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The Borda count is a family of positional voting rules which gives each candidate, for each ballot, a number of points corresponding to the number of candidates ranked lower. In the original variant, the lowest-ranked candidate gets 0 points, the next-lowest gets 1 point, etc., and the highest-ranked candidate gets n − 1 points, where n is the number of candidates. Once all votes have been counted, the option or candidate with the most points is the winner. The Borda count is intended to elect broadly acceptable options or candidates, rather than those preferred by a majority, and so is often described as a consensus-based voting system rather than a majoritarian one.^[1]

The Borda count was developed independently several times, being first proposed in 1435 by Nicholas of Cusa (see History below),^{[2][3][note 1]} but is named for the 18th-century French mathematician and naval engineer Jean-Charles de Borda, who devised the system in 1770.^[4] It is currently used to elect two ethnic minority members of the National Assembly of Slovenia,^[5] in modified forms to determine which candidates are elected to the party list seats in Icelandic parliamentary elections, and for selecting presidential election candidates in Kiribati. A variant known as the Dowdall system is used to elect members of the Parliament of Nauru.^[6] Until the early 1970s, another variant was used in Finland to select individual candidates within party lists. It is also used throughout the world by various private organizations and competitions.

In the Modified Borda count, any unranked options receive 0 points, the lowest ranked receives 1, the next-lowest receives 2, etc., up to a possible maximum of n points for the highest ranked option if all options are ranked. The Quota Borda system is another variant used to attain proportional representation in multiwinner voting.

Voting and counting

Ballot

The Borda count is a ranked voting system: the voter ranks the list of candidates in order of preference. So, for example, the voter gives a 1 to their most preferred candidate, a 2 to their second most preferred, and so on. In this respect, it is the same as elections under systems such as instant-runoff voting, the single transferable vote or Condorcet methods. The integer-valued ranks for evaluating the candidates were justified by Laplace, who used a probabilistic model based on the law of large numbers.

The Borda count is classified as a positional voting system, that is, all preferences are counted but at different values; the other commonly-used positional system is plurality voting (which only assigns one point to the top candidate). By contrast, instant-runoff voting and single transferable voting use Ranked-choice voting (similarly to the Borda count), but in those systems secondary preferences are contingency votes, only used where the higher preference has been defeated.

There are a number of ways of scoring candidates under the Borda system, and it has a variant (the Dowdall system) which is significantly different.

There are also alternative ways of handling ties. This is a minor detail in which erroneous decisions can increase the risk of tactical manipulation; it is discussed in detail below.

Tournament-style counting

Each candidate is assigned a number of points from each ballot equal to the number of candidates to whom he or she is preferred, so that with n candidates, each one receives n – 1 points for a first preference, n – 2 for a second, and so on.^[7] The winner is the candidate with the largest total number of points. For example, in a four-candidate election, the number of points assigned for the preferences expressed by a voter on a single ballot paper might be:

Ranking	Candidate	Formula	Points
1st	Andrew	n − 1	3
2nd	Brian	n − 2	2
3rd	Catherine	n − 3	1
4th	David	n − 4	0

Suppose that there are 3 voters, U, V and W, of whom U and V rank the candidates in the order A-B-C-D while W ranks them B-C-D-A.

Candidate	U Points	V Points	W points	Total
Andrew	3	3	0	6
Brian	2	2	3	7
Catherine	1	1	2	4
David	0	0	1	1

Thus Brian is elected.

A longer example, based on a fictitious election for Tennessee state capital, is shown below.

Borda's original counting

As Borda proposed the system, each candidate received one more point for each ballot cast than in tournament-style counting, e.g., 4-3-2-1 instead of 3-2-1-0. This counting method is used in the Slovenian parliamentary elections for 2 out of 90 seats.^[6]

Applied to the preceding example Borda's counting would lead to the following result, in which each candidate receives 3 more points than under tournament counting.

Candidate	U points	V points	W points	Total
Andrew	4	4	1	9
Brian	3	3	4	10
Catherine	2	2	3	7
David	1	1	2	4

Tournament-style counting will be assumed in the remainder of this article.

Dowdall system (Nauru)

The island nation of Nauru uses a variant called the Dowdall system:^[8][6] the voter awards the first-ranked candidate with 1 point, while the 2nd-ranked candidate receives 1⁄2 a point, the 3rd-ranked candidate receives 1⁄3 of a point, etc. (This system should not be confused with the use of sequential divisors in proportional systems such as proportional approval voting, an unrelated method.) A similar system of weighting lower-preference votes was used in the 1925 Oklahoma primary electoral system.

Using the above example, in Nauru the point distribution among the four candidates would be this:

Ranking	Candidate	Formula	Points
1st	Andrew	1/1	1.00
2nd	Brian	1/2	0.50
3rd	Catherine	1/3	0.33
4th	David	1/4	0.25

This method is more favorable to candidates with many first preferences than the conventional Borda count. It has been described as a system "somewhere between plurality and the Borda count, but as veering more towards plurality".^[6] Simulations show that 30% of Nauru elections would produce different outcomes if counted using standard Borda rules.^[6]

The system was devised by Nauru's Secretary for Justice, Desmond Dowdall, an Irishman, in 1971.^[6]

Properties

Elections as estimation procedures

Condorcet looked at an election as an attempt to combine estimators. Suppose that each candidate has a figure of merit and that each voter has a noisy estimate of the value of each candidate. The ballot paper allows the voter to rank the candidates in order of estimated merit. The aim of the election is to produce a combined estimate of the best candidate. Such an estimator can be more reliable than any of its individual components. Applying this principle to jury decisions, Condorcet derived his theorem that a large enough jury would always decide correctly.^[9]

Peyton Young showed that the Borda count gives an approximately maximum likelihood estimator of the best candidate.^[10] His theorem assumes that errors are independent, in other words, that if a voter rates a particular candidate highly, then there is no reason to expect her to rate "similar" candidates highly. If this property is absent – if the voter gives correlated rankings to candidates with shared attributes – then the maximum likelihood property is lost, and the Borda count is subject to nomination effects: a candidate is more likely to be elected if there are similar candidates on the ballot.

Young showed that the Kemeny–Young method was the exact maximum likelihood estimator of the ranking of candidates. It implies a voting procedure which satisfies the Condorcet criterion but is computationally burdensome.

Effect of irrelevant candidates

The Borda count is particularly susceptible to distortion through the presence of candidates who do not themselves come into consideration, even when the voters lie along a spectrum. Voting systems which satisfy the Condorcet criterion are protected against this weakness since they automatically also satisfy the median voter theorem, which says that the winner of an election will be the candidate preferred by the median voter regardless of which other candidates stand.

Suppose that there are 11 voters whose positions along the spectrum can be written 0, 1, ..., 10, and suppose that there are 2 candidates, Andrew and Brian, whose positions are as shown:

Candidate	A	B
Position	51⁄4	61⁄4

The median voter Marlene is at position 5, and both candidates are to her right, so we would expect A to be elected. We can verify this for the Borda system by constructing a table to illustrate the count. The main part of the table shows the voters who prefer the first to the second candidate, as given by the row and column headings, while the additional column to the right gives the scores for the first candidate.

2nd 1st	A	B		score
A	—	0–5		6
B	6–10	—		5

A is indeed elected.

But now suppose that two additional candidates, further to the right, enter the election.

Candidate	A	B	C	D
Position	51⁄4	61⁄4	81⁄4	101⁄4

The counting table expands as follows:

2nd 1st	A	B	C	D		score
A	—	0–5	0–6	0–7		21
B	6–10	—	0–7	0–8		22
C	7–10	8–10	—	0–9		17
D	8–10	9–10	10	—		6

The entry of two dummy candidates allows B to win the election.

This example bears out the comment of the Marquis de Condorcet, who argued that the Borda count "relies on irrelevant factors to form its judgments" and was consequently "bound to lead to error".^[6]

Other properties

Zdroj:https://en.wikipedia.org?pojem=Borda_count
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