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Assume that H1 and H2 are two given closed subintervals of ℝ and that f2:H1⟶H2 and f1:H2⟶H1 are continuous maps. Let ϒh1,h2=f1h2,f2h1 be a Cournot map over the space H1×H2. In this paper, we study G1,G2-chaos (resp. strong G1,G2-chaos) of such a Cournot map. We will show that the following are true: (1) ϒ is G1,G2-chaotic (resp. strong G1,G2-chaotic) if and only if ϒ2Λ1 is G1,G2-chaotic (resp. strong G1,G2-chaotic) if and only if Γ2Λ2 is G1,G2-chaotic (resp. strong G1,G2-chaotic). (2) ϒ is G1,G2-chaotic (resp. strong G1,G2-chaotic) if and only if ϒ2Λ1∪Λ2 is G1,G2-chaotic (resp. strong G1,G2-chaotic). (3) f1∘f2 is G1,G2-chaotic (resp. strong G1,G2-chaotic) if and only if so is f2∘f1. MR(2000) Subject Classification: Primary 37D45, 54H20, and 37B40 and Secondary 26A18 and 28D20.