New characteristic function for two stage games with spanning tree

Two-stage $n$-player games with spanning tree are considered. The cooperative behaviour of players is defined. After the first stage, a specified player leaves the game with a probability that depends on the actions of all players in the first stage. A new approach to the construction of the characteristic function is proposed. In the game, all players are connected with the source directly or indirectly. Assume that the players in coalition $N\setminus S$ have already connected to the source when the players in coalition $S\subset N$ wish to connect to the source. The players in coalition $S$ could connect to the source with the help of the players in coalition $N\setminus S$. A new characteristic function is defined in the game, and the Shapley value is constructed. Several results based on the new characteristic function in the two-stage stochastic game are given.