For each of the four games described below answer part (a) and part (b). (a) How...

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For each of the four games described below answer part (a) and part (b). (a) How many Nash equilibria (NE) are there in pure strategies? (b) If your answer to (a) is 1, just state the NE. Else, if your answer to part (a) is different from 1, explain briefly. For example, if the number of NE in pure strategies is zero, explain why that is the case. In case of multiple NE, state those NE, and explain briefly why there are multiple NE. Game 1: Two travelers, returning home from a tropical island where they both purchased an identical antique, discover that the airline has smashed these. The airline manager asks each traveler to write down the compensation desired, any whole number between $10 and $100. If both write the same number, they each receive that amount. But if, ni # n2, then the manager gives each the smaller of the two with an adjustment. Suppose ni > n2. Believing that traveler 1 is lying and traveler 2 is being truthful, the manager punishes traveler 1 by giving n2 – 2 as compensation, while rewarding 2 for her supposed honesty by paying her n2 +2. Similarly, if n2 > n1, the manager punishes traveler 2 by giving n – 2 as compensation, while rewarding Traveler 1 for her supposed honesty by paying her nį +2. Game 2: Same as Game 1 except that there is no reward for supposed honesty. If, say, n > n2, then traveler 1 receives n2 – 2 as in Game 1, but traveler 2 only receives n2. Thus Game 2 is Game 1 without the reward (for the traveler who writes the lower number). For example, if Traveler 1 writes 83 and Traveler 2 writes 81, Traveler 1 gets 81 - 2 = 79, Traveler 2 gets 81. Game 3: Same as Game 1 except when n, and n2 differ by 1. In that case, each player gets what they write as the desired level of compensation. For example, if Traveler 1 writes 83 and Traveler 2 writes 82 or 84, then Traveler 1 gets $83 and Traveler 2 gets $82 if she writes 82 and $84 if she writes 84. Game 4 (Tricky): Now consider a game which has elements of both Game 1 and Game 3. If nį – nz is 0 or 1, then, as in Game 3, each traveler i gets nį. The punishment and reward structure is qualitatively similar to Game 1 but differ in magnitude. If n; 2 n; +2, Traveler i gets n; – 3 while Traveler j gets n, + 3 where i, j e {1,2} but i + j. As in Game 3, if Traveler 1 writes 83 and Traveler 2 writes 82, then Traveler 1 gets $83 and Traveler 2 gets $82. However, if Traveler 1 writes 83 and Traveler 2 writes 81, then Traveler 1 gets 81 - 3 = 78 and Traveler 2 gets 81 + 3 = 84. For each of the four games described below answer part (a) and part (b). (a) How many Nash equilibria (NE) are there in pure strategies? (b) If your answer to (a) is 1, just state the NE. Else, if your answer to part (a) is different from 1, explain briefly. For example, if the number of NE in pure strategies is zero, explain why that is the case. In case of multiple NE, state those NE, and explain briefly why there are multiple NE. Game 1: Two travelers, returning home from a tropical island where they both purchased an identical antique, discover that the airline has smashed these. The airline manager asks each traveler to write down the compensation desired, any whole number between $10 and $100. If both write the same number, they each receive that amount. But if, ni # n2, then the manager gives each the smaller of the two with an adjustment. Suppose ni > n2. Believing that traveler 1 is lying and traveler 2 is being truthful, the manager punishes traveler 1 by giving n2 – 2 as compensation, while rewarding 2 for her supposed honesty by paying her n2 +2. Similarly, if n2 > n1, the manager punishes traveler 2 by giving n – 2 as compensation, while rewarding Traveler 1 for her supposed honesty by paying her nį +2. Game 2: Same as Game 1 except that there is no reward for supposed honesty. If, say, n > n2, then traveler 1 receives n2 – 2 as in Game 1, but traveler 2 only receives n2. Thus Game 2 is Game 1 without the reward (for the traveler who writes the lower number). For example, if Traveler 1 writes 83 and Traveler 2 writes 81, Traveler 1 gets 81 - 2 = 79, Traveler 2 gets 81. Game 3: Same as Game 1 except when n, and n2 differ by 1. In that case, each player gets what they write as the desired level of compensation. For example, if Traveler 1 writes 83 and Traveler 2 writes 82 or 84, then Traveler 1 gets $83 and Traveler 2 gets $82 if she writes 82 and $84 if she writes 84. Game 4 (Tricky): Now consider a game which has elements of both Game 1 and Game 3. If nį – nz is 0 or 1, then, as in Game 3, each traveler i gets nį. The punishment and reward structure is qualitatively similar to Game 1 but differ in magnitude. If n; 2 n; +2, Traveler i gets n; – 3 while Traveler j gets n, + 3 where i, j e {1,2} but i + j. As in Game 3, if Traveler 1 writes 83 and Traveler 2 writes 82, then Traveler 1 gets $83 and Traveler 2 gets $82. However, if Traveler 1 writes 83 and Traveler 2 writes 81, then Traveler 1 gets 81 - 3 = 78 and Traveler 2 gets 81 + 3 = 84.

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