Price competition is more intense than quantity competition in private oligopolies, wherein all firms are profit maximizers. However, in mixed oligopolies where one state‐owned public firm competes with profit‐maximizing private firms, price competition may not result in tougher competition than quantity competition. In this study, we introduce common ownership, a distinct feature of recent financial markets, into a mixed oligopoly model and investigate how common ownership affects this ranking. We show that under common ownership, quantity competition is likely to be tougher than price competition. Moreover, we find that common ownership harms welfare regardless of the competition mode. Common ownership enhances private firms' profits under Bertrand competition while these may decline under Cournot competition.