Co-evolutionary dynamics of threshold PGGs with collective-risk environment feedback

At present, the collective-risk dilemma game has recognized the feedback loops between strategies and collective-risk. However, understanding how individual decision-making in complex real-world societies impacts collective-risk, and in turn influences individual strategies through such feedback mechanisms, remains an area that requires further exploration. We consider the co-evolutionary dynamics based on threshold Public Goods Games (PGGs) under collective-risk. First, this co-evolutionary game framework can be divided into two parts: threshold PGGs and the dynamics of collective-risk. The former represents the group dynamics of betrayal and cooperation strategies, while the latter can be seen as the evolutionary dynamics of collective-risk under strategic influences. Next, we examine the presence and stability of possible equilibrium points that could emerge from co-evolutionary dynamics. Subsequently, we introduce the method of manifold control to extend the framework and guide the co-evolutionary dynamics of threshold PGGs to a certain extent. Finally, a particular example is presented to validate the theoretical approaches. In summary, the method of manifold control offers a new perspective for exploring collective cooperation in the context of collective-risk dilemmas.