A Continuous-Time Voting Model with Bounded Confidence: Existence and Convergence of Solutions

We propose a continuous-time bounded-confidence voting model in which each opinion encodes both the agent’s choice between two alternatives (sign) and degree of certainty (magnitude). The dynamics combine bounded confidence with self-reinforcement, a mechanism by which an agent’s conviction strengthens as its opinion approaches the extremes +1 and -1. Upon reaching these values, the agent makes a decision; decided agents keep their opinions fixed yet continue to influence others, while undecided agents average the opinions within their confidence neighborhoods. The resulting hybrid system has a discontinuous right-hand side, motivating the introduction of elementary solutions (with fixed interaction structure) and regular solutions (finite concatenations of elementary ones). We prove order preservation and establish local existence of solutions for all initial conditions. Furthermore, we show that any solution can be infinitely prolonged; the prolongation procedure guarantees regularity of the solution. All equilibria are classified as either fully decided or undecided; undecided equilibria are repelling, so any non-equilibrium infinitely prolonged regular trajectory converges to a fully decided equilibrium.