Social Choice_Lecture 8_SSI

If france is preceded by case 2a germany belgium the

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Unformatted text preview: lgium, the Netherlands, and Luxembourg (4, 2, 2, 1) Case 2b: Italy, Belgium, the Netherlands, and Luxembourg (4,2,2, 1) Case 2c: Germany, Italy, and Luxembourg (4, 4, 1) For each of the three above subcases we can use counting techniques to …gure out the number of ways this can happen. Case 2a: 4!1! = 24 Case 2b: 4!3! = 24 Case 2c: 3!2! = 12 Thus, the total number of pivotal orderings for case 2 (9 votes) is 24 + 24 + 12 = 60. Van Essen (U of A) Power 18 / 22 Case 3: Exactly ten votes precede France How can we get 10 votes? If France is preceded by: Case 3a: Germany, Italy, and Belgium (4, 4, 2) Case 3b: Germany, Italy, and the Netherlands (4, 4, 2) For each of the three above subcases we can use counting techniques to …gure out the number of ways this can happen. Case 3a: 3!2! = 12 Case 3b: 3!2! = 12 Thus, the total number of pivotal orderings for case 3 (10 votes) is 24 Van Essen (U of A) Power 19 / 22 Case 4: Exactly eleven votes precede France How can we get 11 votes? If France is preceded by: Case 4a: Germany, Italy, Belgium, and Luxembourg (4, 4, 2,1) Case 4b: Germany, Italy, the Netherlands, and Luxembourg (4, 4, 2,1) For each of the three above subcases we can use counting techniques to …gure out the number of ways this can happen. Case 4a: 4!1! = 24 Case 4b: 4!1! = 24 Thus, the total number of pivotal orderings for case 4 (11 votes) is 88 Van Essen (U of A) Power 20 / 22 SSI for France The total number of pivotal orderings is 36 + 60 + 24 + 48 = 168 The total number of distinct orderings is 720 Thus, the Shapley-Shubik index for France is SSI (France ) = Van Essen (U of A) Power 168 14 = 720 60 21 / 22 SSI for EEC Country Germany Italy France Belgium Netherlands Luxembourg Van Essen (U of A) Votes 4 4 4 2 2 1 Power SSI 14 60 14 60 14 60 9 60 9 60 0 60 22 / 22...
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