Least Square Solutions Of Multiobjective Cooperative Games And Some Properties

The cooperative game is the theoretical basis of forming coalitions and distributing profits in the cooperation process.For real cooperative games,the multiobjective cooperative games have become a researching hot in the field of cooperative games.In order to study the imputations of the multiobjective cooperative games,many experts extended the imputations of the classical cooperative games.Most of imputations are based on the core solution or Shapley value.The core solution meets the rationality,but it is an interval imputation plan,not the exact one.The Shapley value is an exact imputation but it does not meet the rationality.The least square solution overcomes the shortcomings of the core solution and Shapley value.That is,it satisfies the validity and rationality and is also an exact imputation.The least square solution is an imputation plan minimizing the dissatisfaction of all coalitions.So the least square solution embodies both egalitarian principle and utilitarian principle.The least square solution has been widely studied and applied in the single-objective cooperative game.However,it is seldom seen in the multiobjective cooperative games.This thesis aims to construct the least square solution models for the multiobjective cooperative games,including two cases which the objective is independent or dependent respectively,and gives the solution of the models with no constraint,the effectiveness constraint and the rationality constraints,and the corresponding imputation plan.The structure of this thesis is generally arranged as follows.In the 1st chapter,we summary the significance and background of the least square solution in the multiobjective cooperative game.Then,we review the classical least square value and give some relevant concepts and interpretations,and the algorithms of solving the imputation plans under the rationality are introduced in the second chapter.The main parts of this thesis are Chapter 3 and 4.In the chapter 3 we present the least square solution model in single-objective cooperative games in vector form.Based on this,the least square solution models in multiobjective cooperative games are proposed with no constraint and effectiveness constraint,respectively.To simulate the inequality among coalitions or objections in the real games,we introduce the different weights for different problems,and obtain the corresponding unique or general solution in the vector form.In the first part of Chapter 4,we give a new definition of rationality constraints,and convert the least square solution models in multiobjective cooperative games under rationality constraints into the convex optimization problem,which can be solved by the extension of customized proximal point algorithm.The global convergence and linear convergence rate of the proposed algorithm are proved.Secondly,some numerical results show the validity and the practicability of the proposed algorithm,and hereby the correctness of the imputation plans for the above problems.The last chapter summaries the thesis and gives prospect for future research on multiobjective cooperative games.