In this study, public and private companies are compared in a mixed market. They investigate how these companies choose to compete in terms of prices and quantities while investing in cost-reducing research and development (R&D). Surprisingly, the study reveals that price competition, not quantity competition, tends to be the norm between these firms. The researchers argue that public companies outpace private companies in innovation when both invest in R&D. They also confirm that a public company's profits in Cournot competition exceed those in Bertrand competition in regular settings. Similarly, in mixed markets, a private company's profits in Bertrand competition surpass those in Cournot competition.