayesian model in which people can see part of a signal for free or pay to see the rest. It shows that belief polarization, disconfirmation, and apparent confirmation bias can emerge simply because people with different starting views choose different amounts of information.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4 is false: a numerical counterexample shows polarization in V_alpha^ji, so Proposition 3's necessity proof is invalid as written.","rationale":"The reader's accepted verdict identifies the free/costly two-signal technology as the weakest assumption. My stress-test finds a more serious and more concrete problem: a central lemma used to prove the main necessary-and-sufficient characterization is false as stated. The counterexample is simple and exact, and the proof of Lemma 4 contains a demonstrable algebraic mistake (h<1, not h>1). Because Proposition 3 is the paper's strongest and most cited claim, an invalid proof of a lemma on which its necessity direction relies is a genuine load-bearing concern. I am not claiming the theorem itself is false: the same prior pair also belongs to V_beta^ji, which Proposition 3 includes, so the characterization may survive with a corrected argument. But the text as written cannot be accepted as rigorous without fixing Lemma 4 and re-verifying the necessity direction. Hence CONDITIONAL rather than UNCHANGED: the author should either repair the proof or narrow the claim; a brief numerical verification would settle whether the statement of Proposition 3 remains intact.","tokens_in":1028,"tokens_out":1087,"duration_ms":174075,"concrete_test":"Run a dense grid over theta1,theta2 in (1/2,1), pi,pj in (0,1), and c>0 with pi<pj. For each parameter point, compute the optimal acquisition choices for both agents, the resulting posteriors for all four signal realizations, and check the polarization conditions I(sigma)<0 and D(sigma)<0. Then test whether every polarization point satisfies (pi,pj) in V_alpha^ij union V_beta^ji and theta2>theta1. The point (0.6,0.8,0.3,0.35,0.2) already falsifies Lemma 4; the grid search settles whether Proposition 3 itself needs amendment or only its proof needs repair.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Lemma 4 (Appendix A.4) claims that pairs in V_alpha^ji union V_beta^ij can have inverse updating but never diverging attitudes, hence cannot polarize. This is false. Take theta1=0.6, theta2=0.8, DeltaU=1, pi=0.3, pj=0.35, and c=0.2. The willingness-to-pay values are c_alpha(0.3)=0.191 and c_alpha(0.35)=0.247, so only decision-maker j buys sigma2 after observing sigma1=alpha; thus (pi,pj) lies in V_alpha^ji. For the realized signal (alpha,beta), i's posterior is p_i_alpha=0.3913 and j's posterior is p_j_alpha_beta=0.168. Then |pi-pj|=0.05 is less than |0.3913-0.168|=0.2233, so D(sigma)<0, and the two agents update in opposite directions, so I(sigma)<0. This is polarization by Definition 7. The algebraic error in the proof of Lemma 4 is the claim that h(pi,pj) exceeds 1; in fact h(pi,pj) is less than 1 for all pi in (0,1) and pj>pi. Here h=0.109. Therefore the conclusion that no theta1 allows diverging attitudes is invalid. Since Proposition 3 relies on Lemma 4 to discard the V_alpha^ji and V_beta^ij cases, the necessity direction of the main theorem is not established as written.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies a rational Bayesian decision-maker who observes a first binary signal for free and can pay a processing cost c to observe a second conditionally independent binary signal. It characterizes the willingness to pay for the second signal as a function of prior beliefs and derives conditions under which two agents with different priors, receiving the same signal, exhibit polarization (defined as simultaneous diverging attitudes and inverse updating), as well as conditions under which a single agent appears to exhibit disconfirmation, confirmatory updating, or under/overreaction. The central result is Proposition 3, which states that polarization with positive ex-ante probability occurs for some processing cost if and only if the pair of priors lies in V^ij_alpha ∪ V^ji_beta and θ2 > θ1.","tokens_in":33398,"tokens_out":12090,"duration_ms":107143,"significance":"The formal framework is clean, and the single-agent willingness-to-pay analysis is transparent and largely self-contained: the cost function is derived explicitly, its continuity and symmetry are established, and several propositions are proved by direct calculation. If Proposition 3 were correct, the paper would provide a sharp rational-choice account of apparent confirmation bias and polarization under common evidence. However, the main characterization is false. The numerical counterexample in my report satisfies all of the model's assumptions and produces polarization for a pair of priors outside the union in Proposition 3. The proof of Lemma 4 contains an algebraic error, and the error is load-bearing because Lemma 4 is used to discard the V^ji_alpha ∪ V^ij_beta cases in the necessity direction. Consequently, the paper's central theorem, its ex-ante probability formula, and Corollary 3 are not established and, as stated, are false.","major_comments":[{"comment":"Lemma 4 is false, and Proposition 3 is false as stated. Take θ1=0.6, θ2=0.8, ΔU=1, pi=0.2, pj=0.25, and c=0.1. Direct computation gives cα(pi)=0.0727 < c < cα(pj)=0.1333, so after σ1=α only decision-maker j acquires σ2; hence (pi,pj)∈V^ji_alpha. Both cβ(pi) and cβ(pj) are zero because both priors lie in case 5 for σ1=β, so the pair is not in V^ij_alpha and not in V^ji_beta; condition (i) of Proposition 3 fails. For the realized signal (α,β), decision-maker i's posterior is p_i(α)=0.2727 and decision-maker j's posterior is p_j(αβ)=0.1111. Then |pi−pj|=0.05 < 0.1616=|p_i(α)−p_j(αβ)|, so D(σ)<0, and (0.2−0.2727)(0.25−0.1111)<0, so I(σ)<0. This is polarization by Definition 7, and the event (α,β) has positive ex-ante probability for any interior subjective prior p. The algebraic error in the proof of Lemma 4 is the claim that h(pi,pj)>1 whenever pj>pi; in this example h(0.2,0.25)=0.1739<1, so the conclusion that diverging attitudes are impossible in V^ji_alpha is invalid. Since Proposition 3 relies on Lemma 4 to exclude this case, the necessity direction of the main theorem is not merely unproved; it is false.","section":"§4.2, Lemma 4 and Proposition 3"},{"comment":"Because Proposition 3's necessity direction is false, the companion results that depend on its event characterization are also incorrect. In the same numerical example, with c=0.1 the pair (pi,pj) belongs to B^ji_alpha(c), and the signal realization (α,β) has positive probability, so polarization occurs with positive ex-ante probability. However, formula (4) assigns probability zero to this case: the first indicator requires B^ij_alpha(c), which is empty because cα(pi)<cα(pj), and the second indicator requires B^ji_beta(c), which is empty because neither agent buys σ2 after β. Thus the ex-ante probability Pr(PB) in Proposition 4 is not the actual probability of polarization, and the 'only if' part of Corollary 3 fails as well.","section":"§4.2, Proposition 4 and Corollary 3"},{"comment":"The single-agent results in Propositions 5–7 appear to be derived correctly from the willingness-to-pay function and were not affected by the counterexample to the two-agent polarization theorem. I did not find an independent error in those statements, but they should be rechecked once the polarization characterization is corrected because Corollary 2 and the interpretation of Proposition 3 feed into the paper's overall claims.","section":"§4.3, Propositions 5–7"}],"minor_comments":[{"comment":"There are typographical errors in the introduction, such as 'shereceivesand' in the discussion of Rabin and Schrag (1999); these should be corrected.","section":"§1, p.5"},{"comment":"The definition of polarization allows beliefs to swap order, and the paper explicitly notes this; since Lemma 4 was intended to exclude such swaps but is false, the text should clarify which order-swap cases can actually polarize once the theorem is corrected.","section":"§4.2, Definition 7"},{"comment":"The closed-form thresholds qσ1(c) and q̄σ1(c) should be checked for boundary behavior at c=0 and at c=ΔU[θ2−1/2], since some denominators can vanish for particular parameter values.","section":"Appendix A.3.1"}],"recommendation":"reject","confidential_remarks":"The reader's report recommended acceptance, but I independently verified the numerical counterexample in my major comment; it satisfies every assumption of the model and directly contradicts Proposition 3. The error in Lemma 4 is not a cosmetic gap but a false mathematical claim at the center of the paper's main contribution. A substantially revised version that provides a correct characterization of the polarization event, and that revises Proposition 4 and Corollary 3 accordingly, could be reconsidered, but the current manuscript should not be published as is."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The punchline: Lemma 4 is wrong, and Proposition 3's necessity direction falls with it. I checked the counterexample from the stress-test. With θ1=0.6, θ2=0.8, πi=0.3, πj=0.35, and c=0.2, only j buys σ2 after σ1=α, so (πi,πj)∈V_α^{ji}. For realized signal (α,β), i's posterior is 0.3913 and j's is 0.168, so |πi−πj|=0.05 < 0.2233 = |0.3913−0.168|, giving D(σ)<0; the updates go opposite directions, so I(σ)<0. That is polarization by the paper's own Definition 7, and it happens inside V_α^{ji}, which Lemma 4 tries to exclude. The algebraic error is the claim that h(πi,πj)>1; in this example h≈0.109, and the claimed bound simply does not hold for all πj>πi.\n\nWhat is genuinely good: the setup is clean, the willingness-to-pay analysis is transparent, and Propositions 1–2 appear correct. The sufficiency direction of Proposition 3 — that V_α^{ij}∪V_β^{ji} plus θ2>θ1 yields polarization — is constructive and fine. The individual belief-pattern results (disconfirmation, confirmatory patterns, underreaction) seem to go through. The paper's central idea — apparent biases can be rational under costly information processing — is not destroyed by this bug, but the paper's advertised exact necessary and sufficient conditions are wrong.\n\nThe soft spot is load-bearing, not cosmetic. Because Lemma 4 is false, the claimed iff in Proposition 3 is false. The reader's ACCEPT with high confidence is too generous; soundness should be substantially downgraded. That said, the fix may be tractable: the correct necessary condition likely needs a broader union of prior-pair sets, or an additional condition on signal realizations. This is a major revision, not a rejection of the underlying research question.\n\nWho gets value: researchers working on rational inattention and belief polarization, especially those interested in exact testable conditions rather than qualitative mechanisms. The paper deserves a serious referee — the model is worth engaging, and the flaw is specific and fixable. I would send it to peer review, but with a clear instruction that Proposition 3 and Lemma 4 must be corrected before the paper can be considered publishable.","headline":"The paper's main polarization theorem is false as stated: the stress-test counterexample lands, so Lemma 4 and Proposition 3 need major revision.","tokens_in":784,"tokens_out":770,"would_cite":false,"duration_ms":90703,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that belief polarization, confirmation-bias-like updating, and underreaction to evidence can all arise among fully Bayesian rational decision-makers who simply choose whether to pay a processing cost to scrutinize a…","keywords":["belief polarization","costly information acquisition","Bayesian updating","willingness to pay","confirmation bias","disconfirmation","underreaction","rational inattention"],"falsifier":"Run the model's own comparative statics: with $\\theta_2\\le\\theta_1$, Proposition 3 predicts zero ex-ante probability of polarization for every prior pair and every positive processing cost; observing polarization in that parameter region would refute the necessary and sufficient condition. Alternatively, elicit the willingness-to-pay function directly: the model predicts it is zero for extreme priors, peaks at $p=1-\\theta_1$ after $\\alpha$ and at $p=\\theta_1$ after $\\beta$, and obeys $c_{\\alpha}(p)=c_{\\beta}(1-p)$ for all $p$, so data rejecting any of these shape predictions would falsify the mechanism.","tokens_in":113,"feed_emoji":"🧠","tokens_out":7710,"duration_ms":129758,"temperature":0.7,"pith_summary":"The paper tries to establish that belief patterns usually read as cognitive biases—polarization, confirmatory updating, underreaction—can be produced by fully rational Bayesian agents who differ only in their prior beliefs and face a cost to process additional information. In the model, everyone sees the first component of a two-part signal for free; paying a fixed cost reveals the second component. Because the value of that costly component depends on the prior, decision-makers with different priors optimally make different acquisition choices, and observers who ignore the processing cost see what looks like biased updating. If the paper is right, apparent polarization and confirmation bias are equilibrium responses to incentives rather than deviations from Bayesian rationality.","feed_headline":"Costly information makes rational beliefs polarize","feed_subtitle":"Bayesian agents who skip paid scrutiny can look biased, yet their updating is fully rational.","key_machinery":"The load-bearing object is the willingness-to-pay function $c_{\\sigma_1}(p)$, the maximum processing cost a decision-maker with prior $p$ would pay to observe the second signal component after seeing $\\sigma_1$. The function is piecewise, zero for extreme priors and positive for moderate priors, satisfies the symmetry $c_{\\alpha}(p)=c_{\\beta}(1-p)$, and peaks at $p=1-\\theta_1$ after $\\alpha$ and at $p=\\theta_1$ after $\\beta$. Its shape partitions priors into eight cases. The antagonistic sets $V^{kl}_{\\sigma_1}$ and $B^{kl}_{\\sigma_1}(c)$ record pairs of priors for which one decision-maker has higher willingness to pay than the other; Proposition 3 shows that exactly these asymmetric-acquisition pairs, combined with $\\theta_2>\\theta_1$, drive polarization.","core_discovery":"The central claim is the exact characterization of belief polarization under shared evidence. With priors $p_i<p_j$, polarization occurs with positive ex-ante probability for some processing cost if and only if the prior pair lies in $V^{ij}_{\\alpha}\\cup V^{ji}_{\\beta}$ and the second signal component is more informative than the first, $\\theta_2>\\theta_1$. For a fixed cost $c$, one additionally needs $c<\\max\\{c_{\\sigma_1}(p_i),c_{\\sigma_1}(p_j)\\}$ and $(p_i,p_j)\\in B^{ij}_{\\alpha}(c)\\cup B^{ji}_{\\beta}(c)$; in words, one decision-maker must be willing to pay to see the second component while the other is not. Polarization itself requires the two signal components to disagree ($\\sigma_1\\ne\\sigma_2$), and its ex-ante probability is at most $1/2$. The same framework yields necessary and sufficient conditions for a tendency for disconfirmation, confirmatory belief patterns, and underreaction.","pith_inferences":["The paper leaves implicit that the willingness-to-pay function itself is a directly measurable quantity: an experiment could elicit WTP for the second signal before any belief update and compare the shape to the model's predictions, separating rational inattention from true bias.","Because the necessary and sufficient conditions rely on a fixed cost and binary signals, extending the model to continuous precision or endogenous attention would likely soften the sharp \"if and only if\" characterization; whether polarization survives in that broader class is an open question the paper does not address.","The model suggests a field prediction: in media or advice consumption, individuals with more extreme priors should be less willing to pay for contradicting information, and polarization should be more frequent when the costly information source is more precise than the free one.","The author's observer-neglect framing could be turned into a methodological caution: experiments that make all information freely available may fail to detect the mechanism, while designs with costly optional processing could replicate apparently biased updating in unbiased subjects."],"forward_implications":["Observed polarization under common evidence is not, by itself, evidence of a processing bias; the model predicts it only when one agent pays for extra scrutiny and the other does not, and the paid-for component is the more informative one.","Polarization should be the exception, not the rule: its ex-ante probability is bounded by $1/2$, and only signal realizations with conflicting components can produce it.","Decision-makers with non-extreme priors are more willing to pay to scrutinize evidence that contradicts their prior—a tendency for disconfirmation—so apparent confirmation-biased acquisition is a direct implication of rational Bayesian choice.","Underreaction relative to the full signal occurs when the skipped, costly component agrees with the free component; overreaction occurs only when the costly component is less informative than the free one.","An external observer who ignores processing costs will misclassify optimal behavior as confirmation bias, disconfirmation, or underreaction; accounting for acquisition incentives removes the apparent bias."],"supporting_citations":[{"why":"Provides the rational-choice benchmark in which Bayesian decision-makers optimally demand biased or imperfect advice, which the paper extends to endogenous acquisition of a fixed signal.","marker":"Calvert (1985)"},{"why":"Shows rational agents can choose experts who coarsen information in prior-confirming ways, a mechanism the paper contrasts with its fixed information structure.","marker":"Suen (2004)"},{"why":"Identifies confirmation and complacency effects that produce belief clustering; the paper differs by keeping the information structure fixed and focusing on acquisition choices.","marker":"Nimark and Sundaresan (2019)"},{"why":"Supplies the signal-complementarity result linking public evidence to polarization, used to position the paper's acquisition-driven mechanism.","marker":"Börgers, Hernando-Veciana, and Krähmer (2013)"},{"why":"Models confirmation bias as an exogenous misinterpretation of contradicting signals, serving as the biased-observer benchmark the paper contrasts with optimal costly acquisition.","marker":"Rabin and Schrag (1999)"},{"why":"Provides the definitional discussion of belief divergence and the point that polarization is not always irrational, which the paper's Definition 7 builds on.","marker":"Jern, Chang, and Kemp (2014)"},{"why":"Source of the disconfirmation-bias definition quoted in the paper, anchoring the apparent-bias patterns the model reproduces.","marker":"Lord, Ross, and Lepper (1979)"},{"why":"Documents that people generally underinfer, motivating the paper's analysis of underreaction relative to the full information set.","marker":"Benjamin (2019)"},{"why":"Supplies the rational-inattention interpretation of processing costs that motivates the paper's cost-of-attention setup.","marker":"Maćkowiak, Matějka, and Wiederholt (2023)"}],"fun_headline_variants":["Rational beliefs polarize under costly info","Costly info explains belief polarization","When paying for info drives polarization","Polarization emerges from info costs"],"cache_read_input_tokens":35968,"weakest_assumption_plain":"The load-bearing premise is a very specific information technology: two conditionally independent binary signals with symmetric precisions, the first free and the second available only at a fixed cost $c$, with no partial processing or continuous precision choices.","fun_headline_variants_meta":{"raw":{"variants":["Rational beliefs polarize under costly info","Costly info explains belief polarization","When paying for info drives polarization","Polarization emerges from info costs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000143,"raw_usage":{"total_tokens":1125,"prompt_tokens":852,"completion_tokens":273,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":468,"completion_tokens_details":{"reasoning_tokens":224}},"tokens_in":468,"tokens_out":273,"duration_ms":7859,"temperature":1.0,"reasoning_tokens":224,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:56:55.274258+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the model's own comparative statics: with $\\theta_2\\le\\theta_1$, Proposition 3 predicts zero ex-ante probability of polarization for every prior pair and every positive processing cost; observing polarization in that parameter region would refute the necessary and sufficient condition. Alternatively, elicit the willingness-to-pay function directly: the model predicts it is zero for extreme priors, peaks at $p=1-\\theta_1$ after $\\alpha$ and at $p=\\theta_1$ after $\\beta$, and obeys $c_{\\alpha}(p)=c_{\\beta}(1-p)$ for all $p$, so data rejecting any of these shape predictions would falsify the mechanism.","supporting_citations":[{"cited_title":"APACrefauthors \\ 1985","cited_arxiv_id":null,"evidence_quote":"Provides the rational-choice benchmark in which Bayesian decision-makers optimally demand biased or imperfect advice, which the paper extends to endogenous acquisition of a fixed signal."},{"cited_title":"APACrefauthors \\ 2004","cited_arxiv_id":null,"evidence_quote":"Shows rational agents can choose experts who coarsen information in prior-confirming ways, a mechanism the paper contrasts with its fixed information structure."},{"cited_title":"\\ Sundaresan, S","cited_arxiv_id":null,"evidence_quote":"Identifies confirmation and complacency effects that produce belief clustering; the paper differs by keeping the information structure fixed and focusing on acquisition choices."},{"cited_title":"\\ Schrag, J L","cited_arxiv_id":null,"evidence_quote":"Models confirmation bias as an exogenous misinterpretation of contradicting signals, serving as the biased-observer benchmark the paper contrasts with optimal costly acquisition."},{"cited_title":", Chang, K m K","cited_arxiv_id":null,"evidence_quote":"Provides the definitional discussion of belief divergence and the point that polarization is not always irrational, which the paper's Definition 7 builds on."},{"cited_title":", Ross, L","cited_arxiv_id":null,"evidence_quote":"Source of the disconfirmation-bias definition quoted in the paper, anchoring the apparent-bias patterns the model reproduces."},{"cited_title":"APACrefauthors \\ 2019","cited_arxiv_id":null,"evidence_quote":"Documents that people generally underinfer, motivating the paper's analysis of underreaction relative to the full information set."}],"review_version":1}