The elections with which we are most familiar often involve only two candidates. However, there are real-world situations in which elections must be held to choose a single winner from among three or more candidates, as in the presidential election of 2012 in which, in addition to Barack Obama and Mitt Romney, there were 25 other candidates (although these other 25 candidates garnered only a total of 2,093,848 votes, with more than half of those going to the Libertarian Party candidate Gary Johnson).

There are several methods that can be used to elect a single candidate from a choice of three or more, and we will investigate some of them in this chapter. Most of these methods use a ballot in which a voter provides a rank ordering of the candidates (without ties) that indicates the order in which he or she prefers them.

Preference list ballots allow voters to make a much clearer statement of their preferences than ballots allowing a single vote. Preference list ballots are already used in a wide range of applications, such as rating football teams and scoring track meets.

Although we do not allow ties in a preference list ballot, most voting rules of interest will, in some elections, result in a tie for the win among two or more of the candidates. In the real world, the number of voters is often so large that ties seldom occur. Nevertheless, to avoid excessive annoyances in the theory that we develop, and to simplify what we do in this chapter, we make the following assumption throughout.

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With this in hand, we are ready to begin our study of social choice.