4 - Cascades and Herds

Summary

One million people cannot be wrong.

Each agent observes what others do and takes a zero-one decision in a preordered sequence. In a cascade, all agents herd on a sufficiently strong public belief and there is no learning. In a herd, all agents turn out to take the same decision. A cascade generates a herd, but the converse is not true. Cascades are nongeneric for atomless distributions of beliefs, whereas a herd always takes place, eventually. For that reason, the probability that the herd is broken must converge to zero. Hence, there is some learning in a herd (it is not broken), but the learning is very slow. The stylization of that property is the cascade.

Beliefs converge to the truth only if the distribution of private beliefs is unbounded; but the self-defeating principle in social learning implies that the convergence is slow. Because the filter imposed by discrete actions is coarse, the slowdown of social learning is much more significant than in the previous chapter. Applications for welfare properties and pricing policies by a monopoly are discussed.

A Tale of Two Restaurants

Two restaurants face each other on the main street of a charming Alsatian village. There is no menu outside. It is 6 p.m. Both restaurants are empty. A tourist comes down the street, looks at each of the restaurants, and goes into one of them. After a while, another tourist shows up, sees how many patrons are already inside by looking through the stained-glass windows – these are Alsatian winstube – and chooses one of them. The scene repeats itself, with new tourists checking on the popularity of each restaurant before entering one of them.