This paper attempts to describe an evolutionary game theoretic model of Cournot competition. Every discrete time period a large population of firms is matched in pairs randomly to play a Cournot duopoly game. The authoress assumes the firms to act according to behavioral rules. A behavioral rule specifies the quantity to be produced in current period as a function of past observation. That behavioral rules are costly to operate. The cost depend on the informational and computational requirements of implementing the rule. Each firm chooses a behavioral rule from a finite set of different rules, which are commonly known. A firm takes into account the past realized profits of the costs associated with the behavioral rules. The model of dynamic Cournot duopoly game is described by non-linear evolutionary dynamical system. The authoress focuses on two specifications of Cournot duopoly game: linear and linear-quadratic and shows that bifurcation routes complicating dynamics occur in a linear-quadratic model due to which fact strange atractors arise.