When there are only two candidates in an election, it is very simple to determine who the winner should be: the candidate who receives the most votes is the winner. If the two candidates are called A and B, we say that there are two possible preferences that voters can have:
If a voter has the preference A>B, then certainly they will vote for A, and vice versa. In the window below, click on each box to add voters that have that preference and notice the effect that it has on the outcome.
The situation is decidedly more complex when there are three candidates. Imagine that there are three candidates: A, B, and C. On a standard ballot (like those used in elections in the United States) there would only be three choices, and the voter would have to choose one. However, there are really six possible preferences that the voter can possess. The preference "A>B>C" means that the voter prefers A to B and B to C. So even though on a standard ballot the voter is not given a means to express these preferences, the voters have them nonetheless.
There is a common way to represent these six preferences, developed by Don Saari [reference]. In the diagram below, each vertex of the triangle represents one of the three candidates. In the two regions that are closest to A, candidate A is the top-ranked choice. Of those two regions, in the one closer to B, candidate B is the second-ranked choice. Another way to think about it is that the point in the region labeled "A>B>C" are closest to A, second-closest to B, and farthest from C.
This diagram, called the representation triangle, gives us an easy way to list the numbers of voters with each of the six possible preferences.
For example, suppose that a class of children are trying to decide what kind of drinks they should have during lunch. The teacher takes a poll, and the results are:
Number of People |
Preference |
6 |
Milk > Soda > Juice |
5 |
Soda > Juice > Milk |
4 |
Juice > Soda > Milk |
A list of the number of voters that have each preference is called a voter profile. We do not list the preferences that have zero voters. We represent this profile with the diagram below.
Consider the following voter profile for three candidates, A, B, and C.
Preference Order |
Number of Voters |
8 |
B > A > C |
6 |
C > B > A |
3 |
A > C > B |
1 |
C > A > B |
In the interactive diagram below, enter this profile. Left-click to add voters and right-click to take voters away. You can click on the preference orders on the left or on the corresponding regions of the representation triangle on the right.
Now that you have explored the representation triangle diagram, we will begin to discuss various methods for determining the winner of an election with more than two candidates. The simplest of these methods, the one used in most elections in the United States and other countries, is the plurality method.
Table of Contents |
Plurality |