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HW1_Sol - 14.12 Game Theory Muhamet Yildiz Fall 2010...

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14.12 Game Theory Muhamet Yildiz Fall 2010 Homework 1 Solutions Due on 9/28/2010 (in class) 1. In the following pair of games, for each player, check whether the player°s preferences over lotteries are the same? L M R a 2,-1 1,0 3,-2 b 0,0 1,1 2,0 c 1,-3 2,2 1,4 L M R a 3,-3 0,-1 8,-5 b -1,-1 0,1 3,-1 c 0,-7 3,3 0,7 Solution: The set of outcomes is Z = f a; b; c g°f L; M; R g . By the VNM-representation theorem, the VNM utility functions u i and e u i represent the same preference relation over lotteries for player i i/ e u i ( z ) = au i ( z ) + b for some a > 0 and b 2 R . Clearly, e u 1 ( f a; L g ) | {z } =3 = 3 u 1 ( f a; L g ) | {z } =2 ± 3 and e u 1 ( f a; M g ) | {z } =0 = 3 u 1 ( f a; M g ) | {z } =1 ± 3 , but e u 1 ( f a; R g ) | {z } =8 6 = 3 u 1 ( f a; R g ) | {z } =3 ± 3 , where u 1 and e u 1 are the VNM utility functions of player 1 in the ±rst and second games respectively. Therefore, player 1 has di/erent preferences over lotteries in these two games. It is straightforward to check that e u 2 ( z ) = 2 u 2 ( z ) ± 1 for all z 2 Z , where u 2 and e u 2 are the VNM utility functions of player 2 in the ±rst and second games respectively. Therefore, player 2 has the same preferences over lotteries in these two games. 2. Write the following game in normal form: Solution: First, I would like to repeat the formal de±nition of a strategy given in lecture notes. A strategy of a player is a complete contingent-plan, determining which action 1
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he will take at each information set he is to move ( including the information sets that will not be reached according to this strategy ). Since player 1 moves at 4 information sets and she has 2 moves at each information set, the number of strategies for player 1 is 2 4 = 16 . Therefore, the following reduced representation is not a normal form representation of the extensive game: La 2,1 2,1 1,2 1,2 Lb 1,2 1,2 2,1 2,1 R²A 0,0 1,5 0,0 1,5 R²B 0,0 0,0 0,0 0,0 R³A 5,1 1,5 5,1 1,5 R³B 5,1 0,0 5,1 0,0 Each extensive game has the unique normal form representation. The extensive game given in the homework has the following normal form representation: La²A 2,1 2,1 1,2 1,2 La²B 2,1 2,1 1,2 1,2 La³A 2,1 2,1 1,2 1,2 La³B 2,1 2,1 1,2 1,2 Lb²A 1,2 1,2 2,1 2,1 Lb²B 1,2 1,2 2,1 2,1 Lb³A 1,2 1,2 2,1 2,1 Lb³B 1,2 1,2 2,1 2,1 Ra²A 0,0 1,5 0,0 1,5 Ra²B 0,0 0,0 0,0 0,0 Ra³A 5,1 1,5 5,1 1,5 Ra³B 5,1 0,0 5,1 0,0 Rb²A 0,0 1,5 0,0 1,5 Rb²B 0,0 0,0 0,0 0,0 Rb³A 5,1 1,5 5,1 1,5 Rb³B 5,1 0,0 5,1 0,0 3. Alice, Bob, and Caroline are moving into a 3-bedroom apartment (with rooms, named 1, 2, and 3). In this problem we want to help them to select their rooms. Each roommate has a strict preference over the rooms. The roommates simultaneously submit their preferences in an envelope, and then the rooms are allocated according to one of the following mechanisms. For each mechanism, check whether submitting the
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