As drones are increasingly used in various civilian applications, the security of drone communications is a growing concern. In this context, we propose novel strategies for challenge-response physical layer authentication (CR-PLA) of drone messages. The ground receiver (verifier) requests the drone to move to a defined position (challenge), and authenticity is verified by checking whether the corresponding measured channel gain (response) matches the expected statistic. In particular, the challenge is derived from a mixed strategy obtained by solving a zero-sum game against the intruder, which in turn decides its own positions. In addition, we derive the optimal strategy for multi-round authentication, where the CR-PLA procedure is iterated over several rounds. We also consider the energy minimization problem, where legitimate users want to minimize the energy consumption without compromising the security performance of the protocol. The performance of the proposed scheme is tested in terms of both security and energy consumption through numerical simulations, considering different protocol parameters, different scenarios (urban and rural), different drone altitudes, and also in the context of drone swarms.