A Number Of Countermeasures Problem And The Stability Of The Multi-objective Optimization Problem,

In this thesis, we focus on the stability of solutions of non-cooperative games with infinitely many pure strategies and the stability of solutions of multi-objective optimizations, including the stability of mixed Nash equilibrium points of N-person games with continuous payoff, solutions of games with endogenous sharing rules, pure Nash equilibrium points of general N-person games, and efficient solutions and weakly efficient solutions of multi-objective optimizations with continuous vector-real-valued functions. This thesis is organized as follows..Chapter 1 We recall some notions and results used in our analysis in this thesis, including finite measure, finite signed measure, integral on compact metric spaces, topology on closed subset spaces, continuity of set-valued mappings and some concepts of essential solutions and essential sets.Chapter 2 We study the stability of the set of mixed Nash equilibrium points of N-person non-cooperative games with compact metric spaces of infinite pure strategies and continuous payoff functions. First, we study the stability of mixed Nash equilibrium points robust against the perturbations of payoff functions, and obtain the property of generic stability of mixed Nash equilibrium points, and both the existence of minimal essential sets and the existence of essential components of the set of mixed Nash equilibrium points, the last derived from the existence of essential components of the set of Ky Fan's points, and further, investigate relations between the notion of essential stable sets and the notion of strategic stable sets introduced by Al-Najjar in 1995. Secondly, we study the stability of mixed Nash equilibrium points robust against the perturbations of best reply correspondences, and also obtain the existence of essential components of the set of mixed Nash equilibrium points, which is deduced from the existence of essential components of the set of fixed points on the mixed strategy profile space by defining a new Le vy-Prohorov distance functions on the spaces of mixed strategies. Finally, we argue that the two kinds of notions of essential components for mixed Nash equilibrium points are two sides to reflect the stability of mixed Nash equilibrium points and can not take the place of each other.Chapter 3 We study stability of solutions of N-person games with endogenous sharing rules and obtain the property of generic stability of solutions. As an application, we derive the generic stability of mixed Nash equilibrium points of N-person games with compact metric spaces of infinite pure strategies and continuous payoff functions.Chapter 4 We study the stability of the set of pure Nash equilibrium points of general N-person non-cooperative games with payoff functions having some semi-continuity and some concavity. By applying both the existence of essential components of the set of solutions of Ky Fan Section Theorem and the existence of essential components of the set of solutions of Ky Fan Lemma, we establish the existence of essential components of the set of pure Nash...