We address the challenge of reliable and efficient interaction in autonomous multi-agent systems, where agents must balance long-term strategic objectives with short-term dynamic adaptation. We propose context-triggered contingency games, a novel integration of strategic games derived from temporal logic specifications with dynamic contingency games solved in real time. Our two-layered architecture leverages strategy templates to guarantee satisfaction of high-level objectives, while a new factor-graph-based solver enables scalable, real-time model predictive control of dynamic interactions. The resulting framework ensures both safety and progress in uncertain, interactive environments. We validate our approach through simulations and hardware experiments in autonomous driving and robotic navigation, demonstrating efficient, reliable, and adaptive multi-agent interaction.