Underwater Acoustic Sensor Networks (UASN) plays a crucial role in monitoring and transmitting environmental information for marine resource exploration. However, underwater communication faces significant challenges such as signal attenuation, noise interference, and unstable underwater environments. Moreover, with the deployment of a large number of nodes, inefficient resource allocation strategies in cooperative communication lead to strained communication resources as well as increased energy consumption. This paper introduces a novel relay selection approach based on game theory that takes into account both competition and cooperation among nodes. By building a potential game model and a utility function, we derive the optimal relay selection strategy that can enhance system Quality of Service (QoS). We also define conditions for the feasibility of pure strategy Nash Equilibria (NE) and develop a low-complexity QoS-based Relay Selection Iterative Algorithm (QoS-RSIA) to procure feasible pure strategy NE for the designed game. Through comparative analysis with other relay selection algorithms, simulation results show that the proposed method not only enhances system throughput and average spectral efficiency by at least 8.3%, but also significantly reduces time complexity compared to exhaustive search algorithms.