Social Choice_Lecture 8_SSI

Van essen u of a power 12 22 case 1 exactly eight

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Unformatted text preview: Power 12 / 22 Case 1: Exactly eight votes precede France How can we get 8 votes? If France is preceded by: Case 1a: Germany, Belgium, and the Netherlands (4, 2, 2) Case 1b: Italy, Belgium, and the Netherlands (4,2,2) Case 1c: Germany and Italy (4, 4) For each of the three above subcases we can use counting techniques to …gure out the number of ways this can happen. Van Essen (U of A) Power 13 / 22 Case 1a: Germany, Belgium, and the Netherlands precede France To count the total number of orderings that are included in each subcase we need to count: (1) how many ways can we arrange the countries that precede France; (2) how many ways can we arrange the countries that come after France; and (3) use the multiplication principle to get the total. France is preceded by Germany, Belgium, and the Netherlands The number of distinct ways we can order these three countries is 3! The number of ways we can arrange the countries that come after France (Luxembourg and Italy) is 2! Therefore the total number of pivotal orderings where France is preceded by Germany, Belgium, and the Netherlands is 3!2! = 12 Van Essen (U of A) Power 14 / 22 Case 1b: Italy, Belgium, and the Netherlands precede France France is preceded by Italy, Belgium, and the Netherlands The number of distinct ways we can order these three countries is 3! The number of ways we can arrange the countries that come after France is 2! Therefore the total number of pivotal orderings for case 1b 3!2! = 12 Van Essen (U of A) Power 15 / 22 Case 1c: Italy and Germany precede France France is preceded by Italy and Germany The number of distinct ways we can order these two countries is 2! The number of ways we can arrange the countries that come after France is 3! Therefore the total number of pivotal orderings for case 1b 2!3! = 12 Van Essen (U of A) Power 16 / 22 Case 1 Case 1a: 12 orderings Case 1b: 12 orderings Case 1c: 12 orderings Therefore the total number of pivotal orderings for case 1 (8 votes) is 36 Van Essen (U of A) Power 17 / 22 Case 2: Exactly nine votes precede France How can we get 9 votes? If France is preceded by: Case 2a: Germany, Be...
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This note was uploaded on 04/08/2014 for the course ECON 497 taught by Professor Vanessen during the Spring '12 term at Alabama.

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