Borda count
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The Borda count is a voting system used for single-winner elections in which each voter rank-orders the candidates.
The Borda count was devised 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. It was used by the French Academy of Sciences beginning in 1784 to elect members until quashed by Napoleon in 1800.
This form of voting 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.
The Borda count is used for the election of ethnic minority members of parliament in Slovenia, as well as to elect members of parliament for the south Pacific islands of Nauru and Kiribati in modified versions. It was one of the voting methods employed in the Roman Senate beginning around the year 105. The Borda count and variations have been used in Northern Ireland for non-electoral purposes, such as to achieve a consensus between participants including members of Sinn Féin, the Ulster Unionists, and the political wing of the UDA.
Additionally, 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, and at the University of Missouri Graduate-Professional Council to elect its officers. Both universities are located in the United States.
Borda count is one of the voting methods used and advocated by the Florida affiliate of the American Patriot Party. See here (http://www.patriotparty.us/state/fl/platform.htm) and here (http://www.patriotparty.us/state/fl/bylaws.htm).
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".
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Procedures
Each voter rank-orders all the candidates on their ballot. If there are n candidates in the election, then 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.
An example of an election
Imagine an election for the capital of Tennessee, a state in the United States that is over 500 miles (800 km) east-to-west, and only 110 miles (180 km) north-to-south. In this vote, the candidates for the capital are Memphis, Nashville, Chattanooga, and Knoxville. The population breakdown by metro area is as follows:
- Memphis: 826,330
- Nashville: 510,784
- Chattanooga: 285,536
- Knoxville: 335,749
If the voters cast their ballot based strictly on geographic proximity, the voters' sincere preferences might be as follows:
42% of voters (close to Memphis)
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26% of voters (close to Nashville)
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15% of voters (close to Chattanooga)
| 17% of voters (close to Knoxville)
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| 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 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.
Potential for tactical voting
The Borda count is vulnerable to burying; that is, instead of voting their honest preferences, voters can rank a strong candidate they like first, or rank a strong candidate they dislike last, to increase their chances of getting their prefered candidate elected.
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.
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 satisfies the monotonicity criterion, the consistency criterion, the summability criterion, the participation criterion, 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, which means that 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, thus allowing a highly ranked compromise candidate to be elected.
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 "vote for two", "vote for three", etc.
Variants
- The Borda count method can be extended to include tie-breaking methods. Also, ballots that do not rank all the candidates can be allowed in one of two 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, facilitates 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.
- Another way, called the modified Borda count, is to assign the points up to k-1, 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.
- 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.
See also
- List of democracy and elections-related topics
- Voting system - many other ways of voting
- Voting system criterion
- First Past the Post electoral system
- Instant-runoff voting
- Approval voting
- Plurality voting
- Condorcet method
- Schulze method
Further reading
- Chaotic Elections!, by Donald G. Saari (ISBN 0821828479), is a book that describes various voting systems using a mathematical model, and supports the use of the Borda count.
External links
- The de Borda Institute, Northern Ireland (http://www.deborda.org)
- The Symmetry and Complexity of Elections (http://www.colorado.edu/education/DMP/voting_b.html) Article by mathematician Donald G. Saari (http://www.math.uci.edu/~dsaari/shortcv.pdf) shows that the Borda Count has relatively few paradoxes compared to certain other voting methods.
- Article by Alexander Tabarrok and Lee Spector (http://tinyurl.com/4wly9) Would using the Borda Count in the U.S. 1860 presidential election have averted the american Civil War? (PDF)
- Article by Benjamin Reilly (http://apseg.anu.edu.au/staff/pub_highlights/ReillyB_05.pdf) Social Choice in the South Seas: Electoral Innovation and the Borda Count in the Pacific Island Countries. (PDF)
- A Fourth Grade Experience (http://www.colorado.edu/education/DMP/voting_c.html) Article by Donald G. Saari (http://www.math.uci.edu/~dsaari/shortcv.pdf) observing the choice intuition of young children.
- Consequences of Reversing Preferences (http://hypatia.ss.uci.edu/imbs/tr/Final1.pdf) An article by Donald G. Saari and Steven Barney. (PDF)
- Rank Ordering Engineering Designs: Pairwise Comparison Charts and Borda Counts (http://www2.hmc.edu/~dym/PairwiseComparisons.pdf) Article by Clive L. Dym, William H. Wood and Michael J. Scott. (PDF)
- Arrow's Impossibility Theorem (http://mason.gmu.edu/~atabarro/arrowstheorem.pdf) This is an article by Alexander Tabarrok on analysis of the Borda Count under Arrow's Theorem. (PDF)
- Article by Daniel Eckert, Christian Klamler, and Johann Mitlöhner (http://www.kfunigraz.ac.at/fwiwww/home-eng/activities/pdfs/2003-5.pdf) On the superiority of the Borda rule in a distance-based perspective on Condorcet efficiency. (PDF)
- On Asymptotic Strategy-Proofness of Classical Social Choice Rules (http://www.math.auckland.ac.nz/~slinko/Research/Borda3.pdf) An article by Arkadii Slinko. (PDF)
- Non-Manipulable Domains for the Borda Count (http://www.bgse.uni-bonn.de/fileadmin/Fachbereich_Wirtschaft/Einrichtungen/BGSE/Discussion_Papers/2003/bgse13_2003.pdf) Article by Martin Barbie, Clemens Puppe, and Attila Tasnadi. (PDF)
- Which scoring rule maximizes Condorcet Efficiency? (http://www.math.union.edu/~dpvc/papers/2001-01.DC-BG-BZ/DC-BG-BZ.pdf) Article by Davide P. Cervone, William V. Gehrlein, and William S. Zwicker. (PDF)
- Scoring Rules on Dichotomous Preferences (http://pareto.uab.es/wp/2004/61704.pdf) Article mathematically comparing the Borda count to Approval voting under specific conditions by Marc Vorsatz. (PDF)
- Condorcet Efficiency: A Preference for Indifference (http://www.eco.fundp.ac.be/cahiers/filepdf/c224.pdf) Article by William V. Gehrlein and Fabrice Valognes. (PDF)
- Cloning manipulation of the Borda rule (http://www.hss.caltech.edu/Events/SCW/Papers/seraj.pdf) An article by Jérôme Serais. (PDF)
- Hybrid Voting Protocols and Hardness of Manipulation (http://www.cs.princeton.edu/~elkind/hybrid.pdf) Article by Edith Elkind and Helger Lipmaa. (PDF)
- Cooperative phenomena in crystals and social choice theory (http://allserv.rug.ac.be/~tmarchan/Crystals.pdf) Article by Thierry Marchant. (PDF)
- A program to implement the Condorcet and Borda rules in a small-n election (http://tinyurl.com/7tadt) Article by Iain McLean and Neil Shephard.(PDF)
- The Reasonableness of Independence (http://tinyurl.com/aeloj) Article by Iain McLean.(PDF)
- Flash animation by Kathy Hays (http://ola4.aacc.edu/kehays/umbc/MVP/Modified_BC.html) An example of how the Borda count is used to determine the Most Valuable Player in Major League Baseball.de:Borda-Wahl