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Macroeconomics Exam Review 5

Macroeconomics Exam Review 5 - Solutions for Foundations of...

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1.75 If a 𝑖 is the best element in ( 𝐴 𝑖 , 𝑖 ) for every player 𝑖 , then ( 𝑎 𝑖 , a 𝑖 ) 𝑖 ( 𝑎 𝑖 , a 𝑖 ) for every 𝑎 𝑖 𝐴 𝑖 and a 𝑖 𝐴 𝑖 for every 𝑖 𝑁 . Therefore, a is a Nash equilibrium. To show that it is unique, assume that ¯ a is another Nash equilibrium. Then for every player 𝑖 𝑁 𝑎 𝑖 , ¯ a 𝑖 ) 𝑖 ( 𝑎 𝑖 , ¯ a 𝑖 ) for every 𝑎 𝑖 𝐴 𝑖 which implies that ¯ a is a maximal element of 𝑖 . To see this, assume not. That is, assume that there exists some ˜ 𝑎 𝑖 𝐴 𝑖 such that ˜ 𝑎 𝑖 𝑖 ¯ 𝑎 𝑖 which implies 𝑎 𝑖 , a 𝑖 ) 𝑖 𝑎 𝑖 , a 𝑖 ) for every a 𝑖 𝐴 𝑖 In particular 𝑎 𝑖 , ¯ a 𝑖 ) 𝑖 ( 𝑎 𝑖 , a 𝑖 ) which contradicts the assumption that a is a Nash equilibrium. Therefore, ¯ a is an- other Nash equilibrium, then ¯ 𝑎 𝑖
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