Threshold Cascades

Definition

In a threshold model, each individual acts (joins a riot, adopts a product, changes an opinion) only once some observable aggregate variable — most often the number of other people already acting — exceeds their personal threshold. Because each new joiner can push the running total past other people's thresholds, the model can produce a sudden cascade or "tip" from near-zero participation to mass participation, and — counterintuitively — the shape of the distribution of thresholds across the population determines whether this happens, not the population's average threshold.

In the Book

Page's central case is Granovetter's riot model (Chapter 19): each person i has a threshold T(i), and joins when the number already participating exceeds T(i). He compares three populations of 1,000 people with different threshold distributions: one where everyone's threshold is 10 (no movement ever starts, since no one has threshold zero); one with a small cluster of low thresholds that ignites briefly then stalls at 15 people; and one where thresholds are uniformly spread from 0 to 999, which cascades all the way to full participation as each day's joiners cross exactly one more person's threshold. The first scenario has the lowest average threshold of the three yet produces no movement at all — proof that the full distribution matters more than the mean. Page extends the model to "double riots," where a platform must simultaneously build two interdependent populations past their thresholds, using Airbnb's early failed launches as the case: renters wouldn't visit without enough listings, and hosts wouldn't list without enough renters, until Airbnb manually seeded one side to break the deadlock.

Why It Matters

Threshold cascades explain why interventions aimed at raising the average willingness to act (persuasion campaigns, average price cuts) can fail to trigger a tip, while a small, well-placed group of zero-threshold "instigators" can succeed — because what matters is whether enough people exist below wherever the current count stands, not whether the average person is close to acting. This reframes any adoption, uprising, or platform-launch problem as a question about the shape of a threshold distribution, and explains why identical average sentiment can produce either total inaction or a runaway cascade depending on how that sentiment is distributed.