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To achieve the goal of keeping global warming well below 2 °C, private investors have to shift capital from brown to green infrastructures and technologies and provide additional green investment. In this paper, we present a game-theoretic perspective on the challenge of triggering such investments. The question of climate change mitigation is often related to the prisoner's dilemma, a game with one Nash equilibrium. However, the authors perceive investment for mitigation and adaptation as a coordination problem of selecting among multiple equilibria. To illustrate this, we model a non-cooperative coordination game, related to the stag hunt, with a brown equilibrium with lower payoffs that can be achieved single-handedly and a green equilibrium with higher payoffs that requires coordination. As multiple experiments show, in such games actors often fail to coordinate on a payoff dominant equilibrium due to uncertainty. Thus, we discuss how uncertainty could be reduced along two options: one that concerns a change in the payoff structure of the game and another that concerns subjective probabilities.