  Human society and natural environment form a complex giant ecosystem, where
human activities not only lead to the change of environmental states, but also
react to them. By using collective-risk social dilemma game, some studies have
already revealed that individual contributions and the risk of future losses
are inextricably linked. These works, however, often use an idealistic
assumption that the risk is constant and not affected by individual behaviors.
We here develop a coevolutionary game approach that captures the coupled
dynamics of cooperation and risk. In particular, the level of contributions in
a population affects the state of risk, while the risk in turn influences
individuals' behavioral decision-making. Importantly, we explore two
representative feedback forms describing the possible effect of strategy on
risk, namely, linear and exponential feedbacks. We find that cooperation can be
maintained in the population by keeping at a certain fraction or forming an
evolutionary oscillation with risk, independently of the feedback type.
However, such evolutionary outcome depends on the initial state. Taken
together, a two-way coupling between collective actions and risk is essential
to avoid the tragedy of the commons. More importantly, a critical starting
portion of cooperators and risk level is what we really need for guiding the
evolution toward a desired direction.
