6 Election Math:
Topics Covered
In case you missed something in class, or just want to review a specific topic covered in this Module, here is a list of topics covered:
Simple Majority Election[1]
Elections are ways of helping large groups of people who have different ideas in electing persons, ideas, activities, and more. Elections are used to elect our political representatives and can also be used to decide what food options will be made for a group party or gathering.
Most elections that occur in your everyday life are simple majority elections. Majority elections require a number of votes over 50%, or a majority of the votes cast. This amount can be easily found by simply calculating 50% of the total votes cast in an election.
Simple Majority Election
Janna, Ryan, and Eric are running against each other for the position of class president. Janna is trying to figure out how many votes she would need to win the election. According to her school’s records, there are 356 students in Janna’s class that she is running to represent. If every student votes in the election, how many of the students must vote for Janna to win the election by a simple majority?
Step 1. If the total number of votes cast is 356, then we simply need to calculate 50%,
Setting up an equation with the portion of the votes that are unknown is represented as a variable.
356 * 0.50 = x
Solving for the unknown variable x, gives a value of 178.
Step 2. This is not the actual amount of votes that Janna needs to win the election. Janna needs more than 50%. This means the total necessary votes needed must be rounded up by one, giving a total of 179 votes needed for Janna to win the election by simple majority.
Instant Run-Off Elections[2]
Sometimes a simple majority election cannot be used; usually because an individual does not meet the required greater than 50% of the vote required.
In the case where this happens, an instant-runoff election, or elimination method, might be held. An instant runoff is an election where voting choices are ranked by preference and eliminations are made when no one candidate receives a simple majority.
Try it! – Instant Run-Off Election
The student council election is run by an instant runoff of a preference schedule. Students are asked to rank their preference for each student running for office as either first, second, or third choice. These preference schedules are then combined and presented in a total votes preference schedule to determine the winner of the student body election.
| Rank | 71 voters | 97 voters | 49 voters | 78 voters | 61 voters |
| 1st | Eric | Ryan | Ryan | Janna | Janna |
| 2nd | Janna | Eric | Janna | Eric | Ryan |
| 3rd | Ryan | Janna | Eric | Ryan | Eric |
Step 1: Determine the majority: adding up all votes possible is 356 total votes. 50% is 178 votes, so a majority is 179 votes or more.
After finding the amount needed for a majority, adding up all of the first-place votes for each of the candidates gives the following results:
- Eric: 71 votes
- Janna: 78 votes + 61 votes = 139 votes
- Ryan: 97 + 49 = 146 votes
It is clear that no one candidate received a simple majority of the vote in this election. In this case, the candidate that received the least amount of votes is eliminated: Eric.
Step 2. Once Eric is removed from the ballot, all remaining candidates are moved up in the ranking and first place votes are recalculated.
| Rank | 71 voters | 97 voters | 49 voters | 78 voters | 61 voters |
| 1st | Ryan | Ryan | Janna | Janna | |
| 2nd | Janna | Janna | Ryan | ||
| 3rd | Ryan | Janna | Ryan |
| Rank | 71 voters | 97 voters | 49 voters | 78 voters | 61 voters |
| 1st | Janna | Ryan | Ryan | Janna | Janna |
| 2nd | Janna | Janna | Janna | Ryan | Ryan |
Now, recalculating the total votes shows that:
- Ryan: 97 + 49 votes = 146 votes
- Janna: 71 + 78 + 61 votes = 210 votes
After the instant runoff, Janna has a total of 210 votes and can now be declared the winner of the election because 210 is more than 50% of the total votes in the class.
Borda Count Method of Election[3]
In the instant runoff election, a preference schedule was constructed in order to determine the winner of the student body election. Sometimes a preference schedule can be used to determine the winner of an election using points, in a method called Borda Count method.
The Borda Count Method involves awarding points to candidates based on their positions in the preference schedules collected. The lowest position, or last place, always receives the least amount of points and the highest position, or first place, always receives the most points.
The total amount of points that the candidate receives is the total of all the points awarded for all positions in the preference schedule, instead of just in the first place. The person with the most amount of points wins that election.
Try it! – Borda Count Election
The student council election is run by the Borda Count method from a preference schedule distributed to the student body. Students are asked to rank their preference for each student running for office as either first, second, or third choice. These preference schedules are then combined and presented in a total votes preference schedule to determine the winner of the student body election.
We start with the initial preference schedule that was completed. Each of the places gets a set amount of points. First place gets 3 points, second place gets two points, and third place gets one point. This is because the last place always gets the least amount of points and the first place vote always gets the most amount of points; with the point value ascending upwards between last to first place.
| Rank | 71 voters | 97 voters | 49 voters | 78 voters | 61 voters |
| 1st (3 points) | Eric | Ryan | Ryan | Janna | Janna |
| 2nd (2 points) | Janna | Eric | Janna | Eric | Ryan |
| 3rd (1 point) | Ryan | Janna | Eric | Ryan | Eric |
To calculate the number of points each vote in the preference gets, we multiply the place value of the row with the place value of the column each vote occurs in. So, in the first column, Eric got first place and his points for that vote are calculated by 3 points (for being in first place), times 71 votes because that column had 71 voters assigned to it.
This process is then repeated for each cell in the preference schedule.
| Rank | 71 voters | 97 voters | 49 voters | 78 voters | 61 voters |
| 1st (3 points) | Eric (3 * 71 = 213 points) | Ryan (3 * 97 = 291 points) | Ryan (3 * 49 = 147 points) | Janna (3 * 78 = 234 points) | Janna (3 * 61 = 183 points) |
| 2nd (2 points) | Janna (2 * 71=142 points) | Eric (194 points) | Janna (98 points) | Eric (156 points) | Ryan (122 points) |
| 2nd (1 point) | Ryan (1 * 71 = 71 points) | Janna (97 points) | Eric (49 points) | Ryan (78 points) | Eric (61 points) |
Adding up all of the points each candidate earns for each place vote gives the following totals:
- Eric: 213 points + 194 points + 49 points + 156 points + 61 points = 673 points
- Janna: 142 points + 97 points + 98 points + 234 points + 183 points = 754 points
- Ryan: 71 points + 291 points + 147 points + 78 points + 122 points = 709 points
As can be seen from the totals above, Janna received the highest number of points in this election and thus would be considered the winner in the Borda Count Method election.
Key Concepts[4]
- Winners of elections can be conducted in three common ways
- Simple majority
- Instant run-off
- Borda count method
- Simple majority elections require at least 51% of the votes cast in the election.
- Instant run-off elections are similar to simple majority elections but require eliminating a candidate/option.
- Borda count method elections require calculating and adding up points calculating by place vote and number of votes earned for that candidate/option.
An election in which a candidate is required to achieve at least 51% of the votes cast in order to be considered the winner.
An election in which a candidate/option is eliminated from a ballot or preference schedule in order to achieve a simple majority possibility vote collection for one of the remaining candidates/options.
An election method in which candidates/options in a preference schedule are assigned point values based off ranking position and number of votes cast for that ranking line up.
