The Research On Estimated Value Solutions And Their Extensions Of Cooperative Games

Profit allocation is the core problem in cooperative games, and is also the basis of formation and stability of coalition. Classical cooperative game has three hypothesis:(1) the participation level of players in coalition is complete; (2) the payoffs of coalitions can be accurately obtained by players; (3) the preference information of players are not considered. However, in the practice of cooperation, players often partly take participate in a coalition by considering the cooperation risk and resource constraints. Due to the complexity of social environment and the the limitation of human knowledge, the players can only know the fuzzy information about their payoffs, even some payoffs information is unknown. The preference directly affect the cooperation tendancy and alliance mode, the preference information should be consider in profit allocation. In the above situations, the solutions of classical cooperative games are not suit to allocate the profit. In view of this, this study investigates the cooperative game with fuzzy coalitions and the cooperative game with fuzzy payoffs (such as fuzzy number, linguistic value, qualitative judgement and so on), and considers the situations when the players have individual preference or the payoff information of coalitions is incomplete. In the above non-classical framework of the cooperative games, the corresponding estimated solution and its axiomatic properties are deeply analysed. The main research work is as follows:(1) Research on estimated value of fuzzy cooperative games based on excess valueThe generalized nucleolus of fuzzy cooperative games is proposed based on the generalized excess of fuzzy coalitions. We show that the generalized nucleolus is reasonable and compatible, the generalized nucleolus can be obtained by calculating the f-nucleolus of a constant-sum fuzzy cooperative game. Some linear programming models are employed to generate the generalized nucleolus effectively. Then, a new excess is defined to make distinction between fuzzy coalition and its complement coalition. The least square prenucleolus for fuzzy cooperative games is proposed based on a least square programming model. The uniqueness of the least square prenucleolus for fuzzy cooperative games is proved detailedly. The explicit expression of least square prenucleolus is also present. The least square prenucleolus is extended to the multiplicative setting, and the logarithmic least square prenucleolus for multiplicative fuzzy cooperative game is derived.(2) Research on Shapley value of cooperative games with preference informationA new characteristic function with combined weighted form is developed to fuzzy cooperative games based on the BUM functions. By establishing efficiency, symmetry and additivity axioms, a Shapley function with combined weighted form is presented for fuzzy cooperative games, some desirable properties are introduced. The proposed Shapley function considers not only the weighted average value of the marginal contribution, but also the collective cooperation tendency of players. Moreover, cooperative game is investigated in which the preference information about players are different. A formula of Shapley value for cooperative games with fuzzy coalition worth is proposed based on the preferred mean of fuzzy number. The preferred mean of generalized fuzzy number is defined, and some special properties are presented simultaneously. Based on preferring characteristic function, the Shapley value with preference information is proposed. By considering preferred weight and preferred variance, two continuation methods for cooperative games are derived, respectively.(3) Research on cooperative games with qualitative payoffsThe linguistic Shapley value for cooperative games is proposed under linguistic environment. The 2-tuple linguistic representation model is employed to obtain the valid results and avoid the loss of linguistic information. The linguistic Shapley value is presented based on three axioms. The existence and uniqueness of linguistic Shapley value are discussed in detail. An iterative method for generating consistent imputation is proposed. We give its application in solving risk aversion problems to illustrate the practicality of the proposed method. Then, with respect to cooperative games with judgement worth, the Multiplicative-Shapley is proposed based on the 1-9 judgement scale. The average payoff function is defined, and the adjustment of consistent is introduced simultaneously. Based on the definition of the corresponding characteristic function, three axioms of the M-S value is present.(4) Research on cooperative games with incomplete informationTo explore the cooperation games problems when the payoff information is incomplete, consistency analysis with incomplete information and correction method are proposed. By introducing the concerned definition of incomplete cooperative games, an effective model to examine the superadditivity of such games is given. Based on the criterion for minimizing the deviation of excess value and expected excess, a quadratic programming with upper and lower constraints is established. And then, the L-nucleolus is obtained. Constructing the deviation of imputation and ideal vectors, a single objective programming to derive I-Shapley value is presented. The existence and rationality of L-nucleolus and I-Shapley value are discussed, respectively. Moreover, with respect to cooperative game with fuzzy coalitions, in which the payoffs information are partially known, the definition of incompelte fuzzy games with special structure is introduced. The weighted Shapley value, which satisfies linearity and symmetry, is proposed based on the cardinality set of incomplete coalition value. By considering the marginal contribution between coalitions, the equivalent form of weighted Shapley value is provided. The study showes that the Shapley value for complete fuzzy cooperative games in accordance with the weighted Shapley value.(5) Application of Shapley value in correlative decision makingBased on the Shapley value and fuzzy measure, an interactive interval linguistic aggregation operator is proposed to consider the interactive phenomenon among experts (or attributes). The λ-fuzzy measure is used to determine the Shapley weight of expert set (or attribute set). Some desirable properties of the proposed aggregation operator are present in detail. The MAGCDM algorithm is given to deal with group decision making under interval linguistic environment. The validity of the MAGCDM algorithm is illustrated by solving the facility location selection problems.