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ch04_solutions_solved edit

# ch04_solutions_solved edit - Solutions to Chapter 4...

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Solutions to Chapter 4 Exercises SOLVED EXERCISES S1. False. A dominant strategy yields you the highest payoff available to you against each of your opponent’s strategies. Playing a dominant strategy does not guarantee that you end up with the highest of all possible payoffs. In the prisoners’ dilemma game, both players have dominant strategies, but neither gets the highest possible payoff in the equilibrium of the game. S2. (a) For Row, Up strictly dominates Down , so Down may be eliminated. For Column, Right strictly dominates Left , so Left may be eliminated. These actions leave the pure-strategy Nash equilibrium ( Up , Right ). (b) Row has no dominant strategy, but Right dominates Left for Column (who prefers small numbers, this being a zero-sum game). After eliminating Left for Column, Up dominates Down for Row, so Down is eliminated, leaving the pure-strategy Nash equilibrium ( Up , Right ). (c) There are no dominated strategies for Row. For Column, Left dominates Middle and Right . Thus these two strategies may be eliminated, leaving only Left . With only Left remaining, for Row, Straight dominates both Up and Down , so they are eliminated, making the pure-strategy Nash equilibrium ( Straight , Left ). (d) The game is solved using iterated dominance. Column has no dominated strategies. For Row, Up dominates Down , so Down may be eliminated. Then East dominates North , so it is eliminated. Then Up dominates Low , so it is eliminated. Then East dominates West , so it is eliminated. Then Up dominates High , so it is eliminated, leaving only Up . With only Up remaining, East dominates South , giving the pure-strategy Nash equilibrium ( Up , East ). S3. (a) By Minimax, the minima for Row’s strategies are 3 for Up and 1 for Down . Row wants to receive the maximum of the minima, so Row chooses Up . The minima for Column’s strategies are –2 for Left and –1 for Right . Column wants to receive the maximum of the minima, so Column chooses Right . Again, the pure-strategy Nash equilibrium is ( Up , Right ). Solutions to Chapter 4 Solved Exercises 1 of 7

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(b) Minimax shows that Row has minima of 1 and 2, with 2 for Up being the larger. Column has maxima of 2 and 4, with 2 for Right being the smaller. Then the pure-strategy Nash equilibrium is ( Up , Right ). (c) Minimax shows that the minima for Row’s three strategies are 1, 2, and 1, so Row chooses 2, which is from Straight . Column wants to minimize the maximum, and the maxima for Column’s strategies are 2, 4, and 5, so Column chooses 2, which is from Left . This yields the pure- strategy Nash equilibrium of ( Straight , Left ).
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ch04_solutions_solved edit - Solutions to Chapter 4...

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