Borda count

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 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 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.

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".

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

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 and compromising; that is, instead of voting their honest preferences, voters can rank a strong candidate they somewhat like first, or rank a strong candidate they somewhat dislike last, to increase their chances of getting a result they prefer.

Voters in a Borda count election can usually see the potential for voting strategically: if there are only two candidates with a reasonable chance of winning, they will want to rank the candidate of those two they prefer first, and the other candidate last, to maximize the difference in points they contribute to the total.

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.

Comparison to other methods

Because so much opinion is involved in what makes a good voting system, voting systems are usually 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, and the participation criterion. It does not satisfy the Condorcet criterion, the weak defensive strategy criterion, or various related criteria.

The Borda count differs from most other methods in a significant way: if a majority of voters rank one candidate in first place, that candidate is not guaranteed to win. This is sometimes called the majority criterion. This could be considered a disadvantage for Borda count in political elections and would imply that a second-rated candidate tallied a larger majority of support. Such a popular compromise could also be considered an advantage if the first place majority candidate is contentious with minorities. However, at present, the Borda count is mainly used in competitions, where not satisfying the majority criterion is considered a desirable feature.

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.

Instant Borda Runoff

The Borda count can be combined with an Instant Runoff procedure to create a hybrid election method that satisfies some desirable properties. This method is also called Nanson's method as devised by the mathematician Edward J. Nanson. The method works like this:

Candidates are voted for on ranked ballots as in the Borda count. Then, the points are tallied in a series of rounds. In each round, the candidate with the fewest points is eliminated, and the points are re-tallied as if that candidate were not on the ballot.

Nanson's method is currently used to elect a student society at Trinity College-University of Melbourne, the Assembly and Canonry of the Anglican Diocese of Melbourne, the University Council and academic committees at the University of Melbourne, and the University Council at the University of Adelaide, all in Australia.

Instant Borda Runoff satisfies the Condorcet criterion: since Borda always ranks the Condorcet winner over the Condorcet loser, the Condorcet winner will never be eliminated. As compared with the Borda count, however, Instant Borda Runoff does not satisfy the summability criterion.

See also