Consider a duopoly game in which two firms simultaneously and independently select prices. Assume that prices cannot be negative. Let p1 denote the price set by firm 1 and p2 let the price set by firm 2. Unlike Bertrand competition (see Chapter 10), we assume that products are differentiated. To be precise, once prices are set by both firms, consumers demand 10 − p1 + p2 units from the good that firm 1 produces, and they demand 10 − p2 + p1 units from the good that firm 2 produces. Assume that each firm must supply the number of units demanded. Also assume that the cost of producing qi units is equal to 1/2 qi for firm i = 1, 2.
(a) Write the payoff functions for both players (as functions of their strategies p1 and p2).
(b) Characterize each player’s best response function (as a function of the price set by the other player). That is, characterize BR1(p2) and BR2(p1). Are prices strategic substitutes or complements in this game?
(c) Can you determine the set of rationalizable strategies in this game by inspection of players’ best-response functions? What is the set of rationalizable strategies?