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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.
 The group ranks are then made by adding
  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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