Evolutionary Dynamics of Mixed Rings of Coordinators and Anticoordinators

Decision-making based on peers’ actions often results in one of the following two groups: coordinators who tend to blend in with the population and make trending decisions, and anticoordinators who act against the majority. Only mixed networks of both coordinators and anticoordinators are capable of not reaching an equilibrium state, where every individual is satisfied with her choice. However, the conditions for nonequilibration, and more challengingly, the characterization of the non-equilibrium limit set remain concealed. We answer these problems for ring networks. We show that a mixed ring of coordinators and anticoordinators equilibrates if and only if it does not contain a particular arrangement of consecutive agents, and never equilibrates if and only if it contains a particular arrangement of consecutive agents with particular actions. As a result, a ring may admit both equilibrium and non-equilibrium limit sets. We further investigate the stability of the equilibrium states of the resulting network dynamics.