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We consider the problem of traffic routing in a two-node parallel-link communication network shared by several competing teams of users. Each team has various types of entities (jobs or packets) to be routed from one node to the other on the network. The users in each team cooperate for the benefit of their own team so as to achieve optimal routing over the network links. The teams, on the other hand, compete among themselves for the network resources and each has an objective function that relates to the overall performance of the network. For each team, there is a centralized decision-maker, called the team manager (or leader), who coordinates the routing strategies among all entities in his/her team. A game theoretic approach to deal with both cooperation within each team and competition among the teams, called the noninferior Nash strategy, is introduced. Considering the role of the team managers in this context, the concept of a noninferior Nash strategy with a team leader is introduced. This multi-team solution provides a new framework for hierarchically optimizing the traffic routing over the network while simultaneously addressing complicated coordination problems among the various users. It is shown that the noninferior Nash strategies with team leaders are effective in improving the overall network performance. This solution approach can be extended to more complex networks with more than two nodes.