Cascades, Common Knowledge & Critical Mass

By now you can spot a cascade in the wild. From the earlier lessons you know the shape: a cascade forms when the visible actions of the people ahead of you outweigh your one private signal, so you rationally copy them — and once you copy, you stop adding information of your own, the line freezes, and a confident crowd marches on almost no real evidence. You’ve seen the faces it wears: queues, bestseller lists, runaway apps, bubbles, citation bandwagons, reputational herding.

This lesson does something different. Instead of adding more examples, it puts the cascade in its place — literally. A model is only as useful as the neighbours you can tell it apart from, and a cascade has two neighbours it is constantly confused with: common knowledge (what everyone knows that everyone knows) and critical mass (the threshold of early movers that tips the rest). Get the three straight and you gain a superpower: you’ll see how a single public signal — one thing everyone witnesses at once — can start a cascade, and how another can smash one that had looked unbreakable. That is the whole hinge of the next lesson, so let’s earn it here.

Actions, not beliefs, are the channel

Start with the defining feature, because everything else hangs off it. In an information cascade, the thing travelling from person to person is a choice you can see — not a private signal, not an announced opinion, just the observable action. You watch what the people ahead of you did, treat those actions as evidence, and let them override your own weak signal. The information is inferred, never transmitted. Nobody has to tell you anything.

That last point is easy to miss and worth hammering: a cascade can run with zero communication. The restaurant queue from the opening lesson never spoke to you. The diners didn’t announce “we researched this place.” You inferred quality purely from the visible fact of a line. Strip out every word and the cascade still works, because its channel is behaviour you can observe, full stop.

Contrast that with what an announced belief would be. If each diner handed you a note saying “I think this place is a 7/10,” that’s a different, richer channel — you’d be aggregating stated opinions. A cascade doesn’t get that. It gets actions only, and must reverse-engineer the belief behind each one. And here’s the trap the earlier lessons drilled: once a cascade locks in, the action you observe no longer even reflects the actor’s private belief — they copied too. So you’re inferring information from a choice that contains none.

Cascade vs common knowledge

Now the neighbour that gets confused with a cascade most often. Both involve a whole crowd converging, but they run on different fuel and answer different questions.

A cascade answers “should I copy?” It’s about information: I infer, from your action, that you probably know something, and I update toward doing what you did. I don’t need you to know that I’m watching. I don’t need you to know anything about my beliefs. The inference is one-directional and private.

Common knowledge answers “will we all move together?” It’s about coordination on shared beliefs: a fact is common knowledge when everyone knows it, everyone knows that everyone knows it, and so on up the ladder. Its whole point is higher-order belief — what you know about what I know about what you know. That recursive, mutual quality is exactly what a cascade doesn’t require.

Lay them side by side:

Notice the shared cell at the bottom: a public signal touches both. That’s not a coincidence — it’s the bridge, and the rest of the lesson walks across it.

The two also interact, and the cleanest place to watch them fuse is a bank run. A bank run has both models running at once:

• The cascade layer. You see a line forming at the branch. You infer those depositors might know the bank is shaky, so you join the line and withdraw — copying their action as evidence. Classic herding on visible behaviour.
• The common-knowledge layer. A run only succeeds if enough people withdraw at once, so what you really need is confidence that everyone else will also run — and that they know you’ll run, and so on. A televised report that “the bank is in trouble,” seen by all, doesn’t just inform you privately; it makes the danger common knowledge, flipping everyone from “I’m nervous” to “we’re all running, now.”

So the same event — a queue, a headline — can do double duty: feed the cascade (new evidence to copy) and switch on the common knowledge (now we all know we all know). Same crowd, two machines, one trigger.

Cascade & critical mass

The cascade’s other neighbour is critical mass — the threshold model, where a small share of early movers, once it clears a tipping point, pulls the rest of the population along. And here the relationship isn’t rivalry but identity of mechanism: a cascade is a critical-mass phenomenon in motion.

Recall the threshold picture. Each downstream person is willing to follow once the public tally of prior actions leads by enough — enough that copying beats trusting their own lone signal. That “enough” is a threshold. The instant the running count of aligned early actions clears it, following becomes the optimal move for everyone further back in the line, and the herd tips as one. A tiny early lead — two or three extra actions in the same direction — is all it takes to push the tally past the threshold and cascade the rest.

Work a concrete case. Suppose each person’s private signal is worth “one vote,” and a rational follower copies the crowd whenever visible prior actions outnumber their own signal by two. The first two deciders happen to both choose A (maybe by luck of their private draws). Now decider three has a signal for B — but sees a public tally of A=2, B=0. Two visible A-actions outweigh their single B-signal, so they rationally pick A. The tally is now A=3, B=0, and their action added no new information. Decider four faces an even more lopsided board and copies A too. The threshold of “lead by two” was crossed at person three, and from there the line is locked. A critical mass of two aligned early actions tipped the entire remaining queue.

Public signals start cascades

Here is where the bridge pays off. A public signal is a single observation everyone sees at once and knows everyone else saw too — an official ranking, a regulator’s statement, a celebrity endorsement, a visibly enormous crowd. Because it lands on everyone simultaneously, a public signal is the perfect thing to supply the early alignment a cascade needs to ignite.

Think about why. A cascade struggles to start when everyone’s private signals are scattered and no one dares move first — the line is stalled, each person weighing their own weak evidence. A public signal breaks the stall by handing everyone the same nudge in the same direction at the same instant. Suddenly there’s a visible, shared lean — the equivalent of those first two aligned actions — and the threshold is within reach on the very next decider. From there the cascade self-propels: each follower’s copy thickens the tally for the next.

Worked example. A new novel is sitting flat — decent, unknown, a scatter of mild private opinions among the few who’ve read it. Then a hugely visible book club selects it, and the pick is broadcast to millions at once. That single public signal does two things simultaneously: it’s fresh evidence (“someone credible rates this highly,” so I update toward buying) and it’s seen by everyone, so a wave of buyers appears together. The bestseller rank jumps, which is itself a new public signal — now rank is the visible action people copy — and the cascade is off. The book “takes off” not because its quality changed overnight but because a public signal supplied the aligned early actions that tipped the threshold.

Public signals break cascades

Now the mirror image, and the most useful idea in the lesson. Precisely because a cascade pools so little genuine information — it rests on a couple of early actions and a long echo — it is fragile. There isn’t much real evidence holding it up. So a single credible new public signal can overturn it, flipping a crowd that looked utterly locked in.

The mechanism is the exact inverse of ignition. A locked cascade is a tall stack of copies balanced on a thin base of information. Introduce one observation everyone sees and believes — a debunking report, a visible reversal by a trusted actor, a regulator’s ruling — and you’ve dropped fresh, weightier evidence onto the board. Because the accumulated crowd actions were informationally hollow (mostly echoes), the new public signal can outweigh the entire pile at once. Everyone re-evaluates simultaneously (it’s public, so they all get it together), and the herd can reverse in a single beat.

Worked flip. A hot consumer stock has cascaded upward for months: people buy because others are buying, the rising price is the visible action everyone copies, and almost no independent analysis is happening — a textbook locked cascade. Then a respected short-seller publishes a detailed, credible report showing the revenue was fabricated, and it hits every terminal and feed at once. That single public signal outweighs the whole hollow tower of copy-buying: holders reassess together, the “everyone’s buying” action reverses into “everyone’s selling,” and the price collapses in days. The same fragility that let a thin base support a tall cascade is what lets one credible signal topple it.

Putting it together

Six ideas, one machine. Compress them:

1. Actions are the channel. A cascade travels through observable choices, inferred as evidence — not announced beliefs, and with zero communication required.
2. It’s not common knowledge. A cascade answers “should I copy?” (one-way inference from actions); common knowledge answers “will we all move together?” (mutual, recursive belief). Real crowds — bank runs — run both at once.
3. It’s critical mass in motion. A cascade is what happens after a threshold is crossed: once visible aligned actions outweigh a private signal, copying flips to optimal for everyone downstream, so a small early lead tips the whole line.
4. A public signal can seed it. One shared, authoritative observation delivers the same early lean to everyone at once, supplying the aligned actions that clear the threshold and ignite the herd.
5. A public signal can smash it. Because a cascade pools so little real information, a credible new public signal can outweigh the entire hollow stack and reverse the crowd in a beat.
6. None of this requires the crowd to be right. Every step is individually rational, and the whole thing can still be built on two lucky early draws. A confident herd is not an informed one.

Recap

The lesson in one line: a cascade is inference from actions, ignited when a critical mass of aligned early moves clears the threshold, and lit or extinguished by a single public signal — none of which makes the crowd correct. That last clause is the opening of the finale. In lesson 6, “Breaking a Cascade,” we take the fragility you’ve now seen from the public-signal angle and turn it into a toolkit: how to keep independent signals alive so a crowd can genuinely pool what it knows, and how to engineer the shove that shatters a herd frozen on almost nothing.