Weighted Voting Systems - W.H. Freeman

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$Microscopy Math - W.H. Freeman$

For example,1! 12! 2 1 23! 3 2 1 64! 4 3 2 1 24To continue this list, observe that n! n (n 1)! for n 1. Thus5! 5 4! 5 24 1206! 6 5! 6 120 7207! 7 6! 7 720 50408! 8 7! 8 5040 40,320and so on. You can imagine that n! increases dramatically as n increases—an instanceof the combinatorial explosion. You probably don’t want to calculate 100!.It is a 158-digit number.To justify the formula, suppose that we are listing all of the permutations.There are n voters who could be first; when the first voter is selected, there aren 1 remaining voters who could be in second position, then n 2 who couldbe third, and so on. When it is time to select for the last position, there is onevoter left. By the fundamental principle of counting (see Chapter 1), the numberof permutations is the product of the numbers of choices that we have had ateach stage.The Shapley–Shubik Power Index of a Three-Voter SystemLet us calculate the Shapley–Shubik power index of the voting system [6: 5, 3, 1].We will name the participants A, B, and C, and consider their 3! 6 permutations.Table 11.1 displays all six permutations. Next to each permutation, the totalweights of the first voter, the first two voters, and all three voters are shownin sequence. The first number in the sequence that equals or exceeds the quotais underlined, and the corresponding pivotal voter’s symbol is circled. We seethat A is pivotal in four permutations, while B and C are each pivotal in one.Hence the Shapley–Shubik index of A is 4 , and B and C each have Shapley–Shubik indices of 1 6 6 . The Corporation with Four ShareholdersA corporation has four shareholders, A, B, C, and D, with 40, 30, 20, and 10shares, respectively. The corporation uses the weighted voting system[51 : 40, 30, 20, 10]CHAPTER 11 Weighted Voting Systems 397EXAMPLE 5The 4! 24 permutations of the shareholders are shown in Table 11.2. In 10 ofthe permutations, A is the pivotal voter; B and C are each pivotal voters in 6;EXAMPLE 6
398 PART III Voting and Social ChoiceTABLE 11.1Permutations and Pivotal Voters for theThree-Person CommitteePermutationsWeightsA B C 5 8 9A C B 5 6 9B A C 3 8 9B C A 3 4 9C A B 1 6 9C B A 1 4 9TABLE 11.2Permutations and Pivotal Voters for theFour-Person CorporationPermutations Weights PivotA B C D 40 70 90 100 BA B D C 40 70 80 100 BA C B D 40 60 90 100 CA C D B 40 60 70 100 CA D B C 40 50 80 100 BA D C B 40 50 70 100 CB A C D 30 70 90 100 AB A D C 30 70 80 100 AB C A D 30 50 90 100 AB C D A 30 50 60 100 DB D A C 30 40 80 100 AB D C A 30 40 60 100 CC A B D 20 60 90 100 AC A D B 20 60 70 100 AC B A D 20 50 90 100 AC B D A 20 50 60 100 DC D A B 20 30 70 100 AC D B A 20 50 60 100 BD A B C 10 50 80 100 BD A C B 10 50 70 100 CD B A C 10 40 80 100 AD B C A 10 40 60 100 CD C A B 10 30 70 100 AD C B A 10 30 60 100 B
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Weighted Voting Systems - W.H. Freeman

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