A value for cooperative games with coalition and probabilistic graph structures

In this paper we generalize TU-games with coalition and graph structures to TU-games with coalition and probabilistic graph structures. We introduce the probabilistic graph-partition value and we show that the value is uniquely determined by the axioms of probabilistic graph efficiency, probabilistic balanced contributions and probabilistic collective balanced contributions and the axioms of probabilistic graph efficiency, probabilistic balanced contributions, probabilistic balanced per capita contributions and either probabilistic fairness for joining the grand coalition or probabilistic population solidarity within unions, respectively. Also, we apply this value to China’s railway network and compare it with other values.