This article is devoted to a special class of cooperative multi-stage games, namely, two-stage cooperative network games with pairwise interaction. A feature of these games is that the game is a family of simultaneous bimatrix games between neighbors on the network. The characteristic function and some of its useful properties (convexity) are determined. The vector τ is considered as a cooperative solution. A simplified formula for calculating the components of the vector τ is obtained. For this class of games, it was possible to calculate the value of the coefficient λ, and to show that it does not depend on the number of players and the structure of the network, which significantly reduces the computational complexity of this solution.