Weighted Voting Systems - W.H. Freeman

404 PART III Voting and Social ChoiceNow let’s go back to the original coalition and see what happens if B changeshis vote.A B C Votes Outcome2 1 1 4 Pass2 0 1 3 PassThis time, the outcome doesn’t change, so B is not a critical voter in this coalition.Because C has the same power as B, he is also not a critical voter in thecoalition. EXAMPLE 11Winning and BlockingIn the committee with members A, B, and C, and voting system [3: 2, 1, 1], Aand B have formed a coalition to vote in favor of measure X and to oppose anothermeasure, Y. Member C is voting against X and for Y. Because the coalition{A, B} has 3 votes (2 for A, and 1 for B), it is a winning coalition for X and ablocking coalition for Y.Measure XMeasure YA B C Votes Outcome A B C Votes Outcome2 1 0 3 Pass 0 0 1 1 FailWhen voting for X, both A and B are critical voters. However, it takes only2 votes to block a measure. If A leaves the coalition and joins C in voting for Y,then Y will be approved. Therefore A is a critical voter in the coalition to blockmotion Y. Voter B is not critical in the blocking coalition, because if he decidesto switch his vote and support Y, it will not change the outcome. A can blockmotion Y by herself. The Banzhaf power index was developed in 1965 by an attorney, John F.Banzhaf III, in an analysis of weighted voting that appeared in the Rutgers LawReview. Mr. Banzhaf ’s article was entitled “Weighted Voting Doesn’t Work.” SeeSpotlight 11.3.The Banzhaf Power IndexA voter’s Banzhaf power index is the number of distinct winning or blockingcoalitions in which his or her vote is critical.