We consider a special class of discrete optimal control problems on networks. The dynamics of the system is described by a directed graph of passages. An additional integral-time cost criterion is given and the starting and final states of the system are fixed. The game-theoretical models for such a class of problems are formulated, and some theoretical results connected with the existence of the optimal solution in the sense of Nash are given. A polynomial-time algorithm for determining Nash equilibria is proposed. The results are applied to decision making systems and determining the optimal strategies in positional games on networks.