This simple example illustrates a problem that would perplex social sci-
entists for more than a century: How can one translate a collection of indi-
vidual ballots, consisting of a ranking of preferences, into a ranking of
preferences for the group as a whole? The Condorcet paradox points out
that if we simply look at pairs of candidates and determine which of the
pair the majority prefers, we run into the problem that it is impossible to
maintain what mathematicians call transitivity, which is a property of rela-
tionships such that if A is preferred to B and B is preferred to C, then A is
preferred to C. Individual preferences are transitive; it seems reasonable
to require that whatever method we adopt to discover group preferences,
what we discover should be transitive as well. The Condorcet paradox
highlights the need to determine precisely what properties we want in go-
ing from a collection of individual rankings to a societal ranking.
So Who Really Won?
As Western civilization began its gradual march toward democracy, dif-
ferent methods of translating individual rankings to societal ones were
suggested. It soon became apparent that the outcome of elections could
depend on what voting method was adopted.
Numerous methods have been used to determine the winner of an elec-
tion in which rank-order voting is used. Some of the methods that have
been frequently employed to determine a winner are
- Most first-place votes. The winning candidate is the one who receives
the most first-place votes. This method, which has been described as
winner take all, is used in England to elect members of Parliament,
and is common in the United States as well on all levels. - Runoff between the top-two first-place vote getters. The candidates who
place either first or second in the number of first-place votes are
matched head-to-head, and the winner is the one preferred by a ma-
jority of voters. If voters submit preference lists on the ballot, this
method does not need a second election (which is customarily held in
the real world, and results in additional monetary costs to both candi-
dates and governments), as it is easy to compute which of the two
candidates is preferred by a majority of voters. Nevertheless, top-two
runoffs are used in electing the mayors of many large cities, includ-
ing New York, Chicago, and Philadelphia. - Survivor (after the popular television show). The candidate who re-
ceives the fewest first-place votes is voted off the island. That candi-
date is removed from consideration and stricken off the existing
210 How Math Explains the World