1. Consider a firm (player 1) that produces a unique kind of drug that is used by a consumer (player 2). This drug is regulated by the government so that the price of the drug is p = 6. This price is fixed, but the quality of the drug depends on the manufacturing procedure. The firm (player 1) can choose either to manufacture a "good" (G) or "bad" (B) quality product. The "good" (G) manufacturing procedure costs 4 to the firm, and yields a value of 7 to the consumer (player 2). The "bad" (B) manufacturing procedure costs 0 to the firm, and yields a value of 4 to the consumer. The consumer can choose whether to Purchase (P) or Not Purchase (N) at the price p. If the consumer purchases the product, the firm receives the price p. Suppose the choice of manufacturing procedure and the consumer's decision occur simultaneously. After both players' decisions, the true quality is revealed to the consumer. (a). Draw the game tree and the normal-form matrix of this game, and find all of the Nash equilibria of this game. (b). Now assume that the game described above is repeated twice (T = 2). (For example, in each period a new drug is introduced and the product quality is revealed after each stage-game.) Assume that each player tries to maximize the sum of his stage payoffs (i.e., δ = 1). Find all the subgame-perfect nash equilibria of this game. (c). Now assume that the game is repeated infinitely many times (T = ∞). Assume that each player tries to maximize the discounted sum of his or her stage payoffs, where the discount rate is δ ∈ (0, 1). What is the range of discount factors for which the "good" manufacturing procedure will be used and the consumer will purchase as part of a subgame perfect nash equilibrium in each stage-game? Hint: Construct a Grimm Trigger Conditional Strategy Punishment Mechanism. (d). Use the setting discussed in part (c). Consumer advocates are pushing for a lower price of the drug, say 5. The firm wants to approach the Federal Trade Commission and argue that if the regulated price is decreased to 5 then this may have dire consequences for both consumers and the firm. Can you make a formal argument using the parameters above to support the firm? What about the consumers? 2. Consider the following strategic model of Military Conflict. Two rival countries plan to seize a disputed territory. Each army's general can choose to either Attack (A) or to Not Attack (N). In addition, each army has private information about their strength. Either army can be one of two types: Weak (W) or Strong (S). Assume for each player the type realizations are independent and the probability that they are Weak (or Strong) equals 1 2 . The general learns their army's type before they make their decision and this information is private knowledge to each general. An army can capture the territory if either: (i) it attacks and its rival does not or (ii) it and its rival attack, but it is strong and the rival is weak. If both attack and each army is of equal strength, then neither captures the territory. Assume the value of capturing the territory is m. Further, assume there is a cost of fighting s if the player is strong and w if the player is weak (where 0 < s < w). If an army Attacks and its opponent does not, then no costs are borne by either side. If a player does not capture the territory, they receive zero value (not accounting for the cost of attacking). Identify all Bayesian Nash Equilibrium in Pure Strategies for the following cases. (a). Assume m = 3, w = 2, s = 1. (b). Assume m = 3, w = 4, s = 2. (c). Briefly discuss the intuition for why these two cases differ. 3. (a). Present the Following Bayesian Game in Extensive Form. (b). Characterize the Bayesian Nash Equilibria of this game. (c). Identify if the Bayesian Nash Equilibria identified in part (c) are Perfect Bayesian Nash Equilibria. If there exists Perfect Bayesian Nash Equilibria, what type of strategy does player 1 use in these PBNE? (d). Suppose player 1 used the following mixed strategy: If I am of a High-type choose Advertise with probability σH and Do Not Advertise with probability 1 − σH. If I am of a Low-type choose Advertise with probability σL and Do Not Advertise with probability 1−σL. Discuss player 2's updated system of beliefs in this setting.