The Method of Pairwise Comparisons

For the Method of Pairwise Comparisons, each candidate (or alternative) is matched head-to-head (one-on-one) with each of the other candidates. Each candidate (alternative) gets 1 point for a one-on-one win and a half a point for a tie. The candidate (alternative) with the most total points is the winner.
For instance, in a 4 candidate election, suppose candidate A beats candidates B and D head-to-head, candidate B beats candidate D and ties candidate C head-to-head, candidate C beats candidate A and ties candidate B head-to-head, and candidate D beats candidate C head-to-head. Candidate A would get 2 points (1 each for beating B and D), candidate B would get 1.5 points (1 point for beating D and half a point for the tie with C), candidate C would get 1.5 points (1 point for beating A and half a point for the tie with B), and candidate D would get 1 point (for beating C). Since candidate A has the highest point total, candidate A is the winner by the Method of Pairwise Comparisons.
The Method of Pairwise Comparisons was explicitly designed to satisfy the fairness criterion called the Condorcet Criterion. The Condorcet Criterion addresses the fairness of declaring a candidate the winner even though some other candidate won all possible head-to-head matchups. With the Method of Pairwise Comparisons, any candidate who wins all possible head-to-head matchups always has a higher point total than any other candidate and thus is declared the winner.
The mayor of Smallville is being chosen in an election using the Method of Pairwise Comparisons. The four candidates are Paul (the town barber), Rita (head of the town council), Sarah (Superintendent of Education), and Tim (former District Attorney).

500 registered voters cast their preference ballots. The results are summarized in the preference schedule below.

# of

Voters

Place

130 120 100 150
1st P T T S
2nd R R R R
3rd S S P P
4th T P S T
The Method of Pairwise Comparisons revolves around head-to-head matchups so, to begin, we need an organized way of displaying all possible head-to-head matchups. Perhaps the simplest way to do this is with a table giving each candidate a row and a column.
  Paul Rita Sarah Tim
Paul        
Rita        
Sarah        
Tim        
This table gives us a place to list the winner of each head-to-head matchup. For instance, to record the winner of the head-to-head matchup between Paul and Rita, just go to the Paul row and move over to the Rita column.
  Paul Rita Sarah Tim
Paul   *    
Rita        
Sarah        
Tim        
Of course, there are certain cells in the table that are not needed. For example, the winner of the head-to-head matchup between Paul and Rita could also have been recorded by going to the Rita row and moving over to the Paul column.
  Paul Rita Sarah Tim
Paul        
Rita *      
Sarah        
Tim        
Since the cell in the Paul row and Rita column represents the same matchup as the cell in the Rita row and Paul column, we can just cross one of these cells out. And, since this is true for every row, column pair, we can just cross out all the cells below (or above) the main diagonal (the cells where the row and column names are the same: Paul row, Paul column; Rita row, Rita column, etc.).
  Paul Rita Sarah Tim
Paul        
Rita ----------      
Sarah ---------- ----------    
Tim ---------- ---------- ----------  
We can also eliminate all the cells on the main diagonal since none of the candidates compete against themselves (no Sarah versus Sarah matchup).

 

  Paul Rita Sarah Tim
Paul ----------      
Rita ---------- ----------    
Sarah ---------- ---------- ----------  
Tim ---------- ---------- ---------- ----------
The remaining 6 cells (for the 4 candidate case) can be used to record all the possible head-to-head winners. As a bonus, we can also look at the number of cells above the main diagonal to find out how many head-to-head matchups are possible. Since there are 6 cells above the main diagonal, we know that there are 6 possible head-to-head matchups.
Formula for the total number of head-to-head matchups

Now let's fill in the 6 empty cells in the table above. The first empty cell is for the winner of the head-to-head matchup between Paul and Rita. To determine this winner, we must go back to the preference schedule and see how many voters prefer Paul to Rita and how many prefer Rita to Paul.

Since there are 500 voters, a candidate must get at least 251 votes (a majority) to win a head-to-head matchup.

Remember, we are dealing with the head-to-head matchup between Paul and Rita so we only need to look at Paul and Rita.

# of

Voters

Place

130 120 100 150
1st P T T S
2nd R R R R
3rd S S P P
4th T P S T

Who wins the head-to-head matchup between Paul and Rita? answer

 

Now we move to the head-to-head matchup between Paul and Sarah.

# of

Voters

Place

130 120 100 150
1st P T T S
2nd R R R R
3rd S S P P
4th T P S T

Who wins the head-to-head matchup between Paul and Sarah? answer

 

Now we move to the head-to-head matchup between Paul and Tim.

# of

Voters

Place

130 120 100 150
1st P T T S
2nd R R R R
3rd S S P P
4th T P S T

Who wins the head-to-head matchup between Paul and Tim? answer

 

Next we move to the head-to-head matchup between Rita and Sarah.

# of

Voters

Place

130 120 100 150
1st P T T S
2nd R R R R
3rd S S P P
4th T P S T

Who wins the head-to-head matchup between Rita and Sarah? answer

 

Next we move to the head-to-head matchup between Rita and Tim.

# of

Voters

Place

130 120 100 150
1st P T T S
2nd R R R R
3rd S S P P
4th T P S T

Who wins the head-to-head matchup between Rita and Tim? answer

 

Finally, we move to the head-to-head matchup between Sarah and Tim.

# of

Voters

Place

130 120 100 150
1st P T T S
2nd R R R R
3rd S S P P
4th T P S T

Who wins the head-to-head matchup between Sarah and Tim? answer

 

The completed head-to-head matchup table is shown below.

  Paul Rita Sarah Tim
Paul ---------- RITA SARAH PAUL
Rita ---------- ---------- RITA RITA
Sarah ---------- ---------- ---------- SARAH
Tim ---------- ---------- ---------- ----------

 

Now that we know the 6 head-to-head winners, we can total up points and declare the winner of the election.
Remember, 1 point for a win, half a point for a tie.

What is the point total for Rita? answer

What is the point total for Sarah? answer

What is the point total for Paul? answer

What is the point total for Tim? answer

Who wins the election using the Method of Pairwise Comparisons? answer


see another example


The examples above deal only with 4 candidate elections and require 6 head-to-head matchups to determine a Pairwise Comparisons winner. Using the formula for the total number of head-to-head matchups, N(N-1)/2 (where N is the number of candidates), we can see that the number of candidates in an election tells us immediately how many head-to-head matchups we will need to do in order to determine the winner of a Pairwise Comparisons election. For instance, a 3 candidate Pairwise Comparisons election requires 3 [3(3-1)/2 = 3(2)/2 = 6/2 = 3] head-to-head matchups but a 5 candidate Pairwise Comparisons election requires 10 [5(5-1)/2 = 5(4)/2 = 20/2 = 10] head-to-head matchups.

List all the head-to-head matchups in a Pairwise Comparisons election among candidates Abbott, Barnes, and Collins. answer

List all the head-to-head matchups in a Pairwise Comparisons election among candidates Devers, Evans, Franklin, and Grant. answer

List all the head-to-head matchups in a Pairwise Comparisons election among candidates Henry, Iverson, Jenkins, Kent, and Lawson. answer

List all the head-to-head matchups in a Pairwise Comparisons election among candidates May, Nash, Overton, Pickens, Queen, and Rogers. answer


The Method of Pairwise Comparisons uses all the information from the preference table BUT not all at once. As each pair of candidates is matched up, the information about the remaining candidates is ignored. In this regard, the Method of Pairwise Comparisons is similar to the Method of Plurality with Elimination. With the Method of Plurality with Elimination, only the first place votes are used in each round. However, since lower order preferences (2nd, 3rd, etc.) can make it onto the first place row in later rounds, the Plurality with Elimination Method uses much more of the information contained in the preference schedule than does the Plurality Method. When using the Plurality Method, all of the information in the preference schedule not related to first place is ignored. This is a powerful theoretical argument in favor of the Method of Plurality with Elimination and the Method of Pairwise Comparisons over the Plurality Method. Of course, of the 4 preferential voting methods covered here, the Borda Count Method is the only one that uses all the information in the preference table simultaneously.

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