On symmetric continuum opinion dynamics

This paper investigates the asymptotic behavior of some common opinion dynamic models in a continuum of agents. We show that as long as the interactions among the agents are symmetric, the distribution of the agents' opinions converges. We also investigate whether convergence occurs in a stronger sense than merely in distribution, namely, whether the opinion of almost every agent converges. We show that while this is not the case in general, it becomes true under plausible assumptions on interagent interactions, namely that agents with similar opinions exert a nonnegligible pull on each other, or that the interactions are entirely determined by their opinions via a smooth function.