Co-evolutionary dynamics in optimal multi-agent game with environment feedback

At present, game-based analysis and control problems with environmental feedback significantly contribute to the study of collective cooperation, and poses great challenges for the analysis of multi-agent game systems in complex real world. We consider co-evolutionary dynamics based on an extended multi-agent game with environmental feedback, which is modeled by optional public goods games (OPGGs). Firstly, the co-evolutionary dynamics of OPGGs with environment feedback can be modeled as two parts: the dynamics of OPGGs and synergy coefficient, where the former can represent the group game behavior of three strategies, and the latter can be seen as the evolutionary behavior in changing environments. Subsequently, some possible fixed points will emerge with respect to three types of boundaries for these co-evolutionary dynamics, and the existence and stability for the corresponding boundary fixed points are determined and analyzed. Meanwhile, the results are further extended to interior equilibrium point. It is observed that by combining voluntary participation and environment feedback, cooperation behavior can be still feasible, and the evolving system can oscillatingly converge to the persistent cooperation under some certain threshold conditions. Finally, an example is illustrated to validate the proposed approach. To sum up, environmental feedback mechanism provides a novel perspective to investigate the collective cooperation in a changing world.