ECON 2070 SUMMER SEMESTER, 2020 Practice Questions for Online Test 2 Since by now you already listened to Lecture 7, you know that there are Nash equilibria in pure strategies and Nash equilibria in mixed strategies. As Online Test 2 is based on Lectures 4-6 and Tutorials 4-6, Online Test 2 does not cover Nash equi- libria in mixed strategies. Therefore, in Online Test 2 and in the following practice questions, “Nash equilibrium” always means “Nash equilibrium in pure strategies.” Question 1 Consider the following game. Mary Steve A B C D a 3, 2 1, 0 2, 1 1, 3 b 1,−3 3, 1 3, 0 2,−1 c 3, 1 2, 1 4,−1 −1, 1 d 0, 0 −2,−1 3, 1 −1, 1 Select all of the following that are Nash equilibria in this game. • (3, 1) • (b,B) • (c,A) • (d,C) • (a,D) • (3, 2) 1 ECON 2070 SUMMER SEMESTER, 2020 Question 2 Consider the following 2-player game. The strategy set of each player is {1, 2, 3}. We use x to denote the strategy of player 1 and y to denote the strategy of player 2. The payoff to player 1 is u(x, y) = x− y, and the payoff of player 2 is v(x, y) = |x− y| (the absolute value of the difference between x and y). (A) Write down all Nash equilibria of this game. (B) Write down a strategy profile that is not a Nash equilibrium. Explain why it is not a Nash equilibrium. Question 3 Larry has the utility function u(x, y) over his strategy x and Rachel’s strategy y. Rachel’s utility function is v(x, y): u(x, y) = 2xy2 − y3 − x2 v(x, y) = − 1 y − yx Rachel and Larry can choose any real number strictly greater than zero (not zero itself). (A) Choose all of the following statements which are true. • Larry’s utility function is strictly concave in his strategy. • Larry’s utility function is not strictly concave in his strategy. • Rachel’s utility function is strictly concave in her strategy. • Rachel’s utility function is not strictly concave in her strategy. (B) Write down a formula for Larry’s best response function. (C) Write down a formula for Rachel’s best response function. (D) Write down every Nash equilibrium of the game. Question 4 Consider the function f(x) = x3 − x. Determine all fixed points of f . 2 ECON 2070 SUMMER SEMESTER, 2020 Question 5 Consider two firms that compete in prices. Demands are differentiated and given by Q1 = 20− 5p1 − 4p2 and Q2 = 10 − 4p2 − p1. By law, each firm must set a price of at least $0.10 and a price of at most $2. For each firm, the costs of production are zero. (A) Choose all of the following statements which are true. • Firm 1’s profit function is strictly concave in p1. • Firm 1’s profit function is not strictly concave in p1. • Firm 2’s profit function is strictly concave in p2. • Firm 2’s profit function is not strictly concave in p2. (B) Which of the following statements is true? • Firm 1’s best response function is given by p1 = 2− 0.4p2. • Firm 1’s best response function is given by p1 = 2 + 0.4p2. • Firm 1’s best response function is given by p1 = 4− 0.8p2. • Firm 1’s best response function is given by p1 = 4 + 0.8p2. (C) Which of the following statements is true? • This is a game of cooperation and of strategic complements. • This is a game of cooperation and of strategic substitutes. • This is a game of conflict and of strategic complements. • This is a game of conflict and of strategic substitutes. (D) Determine the Nash equilibrium. 3 ECON 2070 SUMMER SEMESTER, 2020 Question 6 In a 3-player game, S1 = S2 = S3 = [0, 10]. A player obtains a payoff of 1 if 1) her strategy equals the average of the strategies of the other two players and 2) this average is in the interval [4, 6]. In all other cases, the player obtains a payoff of 0. Choose all of the following strategy profiles which are not Nash equilibria. • (4,4,4) • (10,5,0) • (5,1,9) • (2,5,8) • (7,3,5) Question 7 Ann’s payoff is given by u(x, y) = x3y − xy, where x denotes Ann’s strategy, and y denotes Bob’s strategy. Bob’s payoff is given by v(x, y) = 2xy − xy2 (A) Suppose that SAnn = SBob = [0, 1]. Determine all Nash equilibria. (B) Suppose that SAnn = [1, 2] and SBob = [0, 1]. Determine all Nash equilibria. (C) Suppose that SAnn = [−1, 0] and SBob = [0, 1]. Determine all Nash equilibria. 4
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