Lower Bounds on the Rate of Learning in Social Networks

We study the rate of convergence of Bayesian learning in social networks. Each individual receives a signal about the underlying state of the world, observes a subset of past actions and chooses one of two possible actions. Our previous work [1] established that when signals generate unbounded likelihood ratios, there will be asymptotic learning under mild conditions on the social network topology-in the sense that beliefs and decisions converge (in probability) to the correct beliefs and action. The question of the speed of learning has not been investigated, however. In this paper, we provide estimates of the speed of learning (the rate at which the probability of the incorrect action converges to zero). We focus on a special class of topologies in which individuals observe either a random action from the past or the most recent action. We show that convergence to the correct action is faster than a polynomial rate when individuals observe the most recent action and is at a logarithmic rate when they sample a random action from the past. This suggests that communication in social networks that lead to repeated sampling of the same individuals lead to slower aggregation of information.