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The paper considers a Cournot-type duopoly game, in which linear demand and cost functions are used. The two players produce differentiated goods and offer them at discrete times on a common market. In the cost functions of the players, in addition to the production cost, the cost of transporting the products is also included. Each firm does not care only about its profits but also about the percentage of its opponents’ profits, using a generalized relative profit function. In this model, the players follow different strategies. More specifically, the first player is characterized as a bounded rational player while the second player follows an adaptive mechanism. The existence of the Nash Equilibrium is proved, and its stability conditions are found. The complexity that appears for some values of the game’s parameters is shown. A mechanism by which the chaotic behavior of the discrete dynamical system is presented, importing a new parameter m. The algebraic results are verified, and the apparent complexity is shown by plotting bifurcation diagrams and strange attractors, computing the Lyapunov numbers, and checking the system’s sensitivity on its initial conditions. Keywords: Cournot duopoly game, discrete dynamical system, heterogeneous expectations, stability, chaotic behavior, chaos control