Pith. sign in

REVIEW 3 major objections 3 minor 16 references

Belief patterns with information processing

T0 review · 3 major / 3 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 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…

desk verdict The paper's main polarization theorem is false as stated: the stress-test counterexample lands, so Lemma 4 and Proposition 3 need major revision. read the letter →

arxiv 2411.17597 v1 pith:2JMD7WLV submitted 2024-11-26 econ.GN q-fin.EC

Federico Vaccari This is my paper · ORCID
classification econ.GNq-fin.EC
keywords beliefpolarizationcostlyinformationacquisitionBayesianupdatingwillingnesstopayconfirmationbiasdisconfirmationunderreactionrationalinattention
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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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

3 major / 3 minor

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.

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 (3)
  1. [§4.2, Lemma 4 and Proposition 3] 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.
  2. [§4.2, Proposition 4 and Corollary 3] 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.
  3. [§4.3, Propositions 5–7] 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.
minor comments (3)
  1. [§1, p.5] There are typographical errors in the introduction, such as 'shereceivesand' in the discussion of Rabin and Schrag (1999); these should be corrected.
  2. [§4.2, Definition 7] 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.
  3. [Appendix A.3.1] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's belief-pattern results are derived from primitive assumptions rather than assumed into the conclusions.

full rationale

I found no circular step. The model's primitives—theta1, theta2, c, DeltaU, and prior beliefs—are free parameters; none are calibrated to a target quantity, and no fitted parameter is later relabeled as a prediction. Proposition 1 derives the willingness-to-pay function from Bayesian expected utility, and Propositions 3 through 7 derive polarization, disconfirmation, confirmatory patterns, and underreaction as logical consequences of optimal information acquisition given those primitives. The behavioral definitions (polarization in Definition 7, disconfirmation in Definition 8, confirmatory patterns in Definition 9) are explicit outcome criteria, not conclusions embedded in the assumptions. The paper contains no load-bearing self-citation: the cited works are used for terminology, psychological framing, or related literature, not as the source of any uniqueness theorem or of the model's structure. The derivation chain is therefore self-contained. The skeptic's numerical counterexample to Lemma 4, if valid, would be a mathematical correctness defect in a proof, not a circularity; a false lemma does not make the theorem equivalent to its inputs unless the claimed result reduces to a fit or to a self-citation by construction, which is not the case here.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No constant is fitted to data; all results hold for arbitrary parameter values in the stated ranges (theta1, theta2 in (1/2,1), c>0, Delta U>0). The illustrative example uses specific numbers, but the theorems are existence results over regions. The paper introduces no new entities beyond model variables; terms like 'non-extreme beliefs' and 'reciprocal' are definitions of existing concepts.

assumptions (6)
  • standard math Bayes' rule is used for all posterior updating.
    Used throughout, e.g., Appendix A.1; no alternative inference rule is considered.
  • domain assumption Decision-makers maximize expected utility with utility U if correct and Ubar otherwise, minus processing cost c when acquiring the second signal.
    Section 2.1 defines the utility function; this is the standard economic modeling assumption.
  • domain assumption The signal has two conditionally independent components, each binary, with symmetric precisions theta1 and theta2 both greater than 1/2.
    Section 2.1 states 'sigma1 and sigma2 are mutually independent' and the symmetric information structure; these assumptions define the signal space and are not derived.
  • domain assumption The first signal component is freely observed before choosing whether to acquire the second, and the second is available only at fixed cost c.
    Timing structure in Figure 1 and Section 2.1; this ordering is essential for the willingness-to-pay analysis.
  • domain assumption All decision-makers are identical except for their prior beliefs, with the same utility and cost.
    Section 2.2: 'decision-makers differ only in their prior beliefs'; this isolates the effect of priors and is used in the two-DM analysis.
  • domain assumption The external observer knows the signal structure and payoffs but neglects to account for the processing cost when interpreting behavior.
    Introduced in the abstract and Section 1 as the lens through which apparent biases arise; it is a narrative device, not a formal primitive, but it is necessary for the interpretation.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Belief patterns with information processing." pith.science (2026). https://pith.science/paper/2JMD7WLV

@misc{pith2026241117597,
  author       = {Pith},
  title        = {Pith review of: Belief patterns with information processing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2JMD7WLV}},
  note         = {Machine review of arXiv:2411.17597}
}
read the original abstract

This paper presents a model of costly information acquisition where decision-makers can choose whether to elaborate information superficially or precisely. The former action is costless, while the latter entails a processing cost. Within this framework, decision-makers' beliefs may polarize even after they have access to the same evidence. From the perspective of a Bayesian observer who neglects information processing constraints, the decision-makers' optimal behavior and belief updating may appear consistent with biases such as disconfirmation, underreaction to information, and confirmation bias. However, these phenomena emerge naturally within the model and are fully compatible with standard Bayesian inference and rational decision-making when accounting for the costs of information acquisition.

Figures

Figures reproduced from arXiv: 2411.17597 by the authors.

Figure 1
Figure 1. Timing structure selected after observing σ1 but before choosing s, thus a(p, c, σ1) : [0, 1] × R + × Θ → Π. Conversely, s(p, σ1) : [0, 1] × Θ → Ω if a = ¬ρ, and s(p, σ1, σ2) : [0, 1] × Θ2 → Ω otherwise. The decision-maker obtains a utility of v(s = ω, ω) = U when she selects an action s that matches with the state’s realization ω, and obtains v(s ̸= ω, ω) = U otherwise. I denote ∆U := U − U, and assume that ∆U > 0.… view at source ↗
Figure 2
Figure 2. Partition of the conditional posterior belief spaces induced by an information [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. The cost function for different prior beliefs: the blue line represents [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: In both panels, the decision-makers’ prior beliefs are represented on the axes. [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

  1. [1]

    \ Mylovanov, T

    andreoni APACrefauthors Andreoni, J. \ Mylovanov, T. APACrefauthors \ 2012 . Diverging opinions Diverging opinions . American Economic Journal: Microeconomics 4 1 209--232

    work page 2012

  2. [2]

    APACrefauthors \ 2019

    benjamin2019errors APACrefauthors Benjamin, D J. APACrefauthors \ 2019 . Errors in probabilistic reasoning and judgment biases Errors in probabilistic reasoning and judgment biases . Handbook of Behavioral Economics: Applications and Foundations 1 2 69--186

    work page 2019

  3. [3]

    \ Dubins, L

    blackwell APACrefauthors Blackwell, D. \ Dubins, L. APACrefauthors \ 1962 . Merging of opinions with increasing information Merging of opinions with increasing information . The Annals of Mathematical Statistics 882--886

    work page 1962

  4. [4]

    o rgers, T. , Hernando-Veciana, A. \ Kr \

    borgers APACrefauthors B \"o rgers, T. , Hernando-Veciana, A. \ Kr \"a hmer, D. APACrefauthors \ 2013 . When are signals complements or substitutes? When are signals complements or substitutes? Journal of Economic Theory 148 1 165--195

    work page 2013

  5. [5]

    APACrefauthors \ 1985

    calvert1985 APACrefauthors Calvert, R L. APACrefauthors \ 1985 . The value of biased information: A rational choice model of political advice The value of biased information: A rational choice model of political advice . The Journal of Politics 47 2 530--555

    work page 1985

  6. [6]

    \ Tirole, J

    dewatripont APACrefauthors Dewatripont, M. \ Tirole, J. APACrefauthors \ 2005 . Modes of communication Modes of communication . Journal of Political Economy 113 6 1217--1238

    work page 2005

  7. [7]

    \ Smith, E E

    edwards APACrefauthors Edwards, K. \ Smith, E E. APACrefauthors \ 1996 . A disconfirmation bias in the evaluation of arguments. A disconfirmation bias in the evaluation of arguments. Journal of Personality and Social Psychology 71 1 5

    work page 1996

  8. [8]

    , Chang, K m K

    chang APACrefauthors Jern, A. , Chang, K m K. \ Kemp, C. APACrefauthors \ 2014 . Belief polarization is not always irrational. Belief polarization is not always irrational. Psychological review 121 2 206

    work page 2014

Show all 16 references
  1. [9]

    APACrefauthors \ 2003

    kahneman APACrefauthors Kahneman, D. APACrefauthors \ 2003 . A perspective on judgment and choice: mapping bounded rationality. A perspective on judgment and choice: mapping bounded rationality. American psychologist 58 9 697

    2003

  2. [10]

    APACrefauthors \ 2012

    kondor APACrefauthors Kondor, P. APACrefauthors \ 2012 . The more we know about the fundamental, the less we agree on the price The more we know about the fundamental, the less we agree on the price . The Review of Economic Studies 79 3 1175--1207

    2012

  3. [11]

    , Ross, L

    lord APACrefauthors Lord, C G. , Ross, L. \ Lepper, M R. APACrefauthors \ 1979 . Biased assimilation and attitude polarization: The effects of prior theories on subsequently considered evidence. Biased assimilation and attitude polarization: The effects of prior theories on su...

    1979

  4. [12]

    , Matějka, F

    matejka2023 APACrefauthors Maćkowiak, B. , Matějka, F. \ Wiederholt, M. APACrefauthors \ 2023 March . Rational Inattention: A Review Rational inattention: A review . Journal of Economic Literature 61 1 226–73

    2023

  5. [13]

    APACrefauthors \ 1998

    nickerson APACrefauthors Nickerson, R S. APACrefauthors \ 1998 . Confirmation bias: A ubiquitous phenomenon in many guises. Confirmation bias: A ubiquitous phenomenon in many guises. Review of general psychology 2 2 175

    1998

  6. [14]

    \ Sundaresan, S

    nimark2019inattention APACrefauthors Nimark, K P. \ Sundaresan, S. APACrefauthors \ 2019 . Inattention and belief polarization Inattention and belief polarization . Journal of Economic Theory 180 203--228

    2019

  7. [15]

    \ Schrag, J L

    rabin APACrefauthors Rabin, M. \ Schrag, J L. APACrefauthors \ 1999 . First impressions matter: A model of confirmatory bias First impressions matter: A model of confirmatory bias . Quarterly journal of Economics 37--82

    1999

  8. [16]

    APACrefauthors \ 2004

    suen APACrefauthors Suen, W. APACrefauthors \ 2004 . The Self-Perpetuation of Biased Beliefs The self-perpetuation of biased beliefs . The Economic Journal 114 495 377--396

    2004

Pith tools

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