Maximin and minimax strategies in asymmetric duopoly: Cournot and Bertrand

We examine maximin and minimax strategies for firms in asymmetric duopoly with differentiated goods. We consider two patterns of game; the Cournot game in which strategic variables of the firms are their outputs, and the Bertrand game in which strategic variables of the firms are the prices of their goods. We call two firms Firm A and B, and will show that the maximin strategy and the minimax strategy in the Cournot game, and the maximin strategy and the minimax strategy in the Bertrand game are all equivalent for each firm. However, the maximin strategy (or the minimax strategy) for Firm A and that for Firm B are not necessarily equivalent, and they are not necessarily equivalent to their Nash equilibrium strategies in the Cournot game nor the Bertrand game. But, in a special case, where the objective function of Firm B is the opposite of the objective function of Firm A, the maximin strategy for Firm A and that for Firm B are equivalent, and they constitute the Nash equilibrium both in the Cournot game and the Bertrand game. This special case corresponds to relative profit maximization by the firms.