1. The most important weighted voting system in the United States is the Electoral College (see Spotlight 11.1 on page 465). An organization, National Popular Vote (NPV), is attempting to replace the Electoral College with a system in which the president is elected by popular vote. One way to accomplish this would be to amend the U.S. Constitution. NPV has what might prove to be a simpler approach, lobbying state legislatures to pass a bill that pledges their state’s electoral votes to the ticket that receives the plurality of the popular vote. As of this writing, 11 states, with a total voting weight of 165 electoral votes, have passed this bill. The bill includes language that says that it will only take effect when the total voting weight of the states that have enacted it reaches 270, for then these states will make up a winning coalition.

What would happen if the states were to take effect for the next presidential election, without waiting for a winning coalition to form? The NPV website lists the states that have passed the NPV bill. Construct a new voting system in which one voter, NPV, represents all the states that have passed the NPV bill. These states have ceded their votes to NPV. The other voters are the states (or congressional districts in states with the District System) that have not passed the NPV bill, with their electors chosen as usual (see Suggested Websites on the next page). Example 9 on page 473 and Example 16 on page 483 show how a voting system can be altered if some voters combine to act together. Because the voting systems will be sizable, you will need to use the Power Index Calculator.

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.

3. Maine and Nebraska use the District System to select their electors in presidential elections. (The District System is explained in Spotlight 11.1 on page 465.) Discuss the complications involved in determining the power indices when usage of the District System is taken into account. What would happen if all states, except California, adopted the District System for choosing electors?