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Platforms should amplify sources by net accuracy score: information gain minus quadratic bias risk from influence overlap.

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2026-07-14 13:21 UTC pith:AGBEGYYR

load-bearing objection Clean theory paper: exact accuracy score via Katz–Bonacich influence overlap, real demagogue threshold, and honest scoping of the feed/reference coupling. the 3 major comments →

arxiv 2607.10228 v1 pith:AGBEGYYR submitted 2026-07-11 econ.TH

Whom Should a Platform Amplify? Truth, Engagement, and Networked Polarization

Zaruhi Hakobyan This is my paper
classification econ.TH
keywords information designplatformssocial networkscoordinationtransparencypolarizationhigher-order beliefsalgorithmic amplification
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper treats algorithmic amplification as the design of who observes whom in a coordination game where people want the truth, want to match peers, and also lean toward their own biased ideals. Making one private signal commonly observed acts like adding a second public signal and restores the social value of transparency: truth-tracking accuracy then rises with public-signal precision, and under a limited broadcast budget it depends only on total amplified precision, so the budget should go to the most precise sources when public precision is not too low. Once ideal-action biases are present, the same reach becomes a demagogue risk, because an amplified source also becomes a coordination reference. For any finite network the accuracy effect of a network change splits exactly into an information gain and a quadratic bias cost governed by an influence-overlap matrix of Katz–Bonacich profiles, which yields an exact source-ranking score and, in a canonical local topology, a closed-form bias threshold. An engagement objective that tracks users’ own expected utility instead favors validating same-type links, producing more segregated networks and lower accuracy. The value of amplification therefore depends jointly on information, how bias propagates through reference groups, and whether the platform optimizes truth or engagement.

Core claim

For any finite directed network and any bias vector, truth-tracking accuracy equals minus average variance minus a quadratic form in the Gram matrix of R-weighted Katz–Bonacich influence profiles; any intervention’s accuracy change is therefore exactly the information gain minus the change in that quadratic bias risk, so candidate sources are ranked by the exact score information gain net of incremental bias cost, and in the broadcast-with-local-reference benchmark amplification helps only when the hub’s bias is below a closed-form threshold proportional to the square root of the variance gain.

What carries the argument

The influence-overlap matrix Q(M)= (1/N) L(M)' L(M), where L(M)=(I-RΦ(M))^{-1}(I-R) maps source biases into the equilibrium residual-action profile; its diagonal entries are bias-amplification multipliers and its off-diagonals capture reinforcement or offsetting, so the exact accuracy effect of any network change is ΔV - b̃' ΔQ b̃.

Load-bearing premise

The same network links that decide who sees whose signal also decide whom people treat as payoff-relevant peers; without that dual role the polarization results do not operate.

What would settle it

In a finite-network calibration with measured precisions, biases, and coordination weights, compute the exact score Sc for a candidate amplification; if the measured change in truth-tracking mean-squared error has the opposite sign from Sc, or if the engagement platform’s accepted links do not raise polarization relative to the accuracy planner when reference weights are forced to stay uniform, the central claims fail.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper models social-media amplification as feed/reference design in a finite-N linear-quadratic coordination game with Gaussian signals and biased ideal actions. The platform chooses the directed observation network M, which also sets coordination weights ϕij(M). Truth-tracking accuracy W is mean-squared error of actions versus θ. Core analytic results: unique linear equilibrium with loadings independent of biases and bias constants independent of precisions (Theorem 1); broadcast of a private signal restores monotonicity of W in public precision via the identity WS(α)=WMS(α+β) (Theorem 2, Remark 1); under a large-N broadcast budget, accuracy depends only on total amplified precision TH and is maximized by the most precise sources when α≥β/8 (Theorem 3); for any finite network and bias vector, W(M)=-−V̄(M)-b̃′Q(M)b̃ with Q the Gram matrix of R-weighted Katz–Bonacich influence profiles, so any intervention’s accuracy change is exact information gain minus quadratic bias-risk change and sources are ranked by Sc=ΔVc-b̃′ΔQc b̃ (Proposition 2, Corollary 1); a closed-form demagogue threshold in a local-ring broadcast benchmark (Proposition 5). Liberal recommendation dynamics and a reduced-form engagement objective Π then produce a polarization wedge and more segregated networks. Section 5.4 decouples the channels and shows polarization requires the feed/reference coupling.

Significance. If the results hold, the paper supplies a clean, operational design score for algorithmic reach that separates information gain from bias propagation through a positive-semidefinite influence-overlap matrix. The exact finite-network decomposition (Proposition 2) and the source-ranking rule (Corollary 1) are portable objects for network-game targeting and platform design; the transparency-reversal identity and the large-N precision rule give transparent comparative statics that connect Morris–Shin transparency to broadcast reach. The paper is careful about scope: it isolates the accuracy criterion under uniform weights, verifies the finite-N threshold approximation, and shows that the polarization mechanism is a reference-group effect rather than a pure information filter. Strengths include appendix proofs of the linear equilibrium and Neumann-series representation, an explicit decoupling exercise, and reproducible simulation protocols. The contribution is therefore both theoretical (exact quadratic bias cost of reach) and design-relevant (whom to amplify under accuracy versus engagement).

major comments (3)
  1. Section 3.3 and Numerical Result 1: Theorem 3’s precision-based ranking is exact only inside the symmetric large-N broadcast class with vanishing amplified share. The finite-N budget-m problem is acknowledged to have no closed form; exhaustive optimality is established only for N≤6, and for 7≤N≤20 the claim is a local-search regularity at one calibration. The abstract and introduction present the precision rule more broadly. Either restrict the formal claim to the broadcast class and relegate the numerics to a scope check, or supply a general ranking argument (or a counter-example) for coupled finite networks under the exact score of Proposition 2.
  2. Section 2, payoff (1) and ϕij(M); Section 5.4: The demagogue threshold and engagement echo chamber require that the same M sets both information sets and coordination weights. Section 5.4 correctly shows that under uniform ϕ the bias profile is network-invariant and the polarization divergence vanishes. The paper already scopes this, but the abstract and introduction still present polarization as a general consequence of amplification. A short, prominent statement that the polarization results are reference-group effects (and do not operate if platforms can broadcast signals without making sources salient peers) would prevent over-reading of the policy claims.
  3. Section 5.1–5.3 and Proposition 6: Engagement is reduced-form average expected utility. Proposition 6 holds signal/covariance/reference weights fixed; Numerical Result 4 and the two-camp exercise then check that a nonempty ΔW<0<ΔΠ region survives in exact equilibrium. The three-region pattern is useful, but the regulatory λ* cutoff is intervention-specific and normalization-dependent. The manuscript should state more sharply that λ* is not a universal policy instrument and that the engagement-only region is a property of the coupled subjective-utility model, not a prediction about commercial recommenders in general.
minor comments (5)
  1. Figure 1 caption and footnote 2: the plotted window starts above the Morris–Shin dip; a brief note in the caption that the non-monotone region lies to the left of the displayed range would help readers who do not check the footnote.
  2. Proposition 5 and Appendix D.1: the factor-3 local-ring topology is a modeling choice for closed form. Emphasize once more in the main text that b*c is topology-specific, not a universal constant.
  3. Appendix F Algorithm 1 and Table 1: the simultaneous row-update protocol and the fact that joint accepted add/drop pairs need not raise W period-by-period are correctly noted; a one-sentence reminder in Section 4 would reduce the risk that rest points are read as pairwise stability.
  4. Notation: ϕij(M) is defined with qi mass on neighbors; when |Ni|=0 or N-1 the empty component is dropped. A short display of the two edge cases would avoid ambiguity for readers implementing the model.
  5. References: the connection to Sákovics–Steiner (2012) and Galeotti et al. (2020) is well placed; a brief contrast with pure information-design targeting (e.g., which signals to make public without reference-group effects) would further locate the contribution.

Circularity Check

1 steps flagged

No significant circularity: accuracy decomposition, ranking score, and demagogue threshold follow from the linear equilibrium fixed point by algebra, not by fit or self-citation.

specific steps
  1. self citation load bearing [Title footnote / Introduction (p.1); related-work citations]
    "This manuscript substantially revises and supersedes an earlier working paper circulated since 2020 under the title “Can a social planner manipulate network dynamics and solve coordination problems?” (SSRN No. 3587024). … Hakobyan and Koulovatianos, 2020 … Kanik and Hakobyan, 2026"

    Author-overlapping citations appear for lineage and related polarization/expression work. They are not used as an external uniqueness theorem or ansatz that forces Proposition 2, Corollary 1, or the demagogue threshold; those follow from the paper’s own equilibrium algebra. Minor non-load-bearing self-citation only.

full rationale

The load-bearing chain is self-contained. Theorem 1 separates signal loadings from bias constants via a contraction on centered tracking errors; Proposition 1–2 then write W as variance plus the quadratic form b̃′Qb̃ with Q defined as the Gram matrix of the R-weighted Katz–Bonacich columns of L=(I−RΦ)−1(I−R). That is a representation of the equilibrium, not a prediction forced by fitting Q to the demagogue claim. Remark 1’s identity WS(α)=WMS(α+β) is an algebraic substitution of the closed-form MSE expressions, which genuinely moves the system out of the Morris–Shin non-monotone region. Theorem 3’s total-precision rule and Proposition 5’s local-ring threshold b∗c=√ΔV·(3+rq)/(rq) are derived from matching coefficients / solving the residual fixed point in a stated topology; finite-N checks recompute ΔV rather than back out the threshold. Engagement divergence (Proposition 6) holds signal/covariance terms fixed by design and is checked in exact finite-N grids. Section 5.4’s decoupling is a robustness localization, not a circular premise. Self-citations (superseded SSRN working paper; Hakobyan–Koulovatianos 2020; Kanik–Hakobyan 2026) are related-work context and do not supply a uniqueness theorem or ansatz that forces the main scores. No fitted-input-as-prediction or definitional tautology of the central claims was found.

Axiom & Free-Parameter Ledger

6 free parameters · 8 axioms · 2 invented entities

The central accuracy claims rest on a standard linear-Gaussian beauty-contest game plus three modeling choices: diffuse prior, feed/reference coupling through M, and truth-tracking MSE as the planner objective. Polarization and engagement claims add biased ideals and reduced-form average utility. Free parameters appear only in calibrations and the closed-form threshold’s local topology (factor 3 from hub+two neighbors), not as fitted constants that define the theorems. Invented objects (Q, Bc, Sc, b∗c) are derived statistics, not free ontological entities.

free parameters (6)
  • coordination strength r (and heterogeneous ri)
    Primitive in (0,1); simulations use r=0.7 or 0.85. Threshold and transparency region depend on r; not fitted to external data.
  • neighbor reference weight q
    Split of coordination mass on observed neighbors vs non-neighbors; baseline q=0.5, demagogue figure q=0.7. Enters demagogue threshold and coupling intensity.
  • public/private precisions α, βi
    Information primitives; transparency figures use α=β=2.5, network figures α=30, βi~U{10..45}. Chosen for illustration regimes, not estimated.
  • bias magnitudes bi and types σi
    Ideal-action shifts; simulations use ±0.6. Enter quadratic bias cost; demagogue threshold is comparative in bc vs √ΔV.
  • amplification budget m / out-degree
    Attention constraint in Theorem 3 and Numerical Result 1; design parameter, not estimated.
  • engagement weight λ in Oλ=(1−λ)W+λΠ
    Reduced-form interpolation; λ* is intervention-specific. Not estimated from retention data.
axioms (8)
  • domain assumption Linear Bayes–Nash equilibrium in Gaussian signals with diffuse (improper uniform) prior on θ, justified as τ↓0 limit of proper Gaussian prior.
    Section 2 and Appendix A; standard in beauty-contest literature but restricts strategy class and prior.
  • domain assumption Payoff is quadratic tracking of biased ideal plus weighted coordination losses (beauty contest with confirmation bias).
    Utility (1); inherits Morris–Shin structure plus ideal shift θ+b̃i.
  • ad hoc to paper Observation network M also sets coordination weights ϕij(M) via neighbor mass qi.
    Section 2 coupling; load-bearing for demagogue and engagement polarization; relaxed in §5.4.
  • ad hoc to paper Platform accuracy objective is mean squared error of actions vs θ, excluding coordination and confirmation terms.
    W(M) in (3); deliberate non-utilitarian benchmark.
  • ad hoc to paper Engagement equals average expected user utility (retention linear in E[ui]).
    Section 5.1 reduced form; not microfounded from clicks.
  • standard math Spectral radius condition r̄=max ri<1 implies unique linear equilibrium via contraction.
    Theorem 1 / Appendix A Banach fixed-point argument.
  • ad hoc to paper Canonical demagogue topology: follower reference group is hub plus two ring neighbors (factor 3).
    Proposition 5 local-ring benchmark for closed form; not topology-free.
  • ad hoc to paper Liberal planner proposes only accuracy-improving one-add/one-drop links; users accept by private expected utility.
    Section 4 protocol; rest points are protocol-specific, not Jackson–Wolinsky pairwise stability.
invented entities (2)
  • Influence-overlap matrix Q(M)=N^{-1}L(M)'L(M) and bias-amplification multiplier Bc no independent evidence
    purpose: Exact quadratic bias cost of network position; source ranking and demagogue test.
    Derived Gram matrix of R-weighted Katz–Bonacich profiles; independent_evidence false as a named object outside this framework, though Katz–Bonacich centrality is classical.
  • Source amplification score Sc(M;b̃) and demagogue threshold b∗c no independent evidence
    purpose: Operational rule for whom to amplify under accuracy.
    Closed-form objects from the model; falsifiable only after estimating (R,Φ,β,b̃), which the paper does not do.

pith-pipeline@v1.1.0-grok45 · 34652 in / 4256 out tokens · 43166 ms · 2026-07-14T13:21:43.544895+00:00 · methodology

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read the original abstract

Social-media platforms allocate reach, deciding whose content becomes widely visible. We study this as feed/reference design in a networked coordination game where users track an unknown state, coordinate with others, and hold biased ideal actions. Amplification changes both who receives information and who becomes a salient coordination reference. Making a private signal commonly observed adds a second common signal and removes the usual non-monotonicity of truth-tracking accuracy in public-signal precision; under a broadcast budget, accuracy depends only on total amplified precision. For any finite network and biases, a network intervention's effect on accuracy splits exactly into an information gain and a quadratic bias cost governed by a Katz--Bonacich influence-overlap matrix, yielding an exact source-ranking rule and a closed-form amplification threshold. A reduced-form engagement objective instead favors validating, same-type links, producing more segregated networks and lower accuracy. Amplification's value depends jointly on information, bias propagation, and the platform's objective.

Figures

Figures reproduced from arXiv: 2607.10228 by Zaruhi Hakobyan.

Figure 1
Figure 1. Figure 1: Accuracy and equilibrium weights under transparency. [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The demagogue effect in the finite-N local-ring topology (r = 0.85, q = 0.7, β = α = 2.5, N = 16). Left: accuracy as the hub’s own bias bc increases. The central sender is best when the hub is neutral; at the demagogue threshold it falls below the central-receiver benchmark, and at larger bias it becomes the worst of the plotted structures. Right: the polarization channel N −1 P i d 2 i rises quadratically… view at source ↗
Figure 3
Figure 3. Figure 3: Liberal-planner recommendation dynamics over 60 periods ( [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Diagnostics for the recommendation dynamics ( [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Truth versus engagement under the two platform objectives ( [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗

discussion (0)

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