Chaotic triopoly game: a congestion case

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 Nash equilibrium as in the static game. The local stability conditions of the Nash equilibrium are obtained in terms of the reactivity parameters among the firms and the nonlinear costs functions adopted by those firms. The obtained results are supported by a numerical simulation that in turn gives routes where Nash equilibrium may lose its stability. The simulation shows that Nash equilibrium loses its stability via flip and fold bifurcations and then chaos exists.