In the paper, we consider a cooperative version of a network game with pairwise interactions in which connected players play bimatrix games. For a particular type of a network, a simplified formula for the Shapley value based on a constructed characteristic function is derived. We then show the time inconsistency of classical cooperative solutions — the Shapley value and the core. The findings are applied to two important classes of bimatrix games: prisoner’s dilemma and a coordination game.