Problem 3 (A Signaling Game). Consider the incomplete information Bertrand game analyzed in class: There are two firms, firm 2 can be either type A or type B with probability p and 1 - p. The payoff matrices are: . When firm 2 is type A Firm 1 \\Firm 2A | high medium low high 5.5 0.8 0.6 medium 8.0 1.4 0,6 low 6.0 6.3 3.3 . When firm 2 is type B Firm 1 \\Firm 20 | high medium low high 5.5 6,3 10, 1 medium 3.6 4.4 5.2 low 1.10 2.5 3.3 2 Suppose firm 2 makes the price choice before firm 1 and firm I can observe firm 2's choice, (a) Suppose firm I can somehow figure out firm 2's type. When firm 2 is type A, find firm I's best response for all possible pure strategies by firm 2. Show that type A firm 2 can at most get 3 in this case. (b) Again, suppose firm 1 knows firm 2's type. When firm 2 is type B. show that it is a dominant strategy for firm 1 to choose high. What are the possible payoffs for firm 2 in this case? (Notice firm 2 chooses first.) (c) Show that if there is some signal that makes firm 1 to believe that firm 2 is type B, then type A firm 2 would also want to send that signal. Prove informally that there is no separating PBE in this game. (d) Suppose p = 1/5. Verify that both types of firm 2 choosing high and firm I choosing high is a pooling PBE of this game (suppose when a type deviates from high, firm 1 would know perfectly its type)