Games - Borda count

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The Borda count is a voting system used for elections. Each voter is given a preferential ballot where they rank order the candidates. The Borda count can be used as a system for finding a rank-order for every candidate, allowing it to be used as both a simple single-winner election method by selecting the highest ranked candidate and as a multiple-winner method by selecting a larger number of top-ranked 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. This method was devised by Borda to fairly elect members to the French Academy of Sciences and was used by the Academy beginning in 1784 until being quashed by Napoleon in 1800.

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 as a multiplier, 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 as a multiplier.

The points are multiplied by the number of votes and 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, 194, computed as (26*3)+(42*2)+(32*1)+(0*0). 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

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.

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.

If many voters employ such strategies, then the result will no longer reflect the sincere preferences of the electorate.

In response to the issue of strategic manipulation in the Borda count, M. de Borda said "My scheme is only intended for honest men."

Using the above example, if polls suggest a toss-up between Nashville and Chattanooga, citizens of Knoxville might change their ranking to # Chattanooga (compromising their sincere first choice Knoxville) # Knoxville # Memphis (burying their sincere third choice Nashville) # Nashville

However, the effect is not very strong. In order for Chattanooga to win, 62% of all Knoxville voters would have to employ this tactical voting.

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. The teaming or "clone" effect is significant where restrictions are placed on the candidate set.

On the other hand, in 1980, William Gehrlein and Peter Fishburn investigated the likelihood of a positional method to choose the same candidate when one modified the set of candidates by eliminating one losing candidate from a three-candidate election and two losing candidates from a four candidate election. They found that the Borda count was the positional rule which maximizes the probability of electing the same candidate after this modification of the choice set.

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 participation criterion, the Plurality criterion (trivially), reversal symmetry, and the Condorcet loser criterion.

[ Visit the complete Wikipedia entry for Borda count ]

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