Activity III: Pairwise Comparisons (Grades 8-11)

Borda Count Method | Plurality Voting | Pairwise Comparisons | Approval Voting | Career Connections | More Math Concepts

NCTM Standards:

• Number and Operation
• Data Analysis, Statistics, and Probability
• Problem Solving
• Reasoning and Proof
• Communication
• Connections
• Representation

Objectives:

• Develop a formula for the number of pairwise comparisons that will have to be considered
• Use the formula to determine the winner in an election requiring pairwise comparison

For the Pairwise Comparison method, each candidate is matched head-to-head with each other candidate. Each candidate gets one point for a one-on-one win and a half point for a tie. The candidate with the most total points is the winner.

Look again at the election of the student council president used in the previous activities. Determine the winner this time using Pairwise Comparison. One thousand five hundred voters cast preference ballots summarized in the following preference schedule. The four candidates are: Gail, Twanda, Shawn, and Ricco.

 Number of votes received Place 390 360 300 450 1st Shawn Gail Gail Ricco 2nd Twanda Twanda Twanda Twanda 3rd Ricco Ricco Shawn Shawn 4th Gail Shawn Ricco Gail

The method of Pairwise Comparison revolves around head-to-head match-ups, so you need a way to organize and display all possible head-to-head match-ups. Consider the following table, in which each candidate has a row and column assigned.

 Shawn Gail Twanda Ricco Shawn Gail Twanda Ricco

To use this table, select a head-to-head match-up, for example between Shawn and Gail. Go to the Shawn row and move over until you reach the column headed Gail, as shown by the * in the cell.

 Shawn Gail Twanda Ricco Shawn * Gail Twanda Ricco

Clearly, you will be duplicating your effort if you begin in the Gail row and move to the column headed Shawn, because that would also indicate a head-to-head match-up between the same two.

 Shawn Gail Twanda Ricco Shawn Gail * Twanda Ricco

To eliminate this duplication, cross out one of these cells for each candidate.

 Shawn Gail Twanda Ricco Shawn Gail ------------ Twanda ------------ ------------ Ricco ------------ ------------ ------------

Since no candidates will go head-to-head with themselves, you can also cross out the cells on the main diagonal.

 Shawn Gail Twanda Ricco Shawn ------------ Gail ------------ ------------ Twanda ------------ ------------ ------------ Ricco ------------ ------------ ------------ ------------
The remaining 6 cells (because you have a four-candidate race) can be used to record the winner in all possible head-to-head match-ups.

1. Determine a formula for the number of cells in terms of the number of candidates N.

Solution: Consider the table below.

 Shawn Gail Twanda Ricco Shawn ------------ Gail ------------ ------------ Twanda ------------ ------------ ------------ Ricco ------------ ------------ ------------ ------------

For N candidates, the table will be a square with N x N or N2 cells. The table has N2 empty cells that indicate head-to-head match-ups (in this case 16 because there are four candidates). The cells on the main diagonal are eliminated in every case, so the number of match-ups is N2 – N. Half of the remaining cells are duplicates of each other, so divide by 2 to get (N2 – N)/2 for the total number of match-ups for N candidates. Factor out N to get N(N – 1)/2, where N is the number of candidates.

Now begin filling in the table of head-to-head match-ups for the student council election. To do this, you must go back to the preference schedule and determine how many voters prefer Shawn to Gail and how many prefer Gail to Shawn.

 Number of votes received Place 390 360 300 450 1st Shawn Gail Gail Ricco 2nd Twanda Twanda Twanda Twanda 3rd Ricco Ricco Shawn Shawn 4th Gail Shawn Ricco Gail

You can see that 840 voters prefer Shawn to Gail and 660 prefer Gail to Shawn. This indicates that in the head-to-head match-up, Shawn is the winner. Place Shawn in the cell indicating the winner.

 Shawn Gail Twanda Ricco Shawn ------------ Shawn Gail ------------ ------------ Twanda ------------ ------------ ------------ Ricco ------------ ------------ ------------ ------------

2. Who is the winner in the head-to-head match-up between Shawn and Twanda?

3. Who is the winner in the head-to-head match-up between Shawn and Ricco?

4. Who is the winner in the head-to-head match-up between Gail and Twanda?

5. Who is the winner in the head-to-head match-up between Gail and Ricco?

6. Who is the winner in the head-to-head match-up between Twanda and Ricco?

Now that we know the head-to-head winners, we can total all the scores and determine the winner of the election.

7. What is the point total for Shawn?

8. What is the point total for Gail?

9. What is the point total for Twanda?

10. What is the point total for Ricco?

11. Who wins the election using the Pairwise Comparison method?