Game theory exercises

9' Suppose two firms, A, and B, face the demand P=50QAQB Marginal cost for firm A is given by MCA = 5. The marginal cost for firm B can be high, MCBH = 8, or low, MCBL = 2, both with equal probability 0.5. Identify the equilibrium quantities, qA*, qBH*, qBL* under this scenario: Firm A moves first and chooses qA*, Firm B observes qA*, and chooses (1311*, qBL*. Consider the following sequential game: Two firms, 1 and 2, choose their levels of capital investment K as to maximize their profits. At stage 1, firm 1 chooses its optimal level, K1. Firm 2 observes K1 and then chooses its optimal level of investment, K2 ,in stage 2. Suppose the profit functions for each firm are: H1(K1,K2) = K1(1 K1K2)-f H2(K1,K2) = K2(1K1 K2)-f Wherefstands for a fixed cost. Determine the optimal levels of investment, K1 * and K2*. Determine profits at the optimal levels of investment. What level offwould make it unprofitable for firm 2 to enter the market? Explain. Consider the following dynamic game of incomplete information. Firm 1 moves first, then Firm 2 chooses to Accommodate or Fight. Firm 2 Accommodate Fight In with probability Frim 1 p 6,6 0,0 Out with probability 1p 0,1 Illustrate the game in extensive form and find the value of p for which Firm 2 will fight. Knowing that Firm 2 will fight, find the optimal strategy for Firm 1. Now suppose p = 0.5. What would the optimal strategy of Firm 2 be? With p=0.5, illustrate the game in normal form. Identify the Nash equilibria and the subgame perfect equilibria. ape-9