Irrelevance of conjectural variation in duopoly under relative profit maximization and consistent conjectures

We study the equilibrium with quantity setting behavior and price setting behavior of firms in duopoly under relative profit maximization with constant conjectural variations, and show mainly the following results. 1) Conjectural variations of firms are irrelevant to the equilibrium of a duopoly. 2) Quantity setting behavior and price setting behavior are equivalent with any conjectural variation of each firm. 3) Any pair of conjectural variations of firms which satisfies some relation is consistent. In particular, if firms have the same cost functions or the cost functions are linear, and both firms determine the outputs or both firms determine the prices, any conjectural variations which are common to both firms are consistent. Therefore, there are multiple consistent conjectures