Blockchains systems evolve in complex environments that mix classical patterns of faults (e.g crash faults, transient faults, Byzantine faults, churn) with selfish, rational or irrational behaviors typical to economical systems. In this paper we propose a game theoretical framework in order to formally characterize the robustness of blockchains systems in terms of resilience to rational deviations and immunity to Byzantine behaviors. Our framework includes necessary and sufficient conditions for checking the immunity and resilience of games and a new technique for composing games that preserves the robustness of individual games. We prove the practical interest of our formal framework by characterizing the robustness of three different protocols popular in blockchain systems: a HTLC-based payment scheme (a.k.a. Lightning Network), a side-chain protocol and a cross-chain swap protocol.