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Corr Exam Nov 2006

# Corr Exam Nov 2006 - Correction of the Exam of 22 November...

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Correction of the Exam of 22 November 2006 Exercise 1: 1a) This is a strategic game with the following characteristics: Players : Player 1 and Player 2 Actions : Each player’s set of action is { S, C } . Preferences and payo ff s : They are given by the following matrix: Player 2 S C Player 1 S (1 , 1) (1 c, 2) C (2 , 1 c ) (0 , 0) where 0 < c < 1 . 1b) First, it is easy to verify that whatever the values of 0 < c < 1 , there are two pure-strategy Nash equilibria, which are ( S, C ) and ( C, S ). Let us investigate the mixed-strategy Nash equilibria. For that, we calculate the expected utility of each player. Let ( q, 1 q ) be the mixed strategy in which Player 2 plays S with probability q and let ( r, 1 r ) be the mixed strategy in which Player 1 plays S with probability r . For player 1, the expected utility is thus given by: EU 1 = r [ q + (1 q ) (1 c )] + (1 r ) 2 q (1) while for player 2, we obtain: EU 2 = q [ r + (1 r ) (1 c )] + (1 q ) 2 r (2) Player 1 solves the following program: ∂EU 1 ∂r = q + (1 q ) (1 c ) 2 q = 1 c q (2 c ) 1

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which implies that ∂EU 1 ∂r T 0 1 c T q (2 c ) q S 1 c 2 c with 0 < 1 c 2 c < 1 Thus: Player 1 plays the pure strategy S ( r = 1 ) if q < (1 c ) / (2 c ) ; Player 1 plays the pure strategy C ( r = 0 ) if q > (1 c ) / (2
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Corr Exam Nov 2006 - Correction of the Exam of 22 November...

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