Maximin and minimax strategies in symmetric oligopoly: Cournot and Bertrand

We examine maximin and minimax strategies for firms under symmetric oligopoly 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 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 for the firms are all equivalent. However, the maximin strategy for the firms 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 one firm is the opposite of the sum of the objective functions of other firms, the maximin and the minimax strategies for the firms constitute the Nash equilibrium both in the Cournot game and the Bertrand game.