In-Game theory Applications, the 6G-assisted federated learning in continuous monitoring applications with wireless sensor networks (WSN) is a significant concern. With increased applications comes the increased demand for advanced resource allocation and energy management systems. WSN can be determined as a self-configured, infrastructure-less wireless network monitoring physical or other surrounding conditions. In this study, the proposed system is concentrated on applying game theory to 6G-assisted federated learning for continuous monitoring in wireless sensor networks. The techniques imposed by the dual sink, such as Static and dynamic moving nodes approaches, are applied to the tentative node selection based on aggregated data transmission techniques. Based on the Static nodes and trusted nodes, the Aggregated data transmission is achieved high-level data transmission by combining individual-level data, i.e., the aggregate of the output data. This technique is performed with the wireless sensor network (WSN) platform with a future 6G network coordinating with the tool of NS4-Programmable Data Plane simulation. The proposed system simplifies the development of a behavioral model and bridges the gap between simulation and deployment. Finally, the combination of game theory with 6G-assisted federated learning for continuous monitoring applications in WSN solves problems and identifies several future directions. The outcome analysis of the proposed system is to design the wireless sensor network to yield a high network lifetime of more than 20 h and low power (less than 0.2 kJ energy) consumption for efficient communication in the future 6G cellular network.