borda count

The Borda count is a voting system devised by Jean-Charles de Borda in June of 1770 (1), used for single or multiple-seat elections. It was first published in 1781 as Mmoire sur les lections au scrutin. Histoire de l'Acadmie Royale des Sciences, Paris and used by the French Academy of Sciences beginning in 1784 to elect members until quashed by Napolean Bonaparte in 1800. This form of voting is extremely 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, the Eurovision Song Contest, as well as many others. It is used for parliamentary elections in country of Slovenia, the south Pacific islands of Nauru and Kiribati in modified versions, and was one of the voting methods employed in the Roman Senate beginning around 105 C.E. Borda count is used at the University of Michigan College of Literature, Science and the Arts to elect the Student Government and to elect the Michigan Student Assembly. The University of Missouri Graduate-Professional Council uses Borda Count to elect its officers. The Borda count can also be used as a decision-making process.

Procedures for Elections and Decisions

A number n is selected; this number is equal to the number of candidates or options. Each voter lists their top n choices, in order of preference. Borda elections use rank preference ballots. A first-place rank is worth n-1 points, a second-place rank is worth n-2 points, down to an nth rank being worth 0 points. A candidate's score is the sum of the number of points they received. The highest-scoring candidate is elected. An alternative counting procedure is to say a first preference is worth n points, a second-place rank is worth n-1, and an nth rank is worth 1 point. This latter formula has certain advantages in a Modified Borda Count (see below). In the trivial case of n=2, this is mathematically identical to majority voting. If all choices are to be ranked, the number of points given per candidate or option can be reduced by one (so that a first-place rank is worth n-1 points and the last-place ranks is worth no points at all). This variation has the (possibly convenient) property that the number of possible points per candidate will be between 0 and (c-1)*v inclusive, where c is the number of candidates and v the number of voters. Asymmetric ties (a Borda count tie where opponents have the same tally of points, but non-identical point ranking profiles) can be broken with a "Baldwin Elimination", which is a tallying procedure only. The lowest scoring candidate is eliminated along with their vote tally, and the election tally recalculated with one point less for each ranking on every ballot, where the lowest point value is zero. If a winner does not emerge, the process is repeated. Symmetric ties (a Borda count tie where opponents have the same tally of points and identical point ranking profiles) would require a runoff between the tied choices.

An Example of an Election

irst	Second	Third	Fourth	Points
emphis	42	0	0	58	226
ashville	26	42	32	0	294
hattanooga	15	43	42	0	273
noxville	17	15	26	42	207

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.

An Example of a Decision

Most national plebiscites are taken on the basis of only two options. Yet there are very few questions which, if asked properly, are actually dichotomous. Indeed, one example of a strict dichotomy might be the question of which side of the road to drive on. And yet, when Sweden voted on this topic in 1955, there were actually three options: "drive on left," "drive on right" and "blank." It is not known how the "blank" option would have been dealt with had that choice prevailed. Many countries, however, have used multi-option referendums - for example, in Guam, there were six options on a ballot issue. Furthermore, some parliaments also use multi-option voting, notably the Norwegian and Swedish parliaments use two-round voting and serial voting, respectively. So what happens in a multi-option Borda count or a modified Borda count? Firstly, there would be an open discussion among the concerned parties. When the dialogue is completed and if the participants have not managed to achieve a verbal consensus - i.e., if there are still a number of options on the table - then all may proceed to a vote. In a Borda count ballot on n options, a 1st preference gets n-1 points, a 2nd preference gets n-2 points, and so on. In a modified Borda count where a voter casts preferences for only m options, a 1st preference gets m points, a 2nd preference gets m-1 points, and so on. Thus, in a vote on 5 options, he who votes for only his favourite option gives that option just 1 point. She who votes for two options gives her favourite 2 points and her 2nd preference 1 point. He who votes for three options gives his favourite 3 points, his 2nd preference 2 points, and his 3rd choice 1 point. She who votes for four options gives her favourite 4 points, her 2nd preference 3 points, her 3rd choice 2 points, and her 4th preference 1 point. So best of all to list all five options, because then your favourite gets the full 5 points, your 2nd choice gets 4, your 3rd preference gets 3, your 4th preference gets 2, and your last preference gets 1 point. The option with the most points is the winner. In two-option voting, the two options are normally considered to be mutually exclusive (even when they are not). In a five option ballot, it may well be that not all five are mutually exclusive of all the other four, in which case, if the result is close, the most popular option might be composited with aspects of the second most popular. That, too, is up to the parties in the discussion. The Borda count and modified Borda count have been used in Ireland on a number of occasions, although never in an elected chamber. The first 'experiment in consensus' was in 1986, eight years before the cease-fire, with participants who included members of Sinn Fein, the Ulster Unionists, the political wing of the UDA, and pretty well everything else in between. Furthermore, the experiment worked; the participants found a consensus.

Potential for tactical voting

The Borda count relies on honest ballot choices to reflect voter consensus in the tallies. A voter who selects their favored candidate as first choice could rate a strong opponent, but political clone, as the lowest choice in an effort to help their first choice. If done by a large number of supporters of both rival clones, this move could backfire in getting their second or even third choices elected, contrary to their political intentions. Instant Borda Runoff would somewhat discourage such voting behavior, as described in the following section. In an extreme example of burying likely rivals, voters may "bullet vote": vote for a single choice only, thus allocating no points to other choices. One variation of the Borda count, called a Modified Borda Count (see above under decision-making,) addresses this by allocating a number of points for the first choice equal to the number of choices made. In the above example, a partisan for Memphis who listed only Memphis on her ballot would give one point to Memphis, while a voter who listed Memphis first and listed second, third, and fourth choices on the ballot would allocate four points to Memphis. However, this form of Borda count is non-standard and bullet voting can be proscribed by law to invalidate a vote for a particular office. Strategic nomination can be used to manipulate voters into choosing a candidate with a particular political agenda. This is quite common in the United States where two major parties and plurality voting predominate. Borda count voting is vulnerable to a different kind of strategic nomination than is plurality voting: if a group of candidates are ranked in approximately the same way by most voters (for example, they have similar ideologies), then adding more of these candidates to the ballot would increase the odds of one of this group winning. Thus, the method is not cloneproof, a characteristic which it shares most other popular voting methods.

Instant Borda Runoff

While Instant Borda Runoff, also known as Nanson's method (as devised by the mathematician Edward J. Nanson), was developed as an alternative tallying procedure for the Borda count, it can be viewed as a separate election method employing the advantages of the Borda count ballot with the counter-manipulation features of an instant runoff. Not only does this method break asymmetric ties, as stated in a previous section, but discourages tactical voting, as the voter does not know which candidates would be advanced to the runoff. The ranking would not be affected by this tallying procedure except where there is a significant component of a Condorcet cycle in that data, and if there is a Condorcet winner, this will also be the winner using Nanson's method. Note that a choice is eliminated only after losing a Borda count tally and those particular votes are eliminated as well. No choices in the running benefit from the elimination of the lowest ranking choice.

The Quota Borda System

The Borda count and modified Borda count are not proportional. If proportionality is also required, a quota element should be included into the counting procedure, as in the Quota Borda System, which works best in multi-member constituencies of either 4 or 6 members. The procedure is as follows, and we'll take the example of a 4-seater.. The quota is calculated in the usual way: {in a single-seat constituency, the quota would be an absolute majority, i.e., (50% + 1) of the valid vote}; in a 2-seater constituency, it is (33% + 1); in a 3-seater, it's (25% + 1); and in a 4-seater, it is (20% + 1) of the valid vote. Stage i) Any candidate gaining a quota of 1st preferences is elected. Stage ii) Any pair of candidates gaining 2 quotas is elected, both of them. (A pair of candidates, Ms J and Mr M, say, gains 2 quotas when that number of voters vote either 'J-1, M-2' or 'M-1, J-2'.) If seats still remain to be filled, then, ignoring all those candidates who have already been elected: Stage iii) Any pair of candidates gaining 1 quota gains 1 seat, and the seat is given to the candidate of that pair who has the higher MBC score (modified Borda count). Stage iv) Any seats still remaining are given to those candidates with the highest MBC scores.

External links

The de Borda Institute, Northern Ireland For a comparison of all the best known decision-making processes and electoral systems, see Beyond the Tyranny of the Majority. For a description of what might be a good way of doing things, see Defining Democracy. The matrix vote is described in The Politics of Consensus.
The Symmetry and Complexity of Elections Article by mathematician Donald G. Saari shows that the Borda Count has relatively few paradoxes compared to 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 Mitlhner 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.
Scoring Rules on Dichotomous Preferences Article mathematically comparing the Borda count to Approval voting under specific conditions by Marc Vorsatz.
Cloning manipulation of the Borda rule An article by Jrme Serais.
Efficiency in Polling A non-technical article on the Borda count by Courtney Baird.
Flash animation by Kathy Hays An example of how the Borda count is used to determine the Most Valuable Player in Major League Baseball.