Weighted Solutions For Cooperative Games And Their Axiomatizations

Cooperative game theory mainly studies how to distribute the payoff of cooperation fairly and reasonably among all players.A reasonable allocation rule can promote the cooperation among players and maintain the stability of the coalitions.Most of existing solutions are based on the principle of fairness and fail to fully consider the external difference of participants.Targeting at this kind of problem,the thesis is devoted to researching different weighted solutions and their axiomatizations.Through introducing the weight system and players’ weight-related complaint,this thesis constructs a series of optimization models based on the least square criterion and proposes different weighted solutions.On the one hand,we analyze the allocation process through designing procedural implementation mechanism for different weighted solutions.On the other hand,we axiomatically characterize the weighted solutions based on corresponding axioms.The main research results are listed as follows:1.Based on players’ pessimistic complaint,this thesis builds the optimization model and defines the model’s optimal solution as the weighted surplus division value.By analyzing the formation of the grand coalition,this thesis also characterizes the weighted surplus division value from the aspect of procedural implementation.Furthermore,we provide a series of axiomatizations for the weighted surplus division value applying weight-related axioms.Considering the situation that the grand coalition cannot be formed,we extend the weighted surplus division value to a class of weighted surplus division values and axiomatically portray this class of values based on the reduced game property.In addition,we extend the solution concept to the α-weighted surplus division value and axiomatize it with corresponding properties.2.Based on players’ optimistic complaint,we construct the optimization model based on the least square criterion and therefore define the model’s optimal solution as the weighted nonseparable cost value.Then we design the procedural implementation mechanism and algorithm for this value.Through analyzing the dual relationship between the weighted nonseparable cost value and the weighted surplus division value,we characterize the uniqueness of the weighted nonseparable cost value.Nevertheless,we deduce the sufficient conditions of the weighted surplus division value and the weighted nonseparable cost value coinciding with the nucleolus respectively in balanced game through analyzing the anti-dual relationship between solutions.3.We define the harmonic weighted value and a class of harmonic weighted values in cooperative games,then we systematically study the correspondence relationship between the harmonic weighted values and the classical solutions.In addition,we research the properties of the harmonic weighted values and axiomatically characterize a subclass of the harmonic weighted values.4.Based on the normalized weight system,we present a new recursive definition of the weighted Solidarity value and its recursive implementation algorithm.Then we characterize this value from the perspective of algebra.Through compromising the Solidarity value and the Shapley value together in graph-restricted games,we acquire the Efficiency-Solidarity value and axiomatize this value in graph-restricted games.Ultimately,we illustrate the rationality and application of the Efficiency-Solidarity value with a practical example.