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mid1_2008w_z - Answer Key First Midterm Examination Econ...

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Answer Key First Midterm Examination: Econ 101 Richard Buddin Winter 2008 Please answer all questions. The questions are in no particular order. If you do not understand a question, you should skip it and return to it later in the exam. 1. Consider the following payoff matrix between two players: Player 2 Left (L) Middle (M) Right (R) Top (T) 7,3 6,1 3,4 Player 1 Center (C) 6,5 8,2 2,0 Bottom (B) 5,0 4,7 9,4 a) Find the Nash equilibrium value or values for the game if the players act simultaneously. b) Suppose that the game in part a is played sequentially, and player 2 goes first. What strategy should player 2 choose to maximize its payoff? Is player 2 better off playing first or second? c) Now consider the new game in the payoff matrix below. Find the mixed strategies solution to the game. Player 2 Left (L) Right (R) Player 1 Top (T) 6,5 2,9 Bottom (B) 1,8 4,3 d) What is the probability of each outcome occurring in the game in part c (find the probability of TL, TR, BL, and BR)? Find the expected payoff to the mixed strategy solution. Answer: a) There are no Nash equilibrium values in this 3x3 game, if it is played simultaneously. b) If player 2 plays first, then player 2’s best strategy is to play R and earn 4 at BR. a. If player 2 plays L, then player 1 chooses T. The payoff to player 2 is 3. b. If player 2 plays M, then player 1 chooses M. The payoff to player 2 is 2. c. If player 2 plays R, then player 1 chooses B. The payoff to player 1 is 4. If player 2 lets player 1 play first, then player 1 will pick the strategy that maximizes his/her payoff. In this case, player 1 will choose M(see below). Given this choice by player 1, player 2 will choose L and earn a payoff of 5 a. If player 1 plays T, then player 2 chooses R. The payoff to player 1 is 3.
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b. If player 1 plays C, then player 2 chooses L. The payoff to player 1 is 6. c. If player 1 plays B, then player 2 chooses M. The payoff to player 1 is 4. The best strategy for player 2 is to let player 1 play first, so he/she earns 5 at ML (instead of 4 at BR if player 2 plays first). c) P T =5/9 0.555556, P L =2/7 0.285714. d) Pr(TL)=10/63, Pr(TR)=25/63, Pr(BR)=8/63, Pr(BL)=20/63. Payouts: player 1 3.142857, player 2 6.33333. 2. Arnold’s Gym is a new health club in Santa Paula. The club has monopoly power in the local area and deals with two types of consumers. The individual demand for a local woman is q w =264-32P, where q w
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