基于矩阵方法的Banzhaf值的计算及应用

Matrix approach to calculation of Banzhaf value with applications

在合作博弈中,Banzhaf值提供了每个参与者形成大联盟的预期边际贡献,因此Banzhaf值的求解是一个重要的研究内容.本文首先回顾合作博弈及Banzhaf值的定义,并且运用矩阵半张量积,给出合作博弈特征函数的代数表示.然后给出了Banzhaf值的等价的代数形式和简捷的计算方法.最后将所得结果应用于生物网络中,用Banzhaf值度量遗传疾病基因相关性的可能性,确定与遗传疾病发病高度相关的基因.

In cooperative games,Banzhaf value provides the expected marginal contribution of each participant to form a major alliance.Therefore,the calculation of Banzhaf value is an important issue.Firstly,this paper recalls the definitions of cooperative game and Banzhaf value,and establishes the algebraic representation for the characteristic function of cooperative game by using the semi-tensor product of the matrices.Secondly,based on the algebraic representation,the equivalent algebraic form of Banzhaf value is presented,and a simple calculation method is provided for Banzhaf value.Finally,the obtained results are applied to biological networks,and the Banzhaf value is used to determine the genes which are highly associated with genetic diseases.