Demonstrate that, for the Cournot game discussed in this chapter, the only rationalizable strategy is the Bayesian Nash equilibrium ? Consider a differentiated duopoly market in which firms compete by selecting prices and produce to fill orders. Let p1 be the price chosen by firm 1 and let p2 be the price of firm 2. Let q1 and q2 denote the quantities demanded (and produced) by the two firms. Suppose that the demand for firm 1 is given by q1 = 22 - 2p1 + p2 , and the demand for firm 2 is given by q2 = 22 - 2p2 + p1 .
Firm 1 produces at a constant marginal cost of 10 and no fixed cost. Firm 2 produces at a constant marginal cost of c and no fixed cost. The payoffs are the firms' individual profits. (a) The firms' strategies are their prices. Represent the normal form by writing the firms' payoff functions. (b) Calculate the firms' best-response functions. (c) Suppose that c = 10 so the firms are identical (the game is symmetric). Calculate the Nash equilibrium prices. (d) Now suppose that firm 1 does not know firm 2's marginal cost c.
With probability 1/2 nature picks c = 14, and with probability 1/2 nature picks c = 6. Firm 2 knows its own cost (that is, it observes nature's move), but firm 1 only knows that firm 2's marginal cost is either 6 or 14 (with equal probabilities). Calculate the best-response functions of player 1 and the two types (c = 6 and c = 14) of player 2 and calculate the Bayesian Nash equilibrium quantities.