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final2006solutions - MS&E 246 Game Theory with Engineering...

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MS&E 246: Game Theory with Engineering Applications Final Exam Feryal Erhun & Ramesh Johari Winter, 2006 NAME: ......................................................................................... Instructions: 1. This is a closed-book, closed-notes exam. 2. The exam has 5 questions, and will be graded out of 100 points. Each question is worth 20 points. Please show all your work to get credit. 3. You have 3 hours. Questions 1. Aragorn and Arwen would like to go on a date. They have two options: a quick dinner at Wendy’s or dancing at Sofa Lounge. Aragorn first chooses where to go, and knowing where Aragorn went Arwen also decide where to go. Aragorn prefers Wendy’s, and Arwen prefers Sofa Lounge. A player gets 3 out his/her preferred date, 1 out of his/her unpreferred date, and 0 if they end up at different places. All these are common knowledge. (a) (10 points) Find a subgame-perfect Nash equilibrium. Find also a non-subgame-perfect Nash equilibrium with a different outcome. Solutions: SPNE: Arwen goes wherever Aragorn goes, and Aragorn goes Wendy’s. The outcome is both go to Wendy’s. Non-subgame-perfect Nash Equilibrium: Arwen goes to Sofa Lounge at any history, so Aragorn goes to Sofa Lounge. The outcome is each goes to Sofa Lounge. This is not subgame-perfect because it is not a Nash equilibrium in the subgame after Aragorn goes to Wendys. (b) (10 points) Modify the game a little bit: Arwen does not automatically know where Aragorn went, but she can learn without any cost. (That is, now, without knowing where Aragorn went, Arwen first chooses between Learn and Not-Learn; if she chooses Learn, then she knows where Aragorn went and then decides where to go; otherwise she chooses where to go without learning where Aragorn went. As before, the payoffs 1
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depend only on where each player goes.) Find a subgame-perfect equilibrium of this new game in which the outcome is the same as the outcome of the nonsubgame- perfect equilibrium in part (a). (That is, for each player, he/she goes to the same place in these two equilibria.) Solutions: The game tree for part (b) is as follows: Aragorn W SL Arwen Don’t Don’t Learn Learn Arwen Arwen Arwen W SL W SL W SL (3,1) (0,0) (3,1) (0,0) (0,0) (1,3) (0,0) (1,3) W SL The following is a SPE, whose outcome is that each goes to Sofa Lounge. Aragorn W SL Arwen Don’t Don’t Learn Learn Arwen Arwen Arwen W SL W SL W SL (3,1) (0,0) (3,1) (0,0) (0,0) (1,3) (0,0) (1,3) W SL 2. Consider the following network routing game. Four users wish to travel from s to d in the following road network: 2
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s d 1 2 3 4 5 There are three routes available: top (1 2); middle (1 5 4); and bottom (3 4). Each user chooses a route, and each user is interested in minimizing the delay they experience across the entire route they choose. When n i users travel on link i , the delays are as follows: l 1 ( n 1 ) = 10 n 1 ; l 2 ( n 2 ) = n 2 + 44; l 3 ( n 3 ) = n 3 + 44; l 4 ( n 4 ) = 10 n 4 ; and l 5 ( n 5 ) = n 5 + 13. (a) (4 points) Find a pure strategy Nash equilibrium of this game.
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