Quick Review of Preference Schedules
An election is held between four candidates, A, B, C, and D. The preference
ballots of 8 voters are shown.
A
B
C
D
A
C
D
B
A
B
C
D
C
D
B
A
A
B
C
D
C
D
B
A
C
D
B
A
A
C
D
B
Recording each individual ballot can be inefficient. A preference schedule is
a more compact way of representing the same information.
3
A
B
C
D
Paul Koester ()
2
A
C
D
B
3
C
D
B
A
MA 111, Plurality Method
January 18, 2012
1 / 13
Quick Review of Preference Schedules
3
A
B
C
D
2
A
C
D
B
3
C
D
B
A
The ballot [A, B, C, D] appeared in the original list three times. This same
ballot is recorded once in the preference schedule, along with the
multiplicity, 3.
We can easily read information from the preference schedule. For example, A
receives the most first place votes (3 + 2 = 5).
Paul Koester ()
MA 111, Plurality Method
January 18, 2012
2 / 13
Quick Review of Preference Schedules
We can also record the information from the ballots in a different way:
First
Second
Third
Fourth
A
5
0
0
3
B
0
3
3
2
C
3
2
3
0
D
0
3
2
3
There doesn’t seem to be an official name attached to this type of table.
I will refer to this as a Gary Table, in honor of a former student who
suggested using this method instead of preference schedules.
Last time, we saw that the sum across each row, and the sum along each
column, will always be equal to the number of voters.
Paul Koester ()
MA 111, Plurality Method
January 18, 2012
3 / 13
How much information is contained in preference
schedules?
Suppose an election involves 3 candidates, X, Y, and Z. Any preference ballot
must be one of six types. For an election with three candidate, say A, B, and
C, voters can cast one of 6 types of linear preference ballots:
           
X
X
Y
Y
Z
Z
 Y  ,  Z  , X  ,  Z  , X  ,  Y 
Z
Y
Z
X
Y
X
This means that, regardless of the number of voters (even if there are
thousands or voters!), all the information in ballots can be recorded in just 6
numbers which appear along the top row of
X
Y
Z
X
Z
Y
Y
X
Z
Y
Z
X
Z
X
Y
Z
Y
X
(We usually exclude columns if their multiplicity is 0.)
Paul Koester ()
MA 111, Plurality Method
January 18, 2012
4 / 13
How much information is contained in preference
schedules?
Last time, you discovered that in an election involving 4 candidates, there
were 24 distinct preference ballots.
Where did the 24 come from? How many preference ballots are there in a 5
candidate election? 6 candidate election? N candidate election?
You can count the number of ballots in a 4 candidate election as follows:
4:
3:
2:
1:
Choose one of the four candidates to be listed at the top
Choose one of the remaining three to be listed second
Choose one of the remaining two to be list third
The remaining candidate must be listed last.
Thus, the total number of choices is
4 · 3 · 2 · 1 = 24
Paul Koester ()
MA 111, Plurality Method
January 18, 2012
5 / 13
How much information is contained in preference
schedules?
You can count the number of ballots in a 5 candidate election as follows:
5:
4:
3:
2:
1:
Choose one of the five candidates to be listed at the top
Choose one of the four candidates to be listed second
Choose one of the remaining three to be listed third
Choose one of the remaining two to be list fourth
The remaining candidate must be listed last.
Thus, the total number of choices is
5 · 4 · 3 · 2 · 1 = 120
Products of successive whole numbers are called factorials and are denoted
with an !.
3! = 3 · 2 · 1 = 6,
4! = 4 · 3 · 2 · 1 = 24,
5! = 5 · 4 · 3 · 2 · 1
In general, a preference schedule for an N candidate election has N ! columns.
(Again, we usually only list the columns that have multiplicity greater than 0.)
Paul Koester ()
MA 111, Plurality Method
January 18, 2012
6 / 13
An exercise for next time:
It turns out that the Gary Schedules contain LESS information than the
traditional preference schedule. As homework, you are asked to construct two
DIFFERENT preference schedules which give rise to the SAME Gary
schedule:
First
Second
Third
A
1
3
1
B
2
1
2
C
2
1
2
WARNING: Be sure to allot enough time for this assignment. You may get
lucky and get it right away, or you may need to spend a lot of time doing trial
and error.
Paul Koester ()
MA 111, Plurality Method
January 18, 2012
7 / 13
Determining the Winner of an Election:
It turns out that determining the winner of an election depends on BOTH the
ballots and how that ballots are tallied.
For instance, there was disagreement within our class on the first day
concerning which game the ΛΩ fraternity should watch. If you determine the
winner with a point system, then they should watch basketball. If you base
the winner off of which candidate has the most first place votes, then they
should watch football.
Which method is the correct one? I’m going to dodge that question for a little
while.
Before passing judgements on different voting methods, we should try to
classify the different methods.
First, by a voting method, I mean a procedure for determining the winner of
an election. In this course, we will focus primarily on voting methods that use
preference ballots.
Paul Koester ()
MA 111, Plurality Method
January 18, 2012
8 / 13
The Plurarlity Method:
The simplest voting method is to determine the winner based on who has the
most first place votes.
This is the Plurality Method.
Consider the preference schedule from the beginning of class:
3
A
B
C
D
2
A
C
D
B
3
C
D
B
A
In this election, A has 5 first place votes, C has three first place votes, and
neither B nor D have any first place votes. Thus, A is the winner, provided we
determine the winner using the Plurality Method.
Paul Koester ()
MA 111, Plurality Method
January 18, 2012
9 / 13
The Plurarlity Method:
More formally, we say that a candidate is a plurality candidate if that
candidate receives the most number of first place votes.
Notice that plurality is being used in two different definitions: a plurality
candidate is a candidate that receives the most number of first place votes,
whereas the Plurality Method is a method for determining the winner of an
election.
In the above election, not only did A receive more votes than any other
candidate, A actually received more than half of the total first place votes.
More formally, we say that a candidate is a majority candidate if that
candidate receives more than half of the first place votes.
The distinction between a majority and a plurality is very important.
Paul Koester ()
MA 111, Plurality Method
January 18, 2012
10 / 13
How are Majority and Plurality different?
2
A
B
C
2
B
A
C
3
C
B
A
C receives more first place votes than either A or B, so C is a Plurality
Candidate.
No candidate received more than half of the first place votes (exactly half is
7/2 = 3.5, and since a candidate must receive a whole number of votes, a
Majority Candidate would need to receive at least 4 votes)
Paul Koester ()
MA 111, Plurality Method
January 18, 2012
11 / 13
How are Majority and Plurality different?
An election can have multiple plurality candidates (Example?)
An election can have AT MOST ONE majority candidate (Why?)
A majority candidate is automatically a plurality candidate (Why?)
A plurality candidate is not necessarily a majority candidate (like in the
previous slide)
Paul Koester ()
MA 111, Plurality Method
January 18, 2012
12 / 13
Why an election can have at most one majority candidate:
Suppose A is the majority candidate. Thus, A receives more than half of the
first place votes. This means that LESS THAN HALF of the first place votes
are left to be divided amongst the remaining candidates. Thus even if
candidate B receives all of the remaining votes, B will still have fewer than
half of the first place votes.
Why a majority candidate is automatically a plurality candidate:
Suppose A is the majority candidate. Thus, A receives more than half of the
first place votes. This means that LESS THAN HALF of the first place votes
are left to be divided amongst the remaining candidates. Each of the
remaining candidates have less than half of the first place votes, and in
particular, they have fewer votes than A
Paul Koester ()
MA 111, Plurality Method
January 18, 2012
13 / 13