A · Two citizens, the same evidence
Belief after each signal. Higher means more threat.
B · When the weighting attack beats fabrication
Proposition 2. Click or drag to move the point.C · What a defender can do
Proposition 4Bayes’ rule says how much a piece of evidence should change your mind. Start with how likely you thought something was, then ask how much more likely the evidence is if it is true than if it is false.
$$P(\text{threat}\mid\text{report})=\frac{P(\text{report}\mid\text{threat})\,P(\text{threat})}{P(\text{report}\mid\text{threat})\,P(\text{threat})+P(\text{report}\mid\text{no threat})\,P(\text{no threat})}$$An example with round numbers. You think there is a 20% chance a real threat is building. A source reports one. When a threat is real, the source reports it 80% of the time; when there is none, it raises a false alarm 20% of the time. Then the report should move you to \(\dfrac{0.8\times0.2}{0.8\times0.2+0.2\times0.8}=50\%\): a real update, but far from certainty.
The same idea with a number line. When beliefs and evidence are bell curves, Bayes’ rule becomes a weighted average. Your new belief moves from the old one toward the signal by a share \(K\) that depends on how precise the signal is compared with how confident you already were:
$$\mu_1=\mu_0+K\,(s-\mu_0),\qquad K=\frac{\rho_s}{\rho_0+\rho_s}.$$\(\rho\) is precision, one over the variance. A precise signal or a shaky prior means a large \(K\) and a big move. Each signal also adds its precision to your confidence, \(\rho_1=\rho_0+\rho_s\).
Where the attack comes in. Walberg’s citizen treats the signal as if its precision were \(\lambda\rho_s\). Nothing about the signal changes. Only the weight does:
$$K(\lambda)=\frac{\lambda\rho_s}{\rho_0+\lambda\rho_s}.$$Fear sets \(\lambda>1\) and the citizen over-revises toward alarming news. Misplaced trust or distrust sets \(\lambda<1\) and good news stops registering. The panel’s “first update” box shows these numbers for the current draw.
Everything numeric here is a modeling choice of the paper or a teaching default of this tool. No value is estimated from data, and no output is an empirical finding about any real campaign or population.
data/params.js.A latent state \(\theta\), the true level of external threat, has prior \(\theta\sim N(\mu_0,1/\rho_0)\). Each period the citizen sees \(s_t=\theta+\varepsilon_t\) with \(\varepsilon_t\sim N(0,1/\rho_s)\). The emotional citizen updates as if the signal precision were \(\lambda_e\rho_s\):
$$\rho^{\text{post}}=\rho_0+\lambda_e\rho_s,\qquad \mu^{\text{post}}=\frac{\rho_0\mu_0+\lambda_e\rho_s\,s}{\rho_0+\lambda_e\rho_s}=\mu_0+K(\lambda_e)\,\delta,$$with \(\delta=s-\mu_0\). The informal rule “posterior = prior + \(\lambda\cdot\delta\)” is the small-gain linearization of this form; all results use the exact form. Fear raises \(\lambda\) on threatening news (the paper’s signal-contingent gain, applied when \(\delta\ge0\)). Anger is a different operator: it multiplies posterior precision by \(\kappa\ge1\), \(\rho^{\text{post}}_{\text{anger}}=(\rho_0+\rho_s)\kappa\), following the appraisal asymmetry between fear and anger in Lerner and Keltner (2001). Induced affect decays, \(\lambda_t=1+(\lambda_0-1)(1-\rho_e)^t\); a fabricated signal persists.
A foreign sender chooses between a weighting attack and fabrication. A government chooses to reassure (push a captured gain back toward 1), disclose (add truthful signals) or stay silent. The citizen acts on \(P(\theta>c)\).
P1, identification. \(\lambda\) sets the rate of revision and the prior sets its level, so one posterior cannot separate them but a sequence of signals or a panel can. The paper reports ridge curvature \(9.4\times10^{-6}\) at one signal and 1.87 at five, and panel recovery of \(\lambda\) within 3.2%. This tool does not rerun P1.
P2, dominance (Card B). The weighting attack pays \(\tau\,|a_\lambda-a_0|-0.15\times\text{energy}\), where energy is \(\lambda-1\) with no decay plus top-ups when affect decays; fabrication pays \(\tau\,|a_b-a_0|-1/\rho\). The paper finds a monotone, simply connected region that shrinks from 0.581 to 0.467 of the grid as decay rises from 0 to 0.3 at a ten-period horizon, a contraction of 19.6%, where it had predicted at least 50%.
P3, no stealth. A citizen fed only true signals ends with a biased posterior, but the bias is in the mean belief, so it is measurable. The paper’s pre-registered stealth claim failed: the rank correlation between the sender’s payoff and the mean-belief error is 0.978. Card B reports the action and mean-belief shifts together for this reason.
P4, defense (Card C). Disclosure is nearly useless against a heavily captured citizen (\(\lambda_g<0.3\): 95% of the distortion remains); reassurance dominates there. Fear-capture recovers 62.2% under 80 truthful signals; anger-capture recovers 0%. Hardening the gain protects only its own channel (ratio 0.00); affect-awareness transfers (0.79).
P5, cascades. On a network where people catch each other’s emotion, the weighting channel roughly halves the share of panicking neighbours needed to set off a cascade: 0.393 to 0.210 on Erdős–Rényi networks and 0.395 to 0.222 on Watts–Strogatz networks. The transition is smooth, and it needs a grain of real threat in the news. The Misinformation Cascade tool builds on this.
Model
engine.py, fh2_dominance.py, fh3_counterstrategy.py) and the reported numbers.Literature the model builds on