The Borda Count is named for Jean-Charles de Borda, a French mathematician, scientist, and naval engineer who developed this system in 1770. He also helped fight in the American Revolutionary War, during which he was briefly held captive by the British. His name is one of 72 inscribed on the Eiffel Tower.
Section5.2The Borda Count and the Majority Criterion
SubsectionThe Borda Count
Our first method of ranked voting is called the Borda Count. Its rules are fairly easy to state.
The Borda Count
The Borda Count method of voting assigns points to candidates. Each candidate earns 1 point for every voter that ranked them last, 2 points for every voter that ranked them second-to-last, and so on. After adding up each candidate's total points, the candidate with the most points wins.
Let's illustrate it with an example.
Example5.13
Find the Borda Count winner of the election whose preference table is below.
| Number of voters | |||||
| Borda points | Rankings | 6 | 5 | 4 | 2 |
| 4 | 1st choice | A | D | C | B |
| 3 | 2nd choice | B | B | D | C |
| 2 | 3rd choice | D | C | B | D |
| 1 | 4th choice | C | A | A | A |
Notice that we have added a column to the left of the preference schedule indicating how many Borda points are awarded to each ranking. Since there are 4 candidates, a first-place ranking is worth 4 points.
To find the score of Candidate A, we look at the four times A appears on the preference table. The first column says that 6 people ranked A first, giving A \(6(4)=24\) points. Everyone else ranked A last, giving A 1 point each. This adds up to \(5+4+2=11\) points. Therefore, A has a total of \(24+11=35\) points.
We can calculate the Borda points of the other candidates similarly, as follows:
- B: \(6(3) +5(3) +4(2) +2(4) =49\)
- C: \(6(1) +5(2) +4(4) +2(3) =38\)
- D: \(6(2) +5(4) +4(3) +2(2) =48\)
Thus, Candidate B wins this election when the Borda Count is used.
Notice that the plurality winner of the election in Example5.13 was Candidate A, with 6 votes out of 17 (about 35%). Candidate B had only 2 first choice votes, but many others ranked B second. In contrast, Candidate A was ranked last by everyone other than the 6 who ranked A first. We could say that B is a consensus or compromise candidate, while A is a more polarizing figure.
This behavior is common in the Borda Count, and so it is often described as a more consensus-based voting system. For this reason, the 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. In contrast, it is only rarely used in elections for political office. As of 2019, it was used only in two countries: Nauru (a small Pacific island nation) and Slovenia (which only uses it to elect two members of parliament).
Example5.15
The UNL Dairy Store is taking a survey of customers' favorite ice cream flavors. Their preferences are shown in the table below. The three options are Black Walnut Fudge (F), Scarlet & Cream (S), and Bavarian Mint (M). Which flavor would the Borda Count pick as the overall favorite?
| Number of voters | ||||
| Borda points | Rankings | 6 | 3 | 2 |
| 3 | 1st choice | F | S | M |
| 2 | 2nd choice | S | M | S |
| 1 | 3rd choice | M | F | F |
First notice how the number of points assigned to each ranking has changed. The rules for the Borda count state that every last choice vote gets 1 point, and then we count going up. Hence, when there are three candidates, a 3rd choice vote gets 1 point, a 2nd choice vote gets 2 points, and a 1st choice vote gets 3 points. Comparing this to the example with four candidates, note that the number of points awarded for a first choice vote will always be the same as the number of candidates.
We can now calculate the Borda Count winner as usual, and find:
- F: \(6(3) +3(1) +2(1) =23\)
- S: \(6(2) +3(3) +2(2) =25\)
- M: \(6(1) +3(2) +2(3) =18\)
Therefore, Scarlet & Cream wins with 25 points.
Exploration5.2
Let's revisit the election from Exploration5.1.
| Number of voters (in thousands) | |||||||
| Rankings | 44 | 14 | 20 | 70 | 22 | 80 | 39 |
| 1st choice | G | G | G | M | M | B | B |
| 2nd choice | M | B | G | B | M | ||
| 3rd choice | B | M | B | G | G | ||
Find the winner of this election using the Borda Count.
Note: When voters do not rank every candidate, we will treat them as if the candidates who they didn't rank are all in last place. Thus, for example, the 20 voters who ranked only G first will give 3 points each to G, but also give 1 point each to B and M, as if both those candidates were simultaneously ranked last.
Solution
The candidates' points are as follows:
- G: \(132+42+60+140+22+80+39 = 515\)
- M: \(88+14+20+210+66+160+39 = 597\)
- B: \(44+28+20+70+44+240+117 = 563\)
Therefore, M (McCarthy) wins the election when the Borda Count is used.
SubsectionThe Majority Criterion
If you look again at Example5.15 carefully, you might feel uneasy with the result. In that election, there were 11 total voters, and 6 of them voted for Walnut Fudge. That's about 54.5%a clear majority. Yet the winner of the Borda Count was instead Scarlet & Cream! This motivates our first fairness criterion.
The Majority Criterion
- The majority criterion states that if a candidate has a majority of first choice votes, then that candidate should be the winner of the election.
- If no majority candidate exists, then the majority criterion does not apply.
Example5.15 shows that the Borda Count violates the majority criterion: there is an election in which a candidate (Walnut Fudge) had a majority of first choice votes, but another candidate (Scarlet & Cream) won the election. But, does this always happen with the Borda Count? Let's consider another example.
Example5.18
The UNL Dairy Store decides to take another survey on customers' opinions of some different ice cream flavors. Their preferences are shown in the table below. The three options are Cappuccino Chocolate Chip (C), Lemon Custard (L), and Butter Brickle (B). Which flavor would the Borda Count pick as the overall favorite?
| Number of voters | ||||
| Borda points | Rankings | 6 | 3 | 2 |
| 3 | 1st choice | C | L | B |
| 2 | 2nd choice | B | C | L |
| 1 | 3rd choice | L | B | C |
Each flavor earns the following points:
- C: \(6(3) +3(2) +2(1) =26\)
- L: \(6(1) +3(3) +2(2) =19\)
- B: \(6(2) +3(1) +2(3) =21\)
Therefore, Cappuccino Chocolate Chip wins with 26 points. Cappuccino Chocolate Chip also has a majority of first choice votes: 6 out of 11, or about 54.5%. Therefore, the Borda Count does not violate the majority criterion in this election.
We can see that there are really two kinds of question we could ask: whether or not a voting method satisfies a fairness criterion in one specific election, and whether or not a voting method satisfies a fairness criterion in general. What we're ultimately interested in is the second of these questions. Let's make it a little more precise.
Satifying Fairness Criteria
- When we can show that a voting method is guaranteed to fulfill a given fairness criterion in any possible election, we say that the method satisfies the criterion.
- If a voting method does not satisfy a given fairness criterioneven if it fails in only one election!then we say that the method violates the given criterion.
Therefore, Example5.18 does not show that the Borda Count satisfies the majority criterion, because it is one example rather than a general argument. Instead, Example5.15 shows that the Borda Count violates the majority criterion. As soon as we find one example of a violation, we know that the method in question violates the criterion we're looking at.
It sounds like a strong condition to ask that a voting method satisfy a fairness criterion in any possible election, but hopefully it makes sense why. We don't want a system that just happens to be fair in some elections; we want a voting method that is guaranteed to be fair in any possible election that will be held in the future.
This does mean, however, that it can take some work to show that a voting method satisfies a certain fairness criterion. We can't just look at one election, or even many elections. Instead, to show that a voting method satisfies a fairness criterion, we must construct an argument explaining why the criterion will be satisfied in any possible election using that method. Let's see how this works by looking at the plurality method.
Does the plurality method satisfy or violate the majority criterion? Let's first look at the election in Example5.13. The plurality winner of that election was A, with 6 votes out of 17 (about 35%). This is one of many examples we've seen in which a plurality winner did not have a majority. However, this does not represent a violation of the majority criterion. Remember that if no majority candidate exists, then the criterion does not apply: it is neither satisfied nor violated. This election does not tell us anything about whether or not the plurality method satisfies the majority criterion.
What if a majority candidate does exist? By definition, that candidate must have over 50% of the vote. Therefore, all other candidates combined must have less than 50%. No matter what, no other candidate can have as many votes as the majority candidate. Hence, any candidate with a majority must necessarily be the plurality winner, so the plurality method satisfies the majority criterion. This is the kind of explanation we use to show that a method satisfies a criterion: we've demonstrated why it must be true in any election.
So, in spite of its problems with electing candidates without majority support, the plurality method satisfies the majority criterion, because when a candidate does have majority support, they will be elected. In contrast, the Borda Count does not satisfy the majority criterion. This relates to the Borda Count's character, mentioned above, as a consensus-based method. For instance, in Example5.15, Black Walnut Fudge had a majority of first choice votes, but everyone else put it in last place. The Borda Count rejected it in favor of Scarlet & Cream, which had less first choice votes but was not the last choice of anyone. Some people see this as a positive feature of the Borda Count, since it rejects polarizing candidates. Others see its violation of the majority criterion as a drawback. In the following sections, we'll see how other ranked voting methods compare.