This paper presents a game-theoretic path-following formulation where the opponent is an adversary road model. This formulation allows us to compute safe sets using tools from viability theory, that can be used as terminal constraints in an optimization-based motion planner. Based on the adversary road model, we first derive an analytical discriminating domain, which even allows guaranteeing safety in the case when steering rate constraints are considered. Second, we compute the discriminating kernel and show that the output of the gridding based algorithm can be accurately approximated by a fully connected neural network, which can again be used as a terminal constraint. Finally, we show that by using our proposed safe sets, an optimization-based motion planner can successfully drive on city and country roads with prediction horizons too short for other baselines to complete the task.