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In Game Theory, the computation and counting of equilibrium and the structural features of equilibrium are major research topics. The existence of equilibrium naturally forms a decision problem. The Nash equilibrium composed of mixed strategies always exists in the matrix and double matrix games. Therefore, calculating Nash equilibrium is the main task in the matrix and double matrix games. For the double matrix game, the measure of its Nash equilibrium can be based on the stability implied in the Nash equilibrium definition. Besides, a propositional formula can construct a two-player game given as a bi-matrix by specifying a proper finite set of strategies and adding a transformation. In this study, we investigate the development of Nash equilibriums by analyzing the structure-property of bi-matrix. We examine the characterization of Nash equilibrium creation and the relationship between satisfiable assignments of formula and Nash equilibriums of generated Game based on the characterization of bi-matrix formation and the stability principle of Nash equilibriums. The method from analysis and logic helps examine some relationships between combination optimization issues in graph theory and game systems by examining the structure of bi-matrix and combining the stability principle of Nash equilibriums to research the characterization of Nash equilibriums.

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