We revisit the classic discussion comparing price and quantity competition, but in a mixed oligopoly in which one state-owned public firm competes against private firms. It has been shown that in a mixed duopoly, price competition yields a larger profit for the private firm. This implies that firms face weaker competition under price competition, which contrasts sharply with the case of a private oligopoly. Here, we adopt a standard differentiated oligopoly with a linear demand. We find that regardless of the number of firms, price competition yields higher welfare. However, the profit ranking depends on the number of private firms. We find that if the number of private firms is greater than or equal to five, it is possible that quantity competition yields a larger profit for each private firm. We also endogenize the price-quantity choice. Here, we find that Bertrand competition can fail to be an equilibrium, unless there is only one private firm.Keywords Cournot · Bertrand · Mixed markets · Differentiated products · Oligopoly J. Haraguchi and T. Matsumura are indebted to two anonymous referees for their valuable and constructive suggestions. J. Haraguchi and T. Matsumura are grateful to Dan Sasaki and participants of the seminars at The University of Tokyo, Nippon University and annual meeting of JEA 2014 spring for their helpful comments and suggestions. T. Matsumura acknowledges the financial support of the Murata Science Foundation and JSPS KAKENHI Grant Number 15k03347. Any remaining errors are our own. B Junichi Haraguchi