基于半张量积的双合作博弈Shapley值计算  

The Shapley value for bicooperative games based on the semitensor product

作　　者：李志强 李文鸽 何秋锦 宋金利 杨俊起[3] Zhiqiang LI;Wenge LI;Qiujin HE;Jinli SONG;Junqi YANG(School of Mathematics and Information Science,Henan University of Economics and Law,Zhengzhou 450046,China;School of Componter Engineering,Guangzhou City University of Technology,Guangzhou 510800,China;School of Electrical Engineering and Automation,Henan Polytechnic University,Jiaozuo 454000,China)

机构地区：[1]河南财经政法大学数学与信息科学学院,郑州450046 [2]广州城市理工学院计算机工程学院,广州510800 [3]河南理工大学电气工程与自动化学院,焦作454000

出　　处：《中国科学：信息科学》2022年第7期1302-1316,共15页Scientia Sinica(Informationis)

基　　金：国家自然科学基金(批准号:11872175,62073122);河南省高等学校重点科研项目(批准号:20A120003,21A120001,22A880007);河南财经政法大学国家一般项目培育项目和河南财经政法大学青年拔尖人才资助计划资助。

摘　　要：合作博弈中的参与人只将合作、不合作作为自己的策略,而双合作博弈是合作博弈的一般化,参与者以合作、不合作和弃权作为自己的策略,以获得自己所在的联盟利益的最大化,从而使自己的收益达到最优.与合作博弈一样,如何分配参与者联盟获得的总收益是双合作博弈的一个重要研究问题.本文利用矩阵半张量积工具,研究了双合作博弈的Shapley值计算问题.首先构造了双合作博弈的Shapley矩阵,然后将双合作博弈的Shapley值计算转化为双合作博弈的特征函数矩阵与Shapley矩阵乘积形式.本文得到的Shapley值矩阵计算公式形式简洁,不但简化了计算,而且为双合作博弈的研究提供了新的工具.In cooperative games, players can choose whether or not to participate in the coalition based on their own strategies. Bicooperative games are the generalization of cooperative games. In bicooperative games, players have the option of abstention in addition to “yes” and “no”. Therefore, in cooperative games, distributing the total profit obtained by the participant alliance is one of the most important issues in bicooperative game theory.In this paper, the calculation of the Shapley value for bicooperative games is discussed by using the semi-tensor product of matrices. Firstly, the Shapley matrix of bicooperative games is constructed. Secondly, the Shapley value formula for a bicooperative game is transformed into the product of the characteristic function matrix and the Shapley matrix. Finally, an example is given to demonstrate the main results. The matrix form of Shapley value obtained in this paper simplifies the calculation and provides a new tool for researching bicooperative games.

关 键 词：合作博弈 双合作博弈 SHAPLEY值 Shapley矩阵 矩阵半张量积 

分 类 号：O225[理学—运筹学与控制论]