This work assumes a Cournot-type duopoly game where the players produce differentiated goods and decide their productions following different strategies (heterogeneous players). Also, one of the players (the naïve player) has a social profile on the market taking into account not only his profits but also a percentage of the social welfare. The other player (the bounded rational player) manages to have information about the production decision of the other player during the next time period. The authors study the dynamics of the discrete dynamical system of this duopoly game finding the equilibrium positions and formulating the stability condition for the Nash Equilibrium position. They show that the model gives more complex chaotic and unpredictable trajectories and to provide some numerical evidence for this behavior, they present various numerical results including bifurcation diagrams, strange attractors, Lyapunov numbers and sensitive dependence on initial conditions. Finally, an attempt is made to control the chaotic behavior of the system that appears for some values of the main parameters of the system.