False Cascades and the Cost of Truth

TL;DR

This paper analyzes how endogenous evidence and social learning influence the formation and correction of false cascades, highlighting conditions under which truth can overturn misinformation in sequential decision-making.

Contribution

It introduces a framework combining soft and hard information, providing conditions for when false cascades can be corrected through verification and disclosure mechanisms.

Findings

Soft-channel informativeness threshold for public revealing

Verification boundary guarantees overturning false cascades

Policy levers for correcting wrong cascades

Abstract

We study sequential social learning with continuous actions and conformity when agents can endogenously generate hard, publicly verifiable evidence. Actions transmit soft information whose visibility depends on responsiveness to private signals and on observational granularity; investigations produce hard evidence only in the true state and, once found, are disclosed and reset public beliefs. We deliver two primitives. First, a soft-channel informativeness threshold characterizes when actions remain publicly revealing under coarse observation, clarifying why learning can be locally mute at classical cascade boundaries even with continuous actions. Second, a transparent knife-edge for verification guarantees boundary breakability: whenever the expected private return to a one-shot investigation exceeds its cost, occasional disclosures overturn any false cascade with positive probability in finite time. Combining both yields a compact resilience frontier with clean comparative statics in signal quality, responsiveness/conformity, observability, verification quality, and investigation costs. The results provide policy levers -- subsidies to investigation, investments in verification quality, disclosure incentives, and platform design -- that ensure correction of wrong cascades.

Tables

Table 1. Table 1: Notation

Symbol	Meaning
$\theta \in \{0,1\}$	State
${\mu }_t$	Public belief $\mathrm{Pr}\left(\theta =1\mid H_t\right)$ at start of period $t$
$s_t\in \{0,1\}$	Private signal at $t$; precision $q\in (1/2,1)$
${\mu }_t^{+},{\mu }_t^{-}$	Posteriors after $s_t=1$ and $s_t=0$ (Eq. (1))
$\Delta x({\mu }_t,q)$	One-step posterior gap (Eq. (2))
$a_t\in [0,1]$	Continuous action (opinion/intensity)
$\gamma \in [0,1]$	Responsiveness to private information (conformity weight $1-\gamma$)
$a^{\star }(x,{\mu }_t)$	Best response (Eq. (3))
$h\in [0,1]$	Observation granularity; coarsened action ${\tilde{a}}_t=C_h\left(a_t\right)$
$\Delta a({\mu }_t,q,\gamma )$	Signal-induced action separation (Eq. (4))
$i_t\in \{0,1\}$	Investigation decision at cost $c\ge 0$
$p\in (0,1]$	Verification success probability (only if $\theta =1$)
$V\ge 0$	Private disclosure benefit upon evidence
$H_t$	Public history (past coarsened actions and disclosures)

Equations

$$\begin{array}{rl}\mathrm{Pr}(s_t=1\mid \theta =1)& =\mathrm{Pr}(s_t=0\mid \theta =0)=q,\\ \mathrm{Pr}(s_t=1\mid \theta =0)& =\mathrm{Pr}(s_t=0\mid \theta =1)=1-q.\end{array}$$

$${\mu }_t^{+}=\frac{{\mu }_{t}q}{{\mu }_{t}q+(1-{\mu }_t)(1-q)},{\mu }_t^{-}=\frac{{\mu }_t(1-q)}{{\mu }_t(1-q)+(1-{\mu }_t)q},$$

$$\Delta x({\mu }_t,q)\equiv {\mu }_t^{+}-{\mu }_t^{-}.$$

$$\underset{a\in [0,1]}{\min }\gamma \mathbb{E}[(a-\theta {)}^2\mid x]+(1-\gamma )(a-{\mu }_t{)}^2,\gamma \in [0,1],$$

$$a^{\star }(x,{\mu }_t)=\gamma x+(1-\gamma ){\mu }_t.$$

$${\tilde{a}}_t=C_h\left(a_t\right),$$

$$\Delta a({\mu }_t,q,\gamma )\equiv a^{\star }({\mu }_t^{+},{\mu }_t)-a^{\star }({\mu }_t^{-},{\mu }_t)=\gamma \Delta x({\mu }_t,q).$$

$${\sigma }_t:({\mu }_t,s_t,E)⟼(i_t,a_t),$$

$$\Delta a(\mu ,q,\gamma )=a^{\star }({\mu }^{+},\mu )-a^{\star }({\mu }^{-},\mu )=\gamma \Delta x(\mu ,q),$$

$$|\Delta a(\mu ,q,\gamma )|=\gamma \Delta x(\mu ,q)\ge h.$$

$$x=\frac{1}{2}.$$

$$U^{\text{no}}=\frac{1}{2}.$$

$$U^{\text{inv}}=\frac{1}{2}p(1+V)+(1-\frac{1}{2}p)\frac{1}{2}-c.$$

$$U^{\text{inv}}-U^{\text{no}}=\frac{1}{2}p(1+V)-c.$$

$$p(1+V)\ge 2c.$$

$$\mathcal{R}\equiv \{(q,\gamma ,h,p,c,V):\text{the society is misinformation-resilient}\}.$$

$$\underset{s,\Delta p,\Delta V}{\min }$$

$$(p+\Delta p)(1+V+\Delta V)\ge 2(c-s),$$

$$s\in [0,c],\Delta p\in [0,1-p],\Delta V\ge 0.$$

$$G(p,V,c)\equiv \frac{1}{2}p(1+V)$$

$$(1+V+\Delta V)d\Delta p+(p+\Delta p)d\Delta V=2ds.$$

$$\underset{\Delta \gamma ,\Delta h\ge 0}{\min }{\kappa }_{\gamma }(\Delta \gamma )+{\kappa }_h(\Delta h)\text{s.t.}(\gamma +\Delta \gamma )\Delta x(\mu ,q)\ge h-\Delta h\forall \mu \in \mathcal{N}.$$

$$\max \{\alpha s^{\star },\beta \Delta p^{\star },\eta \Delta V^{\star }\}\le \bar{B},$$

$$\begin{array}{rl}{\mu }^{+}(\mu ,q)& =\frac{\mu q}{\mu q+(1-\mu )(1-q)},\\ {\mu }^{-}(\mu ,q)& =\frac{\mu (1-q)}{\mu (1-q)+(1-\mu )q},\\ \Delta x(\mu ,q)& \equiv {\mu }^{+}(\mu ,q)-{\mu }^{-}(\mu ,q).\end{array}$$

$$\begin{array}{rl}{\mu }^{+}& =\frac{OR}{1+OR},\\ {\mu }^{-}& =\frac{O/R}{1+O/R},\\ \Delta x(\mu ,q)& =\frac{OR}{1+OR}-\frac{O/R}{1+O/R}.\end{array}$$

$$\frac{\partial }{\partial R}\Delta x(\mu ,q)=\frac{O}{(1+OR{)}^2}+\frac{O}{R^2(1+O/R{)}^2}>0,$$

$$\Delta a(\mu ,q,\gamma )=a^{\star }({\mu }^{+},\mu )-a^{\star }({\mu }^{-},\mu )=\gamma [{\mu }^{+}(\mu ,q)-{\mu }^{-}(\mu ,q)]=\gamma \Delta x(\mu ,q).$$

$$\gamma \Delta x(\mu ,q)\ge h.$$

$$U^{\text{inv}}=\frac{1}{2}p(1+V)+(1-\frac{1}{2}p)\frac{1}{2}-c.$$

Full text

Keywords: social learning; information cascades; verification; disclosure; conformity; coarse observation

JEL: D83; C72; D85.

1 Introduction

Herds and informational cascades arise when sequentially acting agents rationally overweight public behavior relative to their own private signals. In classic binary-action benchmarks, once beliefs enter the cascade bands, actions cease to react to signals and learning stalls. Two real-world frictions are central but typically modeled separately: (i) conformity dampens how much actions move with private signals; (ii) verification occasionally produces hard, publicly checkable evidence that can collapse a prevailing narrative. This paper integrates both forces in a unified dynamic model and asks: When do false cascades survive, and what minimal conditions make them collapse endogenously?

We consider an infinite sequence of agents who observe a public history, receive a private signal, and choose a continuous action that trades off accuracy and conformity. The public observes a coarsened version of actions (platform granularity),222Granularity captures any lossy mapping from actions to what is publicly observed—e.g., platform discretization, rounding in reporting, or limited attention that bins nearby actions together. so the soft information embedded in actions may be lost when movements are smaller than the observation grid. In addition, any agent may pay a cost to investigate; investigations yield verifiable proof only in the true state with some success probability, after which disclosure is weakly dominant and publicly resets beliefs. These two channels—soft action-based inference and hard disclosure—jointly govern whether wrong beliefs are absorbing or transient.

Our analysis delivers two simple primitives. First, a responsiveness threshold determines when actions remain informative under coarse observation: the signal-induced action separation must clear the platform granularity. This fails locally at classical cascade boundaries, explaining why soft learning can stall even with continuous actions if conformity333Conformity is modeled parsimoniously via a weight $(1-\gamma)$ on proximity to the public belief; microfoundations such as social image or coordination motives yield similar comparative statics. is strong or observation is coarse. Second, a hard-evidence knife-edge pins down when agents optimally choose to verify near a wrong cascade boundary; whenever the expected private return to investigation exceeds cost, occasional disclosures break any false cascade with positive probability in finite time. Combining both margins yields a compact resilience frontier in primitives (signal quality, verification technology and cost, conformity, observation granularity) with clean comparative statics and policy levers.

Relative to the social learning tradition, the paper (i) clarifies when continuous actions do not eliminate cascades once conformity and coarse observation are present, (ii) provides a transparent verification condition guaranteeing boundary breakability via disclosure, and (iii) unifies the two channels into a tractable resilience frontier that is easy to take to applications (rumor correction, fact-checking, replication).

With continuous actions and conformity, the signal-induced action gap equals the product of responsiveness and the posterior gap. We show that actions are publicly informative at belief $\mu$ if and only if this gap exceeds platform granularity. At classical boundaries the posterior gap vanishes, so no finite responsiveness clears the threshold locally; soft learning is therefore locally mute even with continuous actions.

We derive a knife-edge verification condition at the boundary that depends only on verification success and private costs/benefits. When the expected private gain from a one-shot investigation exceeds cost, wrong cascades are overturned with positive probability in finite time via disclosure. The result isolates a minimal mechanism for endogenously collapsing false herds.

We combine both channels to characterize a resilience region: higher verification quality or lower investigation cost expand the hard-evidence region; higher signal precision, higher responsiveness (lower conformity), and finer observation expand the soft region. Either margin suffices for resilience; together they produce clear policy trade-offs.

A planner can guarantee boundary breakability at least cost by subsidizing investigation, investing in verification quality, or improving platform granularity / reducing conformity. We provide tight, closed-form instruments consistent with the primitives.

2 Related literature

This paper is part of the social-learning and information-cascade literature initiated by Banerjee, [2] and Bikhchandani et al., [5] and surveyed in Bikhchandani et al., [4], and Chamley, [8]. In the basic model, agents observe a history of previous actions, receive a private signal, and may end up herding because public actions swamp private information. Smith and Sørensen, [18] showed that even with rich signals learning can be pathological, in the sense that incorrect behavior can persist when actions fail to reveal private information. Much subsequent work has therefore asked when cascades are fragile, for example because observation is limited, signals are coarsened, or payoffs involve interaction effects.

A first branch relevant to our paper studies costly or strategic information acquisition in sequential learning. When agents can choose whether to acquire an extra signal, a new information externality arises: later agents benefit from earlier agents’ information purchases, which in turn reduces earlier agents’ incentives to buy information; this makes cascades more likely rather than less [7, 16, 1]. Recent work has generalized this idea to multi-dimensional or two-stage information choices, showing that the pattern is robust: when private information is costly and social observation is available, equilibrium learning under-invests in verification and can converge to wrong beliefs [6]. Our paper takes this logic one step further by allowing agents to purchase not just a better signal but a piece of verifiable evidence, available only in one state. Precisely because such evidence, once produced, is publicly revealed and benefits all followers, the classic free-riding force is present; we characterize when agents overcome it.

A second branch concerns verifiable disclosure and hard evidence. Since Grossman, [12] and Milgrom, [15] it has been known that, in one-shot environments, certifiable information tends to unravel, because any non-disclosure is interpreted as bad news. More recent contributions have revisited this logic under disclosure costs, partial verifiability, or dynamic arrival of evidence; see for instance Bertomeu and Cianciaruso, [3] for a unified treatment of verifiable disclosure with frictions, and Gratton et al., [11] for the timing of releasing verifiable news. Our model borrows this hard-evidence technology but embeds it in a sequential cascade environment: evidence is not exogenous but produced by private costly investigation, and its arrival resets public beliefs in a way that the soft, action-based channel cannot do when actions are coarsened or heavily conformist.

A third and very recent branch is the economics of fact-checking, misinformation, and platform design. Acemoglu et al., [1] develop a model of online misinformation in which decentralized fact-checking reduces the spread of low-reliability content but interacts in subtle ways with users’ beliefs; follow-up work and empirical studies on fact-check tags (e.g. 9; 17) similarly find that verification is effective on average but subject to attention and participation frictions. Our paper is largely orthogonal in that it keeps agents fully rational and instead asks a more “mechanical” question: in a rational cascade model, when does the availability of a costly fact-checking technology guarantee that false herds cannot persist? Nonetheless, the answer we obtain—an explicit inequality comparing the expected private return to the cost of verification—is directly interpretable as the level of platform-side or policy-side support needed to elicit enough checking.

Finally, there is a growing literature on social learning with non-standard payoff interactions, such as conformity, contrarian motives, or network-topology constraints; see, among others, Lobel and Sadler, [13] on diffusion through networks, Frick et al., [10] on misinterpreting others, and recent work introducing nonconformist preferences into cascade models [14]. Our continuous-action formulation with a conformity weight nests these approaches and shows that once conformity or observation coarsening makes the soft channel locally mute, the only robust way to guarantee correction is to activate the hard-evidence channel—exactly the channel that is underprovided when verification is costly.

Overall, our contribution is to connect these three lines—costly acquisition in sequential learning, dynamic verifiable disclosure, and economic models of fact-checking—by providing a simple, boundary-level condition under which false cascades become endogenously breakable, and by showing how standard policy levers (subsidies to investigation, better verification technology, finer platform observability) move the economy into that region.

The paper proceeds as follows. Section 3 sets up the environment, signals, continuous actions with conformity, coarse observation, and verification with automatic disclosure. Section 4 analyzes soft-channel informativeness and establishes the responsiveness threshold. Section 5 studies verification incentives at cascade boundaries and proves boundary breakability via disclosure. Section 6 combines both channels to characterize the resilience frontier and comparative statics. Section 7 presents welfare and policy instruments. Section 8 concludes.

3 Model

3.1 Environment and timeline

Time is discrete, $t=1,2,\dots$. The state is binary, $\theta\in\{0,1\}$, with common prior $\mu_{1}\in(0,1)$. In each period $t$:

the agent observes the public history $H_{t}$ (all past actions and any past disclosures); 2. 2.

the agent privately observes a binary signal $s_{t}$ about $\theta$; 3. 3.

the agent chooses whether to investigate ($i_{t}\in\{0,1\}$) at cost $c\geq 0$; if $i_{t}=1$ and $\theta=1$, verifiable evidence arrives with probability $p\in(0,1]$; no evidence ever arrives when $\theta=0$;444Asymmetry (evidence only in the high state) is without loss for boundary logic; if evidence arrives in both states with probabilities $(p,\tilde{p})$, the boundary condition becomes $(p-\tilde{p})(1+V)\geq 2c$. 4. 4.

if evidence arrives, disclosure is available and—by our selection—automatic (weakly dominant),555With strictly verifiable evidence and the per-period accuracy payoff used here, disclosure weakly dominates concealment. Small disclosure frictions simply shift the knife-edge by an additive term. conferring a private benefit $V\geq 0$; 5. 5.

the agent chooses a continuous action $a_{t}\in[0,1]$; the public observes a coarsened signal of $a_{t}$ (defined below) and any disclosure.

3.2 Signals and posteriors

Signals are i.i.d. conditional on $\theta$ with precision $q\in(1/2,1)$:

[TABLE]

Let $\mu_{t}=\Pr(\theta=1\mid H_{t})$ be the public belief at the start of $t$, and let $\mu_{t}^{+}$ and $\mu_{t}^{-}$ denote the (private) posteriors666Working with odds $O_{t}=\mu_{t}/(1-\mu_{t})$ and signal likelihood ratio $R=q/(1-q)$ is often convenient: $\mu_{t}^{\pm}=\frac{OR^{\pm 1}}{1+OR^{\pm 1}}$ and $\Delta x(\mu,q)=\frac{OR}{1+OR}-\frac{O/R}{1+O/R}$. after observing $s_{t}=1$ and $s_{t}=0$:

[TABLE]

so that $\mu_{t}^{-}<\mu_{t}<\mu_{t}^{+}$ and both are strictly increasing in $\mu_{t}$. Define the one-step posterior gap

[TABLE]

In the classical binary-action benchmark (without verification or coarsening), cascade bands are $\mu_{t}\leq 1-q$ (lower) and $\mu_{t}\geq q$ (upper); at boundaries $\mu_{t}\in\{1-q,q\}$, one has $\mu_{t}^{+}=\mu_{t}^{-}=\frac{1}{2}$.

3.3 Actions, conformity, and best response

The action $a_{t}\in[0,1]$ is interpreted as an opinion or intensity. Conditional on belief $x\in[0,1]$ about $\theta=1$ and public belief $\mu_{t}$, the agent chooses $a$ to balance accuracy and conformity:

[TABLE]

where $\gamma$ parameterizes responsiveness to private information ($\gamma=1$ truthful actions; $\gamma=0$ pure conformity). With $\Pr(\theta=1\mid x)=x$, the objective is strictly convex and yields the closed-form best response

[TABLE]

3.4 Coarse observation of actions

The public does not observe $a_{t}$ perfectly but instead a coarsened signal

[TABLE]

obtained by rounding $a_{t}$ to the nearest grid of step $h\in[0,1]$ (ties broken by a fixed, common rule). For a fixed $\mu_{t}$ and $q$, the signal-induced separation in best responses is

[TABLE]

Under coarse observation, the two actions are publicly distinguishable at belief $\mu_{t}$ if and only if $|\Delta a(\mu_{t},q,\gamma)|\geq h$; otherwise the soft channel is locally mute.

3.5 Verification and disclosure

If the agent investigates ($i_{t}=1$), she pays cost $c\geq 0$. Investigation yields verifiable evidence only in the true state: with probability $p\in(0,1]$ when $\theta=1$; with probability [math] when $\theta=0$. A disclosure event is publicly checkable and, by our selection, automatic whenever evidence exists.777Disclosure is weakly dominant whenever verifiable proof exists and the per-period accuracy payoff is normalized as below; modeling it as automatic shortens the exposition without changing any results. Upon disclosure, belief jumps to certainty at $\theta=1$ thereafter, and the investigator receives a private benefit $V\geq 0$ in period $t$ (e.g., prize, reputation, priority). If no evidence arrives, play proceeds as without investigation.

3.6 Payoffs and normalization

Per-period accuracy payoff equals $1$ if the chosen action matches the state and [math] otherwise. There is no intertemporal linkage in private payoffs (agents are short-lived). The investigator pays $c$ and, upon disclosure, obtains $V$ in the investigation period; there are no costs or rewards in other periods. Results are unaffected by adding vanishing tie-breaking trembles that (i) select investigation when indifferent and (ii) select disclosure when evidence exists.

3.7 Beliefs, strategies, and equilibrium

The public belief $\mu_{t}$ updates from $(\tilde{a}_{t-1},\text{disclosure at }t-1)$ via Bayes’ rule whenever possible and via standard consistency off the equilibrium path. A (pure) strategy for agent $t$ is a mapping

[TABLE]

where the disclosure choice is suppressed due to automatic disclosure when $E$ exists. An equilibrium is a Perfect Bayesian Equilibrium888Selections at knife-edges (investigate/disclose when indifferent) affect a measure-zero set of histories and simplify the local boundary analysis. Results are robust to vanishing payoff trembles. (PBE): (i) each agent’s strategy is optimal given beliefs, and (ii) beliefs are updated consistently with strategies and the coarse observation technology. We focus on symmetric monotone PBEs and use a canonical selection at measure-zero knife-edges.

3.8 Cascade and breakability

Definition 3.1**.**

Given $(\gamma,h,q)$, a belief $\bar{\mu}$ is a cascade belief if the optimal action choice under $\bar{\mu}$ is the same for both signals; with coarse observation, this is equivalent to $|\Delta a(\bar{\mu},q,\gamma)|<h$ or to the binary-action bands in the special case $h=0$ and $\gamma=0$. A cascade is wrong if the induced action is mismatched with the true state.

Definition 3.2**.**

Fix parameters $(q,p,c,V,\gamma,h)$. A wrong cascade at belief $\bar{\mu}$ is breakable if, conditional on $\mu_{t}=\bar{\mu}$ and the cascade being wrong, there exists a path with positive probability along which public belief exits the cascade region and converges to the truth, either because soft signals become publicly distinguishable or because a disclosure event occurs.

Whenever a decision is (weakly) indifferent, we select (i) investigation and (ii) disclosure if evidence exists; when actions are indifferent, we select the continuous best response (3). These selections only affect measure-zero histories and simplify the characterization of local (boundary) behavior used later.

4 Soft information under coarse observation

This section characterizes when actions remain publicly informative once the platform observes a coarsened signal of behavior. The key object is the signal-induced separation in best responses,

[TABLE]

which must clear the observation granularity $h$ for the public to distinguish whether the agent received $s_{t}=1$ or $s_{t}=0$.

4.1 Posterior-gap geometry

We first collect properties of the one-step posterior gap $\Delta x(\mu,q)=\mu^{+}(\mu,q)-\mu^{-}(\mu,q)$ defined in (2).

Lemma 4.1**.**

Fix $q\in(1/2,1)$ and write $\Delta x(\mu)\equiv\Delta x(\mu,q)$.

For $\mu\in(0,1)$, $\mu^{+}(\mu,q)$ and $\mu^{-}(\mu,q)$ are strictly increasing in $\mu$ and satisfy $\mu^{-}<\mu<\mu^{+}$. Moreover, $\Delta x(\mu)=\Delta x(1-\mu)$. 2. 2.

At the classical cascade boundaries $\mu\in\{1-q,\;q\}$ one has $\mu^{+}=\mu^{-}=\tfrac{1}{2}$, hence $\Delta x(\mu)=0$. 3. 3.

For any fixed $\mu\in(0,1)$, $\Delta x(\mu,q)$ is (weakly) increasing in $q$.

Proof.

The expressions for $\mu^{\pm}$ in (1) are increasing in the odds ratio $\mu/(1-\mu)$, which proves the first part and $\mu^{-}<\mu<\mu^{+}$. Symmetry follows by exchanging state labels, which maps $\mu\mapsto 1-\mu$ and flips the signals. Plugging $\mu=1-q$ or $\mu=q$ into (1) yields $\mu^{\pm}=\tfrac{1}{2}$, proving the boundary zeros. Finally, standard MLRP arguments imply that the likelihood-ratio gap induced by one signal increases in $q$, which translates into a larger posterior gap; a direct derivative check on (1) gives the same conclusion. ∎

4.2 Informativeness threshold under coarsening

Because the platform observes the rounded action $\tilde{a}=C_{h}(a^{\star})$, signal realizations are publicly distinguishable at belief $\mu$ if and only if the pre-rounding gap crosses a grid boundary.

Theorem 4.2**.**

Fix $(q,\gamma,h)$ and a belief $\mu\in(0,1)$. The public can infer the agent’s signal from the observed (coarsened) action if and only if999The threshold depends only on the one-signal posterior gap $\Delta x(\mu,q)$; richer signal structures feed in through this object.

[TABLE]

Equivalently, actions are publicly informative at $\mu$ if and only if the responsiveness $\gamma$ clears $h/\Delta x(\mu,q)$.

Proof.

The coarsening map $C_{h}$ is monotone and piecewise constant with flat segments of width $h$. The two signal-contingent best responses differ by $\Delta a(\mu,q,\gamma)$. After rounding, $\tilde{a}(\mu^{+})=\tilde{a}(\mu^{-})$ if and only if $|\Delta a(\mu,q,\gamma)|<h$, whereas $\tilde{a}(\mu^{+})\neq\tilde{a}(\mu^{-})$ if and only if $|\Delta a(\mu,q,\gamma)|\geq h$. Substituting (4) yields (5). ∎

Corollary 4.3**.**

At the classical boundaries $\mu\in\{1-q,\;q\}$, $\Delta x(\mu,q)=0$ by Lemma 4.1, hence no finite $\gamma$ satisfies (5) for any $h>0$. Therefore the soft channel is locally mute at the boundaries101010At the classical boundaries, a favorable signal moves the posterior exactly to $1/2$, so no finite responsiveness clears a strictly positive grid width. under coarse observation.

Figure 1 visualizes the soft-channel threshold $\gamma\,\Delta x(\mu,q)\geq h$ for a representative configuration $(\gamma,h)=(0.45,\,0.02)$. The dark (negative) region indicates beliefs and signal precisions where coarsened actions are locally mute; the contour traces the knife-edge where informativeness switches on.

The map highlights how higher $q$, larger responsiveness $\gamma$, or finer observability (smaller $h$) expand the informative region, consistent with the comparative statics recorded below.

Remark 4.4*.*

For fixed $\mu$, informativeness in (5) becomes easier to attain when (i) signal precision $q$ increases (Lemma 4.1), (ii) responsiveness $\gamma$ increases (weaker conformity), or (iii) observation granularity $h$ decreases (finer platform observability). These margins are complementary and will enter the resilience frontier in Section 6.

4.3 Implications for belief dynamics

When (5) holds on an interval of beliefs, actions convey signal realizations and public beliefs move away from that region’s boundaries with positive probability. Conversely, when (5) fails in a neighborhood of a boundary, the soft channel alone cannot dislodge beliefs from that boundary: a wrong cascade is locally absorbing absent hard evidence. These observations provide the soft-information half of the resilience frontier; the hard-evidence half is developed in Section 5.

5 Verification and boundary breakability

This section analyzes the hard-evidence channel. Near a cascade boundary of a wrong herd, an agent who receives the pro-truth signal faces a one-shot value comparison between investigating (paying $c$ to possibly obtain verifiable proof available only in the true state) and not investigating. Because disclosure is automatic whenever verifiable proof exists, a successful investigation publicly resets beliefs to the truth and collapses the cascade. The key observation is that at the classical boundaries the pro-truth posterior equals $x=\tfrac{1}{2}$,111111This is the boundary test: at $x=\tfrac{1}{2}$, investigation yields discovery with probability $\tfrac{1}{2}p$; for $x>\tfrac{1}{2}$ the incentive (weakly) increases. yielding a transparent knife-edge for the private return to investigating.

5.1 Boundary calculation

Fix a period $t$ in which the public belief $\mu_{t}$ sits at a boundary of a wrong cascade (e.g., $\mu_{t}=1-q$ when $\theta=1$). If agent $t$ receives the pro-truth signal, then her posterior (before any investigation decision) satisfies

[TABLE]

Without investigation, the optimal action delivers the baseline accuracy value

[TABLE]

If the agent investigates, with probability $xp=\tfrac{1}{2}p$ verifiable evidence arrives and is disclosed, guaranteeing the correct action and a private disclosure benefit $V$ in period $t$; with the complementary probability $1-\tfrac{1}{2}p$, no evidence arrives and the agent reverts to the baseline action yielding $U^{\text{no}}$. Net of cost $c$, the expected payoff from investigating is

[TABLE]

Subtracting (7) gives the net gain

[TABLE]

Hence, at a wrong boundary, investigation is (weakly) optimal after a pro-truth signal if and only if

[TABLE]

5.2 Boundary breakability via disclosure

We now state the main boundary result. It is independent of the soft-channel primitives $(\gamma,h)$ because once disclosure occurs beliefs jump to certainty.

Theorem 5.1**.**

Fix primitives $(q,p,c,V,\gamma,h)$ and consider a classical cascade boundary $\mu\in\{1-q,q\}$. If the knife-edge condition (10) holds, then any wrong cascade at that boundary is breakable with positive probability in finite time.121212The argument is independent of $(\gamma,h)$ because disclosure resets beliefs directly; the soft channel is needed only if verification is too weak to satisfy the knife-edge. In particular, conditional on being at a wrong boundary, the event that some finite-time disclosure occurs has strictly positive probability.

Proof.

At a boundary of a wrong cascade, when the pro-truth signal realizes, the posterior is $x=\tfrac{1}{2}$ by (6). By (9)–(10), investigation is (weakly) optimal following the pro-truth signal. Conditional on investigating, verifiable evidence arrives with probability $p>0$ and, by automatic disclosure, publicly resets belief to the true state in that period, collapsing the cascade. Because the pro-truth signal occurs in each period with strictly positive probability and investigation occurs whenever it does (by our knife-edge selection), the probability that disclosure happens in finite time is strictly positive. Therefore the wrong boundary cascade is breakable. ∎

Remark 5.2*.*

The result uses only the boundary posterior and the verification technology; it does not depend on action responsiveness or observation granularity. Even if the soft channel is locally mute (Corollary 4.3), the hard-evidence channel overturns false herds when (10) holds.

5.3 Conservativeness and heterogeneity

The boundary test is conservative: away from the boundary the pro-truth posterior $x$ exceeds $\tfrac{1}{2}$, so the private return to investigating is (weakly) larger.

Corollary 5.3**.**

If (10) holds at the boundary, then there exists a neighborhood of the boundary in which any wrong belief is also breakable via disclosure with positive probability. Equivalently, the boundary knife-edge is a sufficient (local) condition.

Proof.

For $x>\tfrac{1}{2}$, the probability $xp$ that evidence arrives under investigation is larger than at $x=\tfrac{1}{2}$, so $U^{\text{inv}}-U^{\text{no}}$ in (9) weakly increases. Hence a belief slightly inside the band preserves the investigation incentive and the breakability conclusion. ∎

We also record a robustness implication under heterogeneity of private costs or disclosure benefits.

Corollary 5.4**.**

Suppose $(c_{t},V_{t})$ are i.i.d. across agents with $\Pr\!\big(p(1+V_{t})\geq 2c_{t}\big)>0$.131313Heterogeneity helps: it suffices that a positive mass of agents satisfy the inequality. Rare “high-$V$” types or “low-$c$” opportunities eventually trigger correction with positive probability. Then any wrong boundary cascade is breakable with positive probability in finite time.

Proof.

In any period where the pro-truth signal realizes and $p(1+V_{t})\geq 2c_{t}$, the agent investigates and disclosure occurs with probability $p>0$, collapsing the cascade. The event that such a period occurs in finite time has strictly positive probability. ∎

5.4 Comparative statics and policy levers

Condition (10) becomes easier to satisfy when verification quality $p$ or the disclosure benefit $V$ increase, or when investigation cost $c$ decreases. These margins yield immediate policy instruments:141414Subsidies act on $c$, audit technology on $p$, and prizes or reputational credit on $V$; each moves the knife-edge linearly, enabling least-cost mixes. (i) a per-investigation subsidy $s\in[0,c]$ that lowers the private cost to $c-s$; (ii) investment in verification quality that raises $p$; and (iii) platform or institutional design that increases $V$ (e.g., priority, prizes, reputation). Each instrument expands the hard-evidence region of the resilience frontier in Section 6.

5.5 Selection at knife-edges

Our equilibrium selection assumes agents investigate when (weakly) indifferent at the boundary. This affects a measure-zero set of beliefs and histories and allows us to focus on economically meaningful sufficient conditions. All comparative-statics conclusions remain valid under vanishing payoff trembles that make investigation strictly optimal on the knife-edge and disclosure strictly optimal when evidence exists.

6 Resilience frontier and comparative statics

We now combine the soft-channel condition (Theorem 4.2) with the hard-evidence condition (Theorem 5.1). Intuitively, society is misinformation-resilient if, whenever beliefs approach a wrong cascade region, either actions remain publicly informative (so beliefs move away with positive probability) or verification is attractive enough151515The frontier is local in $\mu$ on the soft side (sharpest at the boundaries) and global in primitives on the hard side (the knife-edge slackens inside the band). that occasional disclosures collapse the herd.

6.1 Definition and characterization

Fix primitives $(q,\gamma,h,p,c,V)$ and a symmetric monotone PBE under the selections described above.

Definition 6.1**.**

The society is misinformation-resilient if, starting from any $\mu_{1}\in(0,1)$, the probability that a wrong cascade persists forever is zero. Equivalently, along any equilibrium path that reaches a wrong cascade region, there is positive probability that beliefs leave that region and converge to the truth (via either soft inference or hard disclosure).

The bottlenecks for correction are the classical cascade boundaries $\mu\in\{1-q,q\}$, where $\Delta x(\mu,q)=0$ and the soft channel is locally weakest. We therefore state a boundary-based sufficient characterization that is easy to verify in primitives.

Theorem 6.2**.**

Fix $(q,\gamma,h,p,c,V)$. Consider the two local conditions at each classical boundary $\mu\in\{1-q,q\}$:

(Soft) There exists $\varepsilon>0$ such that $\gamma\,\Delta x(\mu^{\prime},q)\geq h$ for all $\mu^{\prime}$ in a one-sided neighborhood of $\mu$ that is reached under a wrong cascade (i.e., for $\mu^{\prime}$ just inside the band).

(Hard) The knife-edge condition $p(1+V)\geq 2c$ holds.

If, at every boundary, at least one of (Soft) or (Hard) holds, then the society is misinformation-resilient.

Proof.

At a wrong boundary, if (Soft) holds then by Theorem 4.2 actions are publicly informative in a neighborhood reached by the cascade, and beliefs move away from the boundary with positive probability, breaking the cascade. If (Soft) fails but (Hard) holds, then by Theorem 5.1 occasional investigation yields disclosure with positive probability in finite time, collapsing the cascade. Applying the same argument at both boundaries delivers resilience. ∎

Figure 2 depicts the hard-evidence boundary in $(p,\,c/V)$ space for several disclosure rewards $V$. For each fixed $V$, the knife-edge $c/V=\frac{1}{2}p(1+V)$ separates parameter combinations where wrong boundary cascades are breakable (weakly below the line) from those where verification incentives are too weak.

The frontier provides immediate policy guidance: small increases in $p$ (verification quality) or disclosure incentives $V$ yield large expansions at low $p$, while cost reductions shift the whole region downward.

Remark 6.3*.*

Condition (Soft) is local (one-sided) because $\Delta x(\mu,q)$ is minimized at the boundaries; away from them the inequality becomes strictly easier. Condition (Hard) is global in the sense that once it holds at a boundary, it continues to hold on a neighborhood by Corollary 5.3.

6.2 Comparative statics

Define the resilience region

[TABLE]

The following monotone comparative statics are immediate from Theorems 4.2, 5.1, and 6.2.

Proposition 6.4**.**

If $(q,\gamma,h,p,c,V)\in\mathcal{R}$, then:

Increasing $q$ (better private signals) weakly expands $\mathcal{R}$. 2. 2.

Increasing $\gamma$ (weaker conformity) or decreasing $h$ (finer observation) weakly expand $\mathcal{R}$. 3. 3.

Increasing $p$ or $V$, or decreasing $c$, weakly expand $\mathcal{R}$.

Proof.

(i) From Lemma 4.1, $\Delta x(\mu,q)$ increases in $q$, relaxing the soft threshold; it also increases the posterior mass at which verification is attractive. (ii) From (5), larger $\gamma$ or smaller $h$ relax the soft condition. (iii) From (10), larger $p$ or $V$ or smaller $c$ relax the hard condition. Theorem 6.2 then implies expansion of $\mathcal{R}$. ∎

6.3 Policy instruments

The frontier suggests natural levers:

Verification support: per-investigation subsidies $s$ (effective cost $c-s$) or investments that raise $p$. 2. 2.

Disclosure incentives: prizes or reputational boosts that raise $V$. 3. 3.

Platform design: finer observability (smaller $h$) or features that reduce conformity pressure (raise $\gamma$).

Each instrument weakly enlarges $\mathcal{R}$. In Section 7 we quantify least-cost policies that guarantee boundary breakability.

7 Welfare and policy

We study minimal interventions that guarantee correction of wrong cascades. The resilience frontier in Section 6 suggests two levers: (i) make actions more publicly informative (raise $\gamma$ or reduce $h$), and (ii) make verification attractive (raise $p$ or $V$, or reduce the effective cost of investigation). Because the soft channel is locally mute at the classical boundaries (Corollary 4.3), the sharpest boundary guarantee is delivered by the hard-evidence knife-edge (10).161616Soft-channel policies (e.g., smaller $h$ or larger $\gamma$) do improve resilience away from the boundary and reduce time-to-correction when combined with verification; we record these comparative statics below.

7.1 Planner and instruments

A planner chooses instruments to guarantee boundary breakability at least cost. We consider three canonical hard-evidence instruments:

an investigation subsidy $s\in[0,c]$ that lowers the private cost to $c-s$; 2. 2.

an investment that increases verification quality from $p$ to $p+\Delta p\in(0,1]$; 3. 3.

a disclosure prize that raises the private benefit from $V$ to $V+\Delta V$.

Let the planner’s reduced-form costs be $\kappa_{c}(s)$, $\kappa_{p}(\Delta p)$ and $\kappa_{V}(\Delta V)$, with $\kappa_{\ell}(0)=0$, $\kappa_{\ell}(\cdot)$ continuous, nondecreasing, and convex for $\ell\in\{c,p,V\}$. The planner’s problem is

[TABLE]

Constraint (12) guarantees the knife-edge condition at the boundary and hence boundary breakability by Theorem 5.1. When needed, platform design instruments $\gamma$ and $h$ can be added to the planner’s choice set; see Remark 7.4 below.

7.2 Least-cost guarantees via a single instrument

We first characterize the cheapest single-instrument policy that ensures (12). Let

[TABLE]

denote the private expected gain from a one-shot investigation at the boundary (cf. (9)). The knife-edge reads $G(p,V,c)\geq c$.

Proposition 7.1**.**

Fix primitives $(p,V,c)$ and convex costs. The minimum-cost policy that uses only one instrument171717With convex policy costs, interior mixing across instruments is optimal only if marginal cost slopes differ sufficiently along the iso-knife-edge; otherwise a corner (single-instrument) solution prevails. and guarantees (12) is:

Subsidy only:* $s^{\star}=\max\{0,\,c-G(p,V,c)\}$; cost $\kappa_{c}\!\big(s^{\star}\big)$.* 2. 2.

Quality only:* choose $\Delta p^{\star}=\min\big\{\,1-p,\,\frac{2c}{1+V}-p\,\big\}_{+}$; cost $\kappa_{p}(\Delta p^{\star})$.* 3. 3.

Prize only:* choose $\Delta V^{\star}=\min\big\{\,\text{argmin}_{\Delta V\geq 0}\ (1+V+\Delta V)\geq\tfrac{2c}{p}\,\big\}=\max\{0,\,\tfrac{2c}{p}-1-V\}$; cost $\kappa_{V}(\Delta V^{\star})$.*

In each case, the instrument is just large enough to satisfy (12) with equality whenever possible.

Proof sketch.

With only a subsidy, (12) reduces to $G(p,V,c)\geq c-s$, so $s^{\star}=c-G$ if $G<c$ and [math] otherwise. With only $\Delta p$, the minimum $\Delta p$ that satisfies $(p+\Delta p)(1+V)\geq 2c$ is $\Delta p^{\star}=\frac{2c}{1+V}-p$ if feasible, else the boundary of the domain; analogously for $\Delta V^{\star}$. Convexity implies the corner solution is cost-minimizing. ∎

7.3 Mixing instruments and marginal rates of substitution

When multiple instruments are available, the least-cost policy trades them off along the iso-knife-edge curve. Differentiating (12) at equality gives the local marginal rate of substitution (MRS) between any two instruments.

Lemma 7.2**.**

Along $(p+\Delta p)(1+V+\Delta V)=2(c-s)$, small changes satisfy181818These trade-offs translate policy budgets into equivalent improvements in $p$, $V$, or $c$ along the iso-knife-edge; the least-cost point equates marginal cost to marginal effectiveness.

[TABLE]

Hence, at equality, the MRS between $(\Delta p,\Delta V)$ is $d\Delta V/d\Delta p=-\frac{1+V+\Delta V}{p+\Delta p}$; between $(\Delta p,s)$ is $ds/d\Delta p=\tfrac{1}{2}(1+V+\Delta V)$; and between $(\Delta V,s)$ is $ds/d\Delta V=\tfrac{1}{2}(p+\Delta p)$.

Remark 7.3*.*

Given differentiable costs, the cost-minimizing mix solves $\nabla\kappa_{c}(s)=\lambda\cdot(-2)$, $\nabla\kappa_{p}(\Delta p)=\lambda\cdot(1+V+\Delta V)$, $\nabla\kappa_{V}(\Delta V)=\lambda\cdot(p+\Delta p)$ for some multiplier $\lambda$. With linear costs, any point on the iso-knife-edge with the lowest linear combination is optimal; with convex costs, the optimum is unique.

7.4 Platform design and soft-channel policies

Soft-channel interventions improve informativeness away from the boundary and can shorten time-to-correction, even though they cannot by themselves overturn a boundary cascade under $h>0$.

Remark 7.4*.*

For any interior belief $\mu$, the soft condition is $\gamma\,\Delta x(\mu,q)\geq h$. Thus, given a target interior neighborhood $\mathcal{N}$, the least-cost platform policy solves

[TABLE]

Here $\kappa_{\gamma},\kappa_{h}$ are implementation costs (e.g., reducing conformity pressures or increasing observability). Any policy that lowers $h$ or raises $\gamma$ on $\mathcal{N}$ enlarges the soft region and complements hard-evidence instruments in Theorem 6.2.

7.5 Policy corollaries

We summarize practical choices that follow from Propositions 7.1 and Lemma 7.2.

Corollary 7.5**.**

If the planner faces a budget $\bar{B}$ and linear costs $\kappa_{c}(s)=\alpha s$, $\kappa_{p}(\Delta p)=\beta\Delta p$, $\kappa_{V}(\Delta V)=\eta\Delta V$, a sufficient condition for boundary breakability is

[TABLE]

with $s^{\star},\Delta p^{\star},\Delta V^{\star}$ from Proposition 7.1. The cheapest single-instrument policy is the argmin among these three costs; mixtures are preferred only if costs are strictly convex.

Corollary 7.6**.**

If $\Delta p$ is technologically capped at $\overline{\Delta p}$ and prizes $\Delta V$ are infeasible, a subsidy $s\geq c-\tfrac{1}{2}(p+\overline{\Delta p})(1+V)$ restores boundary breakability provided $s\leq c$. Analogous formulas apply when only prizes are available or only $p$ can be raised.

7.6 Discussion

The planner’s least-cost instrument depends on technology and fiscal costs: if raising $p$ (e.g., better testing, improved audit tools) is cheap, a small $\Delta p$ delivers guarantees quickly; if public prizes are easier to implement (e.g., reputation or priority mechanisms), a modest $\Delta V$ suffices; if neither is feasible, a targeted subsidy $s$ restores incentives directly. Platform design (smaller $h$) and reduced conformity (larger $\gamma$) accelerate soft learning and reduce reliance on verification, but—by Corollary 4.3—cannot alone overturn boundary cascades when $h>0$. In Section 8 we interpret these findings for fact-checking, replication, and platform policy.

8 Conclusion

This paper studied sequential social learning with continuous actions and conformity when agents can endogenously produce hard, publicly verifiable evidence. We separated the two information channels that govern whether societies self-correct191919We use “soft” for information inferred from actions and “hard” for publicly verifiable evidence that, once disclosed, pins down the state independently of behavioral inference. from false narratives: the soft channel, in which actions convey private signals subject to responsiveness and observability, and the hard channel, in which investigation yields verifiable proof only in the true state. The analysis delivered two primitives. First, a soft-channel informativeness threshold shows that actions remain publicly revealing at a belief $\mu$ iff the signal-induced action gap clears platform granularity. This explains why, even with continuous actions, learning can be locally mute at classical cascade boundaries when conformity is strong or observation is coarse. Second, a transparent knife-edge for the hard channel guarantees boundary breakability: whenever the expected private return to a one-shot investigation exceeds its cost, occasional disclosures overturn any wrong cascade with positive probability in finite time. Combining both conditions yields a compact resilience frontier with clean comparative statics in signal quality, responsiveness/conformity, observability, verification quality, and investigation costs.

The frontier translates directly into implementable instruments. Verification can be supported by (i) lowering effective investigation costs (e.g., subsidies or search support), (ii) increasing verification quality (better tests, audit tools), or (iii) raising the private benefit from disclosure (prizes, priority, reputational credit). Platform design complements these by (iv) improving observability (finer granularity) and (v) reducing conformity pressure (interface or institutional changes that raise responsiveness). The hard-evidence lever provides the sharpest boundary guarantee, while soft-channel design accelerates correction away from the boundaries and reduces expected time to truth.

Our model abstracts from several forces that may matter in practice. First, disclosure is weakly dominant and thus modeled as automatic; introducing costs or strategic audience effects would attenuate the effective discovery rate202020Endogenizing disclosure (e.g., audience-dependent benefits or strategic timing) tilts the frontier but preserves its form: the condition becomes $p\cdot\mathbb{E}[1+V\mid\text{disclose}]\geq 2c$. and tighten the condition, but the mechanism survives with an appropriate adjustment. Second, agents are fully rational and short-lived; behavioral frictions (attention, motivated reasoning) or career concerns could either hinder or amplify verification incentives. Third, we use a coarse but reduced-form representation of platform observability and conformity.212121Networked observation or limited attention can be folded into $h$ by treating it as an effective coarsening parameter that enlarges with sparser or noisier visibility of actions. Richer network structures, heterogeneous observation, or endogenous media exposure would sharpen the mapping from actions to beliefs but not the basic frontier logic.

Several extensions are natural. (i) Endogenize disclosure with audience-dependent payoffs or reputational spillovers to study partial unraveling in sequences. (ii) Introduce dynamic complementarities in actions (e.g., coordination or investment) to quantify welfare gains from faster correction. (iii) Embed verification in networks with limited observation and algorithmic amplification to compare decentralized fact-checking to centralized audits. (iv) Allow heterogeneous costs/benefits and targeted policies to characterize least-cost screening rules. Finally, an empirical or experimental calibration of the frontier—mapping platform granularity, conformity proxies, and verification rates to observed correction speeds—would connect the theory to measurable design recommendations.

Appendix: Proofs

Preliminaries

Write $\mu\in(0,1)$ for a generic public belief and fix $q\in(1/2,1)$. Using Bayes’ rule,

[TABLE]

Let $O=\mu/(1-\mu)$ be odds and $R=q/(1-q)>1$ be the signal likelihood ratio. Then

[TABLE]

Proof of Lemma 4.1

Monotonicity and symmetry

From (13), $\mu^{\pm}$ are increasing in $O$, hence strictly increasing in $\mu$; also $O<R$ implies $\mu^{-}<1/2<\mu^{+}$ and therefore $\mu^{-}<\mu<\mu^{+}$. Symmetry follows by exchanging state labels, which maps $\mu\mapsto 1-\mu$ and interchanges signals, yielding $\Delta x(\mu,q)=\Delta x(1-\mu,q)$.

Boundary zeros

At $\mu=1-q$ (or $\mu=q$), the odds equal $O=(1-q)/q=1/R$, hence $OR=1$ and $O/R=1$. Therefore $\mu^{+}=\mu^{-}=\frac{1}{2}$ by (13), so $\Delta x(\mu,q)=0$.

Dependence on $q$

Holding $\mu$ fixed, $R$ increases in $q$. As a function of $R>1$, the difference in (13) satisfies

[TABLE]

so $\Delta x(\mu,q)$ is (strictly) increasing in $q$. ∎

Proof of Theorem 4.2

Let $a^{\star}(x,\mu)=\gamma x+(1-\gamma)\mu$ be the best response. Conditional on the two signal realizations $s\in\{0,1\}$, the pre-rounding action gap equals

[TABLE]

The coarsening operator $C_{h}$ maps $[0,1]$ into a grid with step $h$ (ties resolved by a fixed common rule). Two real numbers $y_{1},y_{2}$ map to distinct grid points iff $|y_{1}-y_{2}|\geq h$, and to the same grid point iff $|y_{1}-y_{2}|<h$. Hence the coarsened actions are publicly distinguishable under belief $\mu$ if and only if $|\Delta a(\mu,q,\gamma)|\geq h$, i.e.

[TABLE]

Equivalently, for a given $(\mu,q,h)$, informativeness holds iff $\gamma\geq h/\Delta x(\mu,q)$. ∎

Proof of Corollary 4.3

At $\mu\in\{1-q,q\}$, Lemma 4.1 gives $\Delta x(\mu,q)=0$. Therefore $\gamma\,\Delta x(\mu,q)=0<h$ for any $h>0$, so the soft threshold fails at the boundary and in any neighborhood where $\Delta x(\cdot,q)$ is sufficiently small. ∎

Proof of Theorem 5.1

Consider a classical boundary of a wrong cascade (say $\mu_{t}=1-q$ when $\theta=1$). Upon receiving the pro-truth signal, the posterior equals $x=\tfrac{1}{2}$ (boundary property). Without investigation, the expected accuracy payoff equals $U^{\text{no}}=\tfrac{1}{2}$. If the agent investigates, evidence arrives with probability $xp=\tfrac{1}{2}p$, is disclosed (automatic), and yields the correct action payoff $1$ plus private benefit $V$; otherwise the agent reverts to the baseline. The expected payoff from investigating is

[TABLE]

Hence the net gain is $U^{\text{inv}}-U^{\text{no}}=\tfrac{1}{2}p(1+V)-c$, which is nonnegative if and only if $p(1+V)\geq 2c$. Under this knife-edge and our tie-breaking selection, the agent investigates after any pro-truth signal. Each such investigation produces disclosure with probability $p>0$, which publicly resets beliefs to the true state and collapses the cascade in that period. Because the pro-truth signal occurs each period with strictly positive probability, the event that disclosure occurs in finite time has strictly positive probability. Thus any wrong boundary cascade is breakable. ∎

Proof of Corollary 5.3

For $x>\tfrac{1}{2}$, the discovery probability under investigation is $xp>\tfrac{1}{2}p$. Therefore $U^{\text{inv}}-U^{\text{no}}=xp(1+V)-c$ is (weakly) larger than at the boundary. If $p(1+V)\geq 2c$ holds at $x=\tfrac{1}{2}$, it continues to hold on a one-sided neighborhood, hence the breakability conclusion extends locally inside the band. ∎

Proof of Corollary 5.4

Suppose $(c_{t},V_{t})$ are i.i.d. across agents with $\Pr\!\big(p(1+V_{t})\geq 2c_{t}\big)>0$. In any period where both the pro-truth signal realizes and $p(1+V_{t})\geq 2c_{t}$, the agent investigates and disclosure occurs with probability $p>0$, collapsing the cascade. The probability that such a period arrives in finite time is strictly positive, so the wrong boundary cascade is breakable. ∎

Proof of Theorem 6.2

At a classical boundary of a wrong cascade, either: (i) the Soft condition holds on a one-sided neighborhood, in which case by Theorem 4.2 the coarsened actions are publicly distinguishable with positive probability and beliefs move away from the boundary, breaking the cascade; or (ii) the Hard condition holds, in which case by Theorem 5.1 disclosure occurs in finite time with positive probability, also breaking the cascade. Applying this argument at both boundaries ensures that along any path reaching a wrong cascade region, there is positive probability of exit and convergence to the truth. Therefore the probability that a wrong cascade persists forever is zero, i.e. society is misinformation-resilient. ∎

Proof of Proposition 6.4

From Lemma 4.1, $\partial\Delta x(\mu,q)/\partial q>0$, so the soft threshold $\gamma\,\Delta x(\mu,q)\geq h$ is (weakly) easier to satisfy as $q$ increases. Likewise, the left-hand side increases in $\gamma$ and decreases in $h$, so higher responsiveness (lower conformity) and finer observability relax the soft condition. For the hard condition, $p(1+V)\geq 2c$ is preserved by increases in $p$ or $V$ and by decreases in $c$. Since resilience requires at least one of the local conditions at each boundary, any parameter change that weakly relaxes either condition weakly expands the resilience region. ∎