Constrained Scalar Valued Dynamic Games And Symmetric Duality For Multi Objective Variational Problem

Source abstract

Early in the development of game theory, it was observed that matrix gamer were equivalent to a dual pair of linear programs (Charnes, 1953; Cottle,1963). Further, the various applications of game theoretic ideas were quite extensive and have recently being applied on various fields, such as linear optimization, control problems networking problems, and so on. Recently, Kawagnchi and Maruyama (1976) formulated dual linear program corresponding to a linearly matrix game. In another development, Husain and Ahmad (2012) considered a linearly constrained matrix game using saddle point theory and they established equivalence between this game and a pair of mutually dual linear programming problem. Consequently, Corley (1976 ), Chandra et al. (1919 ), Prasad and Sreenivas (1997) have studied different scalar-valued games to a certain vector valued games. Basically, dynamic games concerned with the modeling of large scale systems have independent decision makers with respective individual payoff functions. These dynamic games have different applications related to energy managements, environment, and so on. These dynamic system applications have huge application when compared with static games in general. Very recently, Husain and Ahmad (2012) generalized the results of Mond et al. (1987) for scalar games to a pair of symmetric dual variational problem. These formulations were more general than the formulations of Mond and Hanson (1968) . They stated and proved different dual formulations under generalized convexity concepts. Further, Husain and Jain (2013) extended the results of (Husain and Jain, 2013; Mond and Weir, 1981; Mond and Hanson, 1968) to multi-objective setting by formulating constrained vector-valued dynamic game and established equivalence between a pair of multi-objective variational problems under suitable generalized convexity. In this paper, the authors have studied a pair of scalar-valued games using ratio-invexity type of condition on the objective as well as constraint functions as was earlier introduced by Khan and Hanson in 1997. In general, these conditions are Fritz-John type optimality conditions. In general, in a game if the process is controlled by a set of p-player who are interested, suffer from conflicts, then the objective of another set player(s) cannot be expressed in terms of one index, so in such situation, it leads to multi-objective function. So, the set of specific characterization constitute vector-valued/ scalar-valued game problems, where each player(s) wishes to maximize/minimize his objective function or game with two players, in which one wishes to maximize this objective and the