Researches On The Direct Algorithm And Application Of Some Game Theory Problems

The direct algorithm for generalized Nash equilibrium problem, as its objective, comprehensive and openness, has attracted more and more attentions from the theorists. It becomes a hotspot in recent years. In this article, we will design some direct algorithms for finding Nash equilibrium of "a Leader-Followers game", "two-by-two game" and "a class of simultaneous game". In order to make the designed algorithm more reliable, we will regularize the original game model. Besides, we will add the "tolerance" at the same time. In our new model, "tolerance" can reflect the bounded rationality hypothesis of the participants. Under some suitable conditions, the global convergence to Nash equilibrium of all the proposed algorithms is proved.Fisrt, a hybrid operator splitting method is proposed for finding a Nash equilibrium of a class of games. The considered game has three players:one is a leader and the others are followers. The proposed method properly reflects the move sequence of the game encountered, and permits to solve the sub-problem via some inexact style. The latter matches the bounded individual rational of players in practice, which implies all players may make some errors in the game process. Under some suitable conditions, the convergence to Nash equilibrium of the proposed method is proved. Some elementary numerical results indicate the validity of the proposed method.Then, a hybrid splitting method is proposed for finding a Nash equilibrium of the "two-by-two game". In this game, the four players are divided into two groups. Every group has two players, and they act mutual benefit. Similarly, the proposed method properly reflects the move sequence of the considered game, and permits to solve the sub-problem via some inexact style. The latter matches the bounded individual rational of players in practice, which implies all players may make some errors in the game process. Under some suitable conditions, the convergence to Nash equilibrium of the proposed method is proved.At the last, a nonlinear Jacobi method is proposed for finding a Nash equilibrium of a class of simultaneous game. Under some suitable conditions, the convergence to Nash equilibrium of the proposed method is proved.