Characterizing Oscillations in Heterogeneous Populations of Coordinators and Anticoordinators

Oscillations often take place in populations of decision-making individuals that are either a coordinator, who takes action only if enough others do so, or an anticoordinator, who takes action only if few others do so. Populations consisting of exclusively one of these types are known to reach an equilibrium, where every individual is satisfied with her decision. Yet it remains open whether and when oscillations take place in a population consisting of both types, and if they do, what features they share. We take the first step towards answering this question by simulating a well-mixed population of coordinators and anticoordinator, each associated with a possibly unique non-negative threshold and initialized with the strategy A or B. We take the distribution of the actions A over the thresholds as the state of the population dynamics. The dynamics in our example admit two minimally positively invariant sets, where the solution trajectory oscillates, and an equilibrium. We identify the basic properties of the dynamics, based on which, we introduce a class of sets that are positively invariant. Our results highlight the possibility of non-trivial, complex oscillations in the absence of noise and population structure and shed light on the reported oscillations in decision-making populations.