Weighted Voting Systems - W.H. Freeman

414 PART III Voting and Social ChoiceNow we must determine the Banzhaf power index of a weight-3 voter, A.She will be a critical voter in a winning coalition in which the other membershave a combined total of 4, 5, or 6 votes. If she is joined by the other co-chair, the coalition would need 1, 2, or 3of the 5 weight-1 members. The number of coalitions of this sort isC 5 1 C 5 2 C 5 3 5 10 10 25If the other co-chair is opposed, she could be joined by 4 or all 5 of theweight-1 members. The number of such coalitions is C 5 4 C 5 5 6.It follows that A is a critical voter in 31 winning coalitions; her Banzhaf powerindex is 62. The Banzhaf power index of this committee is (62, 62, 18, 18, 18,18, 18). The total number of critical votes in the committee with two co-chairs is (2 62) (5 18) 214. Thus, each co-chair has , or approximately 29.0%, ofthe power, and each weight-1 member has , or about 8.4%, of the power, bythe Banzhaf model. Recall that, according to the Shapley–Shubik model, the cochairseach had 2 (about 28.6%) of the power and the weight-1 members had 7 3 53(about 8.6%). The agreement between the two models is, as in the other examplesthat we have considered, pretty close.There are situations in which the difference in the distribution of power givenby the models is significant. In Spotlight 11.4, the Banzhaf and Shapley–Shubikpower indices of the United States Electoral College are compared. While thedifferences may seem small, by the Shapley–Shubik model, California has about11.0% of the voting power in the college, while by the Banzhaf model, Californiahas 11.4% of the power. When the stakes are high, this difference is significant.The following example presents a situation in which the models give dramaticallydifferent results.EXAMPLE 18The Big ShareholderB holds 100,000 shares of stock in a corporation. There is a total of one millionshares of stock, and the remaining stock is held by 9000 shareholders, each ofwhom has 100 shares. A weighted voting system, in which each shareholder’s votingweight is equal to the number of shares that he or she owns, is used.The Shapley–Shubik index of this system is determined by the same strategythat we used in Example 7 (the seven-person committee). This time, the permutationsof the stockholders are divided into 9001 groups, depending upon the locationof B. Each group has the same number of permutations (9000! of them,to be precise), and B is pivotal when she appears in the 4002nd through the 5001stposition. If she is 4002nd, then there are 4001 100 400,100 shares precedingher, and her 100,000 shares bring the total to a bare majority of 500,100 shares.If there are more than 5000 shares ahead of B, the 5001st, a small shareholder,