This paper presents the dynamic Shapley value for cost-sharing game with spanning arborescence. The cooperative behaviour of players is determined, and a two-stage directed network game is considered. At each stage, a cost matrix associated with the directed network is defined by players adopting strategies, and a minimum cost spanning arborescence on the directed network is determined. After the first stage, a particular player will leave the game with a certain probability, which depends on all players' behaviours in the first stage. The characteristic function is defined. Using the Imputation Distribution Procedure(IDP), the dynamic Shapley value in the game is constructed.