Group-Ranking

1. Group-Ranking How is group ranking accomplished?

2. 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.

3. 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.

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

6. 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 Borda Example A B B C C D D A 8 5 C D B B D C A A 6 7

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 C B B A 16 20 12 7

8. 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.

9. 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

10. Runoff Method Example D A B C 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

11. 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.

12. 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 C B B A 16 20 12 7

13. 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.

14. Sequential Runoff Method Example • B is eliminated since it has the fewest firsts. D A B C B C B B C D D C D A A A 8 5 6 7

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

16. Sequential Runoff Method (cont’d) • The seven votes for D are awarded to C: A: 8 C: 11+ 7 = 18 A C C C A C A A 8 5 6 7

17. 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 C B B A 16 20 12 7

18. Homework • A panel of sportswriters is selecting the best football team in a league, and the preferences are distributed as shown below. C B A B B A C C A 10 52 38