Midterm and Solution

Midterm and Solution - University of British Columbia...

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University of British Columbia Department of Economics Economics 221 (002) Introduction to Strategic Thinking Winter Session Term I, 2008/09 Midterm Examination - Solutions PART A: 1. Find the Subgame Perfect Nash Equilibrium to the following game: U M D 1 2 H (5 ,1 ) (2 ,9 ) 2 2 H (1 0 ,2 ) (4 ,8 ) H L (3,2) (6,4) L L A strategy for player 1 will specify a single action. A strategy for player 2 will specify an action to take at the upper information set and an action to take at the lower decision node. Applying backwards induction, at the lower decision node for player 2 , player 2 will want to play L ( 8 > 2 ). In the upper portion of the game, this is a simultaneous move game Player 2 H L Player 1 U (3 , 2) · (6 , 4) · M · (5 , 1) (2 , 9) · and player 2 will play L in the upper information set. Player 1 will choose between U (giving a fi nal outcome at U, L 6 for player 1) and D (giving a fi nal outcome D, L 4 for player 1) . Player 1 chooses U. The subgame perfect Nash equilibrium is: { player 1 strategy, (player 2 action at upper info. set, player 2 action at lower info. set) } = { U, ( L, L ) } Note that ( L, L, L ) does not describe an appropriate strategy for player 2 since they can’t di ff erentiate between 1 playing U or M . 1
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2. Convert the following game tree to a game matrix and fi nd the Pure Strategy Nash Equilibria of the game: 1 2 2 U D L R L R (2,5) (6,8) (5,2) (7 ,4) Player 2 L,L L,R Player 1 U (2 , 5) (2 , 5) D · (5 , 2) · (7 , 4) · R,L R,R · (6 , 8) · (6 , 8) · (5 , 2) · (7 , 4) · There are 3 Pure Strategy Nash Equilibria: { U, ( R, L ) } , { D, ( L, R ) } , { D, ( R, R ) } 2
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3. What is a dominant strategy? In a 2 player, 2 action (each player has two available actions) game, if player 1 has a dominant strategy, is it possible for the game to have a mixed strategy Nash Equilibrium?
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