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Borda count

The Borda count BC and the Modified Borda count MBC are two voting system used mainly for decision-making and in election systems like the Quota Borda system QBS and the matrix vote. The BC and MBC can also be used for single-winner elections in which each voter rank-orders the candidates.

The Borda count BC was devised by Nicholas Cusanus in 1433, while the Modified Borda Count MBC was proposed by Jean-Charles de Borda in June of 1770. It was first published in 1781 as Mémoire sur les élections au scrutin in the Histoire de l'Académie Royale des Sciences, Paris. This method was devised for decision-making, but mainly for elections, by M. de Borda to fairly elect members to the French Academy of Sciences and was used by the Academy beginning in 1784 until quashed by Napoleon in 1800.

The Borda count is classified as a positional voting system because each rank on the ballot is worth a certain number of points. Other positional methods include first-past-the-post (plurality) voting, and minor methods such as "vote for any two" or "vote for any three".

The BC and MBC Procedures

Each voter rank-orders all the candidates on their ballot. If there are n candidates in the election, then in the BC analysis, the first-place candidate on a ballot receives n-1 points, the second-place candidate receives n-2, and in general the candidate in ith place receives n-i points. The candidate ranked last on the ballot therefore receives zero points.

The points are added up across all the ballots, and the candidate with the most points is the winner.

The MBC procedure is similar, but the difference can be huge. In a ballot of n options or candidates, a voter may cast m preferences, where n > m > 1. In a BC, as outlined above, points are awarded to (1st, 2nd ... last) preferences cast, as per the rule (n, n-1 ... 1) points or (n-1, n-2 ... 0) points. In an MBC, however, points are awarded as per the rule (m, m-1 ... 1) points. Accordingly, in a 5-option (or 5-candidate) vote:

he who casts just one preference gets his favourite only 1 point;

she who casts two preferences gets her favourite 2 points, (and her 2nd choice gets 1 point);

and so on; accordingly

those who cast all 5 preferences get their favourite 5 points, (their 2nd choice 4 points, etc.).

In a BC, he who truncates his ballot and casts only 1 point, gets his favourite an (n-1) points advantage over all the other options/candidates. In an MBC, in contrast, a voter's (x)^th preference always gets just 1 point more than her (x+1)^th preference, regardless of whether or not she has cast that (x+1)^th preference,

An example of a BC election

Tennessee's four cities are spread throughout the state

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 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)
Memphis Nashville Chattanooga Knoxville	Nashville Chattanooga Knoxville Memphis	Chattanooga Knoxville Nashville Memphis	Knoxville Chattanooga Nashville Memphis

City	First	Second	Third	Fourth	Points
Memphis	42	0	0	58	126
Nashville	26	42	32	0	194
Chattanooga	15	43	42	0	173
Knoxville	17	15	26	42	107

Nashville is the winner in this election, as it has the most points. Nashville also happens to be the pairwise champion (aka Condorcet winner) in this case. While the Borda count does not always select the Condorcet winner as the Borda Count winner, it always ranks the Condorcet winner above the Condorcet loser. No other positional method can guarantee such a relationship.

Like most voting methods, The Borda count is vulnerable to compromising. That is, voters can help avoid the election of a less-preferred candidate by insincerely raising the position of a more-preferred candidate on their ballot.

The Borda count is also vulnerable to burying. That is, voters can help a more-preferred candidate by insincerely lowering the position of a less-preferred candidate on their ballot.

For example, if there are two candidates whom a voter considers to be the most likely to win, the voter can maximize their impact on the contest between these candidates by ranking the candidate whom they like more in first place, and ranking the candidate whom they like less in last place. If neither candidate is their sincere first or last choice, the voter is employing both the compromising and burying strategies at once. If many voters employ such strategies, then the result will no longer reflect the sincere preferences of the electorate.

In response to the issue of strategic manipulation in the Borda count, M. de Borda said "My scheme is only intended for honest men."

An Example of an MBC decision

In 2023 in a seminar Queen's University Belfast, the subject was the (forthcoming) border poll. Participants proposed a number of options; everything which did not clash with the UN Charter was allowed 'on the table' and computer screen; a ballot paper of seven options was compiled; all concerned then cast their preferences and the results - the consensus coefficient CC of each option - were as shown. A consensus coefficient is defined a

the option's MBC score = CC

the max possible MBC score

so a CC can very from a maximum of 1.00 to a minimum of 0.00.

A 2-part federal Ireland	0.76
A 4-part federal Ireland	0.71
Anglo-Celtic Federation	0.67
A united Ireland	0.62
An independent NI	0.52
The status quo	0.29
A united British Isles	0.10

Potential for BC tactical voting

The Borda count fails the favorite betrayal criterion and is thus vulnerable to compromising. That is, voters can help avoid the election of a less-preferred candidate by insincerely raising the position of a more-preferred candidate on their ballot. However, it is possible to modify the Borda count to pass this criterion: see Summed-Ranks for an example.

The Borda count is also vulnerable to burying. That is, voters can help a more-preferred candidate by insincerely lowering the position of a less-preferred candidate on their ballot.

For example, if there are two candidates whom a voter considers to be the most likely to win, the voter can maximize their impact on the contest between these candidates by ranking the candidate whom they like more in first place, and ranking the candidate whom they like less in last place. If neither candidate is their sincere first or last choice, the voter is employing both the compromising and burying strategies at once. If many voters employ such strategies, then the result will no longer reflect the sincere preferences of the electorate.

In response to the issue of strategic manipulation in the Borda count, M. de Borda said "My scheme is only intended for honest men."

In majority voting, the two options listed are (regarded as being) mutually exclusive. So it is that, for example, in the Balkans, the 1991 referendum - "Are you Serb or Croat?" - disenfranchised any partner in, or adult child of, a mixed relationship... or anyone who wanted to vote for a compromise, or more importantly, for peace.

In an MBC of, say, five options, not every option will be totally mutually exclusive of all the other options. If, then, the two winning options are 'neck-and-neck', a composite may be formed based on the most popular option, as amended by those parts of the runner-up which are compatible with the winning option. The voter's 2nd preference should therefore best be his/her actual desire. As the old saying goes, "be careful what you wish for."

Effect on factions and candidates

The Borda count is vulnerable to teaming: 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, creating the opposite of the spoiler effect. The teaming or "clone" effect is significant only where restrictions are placed on the candidate set.

Criteria passed and failed

Voting systems are often compared using mathematically-defined criteria. See voting system criterion for a list of such criteria.

The Borda count BC and Modified Borda Count MBC satisfy the monotonicity criterion, the summability criterion, the consistency criterion, the participation criterion, the plurality criterion, reversal symmetry, Intensity of Binary Independence,^{[clarification needed]} and the Condorcet loser criterion.

It does not satisfy the Condorcet criterion, the Independence of irrelevant alternatives criterion, or the Independence of Clones criterion.

The Borda count also does not satisfy the majority criterion, i.e. if a majority of voters rank one candidate in first place, that candidate is not guaranteed to win. This could be considered a disadvantage for Borda count in political elections, but it also could be considered an advantage if the favorite of a slight majority is strongly disliked by most voters outside the majority, in which case the Borda winner could have a higher overall utility than the majority winner. However, Borda does satisfy the majority criterion in the two-candidate case, meaning that it has a spoiler effect when it doesn't elect the Condorcet winner (because the CW would guaranteeably win when it's just them and any other candidate).

Donald G. Saari created a mathematical framework for evaluating positional methods in which he showed that Borda count has fewer opportunities for strategic voting than other positional methods, such as plurality voting or anti-plurality voting.^{[citation needed]}

Variants

Ballots that do not rank all the candidates can be allowed in three ways.
- One way to allow leaving candidates unranked is to leave the scores of each ranking unchanged and give unranked candidates 0 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 points, candidate B receives 8 points, and all other candidates receive 0. This, however, allows strategic voting in the form of bullet voting: voting only for one candidate and leaving every other candidate unranked. This variant makes a bullet vote more effective than a fully-ranked vote. This variant would satisfy the Plurality criterion and the Non-compulsory support criterion.
- Another way, called the modified Borda count, is to assign the points up to k, where k is the number of candidates ranked on a ballot. For example, in the modified Borda count, a ballot that ranks candidate A first and candidate B second, leaving everyone else unranked, would give 2 points to A and 1 point to B. This variant would not satisfy the Plurality criterion or the Non-compulsory support criterion.
- The third way is to employ a uniformly truncated ballot obliging the voter to rank a certain number of candidates, while not ranking the remainder, who all receive 0 points. This variant would satisfy the same criteria as the Borda count.

A proportional election requires a different variant of the Borda count called the quota Borda system.

A voting system based on the Borda count that allows for change only when it is compelling, is called the Borda fixed point system.

A procedure for finding the Condorcet winner of a Borda count tally is called Nanson's method or Instant Borda runoff.

Current Uses of the Borda count

The Borda count is popular in determining awards for sports in the United States. It is used in determining the Most Valuable Player in Major League Baseball, by the Associated Press and United Press International to rank players in NCAA sports, and other contests. The Eurovision Song Contest also uses a positional voting method similar to the Borda count, with a different distribution of points. It is used for wine trophy judging by the Australian Society of Viticulture and Oenology. Borda count is used by the RoboCup robot competition at the Center for Computing Technologies, University of Bremen in Germany.

The Borda count has historical precedent in political usage as it was one of the voting methods employed in the Roman Senate beginning around the year 105. The Borda count is presently used for the election of ethnic minority members of parliament in Slovenia. In modified versions it is also used to elect members of parliament for the central Pacific island of Nauru (using a different positional point system) and for the selection of Presidential election candidates from among members of parliament in neighbouring Kiribati. As managed by the de Borda Institute, the Modified Borda Count MBC, the Quota Borda System QBS, and the matrix vote have been used in Northern Ireland for decision-making, elections, and governance respectively, so as to achieve a consensus between participants including members of Sinn Féin, the Ulster Unionists, and the political wing of the UDA.

In educational institutions, the Borda count is used 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, 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. 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. All these universities are located in the United States.

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.

Borda count is one of the feature selection methods used by the OpenGL Architecture Review Board.

Borda count is one of the voting methods used and advocated by the Florida affiliate of the American Patriot Party. See here and here.

Outside of voting theory

The Borda count is a very common method used to aggregate ranked information in general; it pops up when looking at search algorithms, for example. See https://www.cs.sfu.ca/~jpei/publications/BordaRank-TKDE.pdf and https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4026830/ for examples.

Notes

Borda can be thought of as a variant of Score voting where the scores for each candidate are chosen to some extent for the voter, rather than by the voter. Because of the way Borda chooses those scores, a Condorcet winner can never have the fewest points (i.e. be in last place) in Borda, while this can happen in Score voting (Simple example: if between two candidates, a majority of voters give the first candidate a 1 out of 5, and the minority give the second candidate a 5 out of 5, then the majority's 1st choice, who is the Condorcet winner, would have fewer points than the minority's preference and thus be in last place). As an extension of this property, the instant-runoff form of Borda, Baldwin, is a Smith-efficient Condorcet method, whereas the same is not true for the instant-runoff form of Score, IRNR.

The Borda count can be done in different ways depending on how points are assigned. For certain variants, it may be possible to find the Borda scores for the candidates using pairwise preferences. ^[1]^[2]

The Borda count returns a social order with Kemeny distance at most five times that of the Kemeny-Young method.^[3] Generalizing the Borda count to incomplete rankings takes more care; some such generalizations have a constant approximation factor to Kemeny-Young while others can be arbitrarily bad.^[4]

References

↑ https://rangevoting.org/Wright_Barry.pdf "We can also compute the Borda Count social preference order by summing the rows of the margin of victory matrix. To see why, consider this deconstruction of the Borda Count score. Since even a last place candidate gets 1 point, each candidate automatically gets n points, where n is the number of voters. Then for each pairwise victory, the candidate must be ranked one slot above another candidate on a particular ballot. Thus, the remaining points are exactly equal to the number of pairwise victories the candidate has. Since there is a clear bijection between the total number of pairwise victories and the sum of the entries in a candidate’s row of the margin of victory matrix, we can simply use this value"
↑ https://web.stanford.edu/~jdlevin/Papers/Voting.pdf bottom of p.11-12 "Since every point a candidate receives may be considered a head-to-head vote against some other candidate, Borda scores are equal to the total number of head-to-head votes a candidate receives. This means we can count Borda scores by writing a paired-comparisons matrix and summing the rows to generate the candidates' scores."
↑ Coppersmith, Don; Fleischer, Lisa; Rudra, Atri (2006). "Ordering by weighted number of wins gives a good ranking for weighted tournaments" (PDF). Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithms: 776–782.
↑ Mathieu, Claire; Mauras, Simon (2020). How to aggregate Top-lists: Approximation algorithms via scores and average ranks. Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA). Philadelphia, PA: Society for Industrial and Applied Mathematics. pp. 2810–2822. doi:10.1137/1.9781611975994.171.

External links

The de Borda Institute, Northern Ireland
The Symmetry and Complexity of Elections Article by mathematician Donald G. Saari shows that the Borda Count has relatively few paradoxes compared to certain other voting methods.
Article by Alexander Tabarrok and Lee Spector Would using the Borda Count in the U.S. 1860 presidential election have averted the American Civil War?
Article by Benjamin Reilly Social Choice in the South Seas: Electoral Innovation and the Borda Count in the Pacific Island Countries.
A Fourth Grade Experience Article by Donald G. Saari observing the choice intuition of young children.
Consequences of Reversing Preferences An article by Donald G. Saari and Steven Barney.
Rank Ordering Engineering Designs: Pairwise Comparison Charts and Borda Counts Article by Clive L. Dym, William H. Wood and Michael J. Scott.
Arrow's Impossibility Theorem This is an article by Alexander Tabarrok on analysis of the Borda Count under Arrow's Theorem.
Article by Daniel Eckert, Christian Klamler, and Johann Mitlöhner On the superiority of the Borda rule in a distance-based perspective on Condorcet efficiency.
On Asymptotic Strategy-Proofness of Classical Social Choice Rules An article by Arkadii Slinko.
Non-Manipulable Domains for the Borda Count Article by Martin Barbie, Clemens Puppe, and Attila Tasnadi.
Which scoring rule maximizes Condorcet Efficiency? Article by Davide P. Cervone, William V. Gehrlein, and William S. Zwicker.
Scoring Rules on Dichotomous Preferences Article mathematically comparing the Borda count to Approval voting under specific conditions by Marc Vorsatz.
Condorcet Efficiency: A Preference for Indifference Article by William V. Gehrlein and Fabrice Valognes.
Cloning manipulation of the Borda rule An article by Jérôme Serais.
Democracy and Social Choice: A Response to Saari Article by Mathias Risse.
Cooperative phenomena in crystals and social choice theory Article by Thierry Marchant.
A program to implement the Condorcet and Borda rules in a small-n election Article by Iain McLean and Neil Shephard.
The Reasonableness of Independence Article by Iain McLean.
Variants of the Borda Count Method for Combining Ranked Classifier Hypotheses Article by Merijn Van Erp and Lambert Schomaker
Flash animation by Kathy Hays An example of how the Borda count is used to determine the Most Valuable Player in Major League Baseball.

[1] ttps://rangevoting.org/Wright_Barry.pdf "We can also compute the Borda Count social preference order by summing the rows of the margin of victory matrix. To see why, consider this deconstruction of the Borda Count score. Since even a last place candidate gets 1 point, each candidate automatically gets n points, where n is the number of voters. Then for each pairwise victory, the candidate must be ranked one slot above another candidate on a particular ballot. Thus, the remaining points are exactly equal to the number of pairwise victories the candidate has. Since there is a clear bijection between the total number of pairwise victories and the sum of the entries in a candidate’s row of the margin of victory matrix, we can simply use this value"

[2] ttps://web.stanford.edu/~jdlevin/Papers/Voting.pdf bottom of p.11-12 "Since every point a candidate receives may be considered a head-to-head vote against some other candidate, Borda scores are equal to the total number of head-to-head votes a candidate receives. This means we can count Borda scores by writing a paired-comparisons matrix and summing the rows to generate the candidates' scores."

[3] Coppersmith, Don; Fleischer, Lisa; Rudra, Atri (2006). "Ordering by weighted number of wins gives a good ranking for weighted tournaments" (PDF). Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithms: 776–782.

[Mathieu_Mauras_2020_pp._2810–2822-4] Mathieu, Claire; Mauras, Simon (2020). How to aggregate Top-lists: Approximation algorithms via scores and average ranks. Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA). Philadelphia, PA: Society for Industrial and Applied Mathematics. pp. 2810–2822. doi:10.1137/1.9781611975994.171.

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