2. On November 1, 2014, the Council of Ministers of the European Union (E.U.) replaced its old weighted voting system with a new system, agreed to in the Treaty of Lisbon. A winning coalition now consists of the ministers from at least 55% of the countries in the E.U., comprising 65% of the population. There are currently 28 countries in the E.U., so we may say a winning coalition would have to comprise at least 16 countries that had at least 65% of the population. Show that no country is a dummy voter in this system, even though the largest country, Germany, has a population almost 200 times greater than the population of the smallest, Malta. Do you think that this system is equivalent to a weighted voting system?

Because the European Council is too large for hand calculation, consider the Nassau County Board of Supervisors in 1958, with weights and quota as shown in Example 3 on page 464. If the Board had added a requirement that a winning coalition must include at least four supervisors, show that there would be no dummy voters. Is the resulting voting system equivalent to a weighted voting system? List the winning coalitions, find the critical voters, and determine the Banzhaf and Shapley-Shubik power indices for this system.