Borda count: Difference between revisions
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{{Wikipedia}}
The '''Borda count'''
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
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 ''i''th 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 <u>></u> m <u>></u> 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:
== An example of an election==▼
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)<sup>th</sup> preference always gets just 1 point more than her (x+1)<sup>th</sup> preference, regardless of whether or not she has cast that (x+1)<sup>th</sup> preference,
{{Tenn_voting_example}}
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</div>
Nashville is the winner in this election, as it has the most points.
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.
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In response to the issue of strategic manipulation in the Borda count, M. de Borda said "My scheme is only intended for honest men."
==Potential for tactical voting==▼
<u><big>An Example of an MBC decision</big></u>
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.▼
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
<u>the option's MBC score</u> = CC
the max possible MBC score
so a CC can very from a maximum of 1.00 to a minimum of 0.00.
{| class="wikitable"
|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 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.
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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 [[Strategic nomination|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]].
==Criteria passed and failed==
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Voting systems are often compared using mathematically-defined criteria. See [[voting system criterion]] for a list of such criteria.
The Borda count
It does not satisfy the [[Condorcet criterion]], the [[Independence of irrelevant alternatives]] criterion, or the [[Strategic nomination|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]]
==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]].
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==Current Uses of the Borda count==
The Borda count is popular in determining awards for sports in the [[United States]].
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 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, Illinois|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]].
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Borda count is one of the voting methods used and advocated by the [[Florida]] affiliate of the [[American Patriot Party]]. See [http://www.patriotparty.us/state/fl/platform.htm here] and [http://www.patriotparty.us/state/fl/bylaws.htm 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 [[:Category:Sequential loser-elimination methods|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 preference]]<nowiki/>s. <ref>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"</ref><ref>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."</ref>
The Borda count returns a social order with Kemeny distance at most five times that of the [[Kemeny-Young method]].<ref>{{cite journal|
title = Ordering by weighted number of wins gives a good ranking for weighted tournaments|
last1 = Coppersmith|
first1 = Don|
last2 = Fleischer|
first2 = Lisa|
last3 = Rudra|
first3 = Atri|
journal = Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithms|
pages = 776–782|
year = 2006|
url = http://www.cse.buffalo.edu/faculty/atri/papers/algos/fas-soda.pdf
}}</ref> 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.<ref name="Mathieu Mauras 2020 pp. 2810–2822">{{cite book | last=Mathieu | first=Claire | last2=Mauras | first2=Simon | journal=Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA) | title=How to aggregate Top-lists: Approximation algorithms via scores and average ranks | publisher=Society for Industrial and Applied Mathematics | publication-place=Philadelphia, PA | year=2020 | doi=10.1137/1.9781611975994.171 | pages=2810–2822|url=https://epubs.siam.org/doi/pdf/10.1137/1.9781611975994.171}}</ref>
==See also==
* [[Correlated Instant Borda Runoff]]
==References==
<references/>
==Further reading==
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*[http://www.kfunigraz.ac.at/fwiwww/home-eng/activities/pdfs/2003-5.pdf Article by Daniel Eckert, Christian Klamler, and Johann Mitlöhner] On the superiority of the Borda rule in a distance-based perspective on Condorcet efficiency.
*[http://www.math.auckland.ac.nz/~slinko/Research/Borda3.pdf On Asymptotic Strategy-Proofness of Classical Social Choice Rules] An article by Arkadii Slinko.
*[http://www.bgse.uni-bonn.de/fileadmin/Fachbereich_Wirtschaft/Einrichtungen/BGSE/Discussion_Papers/2003/bgse13_2003.pdf Non-Manipulable Domains for the Borda Count] Article by
*[http://www.math.union.edu/~dpvc/papers/2001-01.DC-BG-BZ/DC-BG-BZ.pdf Which scoring rule maximizes Condorcet Efficiency?] Article by Davide P. Cervone, William V. Gehrlein, and William S. Zwicker.
*[http://pareto.uab.es/wp/2004/61704.pdf Scoring Rules on Dichotomous Preferences] Article mathematically comparing the Borda count to Approval voting under specific conditions by Marc Vorsatz.
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[[Category:Single-winner voting methods]]
[[Category:
[[Category:Monotonic electoral systems]]
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