We consider a model of network formation as a stochastic game with random duration proposed initially in Sun and Parilina (Autom Remote Control 82(6):1065–1082, 2021). In the model, the leader first suggests a joint project to other players, i.e., the network connecting them. Second, the players are allowed to form fresh links with each other updating the initially proposed network. The stage payoff of any player is defined depending on the network structure. There are two types of randomness in the network formation process: (i) links may fail to be formed with different probabilities although players intend to establish them, (ii) the game process may terminate at any stage or transit to the next stage with a certain probability distribution. Finally, a network is formed as a result of players’ decisions and realization of random variables. The cooperative version of the stochastic game is investigated. In particular, we examine the properties of subgame consistency as well as strong subgame consistency of the core. We provide a payment mechanism or regularization of the core elements to sustain its subgame consistency and avoid the player’s deviations from the cooperative trajectory. In addition, the distribution procedure of the core elements is regularized in case there are negative payments to achieve only nonnegative payments to the players at any stage. The sufficient condition of a strongly subgame consistent core is also obtained. We illustrate our theoretical results with a numerical example.