1. Group-Ranking How is group ranking accomplished?

2. NC Standard Course of Study Competency Goal 2: 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: Objective 2.03: Model and solve problems involving fair outcomes: Apportionment. Apportionment. Election Theory. Election Theory. Voting Power. Voting Power. Fair Division. Fair Division.

3. Types of Winners There are several ways that the winner can be chosen from a group-ranking situation. 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. 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. If the winner is chosen because they are first on more than half of the preferences, the winner is known as the majority winner.

4. Group-Ranking There are many methods used to rank preferences. These methods include: There are many methods used to rank preferences. These methods include: The Borda Method The Borda Method The Runoff Method, and The Runoff Method, and The Sequential Runoff Method The Sequential Runoff Method

5. 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. 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. 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. The group ranks are then made by adding each choice’s points.

6. Borda Example For this example, to calculate the Borda winner we would do: 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 A A A A B B B B C C C C D D D D 85 67

7. You try! Determine the plurality and Borda winner for the set of preferences shown below: Determine the plurality and Borda winner for the set of preferences shown below: A A A A BB B B C C CC DD D D 1620127

8. The Runoff Method This is a very popular method, that we currently use (as in runoff elections) This is a very popular method, that we currently use (as in runoff elections) Is expensive and time-consuming. Is expensive and time-consuming. Use preference schedules to avoid hassles. Use preference schedules to avoid hassles.

9. Runoff Method Process To conduct a runoff, To conduct a runoff, Determine the number of firsts for each choice Determine the number of firsts for each choice Eliminate all but the two highest totals 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 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

10. Runoff Method Example 8 56 7 A AAA BBB B C C C C D DD D A A AA D DD D 85 67

11. Runoff Example (continued) Notice that the runoff method eliminates all choices except the two with the most firsts: Notice that the runoff method eliminates all choices except the two with the most firsts: Therefore, B and C were eliminated 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. Because those two are eliminated, the choice that ranks highest of the remaining choices gets the votes for that group. A:8D: 7 + 5 + 6 =18 Therefore, D is the runoff winner. Therefore, D is the runoff winner.

12. You try! Determine the Runoff winner for the set of preferences shown below: Determine the Runoff winner for the set of preferences shown below: A A A A BB B B C C CC DD D D 1620127

13. Sequential Runoff Method The sequential runoff method differs from the runoff method because it eliminates choices one at a time. 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. It eliminates the one that is ranked first the fewest times, and the points are awarded to the next highest choice.

14. Sequential Runoff Method Example B is eliminated since it has the fewest firsts. B is eliminated since it has the fewest firsts. 8 56 7 A AAA BBB B C C C C D DD D

15. Sequential Runoff Method (cont’d) The five votes for B are awarded to C: The five votes for B are awarded to C: A: 8C: 6+5= 11D:7 (D is eliminated) 8 567 A AAA C C C C D DD D

16. Sequential Runoff Method (cont’d) The seven votes for D are awarded to C: The seven votes for D are awarded to C: A: 8C: 11+ 7 = 18 8 567 A AA A C C C C

17. You try! Determine the sequential runoff winner for the set of preferences shown below: Determine the sequential runoff winner for the set of preferences shown below: A A A A BB B B C C CC DD D D 1620127

18. Homework A panel of sportswriters is selecting the best football team in a league, and the preferences are distributed as shown below. A panel of sportswriters is selecting the best football team in a league, and the preferences are distributed as shown below. A A A B B B CC C 5238 10