Correlated Opinion Dynamics in a Model of Collective Learning with Expert Advice

Formation of correlated opinions in a network of interacting agents is a ubiquitous social phenomenon. In a financial market, for instance, traders’ opinions of private information such as a stock price oftentimes tend to be correlated, despite the complexity of opinion exchange mechanisms. Here, we introduce a minimal model of opinion dynamics that naturally exhibits formation of correlated opinions. In this model, each agent (trader) learns an estimate of private information (a stock price) from an expert (broker) while also updating its opinion by taking a weighted average of other opinions. When an expert can at best provide only an estimate of private information, typified by the truth masked with Gaussian white noise, the opinion dynamics is described by Langevin’s dynamics driven by one-dimensional noise. In this case, a stationary Gaussian distribution centered around the truth is developed, and correlated opinions emerge naturally with the correlations encoded in the stationary distribution. In addition, when agents learn from the expert at different rates, the dynamics violates detailed balance. We study in detail the non-equilibrium steady state associated with the correlated opinion dynamics of 2 agents. The case of 3 and higher number of agents is also discussed.