The article explores how private and public companies compete in a mixed market when the public company is less efficient. Depending on the products being sold, competition can be based on prices or quantities. When products are substitutes, competition can be fierce, but when they are complements, competition is always present. By considering a wider range of cost differences, multiple types of competition can exist simultaneously. In particular, both Cournot and Bertrand competition can coexist under certain conditions.