## 2 Related Work  
Research on underwater environmental monitoring was begun in developed countries of Europe and America as early as the 20 th century, with various monitoring instruments and equipment being designed. In 2009, NEPTUNE, which was established jointly by the United States and Canada, was officially launched. It is recognized as the world's largest cabled ocean observatory network, being capable of achieving gigabit-level bidirectional transmission communication monitoring. By June 2022, the Global Ocean Observing System (GOOS) had become the most comprehensive ocean observation system currently available, with its total number of observation devices exceeding 9000, providing reliable data for marine environmental protection and disaster early warning [8]. Currently, the increasing use of UWSNs (underwater wireless sensor networks) to achieve large-scale and flexible monitoring tasks in marine monitoring systems has led to growing attention and research on clustering algorithms for the large number of underwater nodes deployed in UWSNs. For example, the LEACH algorithm for balancing node energy consumption was first proposed by [PERSON] et al. [9], in which CH (cluster head) nodes are rotated to share their load through the setting of random numbers and thresholds, although this also leads to uneven energy consumption. A series of more efficient clustering algorithms have been proposed based on the LEACH algorithm, such as opportunistic particle swarm optimization [10], energy-saving scalable algorithm [11], and correlation clustering algorithm [12], which have significantly improved network performance. Furthermore, [PERSON] et al. [13] proposed a low-energy clustering method that non-uniformizes node density, which involves selecting pre-cluster heads through random number settings and determining the communication range based on their coverage lengths for CH election. This algorithm enhances the performance of UWSNs, but it also increases frequent interactions between nodes, leading to instability in the algorithm. Fixed strategies are used in most of the above solutions to balance node energy consumption, and they are unable to dynamically adjust the node's own strategy to adapt to the constantly changing marine monitoring scenarios in a time-varying, complex underwater environment. As a result, some experts have begun to focus on establishing dynamic competition models between users through game theory to optimize network performance. [PERSON] et al. [14] proposed a game theory-based distributed multi-dimensional clustering protocol, which calculates the probability of Nash equilibrium based on the distance of nodes to CH and the degree of nodes and selects CH based on the size of the remaining energy. However, there is still significant room for improvement in energy consumption optimization with this algorithm. Subsequently, some scholars began to design utility functions using various strategies to enhance the overall performance of the network, such as repeated games [15], a dual-cluster head mechanism [16], and two-level management clustering [17], etc. However, these algorithms only consider the clustering performance of wireless networks in terrestrial environments, and they do not take into account the impact of complex underwater environments on game clustering. Therefore, they are not suitable for marine environments. To address this issue, a multi-dimensional game-based underwater adaptive energy-saving clustering algorithm was proposed in the literature [18], which establishes a multi-dimensional game model and increases the opportunity for historical CH nodes to compete for CH again, further optimizing the strategy for CH election. However, this algorithm takes less consideration of the state of the nodes themselves and environmental factors, resulting in weak applicability in real scenarios.