VOTING SYSTEMS
Section 2.5
Introduction
• The rights and duties of a citizen are captured in a simple one
word mantra – VOTE!
• In this section we discuss different types of ways to count the
votes.
• Counting the votes is the heart of democracy
• Is the counting fair? How well does the counting work? These
are the questions of Voting Theory.
• Not to disappoint you but a mathematical economist by the
name of Kenneth Arrow has a theorem called Arrow’s
Impossibility Theorem.
• The Arrow Theorem states that a method for determining
election results that is democratic and always fair is a
mathematical impossibility.
• Donal Saari a Mathematican and author of Chaotic Elections!
A Mathematician Looks at Voting, has said: “For a price I will
serve as a consultant to your group for your next election. Tell
me who you want to win and I will devise a 'democratic
process' which ensures the election of your candidate.”
Reading the preference schedule
• The total number of ballots cast are
14+10+8+4+1=37.
• We see that Kirk received 14 first place votes.
• Archer received 11 first place votes.
• Pike received 8 first place votes, and Piccard got
4 first place votes.
• Since James T. Kirk received the most first place
votes we can declare him the winner.
• This is an example of using the plurality method
to determine the winner of an election.
Majority vs. Plurality
• Winning an election with a majority is when a candidate
receives 50%+1 first place votes or more.
• The plurality method says that the candidate with the
most first place votes is the winner.
• In the preference schedule from before, we saw that
there were a total of 37 votes.
• A candidate winning with a majority would need 19 first
place votes.
• None of the candidates received a majority.
• Kirk won the election with a plurality of 14 votes.
• President Clinton never won a majority of the popular
vote in his two elections. The only President not to do so.
The problems with Plurality
• The plurality method has many flaws and is
considered a poor method for choosing a winner
of an election.
• The Condorcet Criterion states that if there is a
candidate in head-to-head comparisons is
preferred by the voters over each of the other
choices, then that choice should be the winner.
• The plurality method violates this criterion.
• In the previous example, 14 voters preferred
Kirk, while 23 voters preferred someone other
than Kirk.
Pairwise Comparison Method
(Copeland’s Method)
• The method of pairwise comparison is like
a round-robin tournament. Each candidate
is matched head-to-head with every other
candidate.
• If a candidate wins a head-to-head
matchup, they receive 1 point. If the
matchup is tied each candidate gets ½ a
point.
How many matchups????
Who wins???
• To determine the
correct number of matchups, use the combination
n!
formula C  (n  r )!r! .
• We have four candidates in the election and 2 candidates in a
matchup, so use 4 choose 2. Which is 6.
n
•
r
Here are the matchups
– Kirk vs. Piccard: 23 voters prefer Piccard, 14 prefer Kirk thus Piccard gets the
head-to-head victory. One point for Piccard.
– Kirk vs. Pike: 23 voters prefer Pike, 14 prefer Kirk, thus Pike gets the victory and
the point.
– Kirk vs. Archer: 23 voters prefer Archer, 14 prefer Kirk, Archer wins the matchup
and gets a point.
– Piccard vs. Pike: 28 voters prefer Piccard, 9 prefer Pike, Piccard gets the point.
Piccard has two points.
– Piccard vs. Archer: 19 voters preferred Archer, only 18 for Piccard, thus Archer
gets the point and she now has two.
– Pike vs. Archer: 25 voters preferred Archer, only 12 for Pike, Archer gets a point
and he now has three points.
• Archer got 3 points, Piccard 2 points, Pike 1 point, Kirk 0 points.
Jonathan Archer is your winner.
What’s wrong with Pairwise
Comparisons
• Pairwise Comparison’s violate a criterion
known as Independence-of-irrelevantAlternatives.
• What this means is that if choice X wins an
election, and one or more other choices
are disqualified, then when the ballots are
recounted, choice X should still win the
election.
Borda Count Method
• Each voter ranks all of the candidates. If there
are k candidates, each candidate receives k
points for 1st choice, k – 1 points for second
choice, and so on. The candidate with the most
points is declared the winner.
• The Borda Count method takes into account all
the information from the ballot and it produces
the best compromise candidate. The problem
with Borda Count is that it can violate the
majority criterion. That is a candidate could have
the majority of the 1st place votes but lose with
the Borda Count.
Borda Count Example
• Referring to the preference schedule slide, we see that there are
4 candidates in the election. First choice receives 4 points,
second choice receives 3 points, third choice receives 2 points,
and fourth choice receives 1 point.
• Kirk received 14 first choice votes and 23 fourth choice votes for
a Borda Count total of (14 x 4) + (0 x 3) + (0 x 2) + (23 x 1) = 79
points
• Archer received 11 first choice votes, 8 second choice votes and
18 third choice votes for a Borda Count total of (11 x 4) + (8 x 3)
+ (18 x 2) + (0 x 1) = 44 + 24 + 36 = 104 points.
• Pike’s Borda Count is (8 x 4) + (5 x 3) + (10 x 2) + (14 x 1) = 32 +
15 + 20 + 14 = 81 points.
• Piccard’s Borda Count is (4 x 4) + (24 x 3) + (9 x 2) = 16 + 72 +
18 = 106 points.
• Jean-Luc Piccard wins the election in the Borda Count Method.
Example 2 Continued
• Since there are 11 voters in the preference schedule,
6 votes are needed for a majority.
• Candidate Foster has received 6 votes. Under a
majority rules, Foster should be declared the winner.
• However if Borda Count is used we see the following:
– Foster gets (6 x 4) + (5 x 1) = 29 points
– Winkel gets (2 x 4) + (9 x 3) = 35 points
– Heerey gets (3 x 4) + (2 x 3) + (6 x 2) = 30 points
– LeVarge gets (3 x 3) + (2 x 2) + (6 x 1) = 16 points
• Under the Borda Count rule, Winkel would be declared
the winner.
Ranked-Choice or Instant Run-Off
Method
• Each voter ranks all of the candidates first,
second, third, etc. If a candidate receives a
majority of the first choice votes, that candidate
is declared the winner.
• If no candidate receives a majority, then the
candidate that received the fewest first choice
votes is eliminated. Those votes are given to the
next preferred candidate.
• If a candidate now has a majority, that candidate
is declared the winner. If no candidate has a
majority, then the process continues.
Instant Run-Off Example
• Now we see that Kirk only has 14 first choice
votes and Pike has 23 first choice votes.
• Christopher Pike is declared the winner of this
election by the instant run-off method.
• The problem with Instant run-off is that it violates
the Monotonicity criterion.
• The criterion states that if choice X is the winner
of an election, and in a reelection, the only
changes in the ballots are changes that only
favor X, then X should remain the winner.
Questions
1.
2.
3.
4.
5.
6.
7.
8.
9.
How many votes were cast?
Use the Plurality method to determine the winner.
What percent of the votes did the winner in number 2
receive?
Use instant run-off method to determine the winner.
What percent of the votes did the winner in number 4
receive?
Use Borda Count to determine the winner.
How many points did the winner in number 6 receive?
Use pairwise comparison to determine the winner.
How many points did the winner receive in number 8?