Game Theory

3. Consider a Bertrand duopoly setting in which two firms simultaneously choose prices p1, p2 E R+, at which to sell homogeneous goods.1 Firms' xed costs are zero and marginal costs differ across firms. Firm 1 has a linear cost function ci(qi) = 2q1 (a cost of 2 per unit). Firm 2 has a linear cost function C2(92) = 492 (a cost of 4 per unit). Suppose consumers demand ten units of these goods in total, regardless of the price. Consumers will buy the good from whichever firm has a lower price and will buy half their goods from each firm if prices are equal. Answer the following questions. Explain your answers. (a) Find each firm's payoff function as a function of both firms' prices. (b) Show (by way of contradiction) that there is no pure strategy Nash equilibrium where P1 = P2. (c) Is there a pure strategy Nash equilibrium in this game? If yes, find one. If no, explain why