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Unformatted text preview: Econ414 January 2010 Final Exam Instructor: Jeff Borowitz ANSWERS Wednesday, July 8 2009 1. (10 points) In the course, we studied four types of games: static games of complete information, dynamic games of complete information, static games of incomplete information, and dynamic games of incomplete information. In words, what makes static and dynamic games different, and what makes games of complete and incomplete information different? Be very specific and do not just write down definitions. Answer: I am not looking for a regurgitation of definitions here. The essential difference between games of dynamic and static timing is that in a static game, no player knows any other players strategy at any point before the game is over. In a dynamic game, on the other hand, one player sees the previous action of another player at some point. An answer that only mentions that in a dynamic game, actions are taken sequentially instead of simultaneously will not receive full credit. 1 In games of incomplete information, players do not know each others utility functions. Instead they have beliefs about what each others utility functions might be. Each players utility function is determined from a set of possible utility functions randomly right before the game is played. In games of complete information, players know their opponents utility function, and it is always the same. 2. (30 points) Find all Nash Equilibria in the following games (a) A B L C R T 3 , 2 5 , 2 2 , 3 M 4 , 4 2 , 2 3 , 1 B 2 , 2 4 , 3 1 , 3 (b) A B L C R T 6 , 3 2 , 2 1 , M 1 , 3 , 2 , 1 B 2 , 2 4 , 1 , 1 Answer: (a) This game can be solved either by iterated elimination of underlining. For brevity, I will use underlining here. 1 To see why, think about the Prisoners Dilemma when the prisoners are interrogated sequentially by the same officer, but are locked in separate interrogation rooms. As long as the second prisoner doesnt know what the first has done, the game is exactly the as when the prisoners are interrogated simultaneously. 1 A B L C R T 3 , 2 5 , 2 2 , 3 M 4 , 4 2 , 2 3 , 1 B 2 , 2 4 , 3 1 , 3 The only Nash Equilibrium here is ( M,L ). (b) Lets start to solve this problem by iterated elimination. Using iterated elimination, note that L weakly dominates C for player B. Therefore, the game is: A B L R T 6 , 3 1 , M 1 , 2 , 1 B 2 , 2 , 1 Noting that now, T dominates B for A, we have the game: A B L R T 6 , 3 1 , M 1 , 2 , 1 Iterated elimination cant take us any further. Now, notice that there are two Nash Equilibria in pure strategies. Underlining, we find: A B L R T 6 , 3 1 , M 1 , 2 , 1 So the pure strategy Nash Equilibria are ( T,L ) and ( M,R ). There is also a mixed strategy Nash Equilibrium here. To find it, allow Alice to play T with probability p and Bob to play L with probability q . Equating expected utilities, we find: E [ u A ( T )] = E [ u A ( M )] 6 q + 1(1- q ) =1 q + 2(1- q ) 5 q =1- q q =1 / 6 E [ u B ( L )] = E [ u B ( R )] 3 p + 0(1- p...
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This note was uploaded on 02/28/2010 for the course ECON 414 taught by Professor Staff during the Summer '08 term at Maryland.
- Summer '08
- Game Theory