Macroeconomics Exam Review 3

Macroeconomics Exam Review 3 - Solutions for Foundations of...

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1.68 The 7 unanimity games for the player set 𝑁 = { 1 , 2 , 3 } are 𝑢 { 1 } ( 𝑆 ) = { 1 S = { 1 } , { 1,2 } , { 1,3 } , N 0 otherwise 𝑢 { 2 } ( 𝑆 ) = { 1 S = { 2 } , { 1,2 } , { 2,3 } , N 0 otherwise 𝑢 { 3 } ( 𝑆 ) = { 1 S = { 3 } , { 1,3 } , { 2,3 } , N 0 otherwise 𝑢 { 1 , 2 } ( 𝑆 ) = { 1 S = { 1,2 } , N 0 otherwise 𝑢 { 1 , 3 } ( 𝑆 ) = { 1 S = { 1,3 } , N 0 otherwise 𝑢 { 2 , 3 } ( 𝑆 ) = { 1 S = { 2,3 } , N 0 otherwise 𝑢 𝑁 ( 𝑆 ) = { 1 S = N 0 otherwise 1.69 Firstly, consider a simple game which is a unanimity game with essential coalition 𝑇 and let 𝑥 be an outcome in which 𝑥 𝑖 0 for every 𝑖 𝑇 𝑥 𝑖 = 0 for every 𝑖 / 𝑇 and 𝑖 𝑁 𝑥 𝑖 = 1 We claim that 𝑥 core. Winning coalitions If 𝑆 is winning coalition, then 𝑤 ( 𝑆 ) = 1. Furthermore, if it is a winning coalition, it must contain
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