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Macroeconomics Exam Review 189

# Macroeconomics Exam Review 189 - c 2001 Michael Carter All...

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3.256 By the Minimax theorem, every finite two person zero-sum game has a value. The previous result shows that this is attained at a Nash equilibrium. 3.257 If player 2 adopts the strategy 𝑡 1 𝑓 p ( 𝑡 1 ) = 𝑝 1 + 2 𝑝 2 < 0 if 𝑝 1 > 2 𝑝 2 If player 2 adopts the strategy 𝑡 5 𝑓 p ( 𝑡 5 ) = 𝑝 1 2 𝑝 2 < 0 if 𝑝 1 < 2 𝑝 2 Therefore 𝑣 1 ( p ) = min 𝑧 𝑍 𝑓 p ( z ) min { 𝑓 p ( 𝑡 1 ) , 𝑓 p ( 𝑡 5 ) } < 0 for every p such that 𝑝 1 = 𝑝 2 . Since 𝑝 1 + 𝑝 2 = 1, we conclude that 𝑣 1 ( p ) { = 0 p = p = ( 2 / 3 , 1 / 3 ) < 0 otherwise We conclude that 𝑣 1 = max p 𝑣 1 ( p ) = 0 which is attained at p = ( 2 / 3 , 1 / 3 ). 3.258 1. 𝑣 2 = min z 𝑍 𝑚 max 𝑖 =1 𝑧 𝑖 Since 𝑍 is compact, 𝑣 2 = 0 implies there exists ¯ z 𝑍 such that 𝑚 max 𝑖 =1 ¯ 𝑧 𝑖 = 0 which implies that ¯ z 0. Consequently 𝑍 ∩ ℜ 𝑛 = . 2. Assume to the contrary that there exists
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