Nash Equilibrium And General Equilibrium Under The Conditions Of Weak Punch Algorithm And Application

Awarding Nobel Prize to John Nash and other two theorist in game theory, the theory of Nash Equilibrium began to embrace the unprecedent appreciation in academic. It is believed that the main concentration of modern game theory is on non-cooperative game theory. The core of non-cooperative theory is Nash Equilibrium.Nash Equilibrium now has great impact on eco-nomics,management,public policy and even law. It is undebatably to be one of the most outstanding intellectual achievement in the past century.This paper begins with the introduction of concepts in the field of Nash Equilibrium. By using Nikaido-Isoda function, optimal response function as well as gap function. Nash problem could be transferred to an nonlinear optimization problem, which is a well studied topic. Following with these concepts and propositions, this paper reveals the inherent connection among Nash equilibrium, fixed point and gap function, which is the foundation of convergence proof.Furthermore, this paper introduces relaxation algorithm to find Nash Equilibrium on the condition of weak convex-concave. This paper mainly focuses on designing an efficient relaxation algorithm based on proposing another step-size choice-linear optimizing step size. Comparing with the two well known step size, constant step size and steepest step size, the new method demonstrates more efficiency and stability. Moreover, the proposed algorithm is provided with convergence proof based on fixed point theory and theory of weak convexity. In addition, to know more about the efficiency of the proposed algorithm and its applications to complex situation, the algorithm is further used to solve problems in power market. The applications in both cooperative and non-cooperative cases show that our algorithm could also be better used in complexity situation than the other two.Following with the applications in Nash Equilibrium, this paper extends the algorithm discussed in the above Chapters to general equilibrium Similarly, the convergence proof follows with the algorithm and error bound of this algorithm is also well analyzed. Finally, the algorithm is used to solve the problem in variational inequality.