The Process of Electing a PresidentThe issue of how to equalize the influence of people in different parts of the country in the selection of party candidates is a complex one, which scholars in many disciplines are addressing, using mathematical methods...
Every four years Americans elect a president. Voting and elections are essential for an effective democracy. The people who make up a society get to choose their representatives to govern them and these representatives vote on or administer the laws that affect citizen quality of life on a daily basis. This year, Mathematics Awareness Month calls attention to the role that mathematics and statistics have in understanding the phenomena of voting and elections.
One interesting question is whether or not these numbers are as precise as they appear when reported because most of us, if asked to count many seemingly well-determined numbers, would not get the exact right answer. (Usually this would be because to carry out any large count takes sufficiently long that our attention flags and we would probably make some error.)
The issue of accuracy in vote counts was brought to the fore by the now famous "hanging chads" issues involved in counting votes in Florida for the 2000 presidential election. If you look at different information sources, you will find differing totals for the number of votes that Bush and Gore got in the 2000 election. Whereas many people think in terms of tabulating paper ballots because often day-to-day votes are done with these, the issue of how to vote online, or using electronic/mechanical voting machines, raises many questions about the accuracy of vote counts. Mathematicians and computer scientists are looking into issues related to the security of voting and the accuracy of vote counts. In particular, suppose we have partial results from an election and the accuracy of vote for some specific districts of known size is available, can we determine if the outcome of the election might be affected by the results of these disputed votes? Noteworthy here is the fact that in 2000 the candidate who had the largest popular vote did not become president. Since ballots for president only allow a voter to vote for one candidate, it is not easy to know the consequences of having the election between only the two top vote-getters instead of the many candidates who typically appear on a presidential ballot. Furthermore, we can not be sure if the candidate a voter voted for is that voter's "favorite" candidate. For example, in the 2000 election it is possible that some voters for Gore or Bush would have preferred Nader but voted for the person they did because they felt that, given the way the votes were going to be counted (e.g. plurality voting), they did not want to "throw away" their vote and, hence, voted for someone other than their first choice.
The Electoral College
The reason Gore did not win the election in 2000 is that the president is not elected on the basis of the popular vote but on the vote in the Electoral College. The essence (there are technical details) of the way that the Electoral College works is that each of the 50 states casts a block of votes which represents the total number of Senators (always 2) and the number of members that the state has in the House of Representatives. In the Electoral College the District of Columbia also casts 3 votes. Thus, 100 + 435 + 3 = 538 votes are available and to be elected president (with the vice presidential choice now tied to that of the president) requires 270 votes.
Mathematical insights into elections and voting
Not only do Americans vote for president but we also vote for the representatives who create the laws which form the basis of our daily lives. These representatives use votes to create these laws. Elections and voting behavior are ingrained in democracies. Votes can be taken for what speaker should be chosen for a high school graduate ceremony, what should be served at the union picnic, what athlete should win best pitcher of the year award and what movie actress should be the Academy Award winner for best supporting actress.
In what follows, for simplicity, we will consider voting situations where a single winner is to be selected or a ranking must be constructed. (In "voting" for whom to invite as a college graduation speaker one might want to rank the candidates, since the highest-ranked choice might not be able to attend on the required day.) Now, from a mathematical point of view things get rather more interesting when one considers the ballot given to a voter. The ballot is a way for voters to express their preference for alternatives.
(Photo courtesy of Steven Brams)
After picking a ballot one needs to look for good methods to decide on the winner based on the type of ballot chosen. Again, for simplicity, we will consider elections where there are ordinal (rank) ballots but each voter ranks all of the candidates, with no indifference between two or more candidates. If there are only two alternatives, the situation is straightforward. One votes for that alternative of the two which one likes most, and the decision system is to declare the winner to be the person who gets the largest number of votes. (The role of how to break ties in elections is an important topic. For simplicity in the discussion here I will assume (and use examples) where no ties occur. This is a reasonable assumption in elections where there are large numbers of voters.) When there are two alternatives, the person (or alternative) who gets the larger number of votes will also have a majority. However, as soon as we allow more than 2 alternatives, the usual ballot can result in no candidate getting a majority (as happened in the presidential elections above). The usual decision method in this case is to use plurality voting: the candidate who gets the largest number of votes is the winner.
(Figure 2: An election involving 55 voters)
(Table 1: Entries in row X, column Y indicate the number of voters who prefer X to Y)
(Photo of Donald Saari. Courtesy of Donald and Lillian Saari)
Axiomatic approaches to fairness
If different decision methods for deciding elections, all of which have certain reasonable features in their favor, can lead to different winners, which method should be used in practice? A dramatic breakthrough in how to think about fairness questions was achieved by the mathematically trained economist Kenneth Arrow. Eventually, Arrow went on the win the Nobel Memorial Prize in Economics for his path-finding work. Arrow's basic insight was to inquire about what fairness properties a good election method should obey rather than look at individual methods and study what nice features they had.
(Photo of Kenneth Arrow. Courtesy of Kenneth Arrow)
Manipulation of voting systems
Ideally one would like voting methods to have the property that if the voters know in advance what the method will be to count the ballots as well as information about how other voters might or will vote, the voters will be sincere in how they vote. Sincere means that the way the voter fills out his or her ballot only depends on his/her views about the alternatives. The voter does not take into account trying to get a better outcome (e.g. higher-ranked alternative) by altering his/her ballot based on information not related to the candidates themselves.
the voter realizes that this will contribute 3 points to D and 2 points to C in the election. These 2 points may cause C to win the election and D to lose the election. However, if the voter casts his ballot as below:
(Photo of Allan Gibbard. Courtesy of Allan Gibbard and Molly Mahony)
(Photo of Mark Satterthwaite. Courtesy of Mark Satterthwaite and Donald Saari)
Who studies fairness, voting and elections?
While it might seem that the natural people to be experts on voting and elections would be political scientists, this is only part of the picture, as we have seen . Starting with reformers in the 18th century, many individuals have thought about and made significant contributions to voting and elections. One sees the names of individuals from many countries, times, and backgrounds. Thus, Arrow, Borda, Michel Balinski, Condorcet, C. L. Dodgson (Lewis Carroll), Allan Gibbard, John Kemeny, Hervé Moulin, Donald Saari, Mark Satterthwaite and H. P. Young range over long periods of time, many professions and nationalities, but all sharing the common goal of using a mathematical lens to study fairness questions.
Until democratic societies find the best way to organize and run their elections, and mathematics is doing its part to try to find a way to accomplish this, one important thing you can do to support democracy is to VOTE!
Amy, D., Real Choices/New Voices: The Case for Proportional Representation Elections in the United States, Columbia U. Press, New York, 1993.
Those who can access JSTOR can find some of the papers mentioned above there. For those with access, the American Mathematical Society's MathSciNet can be used to get additional bibliographic information and reviews of some these materials. Some of the items above can be accessed via the ACM Portal , which also provides bibliographic services.
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