 In the 1960s, Nassau County Long Island used a weighted voting system with six members. Hempston, the largest town in the county, actually had two members, Hempston I and II, both of whom cast nine votes. North Hempston, also a large town, cast seven votes. The town of Oyster Bay, much smaller, three votes. Glen Cove, smaller still, one vote. And Long Beach, casting one vote. The lawyer John T. Banzoff III analyzed the voting system by counting the number of times a voter was critical in a winning coalition. What Banzoff found was, well, we'll talk about that later. Let's analyze a simpler system, say this weighted voting system. So remember to commute the Banzoff Index. First, find all winning coalitions and identify critical voters. Since quota is ten, we look for coalitions that cast ten or more votes. The four-voter coalition consisting of all the voters cast a total of 19 votes, so it's a winning coalition. Now let's consider the three-voter coalitions with enough votes. So A, B, and C together cast a total of 18 votes. A, C, and D cast 12 votes. A, B, and D cast 16. And B, C, and D cast 11 votes. So these three-voter coalitions are winning coalitions. The two-person coalitions with enough votes are. And since quota is ten and no single voter has ten votes, no single voter is a winning coalition. So now let's find the critical voters. So the coalition of all four voters, if A leaves the remaining members, still form a winning coalition, so A is not critical. If B leaves, the remaining members are A, C, and D, and they still form a winning coalition, so B is not critical. And we can make a similar observation if C or D leave. This means that no member is critical. How about the coalition A, B, C? So if A leaves, the remaining voters B and C still form a winning coalition, so A is not critical. And likewise, if B or C leave, then we still form a winning coalition, so again, no member is critical. How about the coalition A, C, D? If A leaves, the remaining members are C and D, and they are not one of our winning coalitions, so A is a critical voter. If C leaves, the remaining members A and D, again, do not form a creating coalition, so C is critical, but if D leaves, the remaining members are A and C, and this is a winning coalition, so D is not a critical voter. So for the coalition A, C, and D, the critical voters are A and C. Similarly, for A, B, and D, we can find the critical voters. For B, C, and D, the critical voters are. And since no single member coalition is winning, then both voters in all of the two-person coalitions are critical. And now we count. A is critical four times, B and C are also critical four times each. So even though A casts more votes than B and B casts more votes than C, all three have the same power. And D is critical never. So even though D has a vote, their vote is completely irrelevant. And a similar thing happened in Long Island. Bonzoff showed that Oyster Bay, Glen Cove, and Long Beach, even though they cast votes, those votes didn't matter. And as a result, the weighted voting system was restructured.