Section 11.4 Borda Count
Objectives
Find the winner of an election using the Borda count
Understand whether an election has violated the majority criterion
Subsection 11.4.1 Borda Count
Definition 11.4.1. Borda Count.
In this method, points are assigned to candidates based on their ranking; 1 point for last choice, 2 points for second-to-last choice, and so on. The point values for all ballots are totaled, and the candidate with the largest point total is the winner.
Example 11.4.2.
A group of mathematicians are getting together for a conference. The members are coming from four cities: Seattle, Tacoma, Puyallup, and Olympia.
The votes for where to hold the conference were:
| 51 | 25 | 10 | 14 | |
|---|---|---|---|---|
| 1st Choice | Seattle | Tacoma | Puyallup | Olympia |
| 2nd Choice | Tacoma | Puyallup | Tacoma | Tacoma |
| 3rd Choice | Olympia | Olympia | Olympia | Puyallup |
| 4th Choice | Puyallup | Seattle | Seattle | Seattle |
Which city will be selected with the Borda count method?
Solution.In each of the 51 ballots ranking Seattle first, Puyallup will be given 1 point, Olympia 2 points, Tacoma 3 points, and Seattle 4 points. Multiplying the points per vote times the number of votes allows us to calculate points awarded:
Seattle: \(51 \cdot 4 + 25 \cdot 1 + 10 \cdot 1 + 14\cdot 1 = 253\) points
Tacoma: \(51 \cdot 3 + 25 \cdot 4 + 10 \cdot 3 + 14 \cdot 3 = 325\) points
Puyallup: \(51 \cdot 1 + 25 \cdot 3 + 10 \cdot 4 + 14 \cdot 2 = 194\) points
Olympia: \(51 \cdot 2 + 25 \cdot 2 + 10 \cdot 2 + 14 \cdot 4 = 228\) points
Under the Borda Count method, Tacoma is the winner of this vote.
Problem 11.4.4. Try It Now.
Consider again the election described by the preference schedule. Who will win using the Borda count method?
| 14 | 10 | 8 | 4 | 1 | |
|---|---|---|---|---|---|
| 1st Choice | Alan | Caroline | Dawn | Boris | Caroline |
| 2nd Choice | Boris | Boris | Caroline | Dawn | Dawn |
| 3rd Choice | Caroline | Dawn | Boris | Caroline | Boris |
| 4th Choice | Dawn | Alan | Alan | Alan | Alan |
A gets \(56 + 10 + 8 + 4 + 1 = 79\) points
B gets \(42 + 30 + 16 + 16 + 2 = 106\) points
C gets \(28 + 40 + 24 + 8 + 4 = 104\) points
D gets \(14 + 20 + 32 + 12 + 3 = 81\) points
So the winner is Boris (B).
Subsection 11.4.2 What's Wrong with Borda Count
You might have already noticed one potential flaw of the Borda Count from the previous example. In that example, Seattle had a majority of first-choice votes, yet lost the election! This seems odd, and prompts our next fairness criterion:
Definition 11.4.6. Majority Criterion.
If a choice has a majority of first-place votes, that choice should be the winner.
The election from Example 11.4.2 using the Borda Count violates the Majority Criterion. Notice also that this automatically means that the Condorcet Criterion will also be violated, as Seattle would have been preferred by 51% of voters in any head-to-head comparison.
Borda count is sometimes described as a consensus-based voting system, since it can sometimes choose a more broadly acceptable option over the one with majority support. In the example above, Tacoma is probably the best compromise location. This is a different approach than plurality and instant runoff voting that focus on first-choice votes; Borda Count considers every voter’s entire ranking to determine the outcome.
Because of this consensus behavior, Borda Count, or some variation of it, is commonly used in awarding sports awards. Variations are used to determine the Most Valuable Player in baseball, to rank teams in NCAA sports, and to award the Heisman trophy.