A Novel Integrated Algebraic/Geometric Approach to the Solution of Two by Two Games with Dominance Principle

The classical mixed strategies non-cooperative solution of a two person – two move game is recalled by paying attention to the different proposed methods and to the properties of the so found solutions. The non-cooperative equilibrium point is determined by a new geometric approach based on the dominance principle. Starting from the algebraic bi-linear form of the expected payoffs of the two players in the( x, y) domain of the probabilistic distribution on the pure strategies, the two equations are studied as surfaces in the 3D space on the basis of the sound theory of the quadratic forms. The study of the properties of the quadric is performed by classifying the bi-linear form as pertaining to a classical hyperbolic paraboloid and the relationship between its geometric properties and the probabilistic distribution on the pure strategies is found. The application of the dominance principle allows to choose the equilibrium point among the classical solutions avoiding the ambiguity due to their non-interchangeability and a conjecture about the uniqueness of the solution is proposed in order to solve the problem of the existence and uniqueness of the noncooperative solution of a two-by-two game. The uniqueness of the non-cooperative solution could be used as a starting point to find out the cooperative solution of the game too.