Weighted Voting Systems - W.H. Freeman

410 PART III Voting and Social ChoiceEXAMPLE 14Expressing the Number 49 in Binary NotationThe largest power of 2 less than 49 is 2 5 32. Subtract 32 from 49 to get 17.The largest power of 2 less than 17 is 16 2 4 . Subtract 16 from 17 to get 1, whichis a power of 2 (1 2 0 ). Thus 49 2 5 2 4 2 0 , and hence the nonzero bits of(49) 2 are b 5 , b 4 , and b 0 , while b 3 b 2 b 1 0. Listing the bits in order, (49) 2 110001. Now suppose that we have n voters. A sequence of n bits can represent a votingcombination, where each voter is associated to a particular bit, which is 1 ifthe voter approves and 0 if the voter disapproves. A sequence of n bits also givesthe binary representation of a number between 0 (the sequence with n 0’s) and2 n 1, which, as a binary number, is a sequence of n 1’s. It follows that a set ofn voters can have 2 n different voting combinations.The number of voting combinations with n voters and exactly k “yes” votesis denoted C n k . For example, there is only one combination, 0000 where noone votes “yes,” so C n 0 1. You can show that by the same reasoning, C n n 1.There are n combinations with exactly one “yes” vote:1000, 0100, . . . , 0001,and hence C n 1 n.If each voter in a combination with k “yes” votes and n k “no” votes wereto switch his or her vote to the opposite side, there would be n k “yes” votesand k “no” votes. Thus, the number of combinations of n voters with k “yes”votes is equal to the number of combinations of n voters with n k “yes” votes.In symbols, we can state this fact as follows:Duality Formula for CombinationsC n k C n nkAnother formula that is useful is the addition formula. Suppose that thereare n 1 voters, one of whom is called Z. There are C n k combinations in whichZ votes “no” and k other voters vote “yes,” and C n k1 combinations in which Zand k 1 of the other voters all vote “yes.” This encompasses all C n1 k votingcombinations of the n 1 voters in which there are exactly k “yes” votes. Thuswe have the following:Addition FormulaC n k1 C n k C k n1