This paper tries to investigate the complex characteristics behind quantity competition using generalized relative profit functions. For this purpose, we study the dynamics of a nonlinear Cournot – type duopoly game with differentiated goods and generalized relative profit maximization. We suppose a linear demand and asymmetric cost functions. The game is modeled with a system of two difference equations. Results concerning the equilibria of the economic model and their stability are presented and the occurrence of bifurcations is stated. We show that the model gives more complex, chaotic and unpredictable trajectories. Numerical experiments are presented.