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1. Two government departments interact to provide a public good. For instance, crimi- nal law is enforced by the police department along with the prosecution department. Suppose each department is constrained to choose among four possible strategies. A strategy is characterized by the quality of inputs used by the department, the amount of public consultation undertaken, etc. Each department has its own quantitative as- sessment of outcomes based not only on public welfare but also on other factors such as departmental prestige. These payoffs are summarized by the matrix below. In each cell, department A’s payoff is listed first. Find all Nash equilibria and explain your work. Hint: recall the relationships between iterated elimination and Nash.
2. Two firms (i = 1,2) compete for profits in a homogeneous good industry. In the first stage of competition, each firm simultaneously chooses whether or not to incur a fixed cost F = 2 to advertise the product. After each firm makes its choice, it then observes the other firm’s choice. Then in the second and final stage of competition firms simultaneously choose their output levels, q1 and q2. Their cost of production is Ci(qi) = qi. Inverse demand depends on the first stage advertising decisions. If neither firm paid the fixed advertising cost, inverse demand is P = 1 − 2(q1 + q2). If one of the firms paid this cost, inverse demand is P = 6 − 2(q1 + q2). If both firms paid this cost, inverse demand is P = 8 − 2(q1 + q2). Let Ai ∈ {0, 2} denote firm i’s first stage choice regarding the advertising cost. Each firm seeks to maximize undiscounted profits, denoted Πi(A1, A2, q1, q2).
- (a) Derive expressions for the firms’ profits. (20 marks)
- (b) Solve for the firms’ optimal second stage output levels, conditional on their first stage advertising choices. Explain your work. (30 marks)
- (c) Find all pure strategy subgame perfect Nash equilibria of this two-stage game. Explain your work, and briefly interpret the outcome of the game. (50 marks)
- Two firms, i = 1, 2, form a joint venture to develop new technology to enhance both of their businesses. Let ei ≥ 0 be firm i’s effort toward the venture, which will be successful if and only if e1 + e2 ≥ 1. If the venture fails, firm i gets a payoff normalized to −ei. If the venture is successful, firm i’s payoff is vi − ei where the value vi is firm i’s private information. Assume each vi is independently drawn from a two-state distribution: vi = 1/4 with probability p, and vi = 4 with probability 1 − p where 0 < p < 1. The structure of the game is common knowledge. Find all pure strategy Bayesian Nash equilibria and explain your work.
- A signaling game is illustrated below in which nature selects type t1 or t2 with prob- abilities as indicated. The sender observes nature’s selection and sends the signal L or R. The receiver observes the signal, but does not observe nature’s selection, and chooses u or d. At each terminal node, the sender’s payoff is shown first. Provide a complete description of all pure strategy perfect Bayesian equilibria of this game.
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