Ballots
A ballot is a form for recording a voter’s say in an election.
Ballots can record different types of information.
A ballot could record:
(a) voter’s favorite candidate
(b) voter’s three favorite candidates, in order of preference
(c) voter’s three favorite candidates, in no particular order
(d) voter’s ranking of all candidates, from favorite to least favorite
(e) etc.
In this course, we will be mostly interested in Preference Ballots, in which
voters rank all of the candidates from favorite to least favorite.
Paul Koester ()
MA 111, Ballots and Preference Schedules
January 13, 2012
1 / 6
Assumptions about Preference Ballots
We will make the following assumptions concerning preference ballots:
A preference ballot cannot list two candidates in the same rank. For
example, a voter is not allowed to say “I like A and B equally.” A
preference ballot in which no two candidates are in the same rank is
called a Linear Ballot.
Suppose that a voter ranks A before B and B before C. Then we will infer
that this voter would rank A before C. (The technical term is to say that
voter’s preferences are transitive)
Can you think of a situation involving three items in which the items are NOT
transitively ranked? In other words, A beats B, B beats C, and C beats A?
Paul Koester ()
MA 111, Ballots and Preference Schedules
January 13, 2012
2 / 6
Preference Schedules
Last class, we looked at an election involving 21 voters and 5 “candidates”, or
choices.
Several methods were suggested for determing the winner of the election.
Before we discuss the “correct method for determining the winner,” we should
try to find a better way of recording all of the data from the election.
For instance, it is not easy to quickly read information off of the ballots on the
next slide:
Paul Koester ()
MA 111, Ballots and Preference Schedules
January 13, 2012
3 / 6
The ΛΩ Example from last class
Ballot
B
G
S
F
H
Ballot
F
S
H
B
G
Ballot
H
G
F
B
S
Ballot
F
H
B
S
G
Ballot
B
G
S
F
H
Ballot
S
B
H
G
F
Ballot
F
H
B
S
G
Ballot
F
H
B
S
G
Ballot
B
G
S
F
H
Ballot
F
S
H
B
G
Ballot
H
G
F
B
S
Ballot
B
G
S
F
H
Ballot
F
H
B
S
G
Ballot
B
G
S
F
H
Ballot
H
G
F
B
S
Ballot
F
H
B
S
G
Paul Koester ()
Ballot
S
B
H
F
G
Ballot
S
B
H
G
F
Ballot
F
S
H
B
G
MA 111, Ballots and Preference Schedules
Ballot
S
B
H
F
G
Ballot
S
B
H
G
F
January 13, 2012
4 / 6
How to condense all of this information into a more useable form?
Preference Schedule: Even though 21 votes were cast, many of the voters
cast the “same”
  set of preferences:
 
B
F
G
S 
 
 

 
The ballot 
 S  was cast by five different voters, the ballot H  was cast by
F 
B 
H
G
three different voters, etc.
Thus, instead of keeping track of all 21 ballots, it would be more efficient to
just record the DISTINCT ballots, along with the number of voters that cast
each ballot type.
Paul Koester ()
MA 111, Ballots and Preference Schedules
January 13, 2012
5 / 6
Preference Schedule
21 votes were cast, but there was only 6 distinct types of ballots that were
cast. A preference schedule records the DISTINCT ballots. The top row
records the number of times each type of ballot was cast.
5
B
G
S
F
H
3
F
S
H
B
G
3
H
G
F
B
S
5
F
H
B
S
G
3
S
B
H
G
F
2
S
B
H
F
G
It is much easier to read information from the preference schedule than it is to
read information off of the ballots.
How many first place votes did F get?
Paul Koester ()
MA 111, Ballots and Preference Schedules
January 13, 2012
6 / 6
Preference Schedule
21 votes were cast, but there was only 6 distinct types of ballots that were
cast. A preference schedule records the DISTINCT ballots. The top row
records the number of times each type of ballot was cast.
5
B
G
S
F
H
3
F
S
H
B
G
3
H
G
F
B
S
5
F
H
B
S
G
3
S
B
H
G
F
2
S
B
H
F
G
It is much easier to read information from the preference schedule than it is to
read information off of the ballots.
How many first place votes did F get?
F came in first on the second type of ballot and the fourth type of ballot.
Now, the second type of ballot was cast by 3 voters and the fourth type of
ballot was cast by 5 voters, so F received 3 + 5 = 8 first place votes.
Paul Koester ()
MA 111, Ballots and Preference Schedules
January 13, 2012
6 / 6