Coevolution of cooperative behavior and collective risk via adaptive feedback in finite populations

In collective action, individuals often face a balance between personal costs and collective losses, as observed in contexts such as climate mitigation and public risk management. Such situations are commonly described as collective risk dilemmas, where group members must reach a sufficient level of joint action to avoid shared losses. In this work, we develop a coevolutionary framework in a finite population in which risk is modeled as an endogenous dynamic variable that evolves with individual strategies, allowing for feedback between behavior and risk. We investigate how this feedback shapes the evolution of cooperation under constraints on risk growth. The results show that cooperation can be sustained only when collective actions keep risk within its constraint. When this condition is not satisfied, individuals maintain only the minimal level of cooperation required to avoid losses, and risk rapidly vanishes. By contrast, when the constraint is satisfied, cooperation and risk coexist in a dynamically stable state maintained by their feedback. We further find that increasing the difficulty of cooperation suppresses participation and leads to higher risk levels, while external risk suppression, although effective in reducing risk, weakens incentives for voluntary cooperation. In addition, these patterns persist across heterogeneous and uncertain risk constraints, indicating the robustness of the underlying behavior–risk feedback mechanism. Overall, our results highlight the role of endogenous risk in shaping cooperative dynamics and provide a consistent framework for understanding collective responses in settings where avoiding shared losses requires coordinated action.