Alternating rotation of coordinated and anti-coordinated action due to environmental feedback and noise

The alternating rotation of coordinated and anti-coordinated behaviors is a common phenomenon in real life, for example, consistent and polarizing behaviors in opinion dynamics, innovation and imitation behaviors in the dynamics of product or technology spread. In this paper, we use environmental feedback dynamics to correlate coordination games and anti-coordination games to describe this phenomenon. Interestingly, the dynamic phenomenon of alternating rotation of coordination and anti-coordination is successfully observed in our model, namely the existence of neutral stability centers. We derive theoretically and numerically the conditions for the arising and stability of these stable states. Remarkably, we find that the two regions that do not overlap or intersect and are not affected by the relative feedback parameter. The demarcation of the two regions is determined by the payoff matrix parameter. However, stochastic noise plays a crucial role, and the introduction of a tiny stochastic noise can have a significant effect that the left and right regions can cross each other. The rotation of the emergence and demise of hippies (new products, technology or opinions), and the emergence and dominance of hippies can form a closed trajectory, which can be transformed by each other. The probability distribution of the sojourn time between the system staying in the previous region before entering the next region is well approximated by a power law with an exponential cutoff. Our results have the potential to maintain alternating rotation of coordinated and anti-coordinated behaviors.