Choice of strategic variables under relative profit maximization in asymmetric oligopoly: non-equivalence of price strategy and quantity strategy

We consider choice of strategic variables under relative profit maximization in an asymmetric oligopoly with differentiated goods such that there are three firms, Firm 2 and 3 have the same cost function, but Firm 1 has a different cost function. We show that there are two pure strategy sub-game perfect equilibria. 1) All firms choose the outputs as their strategic variables,  2)Firm 2 and 3 choose the outputs as their strategic variables, and Firm 1 chooses the price as its strategic variable.