Effective resource management and allocation are essential in the field of edge computing to maximize system performance and satisfy the needs of various applications. The current study investigates the formulation of utility functions and the identification of Nash equilibrium methods in resource allocation within edge computing systems using a game theory framework. We examine a deterministic system in which resource providers and customers interact strategically to maximize their utilities. Our objective is to find stable solutions where no participant has an incentive to unilaterally stray from their goal by creating and assessing different resource allocation schemes. The paper discusses the implications of these strategies on system performance, including factors such as latency, resource utilization, and overall efficiency. Our findings provide insights into the stability and effectiveness of different allocation mechanisms, offering a roadmap for future research in optimizing resource distribution in edge computing environments. This work contributes to the advancement of game theory applications in edge computing, highlighting opportunities for enhancing resource management and system performance.