The classic wisdom-of-the-crowd problem asks how a principal can "aggregate" information about an unknown state of the world from agents without understanding the information structure among them. Such aggregation obviously has large social and private value, especially when it concerns important social or economic events. Therefore, it is important to understand the limits of such an exercise: without specific assumptions on the information agents have, how can we aggregate it? Classic results by Prelec et al. [2017] (henceforth PSM) and Arieli et al. [2017] show that that even when agents' signals are i.i.d. conditional on the state, knowing the first-order beliefs of even an infinite set of agents is generally not sufficient to learn the state. In the terminology of econometrics, there is an "identification problem."

We analyze boundedly rational updating from aggregate statistics in a model with binary actions and binary states. Agents each take an irreversible action in sequence after observing the unordered set of previous actions. Each agent first forms her prior based on the aggregate statistic, then incorporates her signal with the prior based on Bayes rule, and finally applies a decision rule that assigns a (mixed) action to each belief. If priors are formed according to a discretized DeGroot rule, then actions converge to the state (in probability), i.e., asymptotic learning, in any informative information structure if and only if the decision rule satisfies probability matching. This result generalizes to unspecified information settings where information structures differ across agents and agents know only the information structure generating their own signal. Also, the main result extends to the case of n states and n actions.

We consider social learning settings in which a group of agents face uncertainty regarding a state of the world, observe private signals, share the same utility function, and act in a general dynamic setting. We introduce Social Learning Equilibria, a static equilibrium concept that abstracts away from the details of the given dynamics, but nevertheless captures the corresponding asymptotic equilibrium behavior. We establish strong equilibrium properties on agreement, herding, and information aggregation.