The speed of sequential asymptotic learning

TL;DR

This paper investigates the rate at which agents in a herding model learn the true state, revealing that learning from actions can be much slower than from private signals, with precise calculations for Gaussian signals.

Contribution

It provides a detailed analysis of the speed of asymptotic learning in herding models, including exact convergence rates for Gaussian signals and conditions for finite or infinite learning times.

Findings

Learning from actions can be arbitrarily close to learning from signals.

Expected time to correct action can be finite or infinite depending on signal distribution.

In Gaussian signals, convergence speed is explicitly calculated and shown to be slower than from signals.

Abstract

In the classical herding literature, agents receive a private signal regarding a binary state of nature, and sequentially choose an action, after observing the actions of their predecessors. When the informativeness of private signals is unbounded, it is known that agents converge to the correct action and correct belief. We study how quickly convergence occurs, and show that it happens more slowly than it does when agents observe signals. However, we also show that the speed of learning from actions can be arbitrarily close to the speed of learning from signals. In particular, the expected time until the agents stop taking the wrong action can be either finite or infinite, depending on the private signal distribution. In the canonical case of Gaussian private signals we calculate the speed of convergence precisely, and show explicitly that, in this case, learning from actions is significantly slower than learning from signals.