# Borda count

Borda count

The Borda count is a single-winner election method in which voters rank candidates in order of preference. The Borda count determines the winner of an election by giving each candidate a certain number of points corresponding to the position in which he or she is ranked by each voter. Once all votes have been counted the candidate with the most points is the winner. Because it sometimes elects broadly acceptable candidates, rather than those preferred by the majority, the Borda count is often described as a consensus-based electoral system, rather than a majoritarian one.

The Borda count was developed independently several times, but is named for the 18th-century French mathematician and political scientist Jean-Charles de Borda, who devised the system in 1770. It is currently used for the election of two ethnic minority members of the National Assembly of Slovenia,[1] and, in modified forms, to select presidential election candidates in Kiribati and to elect members of the Parliament of Nauru. It is also used throughout the world by various private organisations and competitions.

## Voting and counting

Under the Borda count the voter ranks the list of candidates in order of preference. So, for example, the voter gives a '1' to their first preference, a '2' to their second preference, and so on. In this respect, a Borda count election is the same as elections under other preferential voting systems, such as instant-runoff voting, the Single Transferable Vote or Condorcet's method.

The number of points given to candidates for each ranking is determined by the number of candidates standing in the election. Thus, under the simplest form of the Borda count, if there are five candidates in an election then a candidate will receive five points each time they are ranked first, four for being ranked second, and so on, with a candidate receiving 1 point for being ranked last (or left unranked). 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:

Ranking Candidate Formula Points
1st Andrew (n) 5
2nd Brian (n – 1) 4
3rd Catherine (n – 2) 3
4th David (n – 3) 2
5th Elizabeth (n – 4) 1

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 points for a first preference, n – 2 for a second, and so on, with zero points for being ranked last (or left unranked). Another way to express this is that a candidate ranked in ith place receives ni 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:

Ranking Candidate Formula Points
1st Andrew (n – 1) 4
2nd Brian (n – 2) 3
3rd Catherine (n – 3) 2
4th David (n – 4) 1
5th Elizabeth (n – 5) 0

While the first of the above two formulae is used in the Slovenian parliamentary elections (as mentioned, for two out of 90 seats only), Nauru uses a sort of modified Borda count: the voter awards the first-ranked candidate with one point, while the second-ranked candidate receives half of the point, the third-ranked candidate receives one-third of the point, etc. (A similar system of weighting lower-preference votes is used in the Oklahoma primary electoral system.) Using the above example, in Nauru the point distribution among the five candidates would be this:

Ranking Candidate Formula Points
1st Andrew 1/1 1.00
2nd Brian 1/2 0.50
3rd Catherine 1/3 0.33
4th David 1/4 0.25
5th Elizabeth 1/5 0.20

When all votes have been counted, and the points added up, the candidate with most points wins. We have already noted that the Borda count is a preferential voting system; 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 the 'first-past-the-post' (plurality) system, bloc voting, approval voting and the limited vote.

## An example

Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities, and that everyone wants to live as near to the capital as possible.

The candidates for the capital are:

• Memphis, the state's largest city, with 42% of the voters, but located far from the other cities
• Nashville, with 26% of the voters, near the center of Tennessee
• Knoxville, with 17% of the voters
• Chattanooga, with 15% of the voters

The preferences of the voters would be divided like this:

42% of voters
(close to Memphis)
26% of voters
(close to Nashville)
15% of voters
(close to Chattanooga)
17% of voters
(close to Knoxville)
1. Memphis
2. Nashville
3. Chattanooga
4. Knoxville
1. Nashville
2. Chattanooga
3. Knoxville
4. Memphis
1. Chattanooga
2. Knoxville
3. Nashville
4. Memphis
1. Knoxville
2. Chattanooga
3. Nashville
4. Memphis

If the various rankings given to each candidate are added up they are as follows.

City First Second Third Fourth
Memphis 42% 0% 0% 58%
Nashville 26% 42% 32% 0%
Chattanooga 15% 43% 42% 0%
Knoxville 17% 15% 26% 42%

It can be seen above, for example, that Chattanooga is ranked first by 15% of voters, second by 43%, third by 42%, and last by no voters at all. To give points to each candidate for these rankings this example will use the formula, explained above, whereby a candidate receives one point for each time a candidate is ranked lower than them (or ni points). Thus when Chattanooga's votes are added up the results are calculated as: (15*3) + (43*2) + (42*1) + (0*0) = 173. When the points of all candidates are added up, the results are as follows:

City First Second Third Fourth Total points
Memphis 42*3 0 0 58*0 126
Nashville 26*3 42*2 32*1 0 194
Chattanooga 15*3 43*2 42*1 0 173
Knoxville 17*3 15*2 26*1 42*0 107

Result: The winner of the election is Nashville, as it has 194 points, which is more than any other candidate. Since this example was worked purely according to geographical distance, we would expect the "most acceptable" city to be the most central; a glance at the map above confirms that this is indeed the case. However, if voters in both Knoxville and Chattanooga were to put Chattanooga first and Nashville last, the winner would be Chattanooga, a preferable outcome for voters in both those cities. This is an example of tactical voting.

## Variants

As noted above, there is more than one formula for assigning points for each ranking of a candidate. In Nauru, a distinctive formula is used based on increasingly small fractions of points. Under the system a candidate receives 1 point for a first preference, ½ a point for a second preference, ⅓ for third preference, and so on. This method is far more favourable 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 have to complete the full ballot.

In Kiribati, a variant is employed which uses a traditional Borda formula, but in which voters rank only four candidates, irrespective of how many are standing – an example of a truncated ballot.[2]

## 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:

• The simplest method is to allow voters to rank as many or as few candidates as they wish, but simply give every unranked candidate the minimum number of points. For example, if there are 10 candidates, and a voter votes for candidate A first and candidate B second, leaving everyone else unranked, candidate A receives 9 or 10 points (depending on the formula used), candidate B receives 8 or 9 points, and all other candidates receive either zero or 1. However, this method allows strategic voting in the form of bullet voting: voting for only one candidate and leaving every other candidate unranked. This variant makes a bullet vote more effective than a fully ranked ballot.
• Voters can simply be obliged to rank all candidates. This is the method used in Nauru.
• Voters can be permitted to rank only a subset of the total number of candidates but obliged to rank all of those, with all unranked candidates being given zero points. This is the system used in Kiribati.
• In Slovenia legislation does not mention the truncated ballots. Consequently, in the past election bodies dealed with them differently from district to district and from election to election. In 2004 parliamentary election, for instance, in one district unranked candidates received one point while in the other district they received zero points. In 2008, unranked candidates in both districts that use Borda Count received one point.

## Modified Borda count

In a modified Borda count (MBC), the number of points given for a voter's first and subsequent preferences is determined by the total number of candidates they have actually ranked, rather than the total number standing. This is to say, typically, on a ballot of n options/candidates, if a voter casts preferences for only m options (where m is smaller than n), a first preference gets m points, a second preference m – 1 points, and so on. This means, in other words, that if there are ten candidates but a voter ranks only five, then their first preference will receive only five points; their second preference will receive 4 points, their next 3, and so on. This method effectively penalises voters who do not rank a full ballot, by diminishing the number of points their vote distributes among candidates.

Thus he who votes for only one option/candidate exercises only 1 point; while she who casts two preferences will exercise 3 (2 plus 1) points.

In more general terms, an 'x'th preference, if cast, gets one more point than an 'x+1'th preference (whether cast or not). The MBC involves no special weighting: the difference is always just one point.

It is also possible, if specifically stipulated by the body using the MBC as its voting method, that candidates voted for by someone who does not cast preferences for all candidates get even less points than described above. For example, the first preference listed may receive m – 1 points instead of m points, the second preference will then receive m – 2 points, and so on.

The modified Borda count differs from a Borda count only in the preferences of those who submit partial ballots. In a BC on five options, he who votes for all five options gives his first preference 5 points, his second preference 4 points, and so on and she who votes for only one option still gives her first preference 5 points. In effect, therefore, a modified Borda count encourages the voter to submit only a first preference, in which case it degenerates into a plurality vote.

In a five-option MBC, by contrast, she who votes for only one option thus gives her favourite just 1 point; he who votes for two options gives his first preference 2 points (and his second preference 1 point). To ensure your favourite gets the maximum 5 points, therefore, you should cast all five preferences, then your favourite gets 5 points, your second preference gets 4 points, and so on, just like in a Borda count. The MBC thus encourages voters to submit a fully marked ballot.

The modified Borda count has been used by the Irish Green Party to elect its chairperson.[3][4]

## 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 electing those candidates with the most points. 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 most other voting systems, in the Borda count it is possible for a candidate who is the first preference of an absolute majority of voters to fail to 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's method. The Borda count tends to favour candidates supported by a broad consensus among voters, rather than the candidate who is necessarily the favourite of a majority; for this reason, some of its supporters see the Borda count as a method that promotes consensus and avoids the 'tyranny of the majority'. 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. It is also theoretically possible for such a candidate to fail to be elected under approval voting.

### An example

Imagine an election in which 100 voters express the following preferences:

# 51 voters 5 voters 23 voters 21 voters
1st Andrew Catherine Brian David
2nd Catherine Brian Catherine Catherine
3rd Brian David David Brian
4th David Andrew Andrew Andrew

The Borda scores of the candidates are:

• Andrew: 153
• Catherine: 205
• Brian: 151
• David: 91

Under most single-winner voting systems – including 'first-past-the-post' (plurality), instant-runoff and Condorcet's method – Andrew would have been the winning candidate; however, under the Borda count, Catherine has the highest Borda score and so is elected instead. Favouring Andrew as the winner is the fact that he is supported by an unambiguous absolute majority of voters. On the other hand, 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; this seems to suggest that she is broadly acceptable to all voters.

## Potential for tactical manipulation

### Tactical voting

Like all voting systems, the Borda count is vulnerable to tactical voting. In particular, it is highly vulnerable to the tactics of compromising and burying. Compromising: voters can help avoid the election of a less-preferred candidate by insincerely raising the position of a more-preferred candidate on their ballot. 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 above example 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 (compromising their sincere first choice, Knoxville)
2. Knoxville
3. Memphis (burying their sincere third choice, Nashville)
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.

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 in which he claims to show that the Borda count has fewer opportunities for tactical voting than other positional methods such as plurality voting.[citation needed]

### 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.

## 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.

## 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. As noted above, members of the Parliament of Nauru are elected based on a variant of the Borda count that involves two departures from the normal practice: (1) multi-seat constituencies, of either two or four seats, and (2) a point-allocation formula that involves increasingly small fractions of points for each ranking, rather than whole points. In Kiribati, the president (or Beretitenti) 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 (Maneaba). 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 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, and points-based systems similar to it, are often used to determine awards in competitions. The Borda count is a popular method for granting awards for sports in the United States, and is used in determining the Most Valuable Player in Major League Baseball, by the Associated Press and United Press International to rank teams in NCAA sports, to determine the winner of the Heisman Trophy[5], and so on. The Eurovision Song Contest also uses a positional voting method similar to the Borda count, with a different distribution of points: only the top ten entries are considered in each ballot, the favourite 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 favour 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 People's Remix Competition uses a Borda variant where each voter ranks only the top three contestants.

The Borda count is used in a number of educational institutions in the United States, such as at the University of Michigan College of Literature, Science and the Arts to elect the Student Government, to elect the Michigan Student Assembly for the university at large, at the University of Missouri Graduate-Professional Council to elect its officers, at the University of California Los Angeles Graduate Student Association to elect its officers, in the Civil Liberties Union of Harvard University to elect its officers, at Southern Illinois University at Carbondale to elect officers to the Faculty Senate, and at Arizona State University to elect officers to the Department of Mathematics and Statistics assembly. It is also used to elect faculty members to committees at Wheaton College, Massachusetts. Borda count is used to break ties for member elections of the faculty personnel committee of the School of Business Administration at the College of William and Mary.

In professional societies, the Borda count is used to elect the Board of Governors of the International Society for Cryobiology, the management committee of Tempo sustainable design network, located in Cornwall, United Kingdom, and to elect members to Research Area Committees of the U.S. Wheat and Barley Scab Initiative. The Borda count is one of the feature-selection methods used by the OpenGL Architecture Review Board. The Borda count is used to elect the Board of Directors of the X.Org Foundation.

The Borda count is used to determine winners for Toastmasters International speech contests. 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 Finnish Associations Act lists three different modifications of the Borda count for holding a proportional election. All the modifications use fractions, like in Nauru. A Finnish assoction may choose to use other methods of election, as well.[6]

### Application of the term

It is debatable whether or not some of the systems explained above are accurately described as "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.

An example is the case of the voting to determine the winner of the Heisman Trophy: judges vote 3,2,1 for their top three choices – the vote weights are thus (3,2,1,0,0,0, ..., 0,0,0). By contrast, the Borda vote weights in, say, a fifty-candidate election would be (49,48,47, ..., 2,1,0), which is markedly different. Heisman-style voting, when there are more than a handful of candidates, is thus more similar to plurality voting, which has weights (1,0,0,0, ..., 0,0,0), than it is to Borda.

## 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 have been discovered independently by at least three men:

• Ramon Llull (1232–1315), who with the 2001 discovery of his lost manuscripts Ars notandi, Ars eleccionis, and Alia ars eleccionis, was given credit for discovering the Borda count and Condorcet criterion (Llull winner) in the 13th century
• Nicholas of Cusa (1401–1464), who in 1433 unsuccessfully suggested the method as a way of electing the Holy Roman Emperor
• Jean-Charles de Borda, who devised the system in June of 1770, invented his system as a fair way to elect members to the French Academy of Sciences, and first published his method in 1781 as Mémoire sur les élections au scrutin in the Histoire de l'Académie Royale des Sciences, Paris. The method was used by the Academy from 1784 until being quashed by Napoleon in 1800.

## Notes

1. ^ http://www.minelres.lv/NationalLegislation/Slovenia/Slovenia_ElecParl_excerpts_English.htm "Slovenia's electoral law"
2. ^
3. ^ Voting Systems
4. ^ Designing an All-Inclusive Democracy by Peter Emerson, Springer Verlag, 2007, Part 1, pages 15-38 "Collective Decision-making: The Modified Borda Count, MBC" ISBN 978-3-540-33163-6 (Print) 978-3-540-33164-3 (Online)
5. ^ Heisman.com - Heisman Trophy
6. ^ "Finnish Associations Act". National Board of Patents and Registration of Finland. Retrieved 26 June 2011.

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