Group-Ranking
How is group ranking
accomplished?
NC Standard Course of Study

Competency Goal 2: The learner will
analyze data and apply probability
concepts to solve problems.
 Objective 2.03: Model and solve problems
involving fair outcomes:
 Apportionment.
 Election
Theory.
 Voting Power.
 Fair Division.
Types of Winners
There are several ways that the winner can be
chosen from a group-ranking situation.
 When the winner is chosen because they are
ranked first more than any other choice, the
winner is known as the plurality winner.
 If the winner is chosen because they are first
on more than half of the preferences, the
winner is known as the majority winner.

Group-Ranking

There are many methods used to rank
preferences. These methods include:
 The
Borda Method
 The Runoff Method, and
 The Sequential Runoff Method
The Borda Method
In the Borda method, points are assigned
to the choices by the order they come,
this is known as a Borda count.
 To do a Borda count you rank n number of
choices by assigning n points to the first
choice, n-1 to the second, n-2 to the third,
… and 1 point to the last.
 The group ranks are then made by adding
each choice’s points.

Borda Example

A
B
C
D
8
B
C
D
A
5
C
B
D
A
6
D
B
C
A
7
For this example, to calculate
the Borda winner we would
do:
A: 8(4)+5(1)+6(1)+7(1)=50
B: 8(3)+5(4)+6(3)+7(3)=83
C: 8(2)+5(3)+6(4)+7(2)=69
D: 8(1)+5(2)+6(2)+7(4)=58
You try!

Determine the plurality and Borda winner
for the set of preferences shown below:
C
C
D
B
D
A
D
A
A
B
D
C
C
B
16
B
A
20
12
7
The Runoff Method
This is a very popular method, that we
currently use (as in runoff elections)
 Is expensive and time-consuming.
 Use preference schedules to avoid hassles.

Runoff Method Process

To conduct a runoff,
 Determine
the number of firsts for each
choice
 Eliminate all but the two highest totals
 Then consider each preference schedule on
which the eliminated choices were chosen
first and the points from that preference
awarded to the choice that ranked highest
Runoff Method Example
A
B
C
D
B
C
B
B
C
D
D
C
D
A
A
A
8
5
8
6
7
A
D
D
D
D
A
A
A
5
6
7
Runoff Example (continued)

Notice that the runoff method eliminates
all choices except the two with the most
firsts:
 Therefore,
B and C were eliminated
Because those two are eliminated, the
choice that ranks highest of the remaining
choices gets the votes for that group.
A:8
D: 7 + 5 + 6 =18
 Therefore, D is the runoff winner.

You try!

Determine the Runoff winner for the set of
preferences shown below:
C
C
D
B
D
A
D
A
A
B
D
C
C
B
16
B
A
20
12
7
Sequential Runoff Method
The sequential runoff method differs from
the runoff method because it eliminates
choices one at a time.
 It eliminates the one that is ranked first
the fewest times, and the points are
awarded to the next highest choice.

Sequential Runoff Method Example
A
8

B
C
D
B
C
B
B
C
D
D
C
D
A
A
A
5
6
7
B is eliminated since it has the fewest
firsts.
Sequential Runoff Method (cont’d)
8

A
C
C
D
D
A
5
D
C
D
C
A
6
A
7
The five votes for B are awarded to C:
A: 8
C: 6+5= 11
D:7
(D is eliminated)
Sequential Runoff Method (cont’d)
8

A
C
C
C
C
A
A
A
5
6
7
The seven votes for D are awarded to C:
A: 8
C: 11+ 7 = 18
You try!

Determine the sequential runoff winner for
the set of preferences shown below:
C
C
D
B
D
A
D
A
A
B
D
C
C
B
16
B
A
20
12
7
Homework

A panel of sportswriters is selecting the
best football team in a league, and the
preferences are distributed as shown
below.
52
A
B
C
B
A
B
C
C
38
A
10