The Study On Solutions Of Fuzzy Graph Games

As an important branch of game theory,a fuzzy graph game provides an efficient method for the profit allocation of the players in an uncertainty environment.The key issue of fuzzy graph games is the rationality and fairness of the profit allocation.Cooperation is not always possible for players in an outsourcing production model because of many reasons such as different production modes,the competition between service providers,etc.An outsourcing production model is a typical graph game problem.The profit allocation problem of outsourcing production in an uncertainty environment is equal to the solution searching in an fuzzy graph game.In view of the observations above,we study four kinds of games,i.e.,crisp graph game,graph game with fuzzy coalitions,graph game with a fuzzy characteristic function and general fuzzy graph game.The relationship between the Shapley value and the solution in fuzzy graph games are researched.The main results are as follows:(1)The study on solutions of crisp graph games.we extend the average tree solution on the class of crisp graph games.The average tree solution is introduced as a function which derives the value from a given pair of a graph game and a coalition.We proved the average tree solution is an imputation of the crisp graph games.And the imputation is the unique solution satisfying component efficiency and component fairness on the class of cycle-free graph games.The change in payoff can be represented by the average treesolution when a link between two players is deleted for a component that does not hold for Shapley value and the Myerson value..(2)The study on solutions of graph games with fuzzy coalitions.The model of graph games with fuzzy coalitions is proposed based on fuzzy sets and the graph games.We define the fuzzy average tree solution of graph games with fuzzy coalitions.It is shown that the fuzzy average tree solution is equal to the fuzzy Shapley value for complete graph games with fuzzy coalitions.We extend the notion of link-convexity,under which the fuzzy core is non-empty and the fuzzy average tree solution lies in this core.Besides,we give the allocation schemes in an uncertainty environment based on the fuzzy average tree solution.(3)The study on solutions of graph games with fuzzy characteristic values.Graph games with fuzzy characteristic values are introduced based on graph games and cooperative games with fuzzy characteristic values.We propose the M-average tree solution and discuss the additivity and relative component efficiency of the solution.When a graph game with interval payoffs is strictly superadditivity the payoff of every player in coalitions is more than that they do not cooperative.Finally we give the relation between the interval average tree solution and the crisp average tree solution.We propose the H-average tree solution by Hukuhara-difference in graph games with fuzzy characteristic values.We show the relationship between the M-average tree solution and the H-average tree solution.Moreover,based on the fuzzy average tree solutions the practical applications in outsourcing production are shown.(4)The study on solutions of general fuzzy graph games.By fuzzy graph games with Choquet form we propose the graph game with fuzzy coalitions and fuzzy characteristic values,called the general fuzzy graph game.The game is the generalization of the graph game with fuzzy coalitions.The general average tree solution is defined in the general fuzzy graph game.When a general fuzzy graph game is cycle-free the general average tree solution is the unique solution that satisfies component efficiency and component fairness.Moreover,the solution is a fuzzy imputation of the general fuzzy graph game.We discuss the relation between the general average tree solution and the general fuzzy Shapley value.Finally,the profit allocation schemes in outsourcing production is proposed based on the general average tree solution.