Feature Column from the AMS

Voting Games: Part I

2. Voting systems and games

In dealing with voting and elections in a democratic setting we are interested in having the systems be as fair as possible. What does it mean to have a fair voting system? Voting is a complex activity and there are many fairness principles and points of view that might come into play. These include issues as to whether or not a felon should be allowed to vote, the pros and cons of different election decision methods, and using different kinds of ballots.

Most democracies use a direct system of translating voter feelings into the selection of a single winner (e.g. elections for U.S. Senator) or group of winners (e.g. members of a faculty search committee) for the election. However, for a variety of complex reasons, when the U.S. Constitution was created it did not call for the direct election of the President, but for an indirect system using an Electoral College. (Up until 1913, when the 17th Amendment went into effect, even Senators were not directly elected by the American people. They were "chosen" by state legislators.) Thus, strictly speaking, when Americans vote for president they are not voting for a candidate for president but rather for a statewide group of electors committed to voting for the candidate the voter chose. The number of these electors is equal to the total number of senators and members of the House of Representatives for that state. There are also electors for the District of Columbia.

Since the number of members each state has in the House of Representatives can change every 10 years, which reflects the number of seats assigned by the Huntington-Hill Method of Apportionment, the relative power of the states in the Electoral College can change with time. Furthermore, historically the electors who are selected for each state are chosen on a "winner take all" basis. Thus, if in Florida George Bush beat Albert Gore by a single vote in 2000, all of Florida's electors would have been committed to vote for George Bush. Technically, the issue of whom electors vote for is complex, too, and some states have experimented or are considering experimenting with dividing the number of electors between the candidates who run for president on the basis of the popular vote rather than using "winner take all."

Thus, in practice the way the Electoral College works is that there are 51 "players." Each player is either a state or the District of Columbia, and each casts a block of votes: the District of Columbia casts 3 votes and each state casts 2 (the number of senators for the state) plus the number of seats the state has in the House of Representatives, which will be thought of as a weight.

In order to determine if a group of players has "won" the game or been able to achieve its goal, there is a number known as the quota associated with the game. Any group of players that casts blocks of votes, the sum of whose weights exceed the quota, "wins." In the Electoral College there are different groups of states such that if a candidate wins those states, the candidate will be elected president because the weights associated with these states exceed the required quota (270). Thus, the Electoral College is an example involving weighted voting. Sometimes the phrase "weighted voting game" is used. For the case of the Electoral College, weighted voting did not arise directly because of concern with equity issues. Some would argue that it arose for opposite reasons! However, let us look at a variety of settings where equity seems to be the driving force behind using voting games or weighted voting.

In the Electoral College it is natural to describe the way voting proceeds in terms of having players who cast blocks of votes. However, in other situations there are "players" who vote and there are rules which specify when a group of players can "win." For example, the United Nations Security Council consists of players of two kinds: the five so-called permanent members: the United States, the United Kingdom, France, Russia, and China, and ten other members who are chosen on a rotating basis. This makes for a total of 15 countries. To take substantive action requires 9 votes including those of all five permanent members. The Security Council voting game can be represented as a weighted voting game, even though this voting game is not described using weights for the players.

Here are some other voting game examples. Canada is a federation of 10 provinces. For the Canadian Constitution to be amended requires that at least 7 of the 10 provinces approve and that these 7 provinces contain at least one-half of the Canadian population. Voting where complicated rules might want to be implemented also occurs in situations involving organizations with less visibility. The World Bank and the International Monetary Fund are examples where different countries of unequal status are involved with the governance of the organizations. The reason for the unequal status arises from the fact that although in one sense one may say a country is a country, this view oversimplifies the dynamics of the organizations. The U.S. and Japan have relatively small populations but tremendous influence on the world economy. India is a country with a very large population but with a lesser influence when compared to the United States and Japan. Thus, when voting occurs in an organization such as the World Bank or International Monetary Fund, one might want to have complicated rules for passing policies so that the different realities of the countries involved are taken into account. A large group of small countries should not be able to set a policy telling a few rich countries what to do with funds that often come from these richer countries.

In some legislative circumstances, such as some county governments in New York State, one may want each physical person (representing, perhaps, a city in the county) to cast one vote, even though the counties the people represent are very different on many scales (e.g. population, gross product, or wealth). In other cases it may be that one wants to take into account that the representative from a very "small city" (say, as measured by population) does not have the same "influence" as the representative from a very large city within the county government. In New York State, typically, a county legislature is known as a County Board of Supervisors. These considerations explain why many counties in New York use weighted voting.