The actions of an autonomous vehicle on the road affect and are affected by those of other drivers, whether overtaking, negotiating a merge, or avoiding an accident. This mutual dependence, best captured by dynamic game theory, creates a strong coupling between the vehicle's planning and its predictions of other drivers' behavior, and constitutes an open problem with direct implications on the safety and viability of autonomous driving technology. Unfortunately, dynamic games are too computationally demanding to meet the real-time constraints of autonomous driving in its continuous state and action space. In this paper, we introduce a novel game-theoretic trajectory planning algorithm for autonomous driving, that enables real-time performance by hierarchically decomposing the underlying dynamic game into a long-horizon 'strategic' game with simplified dynamics and full information structure, and a short-horizon 'tactical' game with full dynamics and a simplified information structure. The value of the strategic game is used to guide the tactical planning, implicitly extending the planning horizon, pushing the local trajectory optimization closer to global solutions, and, most importantly, quantitatively accounting for the autonomous vehicle and the human driver's ability and incentives to influence each other. In addition, our approach admits non-deterministic models of human decision-making, rather than relying on perfectly rational predictions. Our results showcase richer, safer, and more effective autonomous behavior in comparison to existing techniques.