To address the decision-making and control problem of collaborative hunting of autonomous underwater vehicle (AUV) swarms with multiple dynamic targets, this paper proposes a hunting algorithm integrating auction mechanisms and multi-agent deep reinforcement learning. The method decomposes the hunting task into two stages: target allocation and motion control. Firstly, based on the point-matching method from optimal control theory, training data and bid value labels are generated, taking into account optimization objectives such as hunting posture, minimum time, and minimum energy consumption. The auction neural network is trained using supervised learning, achieving real-time target allocation for the AUVs. Next, the allocated individual state space is constructed, a multi-target hunting reward function is designed, and a multi-agent soft actor-critic algorithm is employed to optimize the hunting strategy. The efficient and adaptive auction algorithm ensures rapid target allocation in dynamic and complex environments, while multi-agent reinforcement learning enhances the swarm's rapid response capability in collaborative control. Finally, hunting experiments are conducted in various scenarios. Experimental results show that the proposed method can significantly improve the performance of the hunting strategy. When dealing with 2, 3 and 4 dynamic targets, the average roundup success rates are 79.04%, 89.78% and 90.43%, respectively. Compared with the baseline method, they are increased by 48.41%, 54.00% and 53.93%, respectively. In other words, the proposed algorithm has better performance in handling hunting tasks of different scales.