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Responsibility is one of the key notions in machine ethics and in the area of autonomous systems. It is a multi-faceted notion involving counterfactual reasoning about actions and strategies. In this paper, we study different variants of responsibility for LTLf outcomes based on strategic reasoning. We show a connection with notions in reactive synthesis, including the synthesis of winning, dominant, and best-effort strategies. This connection provides a strong computational grounding of responsibility, allowing us to characterize the worst-case computa- tional complexity and devise sound, complete, and optimal algorithms for anticipating and attributing responsibility.