The Presidential Election of 2000 provided many challenges to American democracy. When Americans awoke the morning after Election Day, they were surprised to learn that the results of the balloting for president had been indecisive. As the drama unfolded, there were many stories about alleged fraud and the word "chad" entered the mainstream American lexicon. In the recent past it had been possible to use the partial information about the ballots already counted by the morning after Election Day to say with assurance who the next president would be. Knowing who the winner will be is not as easy as counting how many people voted for each of the presidential candidates and saying the person with the largest number of votes must be the winner, so-called plurality voting. In the 2000 election Albert Gore won the popular vote, yet the winner of the United States presidency depends on the results of what happens in the Electoral College. In the simplest case, this involves blocks of votes being cast by electors from each of the 50 states and the District of Columbia. There being 435 seats in the House of Representatives, 3 votes assigned to the District of Columbia, and 100 Senators, the total number of votes to be cast in the Electoral College is 538. In order to be elected President one needs 270 of these electoral votes; note that a tie is possible.
In effect the Electoral College involves "weighted voting," a situation where votes are cast in blocks. Weighted voting systems are more common than many people realize. There are the voting schemes involving the European Union, the United Nations, amending the Canadian Constitution, county governments in New York State, and in many other environments which affect people's lives in many ways. Sometimes there are layers of complexity in a voting system. For example, if one looks at the United States House of Representatives or the Senate, it may seem relatively easy to understand what is going on. However, the passing of a new law in the United States needs the approval of the House of Representatives, the Senate, and the President; when the President vetoes a proposed law, there is a method to override the veto. Is there an easier way to think of this "complex voting" situation and to get insight into the power relationships involved? Can mathematics help in understanding what is going on with these voting systems and in designing systems that are more fair and equitable?