Blockchains systems evolve in complex environments that mix classical patterns of faults with selfish, rational or irrational behaviours typical of economic systems. During the presentation we discuss a new game theoretical framework that formally characterizes the robustness of blockchains systems in terms of resilience to rational deviations and immunity to Byzantine behaviors. We apply the model to known blockchain protocols, including Bitcoin, and we identify the upper and lower bounds of their robustness indices.