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201Midterm2Answers-2008spring-1 - SABANCI UNIVERSITY ECON...

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SABANCI UNIVERSITY ECON 201 – A GAMES AND STRATEGY SECOND MIDTERM EXAMINATION ANSWERS April 3, 2008 (X) Instructor: Mehmet Baç Time allowed: 45 minutes, 100 points READ CAREFULLY. .. 1 ( 20 points ) Olga and Irina, two neighbors, will simultaneously plant flowers in their gardens. Let x denote the number of flowers that Olga plants and y , the number of flowers than Irina plants. The payoffs of Olga and Irina are as follows: U Olga = 10xy – x 2 and U Irina = 10y /x – y 2 /2. The terms x 2 and 0.5y 2 represent the players’ cost of planting flowers. (i) [5p] What kind of externality is there between Olga and Irina? Do they like each other’s flowers? Explain by using the payoff functions above. (ii) [10p] Draw the best response functions of the players on the same figure. (iii) [5p] Find the Nash equilibrium numbers of flowers (x*, y*). Show your calculations. Answer : (i) Olga’s payoff is increasing in y, thus, Olga enjoys Irina’s flowers, whereas Irina’s payoff is decreasing in x, thus, Irina dislikes Olga’s flowers. There is a positive externality from Irina to Olga, a negative externality from Olga to Irina. (ii) Olga’s best response is found by taking the first-order derivative of her payoff with respect to x: 10y – 2x = 0, thus, x = 5y. Similarly, Irina’s best response is: 10/x – y = 0, or y = 10/x. These best response functions look like: y y = 10/x Nash Eq x = 5y 0 x (iii) The Nash equilibrium is found by solving the best response functions: x = 5(10/x) or, x = (50) 0.5 or, x = 5.(2) 0.5 . Given this, y = 10 / 5.(2) 0.5 = 2 0.5 .
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2 ( 40 points ) Consider the following game. Player 2 Player 1 L M R U 0 , 3 4 , 1 1 , 2 D 2 , 0 1 , 4 0 , 1 (i) [10p] Is any of player 1’s or player 2’s strategies not rationalizable? Explain. (ii) [10p] Denoting by p the probability that player 1 plays U, write player 2’s expected payoffs from each of his three pure strategies. (iii) [10p] Obtain player 2’s best response function (denote by q L , q M , and q R the probability with which player 2 plays L, M and R, respectively). Show that player 2’s best response involves q R = 0. (iv)
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201Midterm2Answers-2008spring-1 - SABANCI UNIVERSITY ECON...

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