A dynamic Cournot duopoly game, whose time evolution is modeled by the iteration of a map T:(x,y)→(r₁(y),r₂(x)), is considered. Results on the existence of cycles and more complex attractors are given, based on the study of the one-dimensional map F(x)=(r₁∘r₂)(x). The property of multistability, i.e. the existence of many coexisting attractors (that may be cycles or cyclic chaotic sets), is proved to be a characteristic property of such games. The problem of the delimitation of the attractors and of their basins is studied. These general results are applied to the study of a particular duopoly game, proposed in M. Kopel [Chaos, Solitons & Fractals, 7 (12) (1996) 2031–2048] as a model of an economic system, in which the reaction functions r₁ and r₂ are logistic maps.