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Group-Ranking Methods and Algorithms

Group-Ranking Methods and Algorithms. Common Methods of Determining a Group Ranking from Individual Preferences.

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Group-Ranking Methods and Algorithms

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  1. Group-Ranking Methods and Algorithms Common Methods of Determining a Group Ranking from Individual Preferences.

  2. Look back at the handout from yesterday where choice B is ranked first on 12 schedules, more often than any other choice. If B wins on this basis, B is called the plurality winner.

  3. The plurality winner is based on first-place rankings only. The winner is the choice that receives the most votes. Note, however, that B is first only on about 46.1% of the schedules. Had B been first on over half the schedules, B would be a majority winner.

  4. The table above is another way of representing the preferences schedules we’ve been working with.

  5. TheBorda Count Method a.k.a. Points-for-Preference When applying the Borda count to a ranking of n choices, n points are assigned to a first-place ranking, n-1 points to a second-place ranking, n-2 points to a third-place ranking,….and 1 point to a last-place ranking. The group ranking is established by totaling each choice’s points.

  6. Look back at the rankings on yesterday’s handout. Borda Count: A: 8(1) + 5(2) + 6(1) + 7(1) = 31. B: 8(2) + 5(4) + 6(2) + 7(4) = 76. C: 8(4) + 5(1) + 6(3) + 7(3) = 76. D: 8(3) + 5(3) + 6(4) + 7(2) = 77. C is ranked first by 8 people and fourth by 5 people, and second by everyone else (the remaining 13 people), so C’s points total 8(4) + 5(1) + 13 (3) = 76. Similar calculations give totals of 31, 76, and 77 for A, B, and D, respectively.

  7. The Runoff Method To conduct a runoff, determine the number of firsts for each choice. Eliminate all the choices except the two with the highest first-place totals. Next consider each preference schedule to determine which of the two remaining candidates is ranked higher.

  8. With each preference schedule “shift” the number of votes to the candidate ranked highest among the two remaining. Total the number of voters for each candidate. The candidate with the highest total is the runoff (with plurality) winner.

  9. Sequential Runoff Method: Eliminates one choice at a time.

  10. A is eliminated first because it is ranked first the fewest times. Since A has no first-place votes there is nothing for us to shift.

  11. The point totals are now 5 + 7 = 12 for B, 8 for C, and 6 for D. There are three choices remaining.

  12. Now D’s total is the smallest so D is eliminated next. The 6 votes are transferred to the remaining choice that is ranked highest by these 6.

  13. Thus, C is given an additional 6 votes and so defeats B by 14 to 12.

  14. Now work on the follow-up exercises.

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