Social Learning with Beliefs in a Parallel Network

Consider a social learning problem in a parallel network, where N distributed agents make independent selfish binary decisions, and a central agent aggregates them together with a private signal to make a final decision. In particular, all agents have private beliefs for the true prior, based on which they perform binary hypothesis testing. We focus on the Bayes risk of the central agent, and counterintuitively find that a collection of agents with incorrect beliefs could outperform a set of agents with correct beliefs. We also consider many-agent asymptotics (i.e., N is large) when distributed agents all have identical beliefs, for which it is found that the central agent's decision is polarized and beliefs determine the limit value of the central agent's risk. Moreover, it is surprising that when all agents believe a certain prior-agnostic constant belief, it achieves globally optimal risk as N -> infinity.