Reaching consensus with increasing information

This paper focuses on a model of opinion formation over networks with continuously flowing new information and studies the relationship between the network and information structures and agents' ability to reach agreement. At each time period, agents receive private signals in addition to observing the beliefs held by their neighbors in a network. Each agent then updates its belief by aggregating the information available to it in a boundedly rational fashion. Whether this model results in conformity or disparity of beliefs is contingent on connectivity of the network and identifiability of the unknown state. We show that if the network is strongly connected, agents will eventually reach consensus in their beliefs; if in addition the state is globally identifiable, then agents will be able to learn the unknown state. Agents will also reach consensus and learn the state if the network fails to be connected but the state is locally identifiable. In contrast, agents in a non-strongly connected network will almost never reach agreement in their beliefs about a locally unidentifiable state. We also provide a characterization of the rates of convergence in terms of the top Lyapunov exponent of a set of i.i.d. matrices. The proofs use Oseledets' multiplicative ergodic theorem and recent results on stability of Lyapunov regular dynamical systems.