On consensus and exponentially fast social learning

We analyze a model of social learning in which agents desire to identify an unknown state of the world using both their private observations and information they obtain when communicating with agents in their social neighborhood. Every agent holds a belief that represents her opinion on how likely it is for each of several possible states to be the true one. At each time period, agents receive private signals, and also observe the beliefs of their neighbors in a social network. They then update their beliefs by integrating the information available to them in a boundedly rational fashion. We show that in spite of agents' making new private observations perpetually and the myopic and local updating rule employed by them, agents will eventually reach consensus in their beliefs. This is proved by first showing that agents' beliefs over any state whose truth is inconsistent with their collective observations go to zero exponentially fast.