# Group-Ranking

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```					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.
each choice’s points.
Borda Example
   For this example, to calculate
A           B           the Borda winner we would
B           C
C           D           do:
D           A

8       5               A: 8(4)+5(1)+6(1)+7(1)=50
C               D   B: 8(3)+5(4)+6(3)+7(3)=83
B
D
B   C: 8(2)+5(3)+6(4)+7(2)=69
C
A               A   D: 8(1)+5(2)+6(2)+7(4)=58
6           7
You try!
   Determine the plurality and Borda winner
for the set of preferences shown below:

C          C
A         B

D         D             B          D

C         A             D          A

B         C                        B
A

16        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                   6               7

A               D           D           D
D               A           A           A
8           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
A         B

D         D             B          D

C         A             D          A

B         C                        B
A

16        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          B           C             D

B         C            B             B

C         D            D             C
D         A            A             A

8         5            6            7

   B is eliminated since it has the fewest
firsts.
Sequential Runoff Method (cont’d)

A                     C             D
C

C         D           D             C

D         A           A             A

8         5           6           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)

A        C           C           C

C        A           A           A

8        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
A         B

D         D             B          D

C         A             D          A

B         C                        B
A

16        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.

B            C
A

B             A            B

C             C            A

52            38           10

```
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