Borda count


The Borda count is a family of single-winner election methods in which voters rank options or candidates in order of preference. The Borda count determines the outcome of a debate or the winner of an election by giving each candidate, for each ballot, a number of points corresponding to the number of candidates ranked lower. Once all votes have been counted the option or candidate with the most points is the winner. The Borda count is intended to elect broadly acceptable options or candidates, rather than those preferred by a majority, and so is often described as a consensus-based voting system rather than a majoritarian one.
The modified Borda count is a variant used for decision-making. For multi-winner elections, especially when proportional representation is important, the quota Borda system may be used.
The Borda count was developed independently several times, as early as 1435 by Nicholas of Cusa, but is named for the 18th-century French mathematician and naval engineer Jean-Charles de Borda, who devised the system in 1770. It is currently used to elect members of the Parliament of Nauru and two ethnic minority members of the National Assembly of Slovenia, in modified forms to determine which candidates are elected to the party list seats in Icelandic parliamentary elections, and for selecting presidential election candidates in Kiribati. Until the early 1970s, a variant was used in Finland to select individual candidates within party lists. It is also used throughout the world by various private organizations and competitions.

Voting and counting

The Borda count is a preferential, or 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 the same as elections under systems such as instant-runoff voting, the single transferable vote or Condorcet methods.
Points are then given to each candidate in reverse proportion to their ranking, so that higher-ranked candidates receive more points. When all votes have been counted, and the points added up, the candidate with the most points wins.
Because, from each voter, candidates receive a certain number of points, the Borda count is also classified as a positional voting system. Other positional methods include first-past-the-post voting, bloc voting, approval voting and limited voting.
The number of points assigned for each ranking varies depending on which of several variants of the Borda count is used:

Borda's system (starting at 1)

In Borda's original proposal, the number of points given to candidates for each ranking is determined by the number of candidates standing in the election. If there are five candidates in an election, candidates will receive five points each time they are ranked first, four for being ranked second, and so on, with a candidate receiving one point for being ranked last. In other words, where there are n candidates a candidate will receive n points for a first preference, n − 1 points for a second preference, n − 2 for a third, and so on, as shown in the following example:
RankingCandidateFormulaPointsRelative points
1stAndrewn51.00
2ndBriann−140.80
3rdCatherinen−230.60
4thDavidn−320.40
5thElizabethn−410.20

This system is used in the Slovenian parliamentary elections for two out of 90 seats.

Starting at 0

Alternatively, votes can be counted by giving each candidate a number of points equal to the number of candidates ranked lower than them, so that a candidate receives n − 1 point for a first preference, n − 2 for a second, and so on, with zero points for being ranked last. In other words, a candidate ranked in ith place receives n−i points. For example, in a five-candidate election, the number of points assigned for the preferences expressed by a voter on a single ballot paper might be:
RankingCandidateFormulaPointsRelative points
1stAndrewn−141.00
2ndBriann−230.75
3rdCatherinen−320.50
4thDavidn−410.25
5thElizabethn−500.00

With this second weighting, in a two candidate election, a first rank vote would get 1 point, and second rank vote gets 0 points, just like plurality voting.

Dowdall system (Nauru)

The island nation of Nauru uses a variant called the Dowdall System: the voter awards the first-ranked candidate with 1 point, while the 2nd-ranked candidate receives ½ a point, the 3rd-ranked candidate receives ⅓ of a point, etc. An important difference of this method from the others is that the number of points assigned to each preference does not depend on the number of candidates. Using the above example, in Nauru the point distribution among the five candidates would be this:
RankingCandidateFormulaPointsAbsolute points
1stAndrew1/11.0060
2ndBrian1/20.5030
3rdCatherine1/30.3320
4thDavid1/40.2515
5thElizabeth1/50.2012

This method is far more favorable to candidates with many first preferences than the conventional Borda count; it also substantially reduces the impact of electors indicating late preferences at random because they are required to complete the full ballot. It has been described as a system "somewhere between plurality and the Borda Count, but as veering more towards plurality". Simulations show that 30% of Nauru elections would produce different outcomes if counted using standard Borda rules.

Truncated ballots

A common way in which versions of the Borda count differ is the method for dealing with truncated ballots, that is, ballots on which a voter has not expressed a full list of preferences. There are several methods:
In a modified Borda count, the number of points given for a voter's first and subsequent preferences is determined by the total number of options or candidates they have actually ranked, rather than the total number listed. This is to say, typically, on a ballot of n options/candidates, if a voter casts only m preferences, a first preference gets m points, a second preference m–1 points, and so on. In more general terms, an 'x'th preference, if cast, gets one more point than an 'x+1'th preference. The MBC involves no special weighting: the difference is always just one point.
In a BC on 5 options, voting for all 5 options gives first preference 5 points, second preference 4 points, and so on; whereas voting for only 1 option still gives first preference 5 points. In effect, conventional Borda Count encourages the voter to submit only a first preference, so it degenerates into a plurality vote.
In a 5-option MBC, by contrast, votes for only 1 option thus gives the favorite just 1 point; votes for 2 options give first preference 2 points. Since the effect of a particular ballot on the outcome isn't determined by the absolute value of the votes given, but the difference between the values given to different candidates, there is no need for the voter to rank candidates at the bottom of their list among which they are indifferent.

Multiple winners

The system invented by Jean-Charles de 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. The quota Borda system is a system of proportional representation in multi-seat constituencies that uses the Borda count.

Other systems

A number of voting systems other than the Borda count employ its system of assigning points for rankings. The Nanson and Baldwin methods are single-winner voting systems that combine elements of the Borda count and instant-runoff voting. Unlike the Borda count, Nanson and Baldwin are majoritarian and Condorcet methods.

As a consensual method

Unlike other popular voting systems, in the Borda count it is possible for a candidate who is the first preference of an absolute majority of voters to not be elected; this is because the Borda count affords greater importance to a voter's lower preferences than most other systems, including other preferential methods such as instant-runoff voting and Condorcet methods.
The Borda count tends to favor candidates supported by a broad consensus among voters, rather than the candidate who is necessarily the favorite of a majority; for this reason, its supporters see the Borda count as a method that promotes unity and avoids the 'tyranny of the majority', and the resulting divisiveness and even violence that it can lead to. Advocates argue, for example, that where the majority candidate is strongly opposed by a large minority of the electorate, the Borda winner may have higher overall utility than the majority winner. On grounds such as these, the de Borda Institute of Northern Ireland advocates the use of a form of referendum based on the Borda count in divided societies such as Northern Ireland, the Balkans and Kashmir.
Because it will not necessarily elect a candidate who is the first preference of a majority of voters, the Borda count is said by scholars to fail the majority criterion. Other voting systems that favor consensus rather than majority rule include cardinal methods such as approval voting, score voting, and their variants. These are sometimes called "utilitarian voting methods" because they try to maximize the entire population's utility, rather than maximizing the majority's utility at the expense of the minority's.

Example

In an election in which 100 voters express the following preferences:
No.51 voters5 voters23 voters21 voters
1stAndrewCatherineBrianDavid
2ndCatherineBrianCatherineCatherine
3rdBrianDavidDavidBrian
4thDavidAndrewAndrewAndrew

The Borda scores of the candidates are:
CandidateBase 0Base 1Nauru
Andrew15325363.25
Brian15125149.5
Catherine20530552.5
David9119143.08333...

The first totals are when the Borda system is N-1 such that last place has a point score of 0. The second totals are based on N points for first choice, N-1 second, N-2 third etc. Note that using N for first place or N-1 only changes everyone's totals by the number of voters. In this case, with 100 voters, the differences are 100 for each candidate.
Under most single-winner voting systems – including 'first-past-the-post', instant-runoff, Condorcet methods, and Nauru Borda – Andrew would have been the winning candidate; however, under the standard Borda count, Catherine has the highest Borda score and so is elected instead. Although Andrew is supported by an unambiguous absolute majority of voters, he is the last preference of 49 voters, which suggests that he may be strongly opposed by almost one half of the electorate. Catherine, though she receives only a handful of first-preference votes, is at least the second choice of all voters, implying that she is broadly acceptable to all.
Nauru Borda, relative to normal Borda, places a much stronger emphasis on first choices over second choices, which is why Catherine did so poorly under that system.

Potential for tactical manipulation

Tactical voting

Like many other voting systems, the Borda count is vulnerable to tactical voting. In particular, it is highly vulnerable to the tactics of compromising and burying. In compromising, voters can benefit by insincerely raising the position of their second choice candidate over their first choice candidate, in order to help the second choice candidate to beat a candidate they like even less. In burying, voters can help a more-preferred candidate by insincerely lowering the position of a less-preferred candidate on their ballot.
An effective tactic is to combine these two strategies. For example, if there are two candidates whom a voter considers to be the most likely to win, the voter can maximise his impact on the contest between these front runners by ranking the candidate whom he likes more in first place, and ranking the candidate whom he likes less in last place. If neither front runner is his sincere first or last choice, the voter is employing both the compromising and burying tactics at once; if many voters employ such strategies, then the result will no longer reflect the sincere preferences of the electorate.
Using the example below based on choosing the capital of Tennessee, if polls suggest a toss-up between Nashville and Chattanooga, citizens of Knoxville might change their ranking to
  1. Chattanooga
  2. Knoxville
  3. Memphis
  4. Nashville
If many Knoxville voters voted in this way, it would result in the election of Chattanooga. Citizens of Chattanooga could also increase the likelihood of the election of their city by voting tactically, but would require the assistance of some tactical voters from Knoxville to be successful.
The French Academy of Sciences experimented with Borda's system, but abandoned it, in part because "the voters found how to manipulate the Borda rule: not only by putting their most dangerous rival at the bottom of their lists, but also by truncating their lists". In response to the issue of strategic manipulation in the Borda count, M. de Borda said, "My scheme is intended for only honest men".
The academic Donald G. Saari has created a mathematical framework for evaluating positional methods which shows that, for 3-candidate races, the Borda count is more resistant to tactical voting than other positional methods such as plurality, approval, and cumulative voting.
Tactical voting is common in Slovenia, where truncated ballots are allowed; a majority of voters bullet vote, with only 42% of voters ranking a second-preference candidate.

Strategic nomination

The Borda count is highly vulnerable to a form of strategic nomination called teaming or cloning. This means that when more candidates run with similar ideologies, the probability of one of those candidates winning increases. Therefore, under the Borda count, it is to a faction's advantage to run as many candidates in that faction as they can. For example, even in a single-seat election, it would be to the advantage of a political party to stand as many candidates as possible in an election. In this respect, the Borda count differs from many other single-winner systems, such as the 'first past the post' plurality system, in which a political faction is disadvantaged by running too many candidates. Under systems such as plurality, 'splitting' a party's vote in this way can lead to the spoiler effect, which harms the chances of any of a faction's candidates being elected.
In 1980, William Gehrlein and Peter Fishburn compared the Borda count to other positional methods, such as plurality and approval voting. They investigated the likelihood of a positional method choosing the same candidate when the set of candidates was modified by eliminating one losing candidate from a three-candidate election, and two losing candidates from a four-candidate election. They found that the Borda count was the positional rule which maximises the probability of electing the same candidate after this modification of the choice set.
Strategic nomination is used in Nauru, according to MP Roland Kun, with factions running multiple "buffer candidates" who are not expected to win, to lower the tallies of their main competitors.

Evaluation by criteria

Scholars of electoral systems often compare them using mathematically defined voting system criteria. From among these:
The variant of the Borda count that permits bullet voting satisfies the plurality criterion, but the 'modified Borda count' does not. Variants that oblige voters to rank only a certain specified number of candidates satisfy the same criteria as the conventional Borda count.
Simulations show that Borda has a high probability of choosing the Condorcet winner when one exists.

Example

This leads to the following point counts per 100 voters:
Voters' home cityMemphisNashvilleKnoxvilleChattanooga
Memphis42×3=12642×2=84042×1=42
Nashville026×3=7826×1=2626×2=52
Knoxville017×1=1717×3=5117×2=34
Chattanooga015×1=1515×2=3015×3=45
Total126194107173

Thus Nashville is elected.

Current uses

Political uses

The Borda count is used for certain political elections in at least three countries, Slovenia and the tiny Micronesian nations of Kiribati and Nauru.
In Slovenia, the Borda count is used to elect two of the ninety members of the National Assembly: one member represents a constituency of ethnic Italians, the other a constituency of the Hungarian minority.
Members of the Parliament of Nauru are elected based on a variant of the Borda count that involves two departures from the normal practice: multi-seat constituencies, of either two or four seats, and a point-allocation formula that involves increasingly small fractions of points for each ranking, rather than whole points.
In Kiribati, the president is elected by the plurality system, but a variant of the Borda count is used to select either three or four candidates to stand in the election. The constituency consists of members of the legislature. Voters in the legislature rank only four candidates, with all other candidates receiving zero points. Since at least 1991, tactical voting has been an important feature of the nominating process.
The Republic of Nauru became independent from Australia in 1968. Before independence, and for three years afterwards, Nauru used instant-runoff voting, importing the system from Australia, but since 1971, a variant of the Borda count has been used.
The modified Borda count has been used by the Green Party of Ireland to elect its chairperson.
The Borda count has been used for non-governmental purposes at certain peace conferences in Northern Ireland, where it has been used to help achieve consensus between participants including members of Sinn Féin, the Ulster Unionists, and the political wing of the UDA.

Other uses

The Borda count is used in elections by some educational institutions in the United States:
The Borda count is used in elections by some professional and technical societies:
The OpenGL Architecture Review Board uses the Borda count as one of the feature-selection methods.
The Borda count is used to determine winners for the World Champion of Public Speaking contest organized by Toastmasters International. Judges offer a ranking of their top three speakers, awarding them three points, two points, and one point, respectively. All unranked candidates receive zero points.
The modified Borda count is used to elect the President for the United States member committee of AIESEC.
The Eurovision Song Contest uses a heavily modified form of the Borda count, with a different distribution of points: only the top ten entries are considered in each ballot, the favorite entry receiving 12 points, the second-placed entry receiving 10 points, and the other eight entries getting points from 8 to 1. Although designed to favor a clear winner, it has produced very close races and even a tie.
The Borda count is used for wine trophy judging by the Australian Society of Viticulture and Oenology, and by the RoboCup autonomous robot soccer competition at the Center for Computing Technologies, in the University of Bremen in Germany.
The Finnish Associations Act lists three different modifications of the Borda count for holding a proportional election. All the modifications use fractions, as in Nauru. A Finnish association may choose to use other methods of election, as well.

Sports

The Borda count is a popular method for granting sports awards in the United States. Uses include:
In sailboat fleet racing, the Borda count is used to select the winner of a regatta, with each individual race in the regatta treated as a 'vote'.

Borda-similar systems

Some voting systems described as Borda systems are variants of the Borda count. The scores candidates receive in some of those systems are significantly different from those they would receive using a strict Borda count. Fraenkel and Grofman state that "The system used on Nauru is not merely a ‘modified form of Borda’, but an important rule in its own right."

History

A form of the Borda count was one of the voting methods employed in the Roman Senate beginning around the year 105. However, in its modern, mathematical form, the system is thought to have been discovered independently at least three times: