This study explores the equilibrium outcomes in the contexts of quantity-setting and price-setting mixed duopolies with consistent conjectures of both the social welfare maximizing public firm and the relative profit maximizing private firm. Similar to the private duopoly composed of only relative profit maximizing firms, we show that in the mixed duopoly, the equilibrium outcomes in the quantity-setting competition with the consistent conjectures of both firms are equivalent to those in the price-setting competition with the consistent conjectures of both firms.