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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 ±rst 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 di²erent 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 di²erent 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 ±rst 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 payo²s 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 WS L L W S L (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 L L W S L (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 12 34 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 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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This note was uploaded on 02/02/2009 for the course MS&E 246 taught by Professor Johari during the Winter '07 term at Stanford.

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

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