A formulation is proposed for unifying network design cooperative games which shows that not only the minimum cost spanning tree game satisfies the assumptions of Owen's linear production game. Moreover, since the suggested formulation is polynomial in size, a vector in the core can be computed efficiently by solving a single linear program. Generally, given a formulation for the linear production game, the set of core vectors generated by Owen's scheme from the dual solutions can be a proper subset of the core of the game. In the end, an efficient (polynomial) characterization of the cores of network synthesis games and a strongly polynomial procedure for verifying whether a given vector is in the core of these games is made.