Recently the present author read a few publications which tried to combine Cournot and Bertrand duopoly. The approach, using linear demand functions with appropriate signs for cross derivatives, seemed basically mistaken. First, linear functions never work in global dynamics. Second, linear demand functions are not consistent with utility maximization because they do not fulfil budget constraints. They can locally approximate proper demand functions, but, again, do not work globally. Third, the whole idea is contradictory, because a commodity is either homogenous, or it is not, “tertium non datur”. It depends on how the consumers regard the case, and is nothing for the duopolists to decide. This made the present author think how one might bring some logic in this. Actually, price discrimination is standard in economic theory, and often applied in practice, so one could consider a case with two different groups of consumers, one that regards the oligopolists’ produce as equivalent, and another which regards their supply as different but close substitutes. Then both suppliers sell to both groups and apply price discrimination. In this way the competitors can be Cournot duopolists in one market and Bertrand duopolists in another.