Formal Econometrica Abstract
This paper explores how Bayes-rational individuals learn sequentially from the discrete actions of others. Unlike earlier informational herding papers, we admit heterogeneous preferences. Not only may type-specific `herds' eventually arise, but a new robust possibility emerges: confounded learning. Beliefs may converge to a limit point where history offers no decisive lessons for anyone, and each type's actions forever nontrivially split between two actions.

To verify that our identified limit outcomes do arise, we exploit the Markov-martingale character of beliefs. Learning dynamics are stochastically stable near a fixed point in many Bayesian learning models like this one.
 

Informal Bullet Highlights:

1. This is often called the `cascades' literature (owing to one of its originators), but we show that this is a true malapropism: while herds always occur in single preference models, cascades are a totally nonrobust phenomenon: In generic continuous signal models, cascades simply do not occur (and constructing continuous signal examples of them is stupendously hard). So amazingly, we eventually must have an infinite number of individuals in succession taking the same action (namely, a herd), even though every single one of them has a positive chance of deviating. (See the First Borel-Cantelli Lemma.)
2. We provide a simple standard machine for analyzing any social learning model. This is really the tool of choice if you are working in this area. You simply graph a couple functions and you have a very thorough understanding of what's going to happen. Try it!
3. Even if no one has perfectly revealing private signals, so long as there is no most powerful signal in either dirrection, a correct herd must eventually start. Incorrect herds in the early models owed to the discrete signal structure.
4. Suppose that not everyone has the same ordinal preferences over actions. The existing herding literature has nothing whatsoever to say. We find that an intriguing rational expectations chaos that emerges -- we call it confounded learning -- is a more robust prediction than is herding. (For an incorrect herd cannot occur, while our confounded outcome can.) See our paper for the example of this phenomenon.
5. The appendix of our paper derives a very handy condition for the stability of stochastic difference equations. This is the analogue of the slope less than unity condition for ordinary difference equations. This simple result is applicable totally apart from the social learning model.