A model-independent theory of consensus and dissensus decision making

Abstract: We develop a model-independent framework to study the dynamics of decision-making in opinion networks for an arbitrary number of agents and an arbitrary number of options. Model-independence means that the analysis is not performed on a specific set of equations, in contrast to classical approaches to decision making that fix a specific model and analyze it. Rather, the general features of decision making in dynamical opinion networks can be derived starting from empirically testable hypotheses about the deciding agents, the available options, and the interactions among them. After translating these empirical hypotheses into algebraic ones, we use the tools of equivariant bifurcation theory to uncover model-independent properties of dynamical opinion networks. The model-independent results are illustrated on a novel analytical model that is constructed by plugging a generic sigmoidal nonlinearity, modeling boundedness of opinions and opinion perception, into the model-independent equivariant structure. Our analysis reveals richer and more flexible opinion-formation behavior as compared to model-dependent approaches. For instance, analysis reveals the possibility of switching between consensus and various forms of dissensus by modulation of the level of agent cooperativity and without requiring any particular ad-hoc interaction topology (e.g., structural balance). From a theoretical viewpoint, we prove new results in equivariant bifurcation theory. We construct an exhaustive list of axial subgroups for the action of $\ES_n \times \ES_3$ on $\R^{n-1}\otimes\R^{2}$. We also generalize this list to the action of $\ES_n \times \ES_k$ on $\R^{n-1}\otimes \R^{k-1}$, i.e., for $n$ agents and $k$ options, although without proving that in this case the list is exhaustive.