This paper delves into the intricate relationship between games and the mathematical structures that govern them. By exploring both traditional board games and modern video games, we assess the extent to which mathematical logic plays a pivotal role in determining gameplay and strategies. A central theme is the quest for winning strategies: their existence, their mathematical underpinnings, and their limitations. Through case studies of games like chess and poker, the study elucidates the balance between deterministic gameplay and the unpredictable nature of certain game elements.This comprehensive study offers a bridge between game theory, mathematical logic, and the practical world of gaming.

This paper explores the theoretical foundation of non-expansive mappings in fixed-point iteration and its specific applications in game theory. As a mapping that maintains or reduces the distance between elements, non-expansive mappings provide an effective mathematical tool for finding stable strategies in dynamic games through their inherent convergence properties. The article first introduces the basic concepts of fixed-point theory and defines non-expansive mappings, then analyzes their application in solving Nash equilibria in game theory, with a particular emphasis on the stability and efficiency of iterative methods in the solution process. Additionally, through mathematical modeling and case analysis, this study demonstrates the practical effects and potential applications of non-expansive mappings in multi-stage games and complex strategy updates. This research not only enhances the understanding of fixed-point iteration methods in both theory and practice but also offers new perspectives and methods for addressing high-dimensional strategy spaces in game theory.