Consensus in Communication Networks under Bayesian Updating

This paper studies the evolution of beliefs of a group of Bayesian updaters who are connected through a social network that enables them to listen to the opinions of others. Each agent observes a sequence of private signals about the value of an unknown parameter. In addition, the agent receives private messages from others according to her connections in the network. A message conveys some information about the signal observed by the sender. Both signals and messages are independent and identically distributed across time but not necessarily across agents. Messages cannot be transmitted through indirect connections in the network. We first characterize the long-run behavior of an agent's beliefs in terms of the relative entropies of the conditional distributions of signals and messages available to the agent. Then, under some mild assumptions on the distributions of signals and messages, we identify a condition under which the agents reach a consensus in their opinions even when they begin with different priors. Finally, in contrast with most results in the existing literature, we show that a consensus need not be reached in a strongly connected network.