Symmetric continuum opinion dynamics: Convergence, but sometimes only in distribution

This paper investigates the asymptotic behavior of some common opinion dynamic models. We show that as long as interactions in a continuum of agents are symmetric, the distribution of the agents' opinions converges, but that there exist examples where the opinions themselves do not converge. This phenomenon is in sharp contrast with symmetric models on finite numbers of agents where convergence of opinions is always guaranteed. However, as long as every agent in the continuum interacts with those whose opinions are close to its own (a common assumption in opinion modeling), or that the interactions are uniquely determined by their opinions, the opinions of almost all agents will in fact converge.