A relative approach to opinion formation

Chan, Kit Ming Danny; Duivenvoorden, Robert; Flache, Andreas; Mandjes, Michel

We propose a linear model of opinion dynamics that builds on, and extends, the celebrated DeGroot model. Its primary innovation is that it concerns relative opinions, in that two opinion vectors that differ only by a multiplicative constant are considered equivalent. An agent's opinion is represented by a number in R, and evolves due to positive and negative impacts resulting from the other agents' opinions. As a consequence of the relative interpretation, the model exhibits nonlinear dynamics despite its DeGroot-like linear foundation. This means that, even when agents are"well-connected", the dynamics of the relative opinion model can reproduce phenomena such as polarization, consensus formation, and periodic behavior. An important additional advantage of remaining in a DeGroot-type framework is that it allows for extensive analytical investigation using elementary matrix algebra. A few specific themes are covered: (i) We demonstrate how stable patterns in opinion dynamics are identified which are hidden when only absolute opinions are considered, such as stable fragmentation of a population despite a continuous shift in absolute opinions. (ii) For the two-agent case, we provide an exhaustive description of the model's asymptotic (long-term, that is) behavior. (iii) We explore group dynamics, in particular providing a non-trivial condition under which a group's asymptotic behavior carries over to the entire population.