1 The Mathematics of Voting
1.1 Preference Ballots and Preference
Schedules
1.2 The Plurality Method
1.3 The Borda Count Method
1.4 The Plurality-with-Elimination Method
(Instant Runoff Voting)
1.5 The Method of Pairwise Comparisons
1.6 Rankings
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 1.3 - 2
The Plurality-with-Elimination Method
When there are three or more candidates
running, it is often the case that no
candidate gets a majority. Typically, the
candidate or candidates with the fewest
first-place votes are eliminated, and a runoff
election is held. Since runoff elections are
expensive to both the candidates and the
municipality, this is an inefficient and a
cumbersome method for choosing a mayor
or a county supervisor.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 1.3 - 3
The Plurality-with-Elimination Method
A much more efficient way to implement the
same process without needing separate
runoff elections is to use preference ballots,
since a preference ballot tells us not only
which candidate the voter wants to win but
also which candidate the voter would choose
in a runoff between any pair of candidates.
The idea is simple but powerful: From the
original preference schedule for the election
we can eliminate the candidates with the
fewest first-place votes one at a time until
one of them gets a majority.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 1.3 - 4
The Plurality-with-Elimination Method
This method has become increasingly
popular and is nowadays fashionably known
as instant runoff voting (IRV).
Other names had been used in the past and
in other countries for the same method,
including plurality-with-elimination and the
Hare method.
We will call it the plurality-with-elimination
method - it is the most descriptive of the
three names.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 1.3 - 5
The Plurality-with-Elimination Method
Round 1: Count the first-place votes for each
candidate, just as you would in the plurality
method. If a candidate has a majority of firstplace votes, then that candidate is the winner.
Otherwise, eliminate the candidate (or
candidates if there is a tie) with the fewest firstplace votes.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 1.3 - 6
The Plurality-with-Elimination Method
Round 2: Cross out the name(s) of the
candidates eliminated from the preference
schedule and recount the first-place votes.
(Remember that when a candidate is eliminated
from the preference schedule, in each column
the candidates below it move up a spot.) If a
candidate has a majority of first-place votes,
then declare that candidate the winner.
Otherwise, eliminate the candidate with the
fewest first-place votes.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 1.3 - 7
The Plurality-with-Elimination Method
Round 3, 4 . . . . Repeat the process, each
time eliminating one or more candidates
until there is a candidate with a majority of
first-place votes. That candidate is the
winner of the election.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 1.3 - 8
Example 1.7 The Math Club Election
(Plurality-with-Elimination)
Let’s see how the plurality-with-elimination
method works when applied to the Math Club
election. For the reader’s convenience Table
1-6 shows the preference schedule again. It
is the original preference schedule for the
election first shown in Table 1-1.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 1.3 - 9
Example 1.7 The Math Club Election
(Plurality-with-Elimination)
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 1.3 - 10
Example 1.7 The Math Club Election
(Plurality-with-Elimination)
Round 1.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 1.3 - 11
Example 1.7 The Math Club Election
(Plurality-with-Elimination)
Round 2. B’s 4 votes go to D, the next best
candidate according to these 4 voters.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 1.3 - 12
Example 1.7 The Math Club Election
(Plurality-with-Elimination)
Round 3. C’s 11 votes go to D, the next best
candidate according to these 11 voters.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 1.3 - 13
Example 1.7 The Math Club Election
(Plurality-with-Elimination)
We now have a winner, and lo and behold,
it’s neither Alisha nor Boris. The winner of the
election, with 23 first-place votes, is Dave!
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 1.3 - 14
What’s wrong with the Pluarality-withElimination Method?
The main problem with the plurality-withelimination method is quite subtle and is
illustrated by the next example.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 1.3 - 15
Example 1.10 There Go the Olympics
Three cities, Athens (A), Barcelona (B), and
Calgary (C), are competing to host the Summer
Olympic Games. The final decision is made by a
secret vote of the 29 members of the Executive
Council of the International Olympic Committee,
and the winner is to be chosen using the
plurality-with-elimination method. Two days
before the actual election is to be held, a straw
poll is conducted by the Executive Council just to
see how things stand. The results of the straw
poll are shown in Table 1-9.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 1.3 - 16
Example 1.10 There Go the Olympics
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 1.3 - 17
Example 1.10 There Go the Olympics
Based on the results of the straw poll,
Calgary is going to win the election. (In the
first round Athens has 11 votes, Barcelona
has 8, and Calgary has 10. Barcelona is
eliminated, and in the second round
Barcelona’s 8 votes go to Calgary. With 18
votes in the second round, Calgary wins the
election.)
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 1.3 - 18
Example 1.10 There Go the Olympics
Although the results of the straw poll are
supposed to be secret, the word gets out that
it looks like Calgary is going to host the next
Summer Olympics. Since everybody loves a
winner, the four delegates represented by the
last column of Table 1-9 decide as a block to
switch their votes and vote for Calgary first
and Athens second. Calgary is going to win,
so there is no harm in that, is there? Well,
let’s see.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 1.3 - 19
Example 1.10 There Go the Olympics
The results of the official vote are shown in
Table 1-10. The only changes between the
straw poll in Table 1-9 and the official vote
are the 4 votes that were switched in favor of
Calgary. (To get Table 1-10, switch A and C
in the last column of Table 1-9 and then
combine columns 3 and 4 - they are now the
same - into a single column.)
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 1.3 - 20
Example 1.10 There Go the Olympics
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 1.3 - 21
Example 1.10 There Go the Olympics
When we apply the plurality-with-elimination
method to Table 1-10, Athens gets eliminated
in the first round, and the 7 votes originally
going to Athens go to Barcelona in the
second round. Barcelona, with 15 votes in the
second round gets to host the next Summer
Olympics!
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 1.3 - 22
Example 1.10 There Go the Olympics
How could this happen? How could Calgary lose
an election it was winning in the straw poll just
because it got additional first-place votes in the
official election? While you will never convince
the conspiracy theorists in Calgary that the
election was not rigged, double-checking the
figures makes it clear that everything is on the up
and up - Calgary is simply the victim of a quirk in
the plurality-with-elimination method: the
possibility that you can actually do worse by
doing better!
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 1.3 - 23
The Monotonicity Criterion
Example 1.10 illustrates what in voting theory
is known as a violation of the monotonicity
criterion.
THE MONOTONICITY CRITERION
If candidate X is a winner of an election
and, in a reelection, the only changes in
the ballots are changes that favor X (and
only X), then X should remain a winner of
the election.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 1.3 - 24
What’s Wrong with Plurality-withElimination
• It violates the monotonicity criterion
• It violates the Condorcet criterion
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 1.3 - 25
Plurality-with-Elimination method in
Real Life a.k.a. Instant Runoff Voting
• International Olympic Committee to choose
host cities
• Since 2002, San Francisco, CA
• Since 2005, Burlington, VT
• In process, Berkeley, CA
• In process, Ferndale, MI
• Australia to elect members of the House of
Representatives
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 1.3 - 26