Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: ON EXISTENCE OF EQUILIBRIUM PAIR FOR CONSTRAINED GENERALIZED GAMES | ON EXISTENCE OF EQUILIBRIUM PAIR FOR CONSTRAINED GENERALIZED GAMES P S. SRINIVASAN AND P VEERAMANI Received 28 August 2003 and in revised form 11 November 2003 We obtain sufficient conditions for the existence of an equilibrium pair for a particular constrained generalized game as an application of a best proximity pair theorem. 1. Introduction Consider the following game involving n players. For the zth player a pair Xị Yi of strategy sets is associated. Knowing the choice of strategies xi e Xi nn 1 j iXj of all other players the ith-player choice is restricted to Ai xi Q Yi. Otherwise the choice will be made from Xi. According to these preferences let fi Yi X Xi R be the payoff function associated with the ith player for each i 1 . n. In this situation it is natural to expect an optimal approximate solution which will fulfill the requirement to some extent. Therefore it should be contemplated to find a pair x y where x e X n n 1 Xi and y e Y n n 1 Yi which will behave like an equilibrium point of a generalized game that is yi e Ai xi and maxzeAi xi fi z xi fi yi xi for each i 1 . n and satisfy the optimization constraint namely the distance between x and y is minimum with respect to X and Y. In this case the pair x y is called an equilibrium pair and the game is termed as constrained generalized game. Indeed in this paper sufficient conditions for the existence of an equilibrium pair for this constrained generalized game are obtained as an application of a best proximity pair theorem. The entire edifice of game theory expounds with a mathematical search to strike an optimal balance between persons generally having conflicting interests. Each player has to select one from his fixed range of strategies so as to bring the best outcome according to his own preferences. Following the pioneering work of Debreu 1 the generalized game is one in which the choice of each player is restricted to a subset of strategies determined by the choice of other players. .
