Eulerian Opinion Dynamics with Bounded Confidence and Exogenous Inputs

The formation of opinions in a large population is governed by endogenous (interactions with peers) and exogenous (influence of media) factors. In the analysis of opinion evolution in a large population, decision making rules are often approximated with non-Bayesian "rule of thumb" methods. Adopting a non-Bayesian averaging rule, this paper focuses on an Eulerian bounded-confidence model of opinion dynamics and studies the information assimilation process resulting from exogenous inputs. In this model, a population is distributed over an opinion set, and each individual updates its opinion via (i) opinions of the population inside the individual's confidence range and (ii) the information from an exogenous input in that range. First, we establish various mathematical properties of this system's dynamics with time-varying inputs. Second, for the case of no exogenous input, we prove the convergence of the population's distribution to a sum of Dirac delta distributions. We further derive a simple sufficient condition for opinion consensus under the influence of a time-varying input. Third, regarding information assimilation, we define the attracted population of a constant input. For a weighted Dirac delta input and for uniformly distributed initial population, we establish an upper bound on the attracted population valid under some technical assumptions. This upper bound is an increasing function of the population's confidence bound and a decreasing function of the input's measure (i.e., the integral of input's distribution over the opinion space). Fourth, for a normally distributed input with truncated support, we conjecture that the attracted population is approximately an increasing affine function of the population's confidence bound and of the input's standard deviation; we illustrate this conjecture numerically.