2005_midterm_1_s

# 2005_midterm_1_s - 14.12 Game Theory Fall 2005 Answers to...

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14.12 Game Theory Fall 2005 Answers to Midterm 1, Fall 2005 Answer to Problem 1 a) Player 1 has the same payoff function in both games, so player 1 trivially has the same preference relation over lotteries with strategy profiles as their outcomes. What about player 2? In other words, are the payoffs for player 2 in the game on the right a nonnegative aﬃne transformation of the payoffs in the game on the left? In yet other words, do there exist a 0 and b with 0 a + b = 0 , 1 a + b = 1 , 4 a + b = 3 , and 2 a + b = 2? You can see that we’d need a = 1 and b = 0 in order to satisfy the first two equations, but this does not satisfy the third equation. So there is no such transformation, and player 2 does not have the same preference relation over lotteries with strategy profiles as their outcomes. b) Are the payoffs for player 1 in the game on the right a nonnegative aﬃne transformation of the payoffs in the game on the left? In other words, do there exist a 0 and b with 0 a + b = 1 , 6 a + b = 4 , 2 a + b = 2 , 4 a + b = 3 , 4 a + b = 3 , and 2 a + b = 2? Yes, you can solve the equations and see that a = 1 / 2 and b = 1 are such an a and b. Are the payoffs for player 2 in the game on the right a nonnegative aﬃne transformation of the payoffs in the game on the left? In other words, do there exist a 0 and b with 1 a + b = 0 , 4 a + b = 1 , 4 a + b = 1 , 7 a + b = 2 , 2 a + b = 1 , and 1 a + b = 0? Yes, you can solve the equations and see that a = 1 / 3 and b = 1 / 3 are such an a and b. So yes, both players have the same preference relation on lotteries with strategy profiles as their outcomes.

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