Multi-vehicle autonomous driving couples strategic interaction with hybrid (discrete-continuous) maneuver planning under shared safety constraints. We introduce IBR-GCS, an Iterative Best Response (IBR) planning approach based on the Graphs of Convex Sets (GCS) framework that models highway driving as a generalized noncooperative game. IBR-GCS integrates combinatorial maneuver reasoning, trajectory planning, and game-theoretic interaction within a unified framework. The key novelty is a vehicle-specific, strategy-dependent GCS construction. Specifically, at each best-response update, each vehicle builds its own graph conditioned on the current strategies of the other vehicles, with vertices representing lane-specific, time-varying, convex, collision-free regions and edges encoding dynamically feasible transitions. This yields a shortest-path problem in GCS for each best-response step, which admits an efficient convex relaxation that can be solved using convex optimization tools without exhaustive discrete tree search. We then apply an iterative best-response scheme in which vehicles update their trajectories sequentially and provide conditions under which the resulting inexact updates converge to an approximate generalized Nash equilibrium. Simulation results across multi-lane, multi-vehicle scenarios demonstrate that IBR-GCS produces safe trajectories and strategically consistent interactive behaviors.