HW1_Sol

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) 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 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 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 ²rst 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 ²rst 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 de²nition 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
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This note was uploaded on 10/31/2011 for the course 18 18.445 taught by Professor Liewang during the Spring '11 term at MIT.

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

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