Preventing global warming is a public good requiring overall cooperation. Contributions will depend on the risk of future losses, which plays a key role in decision-making. Here, we discuss a theoretical model grounded on game theory and large-scale population dynamics. We show how decisions within small groups under high risk and stringent requirements toward success significantly raise the chances of coordinating to save the planet's climate, thus escaping the tragedy of the commons. In addition, our model predicts that, if one takes into consideration that groups of different sizes will be interwoven in complex networks of contacts, the chances for global coordination into an overall cooperating state are further enhanced.