In this paper we show how to robustify the computation of equilibria in two integrated assessment models for climate change. Both models deal with the optimal timing of a transition to a 'clean' economy where a technology with low emissions but high energy cost can be used in the production process. The game represents the competition between industrialized and developing countries. A cost-benefit approach is implemented with an economic loss factor that represents the damages due to climate change. In the first model one assumes that both technologies, 'dirty' and 'clean' are available, but the economic loss factor is very uncertain. In the second model one assumes that the 'clean' technology is not yet available and some R&D investment must be made to get the technology breakthrough permitting its penetration. In this second model, formulated in continuous time, the jump rate of the controlled stochastic process describing the effect of R&D investment on the probability of breakthrough, is also considered as very uncertain. In both models we introduce a concept of \(\alpha\)-robust equilibrium, where the robustification is achieved through the use of ambiguous probability distributions with a Kulbach-Leibler divergence cost structure for the worst case choice by Nature.