2003_midterm_1_s

# 2003_midterm_1_s - 14.12 Game Theory-Midterm I Prof Haluk...

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14.12 Game Theory-Midterm I 10/15/2003 Prof. Haluk Ergin Instructions: This is an open book exam, you can use any written material. You have 1 hour and 20 minutes. Each question is 35 points where the bonus question 3(c) accounts for the extra 5 points. Good luck! 1. Consider the following two player game where Player 1 chooses one of the three rows and Player 2 chooses one of the three columns: C 1 C 2 C 3 R 1 2,-1 4,2 2,0 R 2 3,3 0,0 1,1 R 3 1,2 2,8 5,1 (a) What are the strategies that survive IESDS? Solution: Strategy C 3 is strictly dominated by mixture 1 2 C 1 + 1 2 C 2 ;once C 3 is out, R 3 is strictly dominated by R 1 . The answer is R 1 ,R 2 ,C 1 2 . (b) At each step of the elimination what were your rationality and knowledge assumptions? Solution: At the f rst step we assume that player 2 is rational; at the second step we assume that player 1 is rational and knows that player 2 is rational. (c) Find all Nash equilibria, including the mixed one. Solution: By inspection, ( R 1 2 ) and ( R 2 1 ) arepurestrategyNashequi- libria. To f nd the mixed one, assume that player 1 plays αR 1 +(1 α ) R 2 . Playing C 1 gives player 2 the payo f of α +3(1 α )=3 4 α, while playing C 2 gives him 2 α. He will be willing to mix if α = 1 2 . Likewise, assume that player 2 plays βC 1 β ) C 2 . Playing R 1 gives player 1 the payo f of 2 β +4(1 β )=4 2 β , while playing R 2 gives her 3 β. She will be willing to randomize if β = 4 5 . So the mixed Nash equilibrium is ( 1 2 R 1 + 1 2 R 2 , 4 5 C 1 + 1 5 C 2 ) . 2. Consider the following extensive form game with perfect information: (a) Find out the backwards induction outcome. Solution: i. in the last stage of the game, player 1 chooses to play a. ii. in the previous stage, player 2 chooses to play L. 1

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(3,4) (0,1) (1,0) 1 2 2 A B Ll r 1 a b R Figure 1: iii. if we’d been in right hand side part of the game (following a move to the right B in the f rst stage), player 2 would have chosen r. iv. in the f rst stage of the game player 1 choses A. So the backward induction outcome of this game is "player 1 plays A, player 2 responds by playing L; if player 1 had played B in the f rst place, player 2 would have responded by playing r; if A and R had been played, player 1 would have responded by playing a".
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2003_midterm_1_s - 14.12 Game Theory-Midterm I Prof Haluk...

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