Robust Representation of Sequential Social Learning

In the classical herding model, asymptotic learning refers to situations where individuals eventually take the correct action regardless of their private information. Classical results identify classes of information structures for which such learning occurs. Recent papers have argued that typically, even when asymptotic learning occurs, it takes a very long time. In this paper, related questions are referred. We study whether there is a natural information structure representation for which the time it takes until individuals learn is uniformly bounded from above. Indeed, we propose a simple bi-parametric criterion that defines the information structure representation, and on top of that, compute the time by which individuals learn (with high probability) for any pair of parameters. Namely, we identify a family of structure representations where individuals learn uniformly fast. The underlying technical tool we deploy is a uniform convergence result on a newly introduced class of weakly active supermartingales. This result extends an earlier result of Fudenberg and Levine on active supermartingales.