Measure-valued opinion dynamics

In this work, we propose a method of opinion pooling based on pairwise interactions. We assume that each agent has a probability measure on the possible outcomes of some situation, and they try to find a single measure aggregating their estimates. This is a classical problem in Decision Theory, where expert opinions contain some degree of uncertainty, and a Decision Taker needs to pool these estimates. We study this problem using a kinetic theory approach, obtaining a Boltzmann type equation for opinions which are symmetric probability measures defined on the real line. We obtain a non-local, first order, mean field equation as its grazing limit when the parameter in the interaction goes to zero. Also, we prove the convergence to quasi-consensus with explicit estimates on the convergence time depending on the variance of these measures. Let us remark that this model can be interpreted as a noisy model of opinion dynamics. In many models, the opinion of each agent is a point in the real line, the agents interact and observe other agents opinions. We can consider that observed opinions are perturbed or deformed by some noise in the transmission channel or in the interpretation of the agents, so we can think of agents opinions directly as random variables instead of a single point.