Simon Fraser University Fall 2014 Econ 302 D100 Midterm Exam Instructor: Songzi Du Tuesday October 28, 2014, 8:30 – 10:20 AM Write your name, SFU ID number, and tutorial section number both on the exam booklets and on the questionnaire. Hand in both the exam booklets and this questionnaire. But note that only the exam booklets are graded. • Name: • SFU ID number: • Tutorial section number: General instructions 1. This is a closed-book exam: no books, notes, computer, cellphone, internet, or other aids. A scientific, non-graphing calculator is allowed. 2. If you use decimals in calculation, keep two decimal places. 3. Write clearly. Illegible answers will receive no credit. 4. Show your work! Partial credits are given. Answers without proper explanation/calculation will be penalized. 5. A request for regrade can be accepted only if the exam is written with a pen. 1 1. (5 points) Alice and Bob are participants in a televised game show, seated in separate booths with no possibility of communicating with each other. Each one of them is asked to submit, in a sealed envelope, one of the following two requests for the show (requests are guaranteed to be honored): 1. Give me $1000. 2. Give the other participant $4000. Write the payoff matrix for this game. Find the Nash equilibria. Is there any dominated strategy, and why? Why is the Nash equilibrium outcome(s) unfortunate in this game? 2. (10 points) Find and describe the pure-strategy subgame perfect equilibria (if any) in the following game. Explain the steps that you use to find the SPE. The game has random moves by Nature at the beginning of the tree. You can think of the two nodes of Player 1 following the moves of Nature as two distinct players, like the strong and weak types in the Beer-Quiche game. 3. (10 points) The market (inverse) demand function for a homogeneous good is P (Q) = 14 − 2Q, where Q is the total quantity of the good on the market. There are two firms: firm 1 has a constant marginal cost of 4 for producing each unit of the good, and firm 2 2 has a constant marginal cost of 1. Firm 1 is the industry leader, so it sets its quantity of production first. Firm 2 sets its quantity of production after observing firm 1’s quantity. Find (i) the subgame perfect equilibrium of this game. And calculate (ii) the market price and (iii) the firms’ profits in the equilibrium. 4. (15 points) Elizabeth and Mary appear before King Solomon at his palace, along with an infant. Each woman claims that the infant is her child. The child is “worth” 100 dinars to his true mother, but he is only “worth” 50 dinars to the woman who is not his mother (the imposter). The king knows that one of the two women is the true mother of the child, and he knows the “values” that the true mother and the imposter ascribe to the child, but he does not know which woman is the true mother, and which the imposter. To determine which of the two women is the true mother, the king explains to Elizabeth and Mary that he will implement the following steps: 1. He will ask Elizabeth whether the child is hers. If she answers negatively, the child will be given to Mary. If she answers affirmatively, the king will continue to the next step. 2. He will ask Mary if the child is hers. If she answers negatively, the child will be given to Elizabeth. If she answers affirmatively, Mary will pay the king 75 dinars, and receive the child, and Elizabeth will pay the king 10 dinars. (i) Draw the game tree when (a) Elizabeth is the true mother and when (b) Mary is the true mother. (Assume that each woman has a payoff of 0 dinar if she does not receive the child.) (ii) Find and report the subgame perfect equilibrium under (a) and (b), and describe what happens (who gets the child and how much is paid by each woman) in each equilibrium. 5. Part (i) [5 points]: find the Nash equilibria (pure and mixed) of the following simultaneous game, and report the players’ payoffs in each equilibrium: C D A 10, 10 1, 10 B 12, 2 0, 0 Part (ii) [10 points]: now suppose the above game is played twice. In each stage the two players move simultaneously, and in the second stage the players observe the actions taken 3 in the previous stage. Assume that the final payoff of a player is the sum of his payoffs from the two stages. Find and describe a subgame perfect equilibrium (SPE) in which (A, C) is played in the first stage, and report the players’ payoffs in the equilibrium. Explain why it is a SPE. 4