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Macroeconomics Exam Review 195

Macroeconomics Exam Review 195 - c 2001 Michael Carter All...

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3. This immediately implies core( 𝑁, 𝑤 ) = ∅ ⇐⇒ core( 𝑁, 𝑤 0 ) = 3.271 ( 𝑁, 𝑤 ) is 0–1 normalized, that is 𝑤 ( { 𝑖 } = 0 for every 𝑖 𝑁 𝑤 ( 𝑁 ) = 1 Consequently, x belongs to the core of ( 𝑁, 𝑤 ) if and only if 𝑥 𝑖 𝑤 𝑖 = 0 (3.84) 𝑖 𝑁 𝑥 𝑖 = 𝑤 ( 𝑁 ) = 1 (3.85) 𝑖 𝑆 𝑥 𝑖 𝑤 ( 𝑆 ) for every 𝑆 ∈ 𝒜 (3.86) (3.84) and (3.85) ensure that x = ( 𝑥 1 , 𝑥 2 , . . . , 𝑥 𝑛 ) is a mixed strategy for player 1 in the two-person zero-sum game. Using this mixed strategy, the expected payoff to player I for any strategy 𝑆 of player II is 𝑢 ( x , 𝑆 ) = 𝑖 𝑁 𝑥 𝑖 𝑢 ( 𝑖, 𝑆 ) = 𝑖 𝑆 𝑥 𝑖 1 𝑤 ( 𝑆 ) (3.86) implies 𝑢 ( x , 𝑆 ) = 𝑖 𝑆 𝑥 𝑖 1 𝑤 ( 𝑆 ) 1 for every 𝑆 ∈ 𝒜 That is any x core( 𝑁, 𝑤 ) provides a mixed strategy for player I which ensures a
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