Activity III: Pairwise Comparisons (Grades 811)
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 headtohead with each other candidate. Each candidate gets one point for a oneonone 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 headtohead matchups, so you need a way to organize and display all possible headtohead matchups. 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 headtohead matchup, 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 headtohead matchup 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 headtohead 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 fourcandidate race) can be used to record the winner in all possible headtohead matchups.
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 N^{2} cells. The table has N^{2} empty cells that indicate headtohead matchups (in this case 16 because there are four candidates). The cells on the main diagonal are eliminated in every case, so the number of matchups is N^{2} – N. Half of the remaining cells are duplicates of each other, so divide by 2 to get (N^{2} – N)/2 for the total number of matchups 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 headtohead matchups 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 headtohead matchup, 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 headtohead matchup between Shawn and Twanda?
3. Who is the winner in the headtohead matchup between Shawn and Ricco?
4. Who is the winner in the headtohead matchup between Gail and Twanda?
5. Who is the winner in the headtohead matchup between Gail and Ricco?
6. Who is the winner in the headtohead matchup between Twanda and Ricco?
Now that we know the headtohead 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?
