| The research into cooperative game theory has been developed very quickly inrecent decades, and a lot of results had been obtained. The main objects of this theoryare to evaluate the strategic position of each player as well as the effects of coalitionalstructures. The cooperative game can be applied in many fields, such as the profit andcost sharing in economics, as well as the power distribution of voting in politics, andso on. In this case, the Shapley value is a well-known concept, and its axiomaticcharacterizations of this value introduce a new style for game theories, and a lot ofimportant research findings have been obtained.The main contribution of this thesis is to study the solution of the monotonicgame with coalitional structures, which is an extension of Shapley value (coalitionalvalue). First, bases on relative reference reviews, a new solution called proportionalcoalitional Shapley value is put forward in this thesis. The basic method is: the unionsformed by the players in game v on the set N obtain their payoffs in terms of thequotient game, and then the members of each union equally share the payoffsproportionally according to the Shapley values in the original game v. The mean of“equally” is that the proportional coalitional Shapley value keeps the relativesignificances of players in the original game the same. Second, this thesis provides twodifferent axiomatic characterizations of the proportional coalitional Shapley value. Theproportional coalitional Shapley value is the unique coalitional value in monotonicgame that has the efficiency, dummy player, symmetry within unions, symmetry in thequotient game and weighted additivity, and also the unique coalitional value inmonotonic game that has the nonnegativity, coalitional Shapley value property,quotient game property and proportionality within unions. Final, the proportionalcoalitional Shapley value is applied to two examples, bankruptcy and voting problems,showing that the value is reasonable. |
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