The preference schedule below represents an election among George (G), Holly (H), James (J), and Inez (I).

   

# of Voters

 

Place

13 12 10
1st G I J
2nd H H G
3rd J J H
4th I G I

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.

  Feorge Holly James Inez
George        
Holly        
James        
Inez        

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 Holly and James, just go to the Holly row and move over to the James column.

  George Holly James Inez
George        
Holly     *  
James        
Inez        

Of course, there are certain cells in the table that are not needed. For example, the winner of the head-to-head matchup between Holly and James could also have been recorded by going to the James row and moving over to the Holly column.

  George Holly James Inez
George        
Holly        
James   *    
Inez        

Since the cell in the Holly row and James column represents the same matchup as the cell in the James row and Holly 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: George row, George column; Inez row, Inez column, etc.).

  George Holly James Inez
George        
Holly ----------      
James ---------- ----------    
Inez ---------- ---------- ----------  

We can also eliminate all the cells on the main diagonal since none of the candidates compete against themselves (no Holly versus Holly matchup).

  George Holly James Inez
George ----------      
Holly ---------- ----------    
James ---------- ---------- ----------  
Inez ---------- ---------- ---------- ----------

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 George and Holly. To determine this winner, we must go back to the preference schedule and see how many voters prefer George to Holly and how many prefer Holly to George.

Since there are 35 voters, a candidate must get at least 18 votes (a majority) to win a head-to-head matchup.

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

   

# of Voters

 

Place

13 12 10
1st G I J
2nd H H G
3rd J J H
4th I G I

Who wins the head-to-head matchup between George and Holly? answer

 

Now we move to the head-to-head matchup between George and James.

   

# of Voters

 

Place

13 12 10
1st G I J
2nd H H G
3rd J J H
4th I G I

Who wins the head-to-head matchup between George and James? answer

 

Now we move to the head-to-head matchup between George and Inez.

   

# of Voters

 

Place

13 12 10
1st G I J
2nd H H G
3rd J J H
4th I G I

Who wins the head-to-head matchup between George and Inez? answer

 

Next we move to the head-to-head matchup between Holly and James.

   

# of Voters

 

Place

13 12 10
1st G I J
2nd H H G
3rd J J H
4th I G I

Who wins the head-to-head matchup between Holly and James? answer

 

Next we move to the head-to-head matchup between Holly and Inez.

   

# of Voters

 

Place

13 12 10
1st G I J
2nd H H G
3rd J J H
4th I G I

Who wins the head-to-head matchup between Holly and Inez? answer

 

Finally, we move to the head-to-head matchup between James and Inez.

   

# of Voters

 

Place

13 12 10
1st G I J
2nd H H G
3rd J J H
4th I G I

Who wins the head-to-head matchup between James and Inez? answer

 

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

  George Holly James Inez
George ---------- GEORGE JAMES GEORGE
Holly ---------- ---------- HOLLY HOLLY
James ---------- ---------- ---------- JAMES
Inez ---------- ---------- ---------- ----------

 

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 George? answer

What is the point total for James? answer

What is the point total for Holly? answer

What is the point total for Inez? answer

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

 

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