This paper reconsiders the literature on the irrelevance of pri- vatization on mixed markets, addressing competition in a duopoly with both quantity and price competition. We assume partial privatization under differ- ent time structures and derive the conditions on firm ownership under which an optimal subsidy allows to achieve maximum efficiency. We find that, indepen- dently of the mode of competition, the firms’ ownership structure is irrelevant when firms play simultaneously and conversely matters when they compete se- quentially, requiring the state-controlled firm to be publicly-owned in order to let the optimal subsidy restore first best. The paper also focuses on the extent to which a subsidy is necessary to attain the social optimum, highlighting the equivalence between a price (quantity) game with public leadership or simulta- neous moves and a quantity (price) game with private leadership.