Weak Sharpness And Its Properties In Nash Equilibrium Problem

Game theory is the main study of the competitive or belligerent mathematical theory method. It has a broad application value. The study of modern game theory including cooperative game theory and non-cooperative game theory, the main content of the research is the non-cooperative game theory. Rather than the non-cooperative game theory is the core issue of Nash equilibrium problem. In recent years to the problem of Nash equilibrium theory and algorithm research has many, but the solution of the duality theory or feasible solution sequence converges problem to Nash equilibrium problem has rarely been studied.In this paper, by the concept of weak sharp minima of solution set in the mathematical programming problem, in Nash equilibrium problem, we defined the concept of weak sharpness of solution set, respectively for unconstrained problems and problems with constraints, we discussed some properties of weakly sharp set, and got the necessary and sufficient conditions of that the solution set is weak sharpness. Under the condition that the solution set is weak sharpness, we got the necessary and sufficient conditions of the convergence of the feasible solution sequence in Nash equilibrium problem.Main work of this paper, the first chapter introduced the background and current research status of Nash equilibrium problem, and then introduced the main content of this paper. The second chapter introduced the basic knowledge and the basic definitions of the Nash equilibrium problem, the definition of the weak sharp minima was introduced in the problems of mathematical programming and the definition of weak sharpness was introduced in variational inequalities problems. In the third chapter, weak sharpness of the solution set was introduced to Nash equilibrium problem.Respectively for unconstrained problems and problems with constraints, we discussed some properties of weak sharpness. In Nash equilibrium problem without constraint, we discuss the relationship between weak sharpness of the solution set and the directional derivative of the objective functions on the solution set.In the cases of conditional constraints, under the differentiability assumption, we discuss some properties of the weak sharpness to obtain the necessary and sufficient conditions of weak sharpness of the solution set in Nash equilibrium problem.The fourth chapter, under the condition that the solution set is weak sharpness, we got that feasible solution sequence was convergent in Nash equilibrium problem.