We analyze the effect of strategic ambiguity and heterogeneous attitudes towards such ambiguity on optimal mitigation and adaptation. Pessimistic players tend to invest more in mitigation, while optimists favor adaptation. When adaptation is more expensive than mitigation, three types of equilibria can obtain depending on the level and distribution of ambiguity aversion: (i) a mitigation equilibrium, (ii) an adaptation equilibrium and (iii) a mixed equilibrium with both adaptation and mitigation. The interaction between ambiguity attitudes and wealth distribution plays a crucial role for the aggregate environmental policy: a wealth transfer from pessimistic to optimistic agents increases total mitigation. A similar result applies to the choice of an optimal tax on consumption, which is shown to increase in optimism, but decrease following a transfer of income towards the more optimistic players. Finally, we show that under strategic ambiguity, the introduction of a non-binding standard can impact agents' beliefs about their opponents' behavior and as a result lower total equilibrium mitigation. Our results highlight the necessity to consider attitudes towards strategic ambiguity in the design of economic policies targeting climate change. They might also shed some light on the slow rate of convergence of environmental policies across countries.