A Unified Framework for Opinion Dynamics

We develop social learning environments as a unified framework for modeling and analyzing opinion dynamics. A social learning environment consists of a finite set of individuals and, for every individual, a set of possible opinions and an opinion updating rule. We establish conditions that guarantee the existence of an equilibrium and the convergence of learning paths to an equilibrium. Our framework unifies a wide range of models from the literature, including the voter, DeGroot, and Sznajd models, as well as the majority rule and a variation we call individual majority rule. This unification reveals a common underlying structure that explains the stability of these seemingly distinct models. Building on this foundation, we extend the theory to adaptive networks, in which network topology and opinion dynamics change simultaneously through a feedback loop. In particular, we introduce a novel class of adaptive networks called selective diffusion networks, where opinions are about the benefits provided by other individuals, and link formation is costly. We show that equilibria are absorbing when stability of opinions coincides with consensus within network components. Lastly, we develop potential functions as an alternative method for analyzing convergence. We provide an order-theoretic characterization of potential functions, which leads to a systematic method for their construction, and establish formal connections between potentials and conditions on social learning environments.