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University of California, Spring 2012
Irvine Priya Ranjan
Econ 147B: Economics of Strategy Answer key to Midterm Examination
The time for the exam is 50 minutes. There is a total of 45 points.
Please answer all questions and write legibly. Credit will be given for partial answers.
Good Luck. Problem 1: (10 points) Consider the following extensive form game
a) (1 points) How many subgames does this game have? Answer: 3 subgames b) (5 points) Write down all the strategies of both players in this game. Answer: Player 1: {A, E if BD}, {A, F if BD}, {B, E if BD}, {B, F if BD} Player 2: C if B, D if B
c) (4 points) Find the subgame perfect equilibrium of the game clearly specifying the strategies that lead to the subgame perfect equilibrium. Answer: Subgame perfect equilibrium is when player 1 plays A. Strategies leading to SPNE: {A, E if BD} for player 1 and C if B for player 2.
A B
C D
E F
Problem 2: (18 points) In the following normal form game, both players have two strategies: C (Cooperate) and D (Defect). The number on the left in each box represents player I’s payoff while the number on the right represents player II’s payoff.
C D C 4, 4 0, D 6,0 1,
a. (2 points) Does either player have a dominant strategy? If so, what is it? Answer: Yes. D is a dominant strategy for both players. b. (2 point) Find all the Nash equilibria of the game above. Answer: The unique Nash equilibrium of the game is (D,D). c. (5 points) Suppose the above game is repeated infinitely between two players. For what discount factor δ can cooperation (outcome (C,C) every period) be sustained using a trigger strategy?
Answer: δ must satisfy the following inequality:. The solution is.
Answer the remaining questions based on the following information. A Rigid player is defined as someone who always plays D. A Tit for Tat player is one who uses the following strategy. Play C in period 1. Play in period t what the other player played in period t-1, where t= 2,3,4,5,6 and t-1 is 1,2,3,4,5. (For the questions below, ignore the discounting of future payoffs. Simply add the payoffs of each period.) d. (2 points) Suppose the game above is played 6 times between two Rigid players. What is the combined payoff of each Rigid player? Answer: 1 every period for a combined payoff of 6 e. (4 points) Suppose the game above is played 6 times between a Rigid player and a Tit for Tat player. What is the combined payoff of the Rigid player? What is the combined payoff of the Tit for Tat player? Answer: Rigid player: 6 in period 1 and 1 thereafter for a combined payoff of 11. Tit for Tat player: 0 in period 1 and 1 thereafter for a combined payoff of 5. f. (3 points) Suppose the game above is played between two Tit for Tat players 6 times. What is the combined payoff of each Tit for Tat player? Answer: 4 every period for a combined payoff of 24.
- (17 points) Consider a duopoly market with two horizontally differentiated firms, X and Y. Each has a constant marginal cost of $20. Demand functions are: Qx = 100 – Px + 0.5Py Qy = 100 – Py + 0.5Px a) (6 points) Calculate the Bertrand equilibrium in prices in the market. Calculate the profit of each firm in a Bertrand equilibrium.
Answer: The first order condition for firm X is given by 100+20 -2Px +.5Py = Simlarly, for firm-Y the first order condition is
