ballots, which are then reexamined. The whole process is repeated,
until only two remain. The winner of this two-person contest is the
one preferred by the majority. The fact that it is possible to avoid run-
offs with this method has resulted in it being referred to as instant
runoff voting (IRV). This method is used by Australia in its national
elections, and was adopted in Oakland, California, in 2006.- Numerical total. Each voter ranks the candidates on a ballot, and each
rank is assigned a point value: for example, a first-place vote could
give a candidate 5 points, a second-place vote 4 points, and so on. The
winner is the candidate who receives the most total points. This
method was suggested by the fifteenth-century mathematician Nich-
olas of Cusa for electing the Holy Roman Emperors,^5 but today it is
used only for major elections in a few small countries. However, it is
widely used in nonpolitical elections; the American and National
League most valuable players are chosen this way. - Head-to-head matchups. Each candidate is matched head-to-head with
each other candidate. The candidate with the most number of head-to-
head victories is the winner. One major advantage of this method, as
we shall see later, is that it avoids the Condorcet paradox. However,
this method will frequently not give a clear winner; if two candidates
tie for first using this method, the winner is determined by the result
of the head-to-head matchup between these two candidates. The most
widespread use of this method is in round-robin tournaments, which
frequently occur in sports, or in games such as chess.
Each of these methods has its advocates, and each has been used in
many elections. However, the hypothetical election shown in the table
gives us a sense of how difficult it may be to find a good method of select-
ing a winner from a collection of preference rankings. Five candidates—
A, B, C, D, and E—are running for office. Fifty-five ballots were cast, but
only six different preference rankings occurred. Here are the results.^6
Number First Second Third Fourth Fifth
of Ballots Choice Choice Choice Choice Choice
18 A D E C B
12 B E D C A
10 C B E D A
9 D C E B A
4 E B D C A
2 E C D B ACracks in the Foundation 211