The Borda method or order of merit is a positional voting rule that gives a candidate a number of points equal to the number of candidates ranked below them: the lowest-ranked candidate (the candidate least favoured by the voter) gets 0 points, the second-lowest gets 1 point, and so on. The points on all the ballots are calculated, and the candidate with the most points wins.
The Borda count has been independently reinvented several times, with the first recorded proposal in 1435 being by Nicholas of Cusa (see History below), but is named after the 18th-century French mathematician and naval engineer Jean-Charles de Borda, who re-devised the system in 1770.
The Borda count is well-known in social choice theory both for its pleasant theoretical properties and its ease of manipulation. In the absence of strategic voting and strategic nomination, the Borda count tends to elect broadly-acceptable options or candidates (rather than consistently following the preferences of a majority); when both voting and nomination patterns are completely random, the Borda count generally has an exceptionally high social utility efficiency. However, the method is highly vulnerable to spoiler effects when there are clusters of similar candidates; because the effects of more candidates on the election are unbounded, it is possible for any political party to win an election by running enough clones. Common implementations of equal-rank or truncated ballots can also incentivize extreme burial when voters are strategic, which allows deeply unpopular dark horse candidates to win by avoiding any attention. This problem arises because under the Borda count, a marked lesser preference may cause a voter's first preference to fail election. Under Borda, lesser preferences are given less weight than higher preferences so this problem is less severe than under the Bucklin system, but it still exists.
The traditional Borda method is currently used to elect two ethnic minority members of the National Assembly of Slovenia, and in modified forms to determine which candidates are elected to the party list seats in Icelandic parliamentary elections. A variant known as the Dowdall system is used to elect members of the Parliament of Nauru. Until the early 1970s, another variant was used in Finland to select individual candidates within party lists. From 1979 until 2002 the method was used to select presidential election candidates in Kiribati. It is also widely used throughout the world by various private organizations and competitions.
The Quota Borda system is a proportional multiwinner variant.
Voting and counting Ballot
The Borda count is a ranked voting system: the voter ranks the list of candidates in order of preference. So, for example, the voter gives a 1 to their most preferred candidate, a 2 to their second most preferred, and so on. In this respect, it is similar to other ranked voting systems such as instant-runoff voting, the single transferable vote or Condorcet methods. The integer-valued ranks for evaluating the candidates were justified by Laplace, who used a probabilistic model based on the law of large numbers.
The Borda count is classified as a positional voting system, that is, all preferences are counted but at different values. The other commonly used positional system is plurality voting, which only assigns one point to the top candidate.
Each candidate is assigned a number of points from each ballot equal to the number of candidates to whom he or she is preferred.
In Borda's original 1784 paper, he assigned 1 point to the lowest-ranked candidate, 2 to the second-lowest, and so on, so that with n candidates, the highest-ranked candidate on a ballot received n points. He noted that if every voter ranks all candidates, it does not matter how many points are assigned to the lowest-ranked candidate.
In modern usage, the lowest-ranked typically receives 0 points, the second-lowest 1 point, and so on, so that with n candidates, each one receives n – 1 points for a first preference. The winner is the candidate with the largest total number of points. For example, in a four-candidate election, the number of points assigned for the preferences expressed by a voter on a single ballot paper might be:
Ranking Candidate Formula Points 1st Andrew n − 1 3 2nd Brian n − 2 2 3rd Catherine n − 3 1 4th David n − 4 0 Suppose that there are 3 voters, U, V and W, of whom U and V rank the candidates in the order A-B-C-D while W ranks them B-C-D-A. Candidate U Points V Points W points Total Andrew 3 3 0 6 Brian 2 2 3 7 Catherine 1 1 2 4 David 0 0 1 1 Thus Brian is elected.
A longer example, based on a fictitious election for Tennessee state capital, is shown below.
Properties Elections as estimation procedures Condorcet looked at an election as an attempt to combine estimators. Suppose that each candidate has a figure of merit and that each voter has a noisy estimate of the value of each candidate. The ballot paper allows the voter to rank the candidates in order of estimated merit. The aim of the election is to produce a combined estimate of the best candidate. Such an estimator can be more reliable than any of its individual components.
Peyton Young showed that the Borda count gives an approximately maximum likelihood estimator of the best candidate. His theorem assumes that errors are independent, in other words, that if a voter rates a particular candidate highly, then there is no reason to expect her to rate "similar" candidates highly. If this property is absent – if the voter gives correlated rankings to candidates with shared attributes – then the maximum likelihood property is lost, and the Borda count is highly subject to nomination effects: a candidate is more likely to be elected if there are similar candidates on the ballot.
Effect of irrelevant candidates
The Borda count is particularly susceptible to distortion through the presence of candidates who do not themselves come into consideration, even when the voters lie along a spectrum. Voting systems which satisfy the Condorcet criterion are protected against this weakness since they automatically also satisfy the median voter theorem, which says that the winner of an election will be the candidate preferred by the median voter regardless of which other candidates stand.
Suppose that there are 11 voters whose positions along the spectrum can be written 0, 1, ..., 10, and suppose that there are 2 candidates, Andrew and Brian, whose positions are as shown: Candidate A B Position 5 6 The median voter Marlene is at position 5, and both candidates are to her right, so we would expect A to be elected. We can verify this for the Borda system by constructing a table to illustrate the count. The main part of the table shows the voters who prefer the first to the second candidate, as given by the row and column headings, while the additional column to the right gives the scores for the first candidate. A B score A 0–5 6 B 6–10 5 A is indeed elected.
But now suppose that two additional candidates, further to the right, enter the election. Candidate A B C D Position 5 6 8 10 The counting table expands as follows: A B C D score A 0–5 0–6 0–7 21 B 6–10 0–7 0–8 22 C 7–10 8–10 0–9 17 D 8–10 9–10 10 6 The entry of two dummy candidates allows B to win the election. Similar examples led the Marquis de Condorcet to argue that the Borda count is "bound to lead to error" because it "relies on irrelevant factors to form its judgments".
Other properties There are a number of formalised voting system criteria whose results are summarised in the following table.
Simulations show that Borda has a high probability of choosing the Condorcet winner when one exists, in the absence of strategic voting and with ballots ranking all candidates.
Equal ranks Several different methods of handling tied ranks have been suggested. They can be illustrated using the 4-candidate election discussed previously. Ranking Candidate Points 1st Andrew 3 2nd Brian 2 3rd Catherine 1 4th David 0
Traditional Borda: In Borda's system as originally proposed, each tied candidate was given the minimum number of points. So if a voter marks Andrew as his or her first preference, Brian as his or her second, and leaves Catherine and David unranked, then Andrew will receive 3 points, Brian 2, and Catherine and David none. This is an example of what Narodytska and Walsh call "rounding up". Tournament Borda: each candidate receives half a point for every other candidate they are tied with, in addition to a whole point for every candidate they are strictly preferred to. In the example, suppose that a voter is indifferent between Andrew and Brian, preferring both to Catherine and Catherine to David. Then Andrew and Brian will each receive 2 points, Catherine will receive 1, and David none. This is referred to as "averaging" by Narodytska and Walsh. Modified Borda: again allows ties only at the end of a voter's ranking. It gives no points to unranked candidates, 1 point to the least preferred of the ranked candidates, etc. So if a voter ranks Andrew above Brian and leaves other candidates unranked, Andrew will receive 2 points, Brian will receive 1 point, and Catherine and David will receive none. This is equivalent to "rounding down". The most preferred candidate on a ballot paper will receive a different number of points depending on how many candidates were left unranked.
Effects on strategy The modified Borda and tournament Borda methods, as well as methods of Borda that do not allow for equal rankings, are well-known for behaving disastrously in response to tactical voting, a reaction called a turkey election. The French Academy of Sciences (of which Borda was a member) experimented with Borda's system but abandoned it, in part because "the voters found how to manipulate the Borda rule". In response to the issue of strategic manipulation in the Borda count, M. de Borda said:Despite its abandonment, the rounded-down Borda rule has a substantially less severe reaction to tactical voting than the traditional or tournament variants. Tactical voting consists of the relatively mild bullet voting, which only causes the race to behave like a cross between a plurality vote and an honest Borda count, rather than producing a potential turkey-election. In Slovenia, which uses this form of the rule, roughly 42% of voters rank a second preference.
Forced truncation Some implementations of Borda voting require voters to truncate their ballots to a certain length:
In Kiribati, the variant employed between 1979 and 2002 used a traditional Borda formula, but with voters ranking only four candidates, irrespective of how many were standing. This approach was abandoned with the passage of the Election of the Beretitenti (Amendment) Act 2002, which commenced on 18 October 2002. In Toastmasters International, speech contests are truncation-scored as 3, 2, 1 for the top-three ranked candidates. Ties are broken by having a special ballot that is ignored unless there is a tie.
Multiple winners The system invented by Borda was intended for use in elections with a single winner, but it is also possible to conduct a Borda count with more than one winner, by recognizing the desired number of candidates with the most points as the winners. In other words, if there are two seats to be filled, then the two candidates with most points win; in a three-seat election, the three candidates with most points, and so on. In Nauru, which uses the multi-seat variant of the Borda count, parliamentary constituencies of two and four seats are used.