Opinion dynamics with bounded confidence in the Bayes risk error divergence sense

Bounded confidence opinion dynamic models have received much recent interest as models of information propagation in social networks and localized distributed averaging. However in the existing literature, opinions are only viewed as abstract quantities rather than as part of a decision-making system. In this work, opinion dynamics are examined when agents are Bayesian decision makers that perform hypothesis testing or signal detection. Bounded confidence is defined on prior probabilities of hypotheses through Bayes risk error divergence, the appropriate measure between priors in hypothesis testing. This definition contrasts with the measure used between opinions in the standard model: absolute error. It is shown that the rapid convergence of prior probabilities to a small number of limiting values is similar to that seen in the standard model. The most interesting finding in this work is that the number of these limiting values changes with the signal-to-noise ratio in the hypothesis testing task. The number of final values or clusters is maximal at intermediate signal-to-noise ratios, suggesting that the most contentious issues lead to the largest number of factions.