The Borda count
A voting method in profile
A voting method in profile
The Borda count is one of the more mathematically elegant alternatives to a plurality vote. Its inventor, Jean Charles de Borda, is one of the two Frenchmen who arguably started the modern field of voting theory. For the entire history of the field, the Borda count has stayed relevant as a theoretically promising (but rarely implemented) voting system. The system begins with asking voters to rank candidates.
There are two main drawbacks to a Borda count: First, ranking candidates can be fairly difficult when there are more than a handful of candidates. Second, voters can easily have a strategic incentive to misrepresent their preferences. Neither of one these drawbacks is unique to a Borda count; many advanced voting systems require that voters rank candidates, and all voting systems have some degree of vulnerability to strategic voting.
Interestingly, while a Borda count requires ranking, it is structurally and mathematically more closely related to approval voting than ranked choice voting. This is also true in the sociology of support for voting systems; ranked choice voting has more support among activists than theorists, while the Borda count has more support among theorists than activists.
How a Borda count works
There are several ways to describe a Borda count. As described by Borda in a 1781 paper, a voter ranks candidates on a ballot, and the candidates receive points in reverse order — for example, out of three candidates, the first-ranked candidate earns 3 points, the second-ranked candidate earns 2 points, and the third-ranked candidate earns 1 point. Each ballot adds <3,2,1> points to the candidates.
A second way to reinvent the Borda count is to compare candidates in pairs. The ranked ballots are used to give the results of a majority vote between the two candidates. All votes from these pairwise contests are added together. A single ballot effectively adds <2,1,0> pairwise votes to the three candidates.
A third way to think of this is that a voter casts multiple “place” votes. With three candidates, their second place vote is worth half as much as a first place vote. In this case, a Borda ballot with three candidates represents giving <1,½,0> votes to the three different candidates. This makes it easiest to compare Borda results to the results of systems like plurality voting.
It may not be immediately obvious, but all three of these descriptions are mathematically equivalent. To go between the different ways of counting Borda ballots involves multiplying all candidates’ totals by the same amount, or adding or subtracting the same amount from each candidate’s total.
A brief history of use
Borda devised the system in 1770. His goal was to improve the selection of new members by the French Academy of Sciences. The Academy adopted it in 1781, and continued to use it until Napoleon decided to take direct control of appointments in 1800.
Variations of the Borda count have been invented and re-invented for many different situations, arguably including a number of sporting competitions using total wins and losses to rank teams or individuals. To my knowledge, the most notable political use of the Borda count has been for the Slovenian parliament.
How to handle ballots that do not rank all candidates
One of the difficulties presented by ranking systems is the question of what to do with ballots which do not provide a ranking of all candidates. Rejecting them is less than ideal, because ranking many candidates is a difficult task and not all voters may have formed opinions of all candidates. For a Borda count, there are two good methods for counting incompletely ranked ballots, and several less ideal methods. I will discuss the two good methods here.
Averaging is the mathematically elegant method. This means averaging the point values that each candidate would have received on average if the indecisive voter had simply flipped a coin or rolled a die to decide which candidate to rank above the other. This encourages voters who are unable to see a difference between two candidates to simply express their sincere preferences.
For example, if a voter marks a ballot for four candidates with two candidates tied in the middle, this is counted as one first place ranking, two tied between 2nd and 3rd (⅔ and ⅓ of a vote), and one in 4th place (their zero vote). Averaging the middle place rankings, the ballot is counted as <1,½,½,0> votes for the four candidates.
Similarly, a voter who marks their ballot for only a first place candidate, as if casting a plurality ballot, gives the other three candidates the average of 2nd, 3rd, and 4th place — meaning their ballot is counted as <1,⅓,⅓,⅓>. This has the same impact on the election as a <⅔,0,0,0> ballot, and the overall magnitude of the impact of the ballot is just the same as a fully-ranked ballot.
Counting from the bottom up encourages voters to rank candidates more completely, and rewards informed voters who have paid close enough attention to have a complete ranking of candidates. All bottom-ranked candidates receive the last place value; candidates sharing the next-highest ranking receive the second-to-last-place value; etc.
Under this version of the rule, the voter who is uncertain about 2nd and 3rd place would cast <⅔,⅓,⅓,0> votes. The voter who tries to cast a plurality vote simply gives <⅓,0,0,0> points — half as much of an impact on the election as a normal Borda ballot. This is not as strong of a penalty as throwing out the ballot, but is a penalty nonetheless.
The strategic voting problem
Like all other voting methods, Borda counts have some degree of vulnerability to strategic action, and this degree of vulnerability increases with a larger number of candidates. Whether this is a large degree or a small degree depends on the details of how the system is implemented. In particular, it matters how incomplete ballots are handled and what degree of support a candidate needs to have before being listed on the ballot.
Using the wrong method for ballots with ties
Using an inappropriate method for scoring incompletely ranked ballots can lead to problems with strategic voting. For example, if Borda’s classic <3,2,1> point scheme is replaced with <3,0,0> points for a ballot with one ranked candidate, then voters have an incentive to just vote for one candidate.
This happened historically at the French Academy of Sciences during Borda’s lifetime. Borda responded to the ensuing criticism by saying that his method was intended only for use by “honest men.”
Separating the top two candidates
If a voter has good information about which candidates are likely to place first and second in an election that only produces a single winner, they can maximize their impact by ranking one of those candidates in first place and the other in last place. If enough other voters behave similarly, a less-preferred third candidate could emerge as a surprise winner.
This is not to say that the Borda count is exceptionally vulnerable to strategic action. In raw mathematical terms, strategic voting decisions in a plurality vote or approval vote by an individual or group have more than twice the impact on vote tallies. The reason why experts worry about strategic voting in a Borda count is that the technique is obvious, conceals voters’ true preferences, and can potentially result in electing by surprise a candidate who is widely seen as a worse option than either of the top two contenders.
Preventing this behavior requires either making sure that all candidates on the ballot are seen as potentially viable, or using the Borda count as a preliminary round, followed by one or more additional election rounds. (It is far more difficult and complex to figure out how to vote strategically in Nanson’s method, which uses a series of Borda counts.)
Relationships to other voting systems
In this series thus far, I have discussed plurality, approval voting, and the methods known as instant runoff, single transferable vote, or ranked choice voting. Like ranked choice voting, a Borda count requires voters to rank candidates. However, the counting process and results are much more similar to an approval vote.
Positional methods
The Borda count can be classified as a positional voting method. That is, it assigns a point value to each rank. Not all positional voting methods are Borda counts; a positional voting method is a Borda count only if it its weights are linear. A <4,2,1> voting method is not a Borda count. Plurality voting is also a type of positional voting method, e.g., <1,0,0> over three candidates.
Positional voting methods as a group have some desirable properties. One is that ballots only need to be counted or processed once, rather than iteratively. This means that the counting process is short, simple, and can be decentralized. Another is that they are always monotone —candidates cannot lose by gaining support, or win by losing support.
An undesirable property of almost all positional voting methods is that they can elect a Condorcet loser, because they do not include an explicit or implicit majority vote. In a Borda count, a Condorcet winner will always rank above average, while a Condorcet loser will always rank below average; this makes the Borda count the only positional method that fulfills the Condorcet loser criterion. A Condorcet winner is not guaranteed to win a Borda count, but it is very likely to happen.
Approval voting
One of the other major contenders for best voting system in theoretical circles is approval voting. Approval voting is a multiple positional method. In a multiple positional method, a voter with the same preferences may choose different weights, e.g., with three candidates, a voter can cast a <1,1,0> or <1,0,0> approval ballot.
Interestingly, if tied and truncated ballots are allowed, it is possible for voters to decide to effectively cast an approval ballot in a Borda count system by ranking all their approved candidates in first place and leaving other candidates unranked or bottom-ranked.
In an approval voting system, if voters are roughly equally likely to approve of any number of candidates and have underlying ranked preferences, then a Borda count gives the average result of an approval vote. Conversely, if voters only really approve or disapprove of candidates, and rank all candidates they approve of randomly, the average result of a Borda count corresponds to an approval vote.
Differences emerge when strategic voting comes into play. Strategic voting in an approval vote will tend to push the expected leaders further ahead, with the potential for self-fulfilling prophecies. Strategic voting in a Borda count will tend to drag expected leaders towards the average, with the potential for surprises if a large number of voters vote at a basic strategic level.
From the perspective of a voter or small group, strategic action in an approval vote has a larger per-vote impact on margins, but never involves misrepresenting the order of one’s true preferences. In a Borda count, a strategic voter is likely to assign less than a full-value first-place vote to their favorite candidate, but will have less of an impact on final results.
Summary
From the perspective of voting theorists, the Borda count has been a perennial major contender from 1770 to the present. In spite of the amount of theoretical analysis that has been performed over the last two and half centuries, it has mostly been ignored by activists interested in political reform. It is a promising system that has not been tested very often in the political arena.
Mathematically, Borda counts have many attractive properties; it is a simple positional method that coincidentally has a strong connection to pairwise majority voting. As long as voters use it to express preferences sincerely, it is likely to elect a Condorcet winner and will not elect a Condorcet loser. It performs especially well when it comes to trying to fully rank a set of candidates as opposed to coming up with a single winner.
There are two main drawbacks to a Borda count. The first is that ranking candidates can be very difficult for voters, especially when there are more than five candidates. Care needs to be taken with how to count ballots that do not The second major issue is that it’s relatively easy for voters to figure out how to vote strategically by misrepresenting the strength of their preferences over the top two candidates — and that this can backfire if many voters on both sides make similar strategic decisions.