# The Pairwise Comparison Method in Elections

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• 0:02 Pairwise Comparison Method
• 1:39 Pairwise Comparison Process
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Lesson Transcript
Instructor: Maria Airth

Maria has a Doctorate of Education and over 15 years of experience teaching psychology and math related courses at the university level.

The pairwise comparison method in elections is a method of comparing candidates to each other in head-to-head contests. This lesson reviews the pairwise comparison method.

## Pairwise Comparison Method

Hi. Have you ever wondered what would happen if all candidates in an election had to go head to head with each other? I mean, sometimes I wonder what would happen if all the smaller candidates weren't available and voters had to choose between just the major candidates. Would that change the results? Would the smaller candidates actually perform better if they were up against major candidates one at a time?

This is exactly what a pairwise comparison method in elections does. It compares each candidate in head-to-head contests. In each comparison, the winner receives 1 point and tying candidates receive half a point each. The candidate with the most points after all the comparisons are finished wins.

You can think of it like a round-robin in boxing matches. Each candidate must fight each other candidate. The winner of each match gets a point. Ties earn the boxers half a point each. No one is eliminated, and all the boxers must match up against all the others. Only at the end of the round-robin are the results tallied and an overall winner declared. Join me as we investigate this method of determining the winner of an election.

But, before we begin, you need to know that the pairwise comparisons are based on preferential voting and preference schedules. If you're not familiar with these concepts, it may be difficult for you to follow this lesson. Please review the lesson on preferential voting if you feel you may need a refresher.

## Pairwise Comparison Process

Okay, so, a pairwise comparison starts with preferential voting, which is an election method that requires voters to rank all the candidates in order of their preference. A preference schedule is the chart in which the results from preferential voting are listed.

If we imagine that the candidates in an election are boxers in a round-robin contest, we might have a result like this:

Now, we'd start the head to head comparisons by comparing each candidate to each other candidate. We can start with any two candidates; let's start with John and Roger. So, we count the number of votes in which John was chosen over Roger and vice versa.

We see that John was preferred over Roger 28 + 16, which is 44 times overall. And Roger was preferred over John a total of 56 times. So, Roger wins and receives 1 point for this head-to-head win. If we continue the head-to-head comparisons for John, we see that the results are:

John / Bill - John wins 1 point
John / Gary - John wins 1 point
John / Roger - John loses, no points

Against Bill, John wins 1 point. Against Gary, John wins 1 point. Against Roger, John loses, no point. So, John has 2 points for all the head-to-head matches. In the same way, we can compare all the other matches and come out with the following information:

On this chart, we see the results for all the individual match-ups. John received a total of 2 points and won the most head-to-head match-ups. In pairwise comparison, this means that John wins.

## Formula for Matches

In our current example, we have four candidates and six total match-ups. How many head-to-head match-ups would there be if we had 5 candidates?

Let's look at the results chart from before.

It looks a bit like the old multiplication charts, doesn't it? Remember the ones where you multiplied each number on top by each number on the side and put the result in the corresponding square? Yeah, this is much the same and we can start our formula with that basis. The number of comparisons is N * N, or N^2.

But, that can't be right. The diagonal line through the middle of the chart indicates match-ups that can't happen because they are the same person. It is clear that no matter how many candidates you have, you will always have that same number of match-ups that just aren't possible. So, we modify our formula to take this into account. The total number of comparisons equals N^2 - N, which can be simplified to N*(N - 1).

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