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# ch08_solutions_solved edit - Solutions to Chapter 8...

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Solutions to Chapter 8 Exercises SOLVED EXERCISES S1. False. A player’s equilibrium mixture is devised in order to keep her opponent indifferent among all of her (the opponent’s) possible mixed strategies; thus, a player’s equilibrium mixture yields the opponent the same expected payoff against each of the player’s pure strategies. Note that the statement will be true for zero-sum games, because when your opponent is indifferent in such a game, it must also be true that you are indifferent as well. S2. Sally’s expected payoff from choosing Starbucks when Harry is using his p-mix is p; her expected payoff from choosing Local Latte when Harry is mixing is 2 – 2p. Similarly, Harry’s  expected payoff from choosing Starbucks when Sally is using his q-mix is 2q; his expected  payoff from choosing Local Latte when Sally is mixing is 1 – q. These expected payoffs are  graphed below. Sally’s best response to Harry’s p-mix is to choose Local Latte for values of p below 2/3  and to choose Starbucks for values of p above 2/3. Sally is indifferent between her two choices  when p = 2/3. Similarly, Harry’s best response to Sally’s q-mix is to choose Local Latte for  values of q below 1/3 and to choose Starbucks for values of q above 1/3. He is indifferent  between his two choices when q = 1/3. The mixed-strategy equilibrium occurs when Harry  chooses Starbucks two-thirds of the time and Local Latte one-third of the time (p = 2/3) and  when Sally chooses Starbucks one-third of the time and Local Latte two-thirds of the time (q =  1/3). Best-response curves are shown below. Solutions to Chapter 8 Solved Exercises 1 of 17

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Expected payoffs for Sally and Harry are 2/3 each. Both players would prefer either of the pure- strategy Nash equilibria. If they can coordinate their randomization in some way so as to alternate between the two pure-strategy Nash equilibria, they can achieve an expected payoff of 1.5 rather than the 2/3 that they achieve in the mixed-strategy equilibrium. S3. (a) Nash equilibria occur where the best-response curves intersect. In this game, the pure-strategy Nash equilibria occur at points, (0, 0) and (1, 1). The mixed-strategy Nash equilibrium occurs at point (1/3, 1/3). (b) Solutions to Chapter 8 Solved Exercises 2 of 17
In this game, the pure-strategy Nash equilibria occur at points (0, 0) and (1, 1). There are no mixed-strategy Nash equilibria. S4. (a) Solutions to Chapter 8 Solved Exercises 3 of 17

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(b) Navratilova wants to minimize Evert’s success rate, which—as seen from the upper envelope of the graph—will be 70% in equilibrium. The graph also illustrates that Navratilova can use many mixtures of q in equilibrium: any q -mix between 1/3 and 5/7 (inclusive) will work in a mixed- strategy equilibrium (while Evert plays Lob). There are then an infinite number of mixed-strategy
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ch08_solutions_solved edit - Solutions to Chapter 8...

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