Novel Differential Dominance Principle based Approach to the Solution of More than Two Persons n Moves General Sum Games

In a previous paper the application of the dominance principle was proposed to find the non-cooperative solution of two persons two by two general sum game with mixed strategies; in this way it was possible to choose the equilibrium point among the classical solutions avoiding the ambiguity due to their non-interchangeability, moreover the non-cooperative equilibrium point was determined by a new approach based on the dominance principle. Starting from that result it is here below proposed the extension of the method to more than two persons general sum games with n by n moves. Generally speaking the dominance principle can be applied to find the equilibrium point both in pure and mixed strategies. In this paper in order to apply the dominance principle to the mixed strategies solution, the algebraic multi-linear forms of the expected payoffs of the players are studied. From these expressions of the expected payoffs the derivatives are obtained and they are used to express the probabilities distribution on the moves after the two defi nitions as Nash and prudential strategies. The application of the dominance principle allows to choose the equilibrium point between the two equivalent 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 non-cooperative solution of a many persons n by n 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. Some games from the sound literature are discussed in order to show the eff ectiveness of the presented procedure.