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In this paper, we propose a simple network consisting of only two nodes and two paths. The first node, which is called the source, has three competing firms that send their quantities of load via the two paths to the second node, called the destination node. The static game that describes the reaction among the three firms is constructed. The Nash equilibrium point of the static game is discussed. Assuming a gradient firm based rule we investigate the dynamic game which has the same Nashdoi:10.1186/s13662-020-02683-0 fatcat:7hyturrgabfdbnsar4uzhuo7hm